Higher Order Dynamics:
Relating Operational and Denotational Semantics for -DRT
Susanna Kuschert
1 Introduction
In the formulation of semantic representation formalisms for NLP applications two aims
have substantially shaped research: rstly, to nd a compositional algorithm for semantic
construction and secondly, to be able to interpret discourse. Both Montague's idea in the
early seventies of using the well-known typed -calculus to formulate a rule system for semantic construction, and Kamp's Discourse Representation Theory (DRT), rst presented
in 1981, even today play a kind of paradigm role in these two aims. In particular, because of the (deliberate) absence of compositionality of Discourse Representation Theory,
a number of approaches dening compositional discourse semantics have been proposed,
such as e.g. [GS90], [Zee89], [EK95] and [Mus94].
This paper presents -DRT, a formalism that combines these two paradigms very
straightforwardly: put simply, -DRT expressions look like type-theoretic expressions with
DRSes in their body, or DRSes that have been -abstracted over | the perspective will
largely depend on the personal point of departure on the encounter of -DRT. -DRT's
distinctive feature is that -reduction is used as a convenient tool for the semantic processing, just as in Montague semantics, while at the same time it is able to work explicitly
with the structural properties that DRT builds on, rather than eliminating its structure.
This combination indeed allows for a great exibility and expressivity in practical applications; -DRT has already stood test for several years as the basic semantic representation
formalism of the dialogue translation project Verbmobil [BMM+ 94], just to name one.
Though -DRT has been used in several applications already, its semantics and reduction system have not yet been formally described; in particular the central -reduction has
not yet been proved to be semantically correct. Straightforwardly combining -calculus
and DRT at rst sight seems to be a smooth and trivial matter. The main contribution of
this paper is to pinpoint where this combination meets its limits, that is, which constraints
do we need to place on the syntax? We further suggest a direct, denotational semantics
of the language and formally prove that it correctly relates to the operational semantics
used for linguistic semantic construction.
However, this is not the only aim of this paper. We shall also be interested in the
interplay of type theory and discourse semantics in general, and its consequences. Typetheoretic languages are well understood through a vast amount of research that has looked
at them. Indeed, the assumption of built-in alphabetical variance (meaning that the name
of a bound variable is of no importance for the semantics) and the property that a free
variable must not be captured by a -abstraction at -reduction are at the very core of the
This paper developed from the author's Diplomarbeit (Master's Thesis [Kus95]). I would very much like
to thank Michael Kohlhase and Manfred Pinkal for the many discussions and help during the supervision
time and beyond, as well as Patrick Blackburn, Hans Kamp and Peter Ruhrberg for their helpful comments.
The work was in part supported by the DFG project -II A 2 - Pi 154/5 - 3 (SAMOS).
1
1 INTRODUCTION
-calculus. These two properties do not hold for discourse semantical expressions, though.
DRT's key idea is that the utterance of an indenite noun phrase introduces a discourse
referent | that is it introduces a variable by declaring its name | and that this discourse
referent may bind variables with the same name. We will thus nd that in -DRT some of
the variables are such that their names are important and that there is a binding operator
that binds free variables across -reduction (variable capture!).
-calculus uses the -abstraction operator to create the binding power by which functional, higher order expressions are constructed. In the same way, -DRT has two kinds
of abstraction operators which allow for two kinds of bindings, namely this standard functional binding and the above mentioned name-sensitive binding. The new, name sensitive
abstraction operator which is the means to construct a DRS, shall be called -operator. As
a result, we view -DRT as a marriage of -calculus and discourse semantics (in particular
DRT) in this paper, rather than as an extension of one concept by the other.
A couple of approaches aiming for compositional discourse semantics take quite a
contrary point of view. Reinhard Muskens [Mus94] shows that given some special type
objects, DRSes can be viewed as abbreviations of type-theoretic expressions. His semantic
representations look very much like those of -DRT, but since the DRSes really are -terms,
Muskens stresses that all we need is type theory. Peter Ruhrberg [Ruh95] does not take
quite such a radical view of merging the two ideas into one, but stays in the realm of abstraction by using simultaneous -abstraction of several variables at a time. We decided
to distinguish clearly between functional and dynamic behaviour by attributing them to
distinct abstraction operators and believe doing so is desirable for two reasons: We hope
to come to a better understanding of rstly, which of the discourse referents' properties
enable the correct treatment of anaphora, and secondly, how the two binding mechanisms
interact with each other. We reckon that this second point plays an important part in
the denition of a unication algorithm for -DRT, and we also have the suspicion that
discourse aspects may substantially guide inference processes.
The -DRT expressions will be interpreted directly in a standard type-theoretic fashion. -abstractions will get a base type, t1 , in contrast to -abstractions which get functional types like as usual. This follows the intuition that -abstracted expressions are
DRSes, regardless of how many variables are -abstracted. Denotations of type t will be
quite similar to Zeevat's denotations for DRSes [Zee89], that is, pairs of a set of abstracted
variables and a set of partial assignments on the set of those variables that may act as
discourse referents | we shall call these assignments states.
We will also allow intensionalised expressions ^A which are of type (s; ) if A is of
type , just as in Intensional Type Theory. However, semantically we will abstract over
states rather than over possible worlds. The accessibility of states within the denotations
of intensionalised expressions and of DRSes is of central importance in coping with the
fact that binding through the -operator may be delayed through -reduction. This delay
means that when we interpret an unreduced expression we may not have the relevant state
at hand. The accessibility of the state enables us to choose the relevant state or to check
that our choice was felicitous after we get access to the relevant state.
The need to compositionally construct texts from single sentences, as well as sentences
from words and phrases, asks for an operator to conjoin type-t expressions | the basic
building blocks for discourse representations | such that the dynamic bindings extend
across this conjunction. We call this dynamic conjunction or merge, and use the operator
`
'. In accordance with the standard conjunction ^, is a symmetric operator. The
symmetry allows for the treatment of some (admittedly marginal) linguistic features2. Its
1
2
The truth type will be named o.
Such as Bach-Peters-sentences in which two phrases are connected by both an anaphor and a cataphor.
2
2 A SYNTAX FOR -DRT
main motivation, however, is of methodological (e.g. to allow for incremental processing)
and technical (i.e. computational/implementational) nature. Most of the above mentioned compositional discourse representation formalisms join their dynamic objects by
means of an asymmetric conjunction operator, often called `;'. We will show that `;' is a
specialization of our -operator.
The paper is set up as follows. We will rst dene the syntax of -DRT, including
the notion of bound and free formulae, of substitution and the denition of the reduction
rules. A fragment is given in section 3 to illustrate the workings of the reduction system
for semantic construction of natural language. In section 4 we eventually present the
semantics of -DRT and show by means of examples both how dynamic binding works
in the semantics and how the discourse semantical values of functor and arguments are
coordinated at -reduction. Section 4.2 explores what validity and equivalence could
mean within -DRT. Sections 5 and 6 provide the formal proofs that the operational
and denotational semantics indeed relate in a correct way and that the reductions have
unique normal forms. Finally, we conclude with the investigation of completeness results
of -DRT and further prospects.
2 A Syntax for -DRT
From what has been said in the introduction, it is not a dicult task to write down the
denition of -DRT's well-formed formulae; we recycle the type-theoretic denition of
well-formed formulae in typed -calculus (i.e. constants, variables, functional abstraction
and application) and add to it the -abstraction operator, the dynamic conjunction and
the intensionality operators ^ and _. However, as hinted above, we shall not be done
with a straightforward denition: we need some further restrictions on the distribution
of variables due to the name sensitivity of -abstraction, and therefore take two steps to
dene well-formed formulae.
In the due course of this section we shall discover that we need even further restrictions
and therefore further classes of formulae. Here is an overview of the dierent classes of
formulae and their role in the whole of this paper. The well-formed formulae are the basis
for the denition of the interpretation function, that is all well-formed formulae can be
interpreted in -DRT. The interference of the two abstraction operators at (standard) reduction will lead to an ambiguity of the notion of binding of variables, and consequently
to an incorrectness of -reduction itself. We need a restriction to avoid such ambiguities
and call formulae in which such an ambiguity cannot occur sensible formulae. Secondly, we
will dene the class of contextually closed formulae which will guarantee a sucient degree
of intensionalisation to avoid undened and | with respect to -reduction | incorrect
interpretations in the semantics. Formulae which are both sensible and contextually closed
are called safe. Thus it is the safe formulae that have a model and on which we shall
perform all proofs in the later sections, such as the correctness of -reduction.
2.1 Types and Well-Formed Formulae
For -DRT we shall extend Church's type system (using e for individuals and o for truth
type) by two types: the base type t for -abstractions and intensionalised such as (s; )
for any type .
Denition 2.1 (Type Symbols). The set of type symbols, or types for short, of DRT, T , is dened inductively as follows:
1. e is a type symbol (denoting the type of individuals).
3
2 A SYNTAX FOR -DRT
2.1 Types and Well-Formed Formulae
2. o is a type symbol (denoting the type of truth values).
3. t is a type symbol (denoting the type of DRSes).
4. If and are type symbols, then so is (; ) (denoting the type of functions with
argument of type and values of type ).
5. If is a type symbol, then so is (s; ) (denoting the intension of type ).
We call the set BT = fe; o; tg the set of base types.
For brevity we will sometimes write (s; e) for d.
Denition 2.2 (Intensional Types). The functional closure of the set ft; (s; e); (s; o)g
is called the set of intensional types, abbreviated IT .
The signature is straightforward; it denes the standard logical operators used in basic
DRT, plus an additional conjunction.
Denition 2.3 (Signature). For Severy type 2 T we have a non-empty set of
constants of type and call = 2T the signature.
In particular, we include in the signature the identity relation =(;;o) for every type .
Furthermore, the signature must contain the following distinguished constants, the logical
constants:
: of type (t; o)
_ of type (t; (t; o))
! of type (t; (t; o))
^ of type (o; (o; o))
of type (t; (t; t))
The signature must also contain the standard truth constants (T)o and (F)o = ?, as
well as the analogous constants of type t, (T)t and (F)t, which denote the universally valid
and the unsatisable DRS respectively.
We use the ^ to join the conditions of DRSes. We can thus write the body of a DRS
as one single, possibly complex condition rather than as a set of conditions, as is common
in standard DRT.
Remark 2.4. For every typeS we have a countably innite set V of variables and call
the set of all variables V = 2T V . Within the set Ve we have a distinguished innite
set of variables VD Ve . We usually denote variables of type e by X; Y; Z , variables of
type d by U; V; W and variables of any other type by P; Q; R.
Remark 2.5. Well-formed formulae have two kinds of restrictions on the straightforward
combination of - and -abstractions: rstly, -abstractions (the declarations of discourse
referents in DRT) should use dierent names than -abstractions, and secondly, no two
discourse referents should have the same name. The rst of these restrictions is eected
by using the distinguished set of variables VD for -abstraction (clause 3 of denition 2.6)
and excluding VD in clause 5. The second restriction is placed in denition 2.8.
The restrictions reect the idea that discourse referents declare a name which should
be unique and not used elsewhere. Incidentially, the uniqueness requirement of discourse
referents can be found in DRT's syntax as well.
Denition 2.6 (Raw Formulae). Given a distinguished set of variables of type e, VD.
For every type 2 T we inductively dene the set of raw formulae of type , rf :
4
2 A SYNTAX FOR -DRT
2.1 Types and Well-Formed Formulae
A 2 rf. (Constants)
A 2 V rf. (Variables)
If X VD and B 2 rfo , then A = X :B 2 rft . (Construction of a DRS)
If X 2 V n VD and B 2 rf , then A = X:B 2 rf ; . (-Abstraction)
If B 2 rf and B 2 rf ; , then A = B (B ) 2 rf . (Application)
If A 2 rf , then ^A 2 rf s; . (Up-Operator)
If U 2 rf s; , then A = _U 2 rf . (Down-Operator)
Denition 2.7 (Sets of Discourse Referents of a Raw Formula A). The set of discourse referents of a raw formula A, DR(A), is dened inductively in parallel to A's
inductive denition. It collects all of A's discourse referents on all levels.
1./2. For A 2 [ V , then DR(A) = ;.
3. If A = X :B, then DR(A) = X [ DR(B).
4. If A = X:B, then DR(A) = DR(B).
5. If A = B (B ), then DR(A) = DR(B ) [ DR(B ).
6./7. If A = ^X or A = _U, then DR(A) = ;.
Denition 2.8 (Well-Formed Formulae). A raw formula A is a member of the set
w of well-formed formulae of type , if for every sub-expression A (A ) we have
that DR(A ) \DR(A ) = ;, and for every subexpression X :A we have X \DR(A ) = ;.
The well-formed formulae of type t, wt , are called discourse representation structures (DRSes).
Notation 2.9. We will use the convention that brackets ( and ) associate to the left,
i.e. ABC = (AB)C.
For legibility we shall sometimes use the application operator @ : w ; w ?!
w with A@B := AB for functional application, which, too, associates to the left.
1.
2.
3.
4.
5.
6.
7.
(
2
1
(
(
(
1
1
)
)
2
)
)
2
1
2
1
1
2
2
1
1
(
)
Furthermore, -abstraction binds more strongly than the merging operators.
Example 2.10. Since the habit of writing DRSes as boxes eases readibility, we will sometimes use the Kamp's box notation for the dynamic part of -DRT expressions. The
following examples represent the same expression. The reader accustomed to DRT will
recognise how DRSes are intermingled with -abstraction in the rst example. Also, the
-DRT notation for DRSes should be illustrative from the second example in comparison
to the rst.
(1) (Q:
X
student(X )
Q(X ) )@
U:
Y
book(Y )
read(Y; U )
(2) (Q: fX g: student (X )
Q(X ))@U: fY g: book(Y )^read(Y; U ) with DR = fX; Y g:
5
2 A SYNTAX FOR -DRT
2.2 Substitution and Reduction
2.2 Substitution and Reduction
The two abstraction operators in -DRT induce two notions of binding. We shall look
at the binding properties of the -operator in the following section, and concentrate on
-binding in this section. It turns out that both, the notion of binding and the reduction
rules which -DRT inherits from standard -calculus are just like in -calculus, and they
are independent of the -operator.
If an occurrence of a variable X 2 V n VD is bound by a -operator, we will call it
functionally bound. Remember that variables in V nVD can be used for -abstraction only,
not for -abstraction (cf. denition 2.6, clauses (3) and (5)).
On dening the notion of functional binding, we notice that a variable occurrence
which is in the scope of ^ can be bound by neither the - nor the -abstracted variable |
provided that the intensionalized expression is not preceeded by an adjacent _, of course
(for the elimination of _^, as in intensional type theory, cf. also denition 2.19). We shall
call such a variable protected.
Denition 2.11 (Protected Variables). An occurrence of a variable X in a -DRT
expression A is called protected, if it occurs in a sub-term ^B of A which in turn is not
immediately preceeded by an adjacent _.
Denition 2.12 (Functionally Bound and Free Variables). An occurrence of the variable X in a -DRT-expression A is functionally bound, if it is not protected and
X occurs in D in a sub-expression X:D of A.
An occurrence of a variable X 2 V n VD is called (-)free, if it is neither functionally
bound nor protected.
A well-formed formula is called functionally closed, if it contains no -free variables.
As the basis for the syntactic reduction rules to be used for the semantic processing in
-DRT let us rst dene substitution. We shall only dene substitution for variables in
V nVD, that is those that can be -abstracted. The denition of substitution presupposes
that of substitutability.
Denition 2.13 (Substitutability). An expression B is substitutable for Y 2 V nVD
in an expression A, if and only if:
If X 2 FV (B), then Y is not (-)free in the sub-expression of A which is of
the form X:C.
This denition rules out variable capture for functionally bound variables, just as
in standard -calculus. However, we will not introduce a respective notion for -bound
variables. In fact, a free variable X 2 VD may be captured through the -abstraction
operator, as we shall observe in the next section.
Denition 2.14 (Substitution). The substitution of B for free occurences of (variable) Y 2 V n VD in A, written [B=Y ]A, where B is substitutable for Y in A, is dened
inductively on the structure of A.
1. If a 2 , then [B=Y ]a = a.
2. If A 2 V , then [B=Y ]A = B, if A = Y and
[B=Y ]A = X , if A = X 6= Y .
3. If A = X :C, then [B=Y ]A = X :([B=Y ]C).
6
2 A SYNTAX FOR -DRT
4.
5.
6.
7.
2.2 Substitution and Reduction
If A = X:D, then [B=Y ]A = X:([B=Y ])D
If A = C(D), then [B=Y ]A = [B=Y ]C([B=Y ]D).
If A = ^X, then [B=Y ]A = ^X.
If A = _U, then [B=Y ]A = _[B=Y ]U.
In short, this denition is quite conservative in that simply nds all occurrences of the
variable Y recursively and replaces those occurrences, except that the ^-operator acts as
a barrier for substitution. Note that we demanded that Y was not a member of dynamic
variables and Y was free in A. Thus the conditions that Y 62 X and Y 6= X which we
would expect in cases 3 and 4 respectively are implicit in the denition.
We are now ready to give the syntactic reduction rules. The rst three rules are just
like in the -calculus. To complete the reduction system in this section, we also give
another two rules which are specic to -DRT, the - and -reduction rules.
Denition 2.15 (-Conversion). A -DRT-expression B results from A in an -conversion (A ?! B) by replacing a sub-expression (X:C) in A by (Y:[Y=X ]C) where
Y 2 V n VD and Y is not free in C.
Denition 2.16 (-Reduction). A -DRT-expression B results from A in a -reduction
(A ?! B) by replacing a sub-expression (X:D)C in A by [C=X ]D, if C is substitutable
for X in D.
Denition 2.17 (-Reduction). If X 62 FV (C), then the -DRT-expression B results
from A in an -reduction (A ?! B)by replacing a subexpression X:CX in A by C.
Denition 2.18 (-Reduction). The merge of two DRSes A1 = X :C1 and A2 =
Y :C2 reduces to a DRS by:
X :C1 Y :C2 ?! (X [ Y ):C1 ^ C2
Denition 2.19 (-Reduction). Adjacent _- and ^-operators cancel:
_^X ?! X.
In connection with the denitions of reductions we have the following standard denitions and notations:
Notation 2.20. We write a sequence of -reductions A ?! : : : ?! B as A ?! B
(for = ; ; ; ; ).
Denition 2.21 (Redex). The subexpression of A to be reduced by one of the above
reduction rules is called the -/ -/etc.-redex, or, more generally, a redex.
Denition 2.22 (Normal Form). A formula which does not contain a redex and therefore cannot be reduced by a -, ?, or -reduction, is called normal form.
Denition 2.23 (-Equality). Two expressions A and B are called -equal,
A = B, i there exists a sequence of -,?, - and -conversions, A ! : : : !# B,
where and # are , , or .
7
2 A SYNTAX FOR -DRT
2.3 Dynamic Binding Property
2.3 Dynamic Binding Property
In parallel to functionally bound variables we will call the occurrence of a variable X 2 VD
dynamically bound, if it is bound by a -operator.
The denition of -binding turns out to be not as straightforward as it seems from
this description, though. Our main focus in this section will be the aforementioned ability
of the -operator to capture free variables through -reduction, so we will have to dene
dynamic binding using -reduction. However, let us rst look at what dynamic binding
means if no -reduction is possible, i.e. if the expression in question is in -normal form.
We need to capture DRT's notion of accessibility (cf. [KR93]) that determines the scope
of discourse referents, in that it restricts the resolution of pronouns to those discourse
referents that are accessible to the condition the pronoun occurs in.
[KR93] denes accessibility as a relation between a discourse referent and a condition
with respect to a certain (complex) DRS. The relation says that a discourse referent is
accessible to all conditions on a lower level and, if the discourse referent is introduced
in the antecedent DRS of an implication, also to the conditions in the consequent DRS.
Note that this means that a discourse referent introduced in a negation or an argument
of a disjunction is not accessible to conditions outside it. The denition takes a rather
top-down view. We are about to dene a bottom-up variant now, the name context of the
occurrence of an expression Ai (not necessarily a condition) in a DRS E, which gives the
set of discourse referents (DRT-)accessible to Ai in E. The denition of top level discourse
referents TLDR(A) of an expression A is needed for the denition of the name context,
and others.
Denition 2.24 (Top Level Discourse Referents). The set of top level discourse
referents of A, TLDR(A), is dened by
TLDR(A) = S Xi
i
if A = A1 A2 : : : An , n 1, and Ai = Xi :Di .
Denition 2.25 (Name Contexts). Let Ai be the i-th occurrence of some sub-expression
A within a complex type-t expression E. The set of all discourse referents of E which are
accessible at Ai in E, is called the name context of Ai in E, written NC (E; Ai) and
dened by:
IfS E = D1 D2 : : : Dn , n 1, and Ai occurs in Dj , then NC (E; Ai) =
TLDR(Dr) [ NC (Dj ; Ai)
r6=j
If E = X :C ^ C ^ : : : ^ Cn , n 1, and if some Cj is of the form :B , B ! D,
B ! B , B _D or D_B , and Ai occurs in B , then NC (E; Ai) = X [TLDR(B )[
NC (B ; Ai).
If E = X :C ^ C ^ : : : ^ Cn , n 1, and if Ai occurs in some Cj which is not of
one of the above forms, then NC (E; Ai) = X .
1
2
1
2
1
1
1
1
2
1
1
2
Example 2.26. For E =
X
p(X )
Z
r(Z; X ) A
!
8
Y
1
:
S
r(S; Y )
2 A SYNTAX FOR -DRT
2.3 Dynamic Binding Property
we have NC (E; A) = fX; Y; Z g, in particular S 62 NC (E; A).
Denition 2.27. The occurrence of a variable X in a -DRT expression A which is in
-normal form is in the scope of a -operator, if there is some type-t sub-expression
B of A such that X occurs in B and X 2 NC (B; X ).
Thus a variable occurrence is in the scope of a -operator if in DRT parlance we would
say that a discourse referent of the same name is accessible from the condition the variable
occurs in.
Let us now look at an example of how a variable can be captured by the -operator.
Remember denition 2.16 of -reduction in -DRT.
X
(3) A(B) = P:
Y
X
!P
@
r(Y ) ?!
Y
!
r(Y )
In the argument expression B, which is itself in -normal form, the occurrence of
variable Y in B is clearly neither functionally bound, protected or dynamically bound.
However, in the reduced expression the occurrence of variable Y in the consequent of the
implication is bound by the -abstraction in the antecedent. In the unreduced expression
A(B) we must therefore consider the occurrence of variable Y in B to be bound by the
-abstraction of Y in A | through -reduction. Thus here for one the binding power of
the - and -operators interfere.
Before we nally embark on dening dynamically bound variables, we notice that
with the syntax as dened as yet, the notion may be ambiguous. Consider the following
example, a small variation of (3):
X
(4) A(B) = P:
Y
P
!P
@
r(Y)
_B
In this example, the argument will be duplicated upon -reduction and the two resulting occurrences of the variable Y will be of dierent status: in the consequence of the
implication the occurrence of Y will be bound by the discourse referent in the antecedent
in the implication, while the occurrence in the disjunction will not be bound. Thus, we
are faced with the question, what is the status of the occurrence of Y in the argument of
the unreduced expression?
We shall not be content with any classication of the occurrence of variable Y in the
argument of example (4). Thus any denition of dynamic binding must be partial on
the set of well-formed formulae. We will, however, proceed by excluding the problematic
cases in the following denition. On the restricted class we can then give a total classication, attributing each variable occurrence as either protected, functionally or dynamically
bound, or free. Thus we will be able to dene free variables as the complement of bound
variables as usual. The decision to base all further denitions on the restricted class of
9
2 A SYNTAX FOR -DRT
2.3 Dynamic Binding Property
expressions is further backed in view of a possible semantics: Any interpretation of the
above expression will be such that the two occurrences of Y in the reduced expression may
be assigned dierent values while, of course, the single occurrence in the argument B can
only receive one value at a time | inconsistency problems are inevitable.
In this determination we shall now zoom from well-formed formulae to those that are
unambiguous in the above respect, which shall be called sensible expressions.
Denition 2.28 (Sensible Expressions). A well-formed formula A is called sensible,
if either no variable occurrence is multiplied in any number of -reduction steps, or if it
is, then in the normal form of A all of the copies of this occurrence are either in the scope
of a -operator or none of them are.
Denition 2.29 (Dynamically Bound and Free Variables). An occurrence of a variable X in a sensible -DRT expression A is called dynamically bound, if it is not
protected and if one of the following conditions is met:
1. X is in the scope of a -operator.
2. X gets into the scope of a -operator through some number of -reductions.
An occurrence of a variable X 2 VD is called ( -)free, if it is neither dynamically
bound nor protected.
Denition 2.30 (Bound Variables). An occurrence of the variable X in a sensible DRT-expression A is bound, if it is either functionally or dynamically bound.
Denition 2.31 (Free Variables). An occurrence of the variable X in a sensible expression of -DRT, A, is called free, if it is neither bound nor protected. A variable X
is free in a sensible expression A, if at least one occurrence of X is free in A.
The set FV (A) of free variables of A is dened by
FV (A) = fX j X occurs free in Ag
Example 2.32. In the following, P and U are functionally bound. Z is dynamically
bound in A but free in C; likewise, Y is dynamically bound in A and free in B. The
occurrence of X in the second argument of the merge in B is protected.
X; Z
A = B(C) = (P: . . . Y; X . . .
P ( ^X ) )
Y
@ U: . . . Z; U . . .
Remark 2.33. Coming back briey to sensible expressions, we observe that expressions
that are not sensible, i.e. in which the same object is both free and bound, intuitively
do not make sense both logically and linguistically. So, all linguistic processing should
be possible with sensible expressions. Though for the general case it may be tedious to
check whether an expression is sensible or not, for linguistic purposes we have not much
worry since we shall have not more than two or three -reductions up to normal form.
Yet still we may like to have a classication of expressions that are expressive enough
for linguistic purposes and still sensible, without using the notion of -reduction. One
solution, proposed by Reinhard Muskens in personal communication, is to restrict well
formed formulae to linear terms, that is, terms in which every -abstracted term occurs
exactly once in the abstracted expression. However, the treatment of coordination and
10
3 A FRAGMENT FOR ENGLISH
ellipsis use non-linear terms, so this seems to be too strong a restriction. We leave this
matter to research outside this paper and shall be content with the conjecture, so far only
based on intuition and on studying a number of semantic constructions, that for linguistic
purposes we can restrict ourselves to sensible expressions.
Finally, we will dene a syntactic restriction that is motivated by the semantics to be
dened in section 4. This restriction serves a two-fold purpose: it rules out those formulae
that might lead to undened values in the proposed semantics and, most importantly, it is
necessary to prove the -reduction rule correct despite the possibility of variable capture.
The semantics of -DRT will be dened so that both the denotations of type t expressions and expressions of type (s; e) and (s; o), will allow access to states. This access
to states is the key for the coordination of the functor's and the argument's denotation
at function application (in fact, the interpretation function can only coordinate states)
and thus the key to the semantics of variable capture. We thus have to ensure that both
functor and argument of those expressions we want to look at are of types t, (s; e), (s; o)
or functional constructions thereof.
Denition 2.34 (Contextually Closed). A well-formed formula A is contextually
closed, denoted by A 2 ccf, if it is of a type 2 IT .
Example 2.35. A = fX g:r(X; Y ) is contextually closed, since it is of type t, while
B = P:(fX g:r(X; Y ) P (X )) is not since it is of type ((e; t); t).
Remark 2.36. We claim that we need no more than contextually closed expressions for
linguistic semantic construction. This means that the lexicon must consist of contextually
closed expressions. It is immediately obvious from the denition 2.34 that this property
is preserved through composition.
We merge the classications of sensible and contextually closed expressions into the
notion of safe expressions; these form the basis of our technical results in sections 5 and 6.
Denition 2.37 (Safe Expressions). We will call sensible, contextually closed expressions A safe and write A 2 sf.
3 A Fragment for English
In this paper we have no ambition to study the use of -DRT in some specic grammar. For this the reader is referred to descriptions of the Verbmobil semantic formalism,
e.g. [BMM+94], which use -DRT. However, for illustrative purposes we will present a
very small English fragment here and show how semantic construction works compositionally from the lexical entries in -DRT by using the tools of reduction rules dened in
the previous section.
For simplicity the fragment does not consider the temporal relations of verbs in dierent
tenses. However, it uses an event variable for the representation of verbs, similar to the
account of Davidson, [Dav67], so that modicators like in-the-park can be represented.
The following elaboration assumes that the syntactic analysis assigns indices to all
names, pronouns, determiners and events. In this respect, the lexical entries really are patterns and well-formedness of the representations is guaranteed (i.e. no duplicates among
the discourse referents), if every denite NP | or more generally: every antecedent |
uses a new index.
The indices will be written as super- and subscripts: a superscript denotes the index of
a word that acts as an antecedent, a subscript is the index of an anaphora. This fragment
uses the letters i; j; k, etc. as indices for NPs, and a; b; c, etc. as indices for events.
11
3 A FRAGMENT FOR ENGLISH
The fragment also uses the standard functional application for simplicity instead of
the generalised functional application of [BMM+ 94]; thus the representations of transitive
verbs look dierent to those given there.
A Fragment of the English Language
lexical entry representation in -DRT
type of expression
ai
P:Q:(fXig:T P ( ^Xi) Q( ^Xi))
((d; t); ((d; t); t))
P:Q:(fg:(fXig:T P ( ^Xi)) ! Q( ^Xi))
((d; t); ((d; t); t))
every
no
P:Q:fg::(fXig:T P ( ^Xi) Q( ^Xi))
((d; t); ((d; t); t))
i
^
^
P:Q:((fXig:Xi = Xj ) P ( Xi) Q( Xi))
((d; t); ((d; t); t))
thej
hisij
P:Q:((fXig:poss(Xj ; Xi)) P ( ^Xi) Q( ^Xi)) ((d; t); ((d; t); t))
P:(fXig:Xi = j P ( ^Xi))
((d; t); t)
johni
man
U:fg:man( _U )
(d; t)
U:fg:cat( _U )
(d; t)
cat
P:P ( ^Xi)
((d; t); t)
hei
walka
U:fEag:walk(Ea; _U )
(d; t)
whistle a
U:fEag:whistle(Ea; _U )
(d; t)
a
P:U:(P (V: _U = _V ))
(((d; t); t); (d; t))
be
_
_
a
P:U:(P (V:fEag:carry(Ea; V; U )))
(((d; t); t); (d; t))
carry
black
P:U:(fg:black( _U ) P (U ))
((d; t); (d; t))
P:U:(fg:happy( _U ) P (U ))
((d; t); (d; t))
happy
((d; t); (d; t))
in-the-park a P:U:(P (U ) fg:loc(Ea) = in-park)
.
Pt:Qt:(P Q)
(t; (t; t))
The rst example (5) is a very simple case of semantic construction. Note examples (6)
and (8) where free variables are caught in the course of semantic construction. In (6), the
free variable Ea in the representation of in the park is caught by the discourse referent
in the representation of walk. The same happens when the representation of He whistles
in example (7) is joined with the representation of (6) to form a text in (8).
The following presentations use =b for the equivalence to the graphical notation of an
expression.
(5) (a) John carries a cat.
(b) Johnj @ (carrya @ (ai @ cat))
1. = Johnj @ (carrya @ (P:Q:( fXig:T P ( ^Xi ) Q( ^Xi )))@(U: fg:cat( _U )))
?! Johnj @ (carrya @ (Q:(fXig:T (U:fg:cat( _U )@( ^Xi)) Q( ^Xi))))
?! Johnj @ (carrya @ (Q:(fXig:T fg:cat( _^Xi) Q( ^Xi))))
?! Johnj @ (carrya @ (Q:(fXig:T fg:cat(Xi) Q( ^Xi))))
=b
Johnj
@ (carrya
@ ( Q:
Xi
cat(Xi )
Q( ^Xi )
))
2. = Johnj @ (P:U:(P (V: fEag:carry(Ea; _V; _U )))
@ Q:( fXig:T fg:cat(Xi) Q( ^Xi )))
?! Johnj @ (U: (Q:(fXig:T
fg:cat(Xi)
Q( ^Xi))@(V:fEag:carry(Ea; _V; _U ))) )
?! Johnj @ (U:(fXig:T fg:cat(Xi) fEag:carry(Ea; _^Xi; _U )))
?! Johnj @ (U:(fXig:T fg:cat(Xi) fEag:carry(Ea; Xi; _U )))
12
3 A FRAGMENT FOR ENGLISH
=b
Johnj
Xi
@ ( U:
cat(Xi )
Ea
carry(Ea; Xi; _U )
)
3. = (P:( fXj g: Xj = j P ( ^Xj )))
@ (U:( fXig:T fg:cat(Xi) ( fEag:carry(Ea; Xi; _U ))))
?! fXj g: Xj = j ((U:(fXig:T
fg:cat(Xi)
(fEag:carry(Ea; _^
Xi; _U ))))( ^Xj ))
?! fXj g: Xj = j fXig:T fg:cat(Xi) fEag:carry(Ea; Xi; Xj )
?! fXj g: Xj = j fXig:T fg:cat(Xi) fEag:carry(Ea; Xi; Xj )
=b
Xj
Xj = j Xi
cat(Xi )
Ea
carry(Ea; Xi; Xj )
?! fXj ; Xi; Eag:Xi = j ^ cat(Xi) ^ carry(Ea; Xi; Xj )
=b
Xj Xi Ea
Xi = j cat(Xi )
carry(Ea; Xi; Xj )
(6) (a) John walks in the park.
(b) Johnj @ (in-the-parka @ walka )
1. Johnj @ (P:U:(P (U ) fg:loc(Ea) = in-park)@ (Y: fEag:walk(Ea; Y )))
?! Johnj @ (U:(((V:fEag:walk(Ea; _V ))@ (U )) fg:loc(Ea) = in-park))
?! Johnj @ (U:((fEag:walk(Ea; _U )) fg:loc(Ea) = in-park))
=b
Johnj
Ea
walk(Ea; _U )
@ ( U:
loc(Ea) = in-park
)
2. = (P:( fXj g: Xj = j P ( ^Xj )))
@ (U:(( fEag:walk(Ea; _U )) fg:loc(Ea) = in-park))
?! fXj g: Xj = j (U:((fEag:walk(Ea; _U )) fg:loc(Ea) = in-park)( ^Xj ))
?! fXj g: Xj = j fEag:walk(Ea; Xj )) fg:loc(Ea) = in-park
=b
Xj
Xj = j Ea
walk(Ea; Xj )
loc(Ea) = in-park
?! fXj ; Eag:Xj = j ^ walk(Ea; Xj ) ^ loc(Ea) = in-park
=b
Xj Ea
Xj = j walk(Ea; Xj )
loc(Ea) = in-park
13
4 SEMANTICS FOR -DRT
(7) (a) He whistles.
(b) hej @ whistleb
1. = P:P ( ^Xj ) @ U: fEbg:whistle(Eb; _U )
?! (U:fEbg:whistle(Eb; _U ))( ^Xj )
?! fEbg:whistle(Eb; Xj )
=b
Eb
whistle(Eb ; Xj )
(8) (a) John walks in the park. He whistles.
(b) . @ (Johnj @ (in-the-parka @ walka )) @ (hej @ whistleb )
1. P:Q(P Q)@( fXj ; Eag:Xj = j ^ walk(Ea; Xj ) ^ loc(Ea) = in-park)
@( fEbg:whistle(Eb; Xj ))
?! fXj ; Eag:Xj = j ^walk(Ea; Xj )^loc(Ea) = in-park fEbg:whistle(Eb; Xj )
=b
Xj Ea
Xj = j walk(Ea; Xj )
loc(Ea) = in-park
Eb
whistle(Eb ; Xj )
?! fXj ; Ea; Ebg:Xj = j ^ walk(Ea; Xj ) ^ loc(Ea) = in-park ^ whistle(Eb; Xj )
=b
Xj Ea Eb
Xj = j walk(Ea; Xj )
loc(Ea) = in-park
whistle(Eb ; Xj )
4 Semantics for -DRT
The examples of the previous section show that the -DRT language dened in section 2.1
and the operational semantics in section 2.2 are appropriate to compositionally construct
representations for natural language texts. Our aim now is to give a direct denotational
semantics for -DRT such that the interpretation function works compositionally, in the
sense of Montague. Of course, the fundamental question is the layout of the carrier sets,
in particular here the carrier set for type-t expressions | what amount and what kind of
information must the denotation of a DRS contain in order to allow for compositionality?
Let us look at a few motivational points for the semantics given below.
First and foremost, the central question of -DRT is the interaction of functional and
dynamic properties. This was our main focus in the denition of an operational semantics
in sections 2.2 and 2.3, and this shall also be our main theme as we embark on dening
the denotational semantics. Depending on the point of departure, -calculus or dynamic
14
4 SEMANTICS FOR -DRT
4.1 A Model for -DRT
semantics, the essence of this interaction will be described either as the departure from
the one of the core properties of -calculus by allowing variable capture, or as the need
to build contexts | the semantic content of discourse structures | non-locally. Two
approaches to model this come to mind. We may either delay the specic building of
contexts until we have all relevant information to build the context locally, for example
by using abstraction through ^ on both the functor and the argument of each application.
Or we may build the contexts with the information available non-locally and prune them
when the local information is at hand, that is after -reduction. We shall take the latter
option here wherever possible, in view of retaining transparency where we do not use
the ^-operator. This section will show that we need not delay the interpretation of an
expression by intensionalisation if this expression has result type t; we can interpret locally
and then use pruning to correct the overgeneration of semantic values for those variables
that get captured.
Second, the semantics has to pay attention to the commutativity of the -operator
which allows a crosswise binding situation as in the following example:
A
B =
X; ZX
XXXX
XY
9 XXXz : : : (Y; Z )
: : : (X; Y ) We want to interpret the subexpressions A and B independently rst, and let them
contribute to the denotation of the whole expression only later. This presents a diculty in
the following sense. In the interpretation of A the variable Y appears to be a free variable,
and, at merging, it turns out to be a bound variable. Considering that (existentially) bound
and free variables in the classical logical sense describe two quite dierent properties, this
is a semantically very interesting situation to model.
Again, two options are possible. Henk Zeevat's denotations contain the full range
of total assignments which render the DRS's conditions true (cf. [Zee89]). This set is
generated independent of some incoming assignment, i.e. it does not capture a change
notion, but rather a declarative notion. For this reason such semantics have been called
static semantics in the literature (e.g. [EK95]). They are in contrast to so-called dynamic
semantics which try to grasp more of the notion of input and output assignments or
changes of assignments.
The semantics dened below attempts to capture some degree of dynamicity by describing the change from an incoming assignment to those assignments that make the
conditions of the DRS true. We shall build much on the observation that the interpretation of a variable X has the same eect whether X be free or bound, if the incoming
assignment does not assign a value to X . For this reason we shall use partial assignments
in this paper.
Third, we would like to dene the -operator simply by means of set intersection. For
this the denotations must in a way foresee the occurrence of all variables in the expressions
they are about to be merged with. In case of the above example, the denotation of B
must not only contain information about Y and Z , the variables occurring in B, but also
about X , the variable occurring in A.
4.1 A Model for -DRT
With these considerations in mind, what shall the denotations for the dierent typed
expressions be like? The denotations for expressions of type e, o and the function types
15
4 SEMANTICS FOR -DRT
4.1 A Model for -DRT
are just as is usual in standard type theory. For the dynamic part of -DRT we need
semantic objects that include some notion of context. We may, for example, take the
simplied view of a context to be an assignment of values (denotations) to objects which
we currently talk about, and possibly other objects. Following the intuition that discourse
referents are those objects that we introduce into the context (in order to talk about
them), a context may be described by an assignment on discourse referents. Indeed, we
shall dene states which are partial assignments on those variables that are (and may
potentially get) -abstracted, i.e. partial assignments on the set VD (cf. remark 2.4 and
denition 2.6). Consequently, denotations of type t expressions shall contain a set of
states, namely those states that make the conditions in the expression true. Only to be
able to dene the semantics of the implication we dene the denotations as pairs where
the rst component remembers the top level discourse referents3. These states are also
used to abstract over in the denotations of intensional expressions.
The denition of the carrier set for type d follows directly from the interpretation of
type d expressions as abstractions over states.
Denition 4.1 (Carrier Set). Let U be a set of individuals. We call the set Be =
Fp(VD; De) of partial assignments on the distinguished set of variables VD the set of
states. We dene the following typed family of sets D = fDg the carrier set of -DRT:
De = U
Do = fT; Fg
Dt }(VD) }(Be)
D ; F (D; D)
D s; = F (Be; D)
A -DRT expression A will be interpreted with respect to a model M and two as1.
2.
3.
4.
5.
(
)
(
)
signment functions; one | the subscript assignment function, written as ' or | is
responsible for the -bound variables, the other | the superscript assignment function
s, or t or r | will assign a value to -bound variables and therefore is a state. The
s , or rather I s for short.
interpretation function will be called, e.g. IM;'
'
We will simplify the representation of the partial assignment functions by using the
special symbol ? and implicitely extending the denition of any partial function t by
t(a) := ?, if a 62 Dom(t).
For legibility of what follows we dene functions to retrieve the two components of a
DRS denotation.
Denition 4.2 (Projection Functions for DRS Denotations). Given the denotation
A = hX ; Mi of a DRS, we dene the following two projection functions for DRS denotations:
VAR := Dt ?! } (V ) with VAR(hX ; Mi) = X
FUN := Dt ?! } (Be) with FUN (hX ; Mi) = M
Denition 4.3 (Components of a DRS's Denotation). Likewise, given a DRS D,
we dene two functions which return the respective component of a DRS's denotation.
IV's := wt ?! } (V ) with IV's (D) = VAR(I's (D))
IF's := wt ?! } (Be) with IF's (D) = FUN (I's (D))
3
Zeevat [Zee89] uses the same format for the denotation of DRSes, for the same reason.
16
4 SEMANTICS FOR -DRT
4.1 A Model for -DRT
Since s is responsible for -bound variables, it is restricted to the assignment of individual variables only. However, ', being responsible for -bound variables, may assign values
to variables of every type, indeed to variables of type t. The values for these variables
themselves contain assignments to individual variables | recall that Dt }(Ve) }(Be )
| which should, of course, not be in conict with the assignment of s and ' to any
variables of type e. Take, for example,
(9)
X
man(X)
:P
and suppose that we had ' and s such that '(P ) = h: : :; f: : :; fX ! mary; : : :g; : : :gi
and s(X ) = john. What then does the variable (name), the discourse referent X , denote
| john or mary?4 Inconsistencies like these must be excluded using the following denition
of a consistency relation.
Denition 4.4 (Consistency Relation on s and '). Let s and ' be two assignment
functions for variables, i.e. s; ' : V ?! D . Then the two functions are mutually
consistent, (s; '), i5: for all type-t variables Zt 2 Dom(') the assignments r which
are in the values of Zt , i.e. r 2 FUN ('(Z )), agree with s, r k s.
In the above example (9), '(P ) may include some states that assign X the value john,
just as s does. Thus we need a function that prunes the assignment of type-t variables in
' so that s and the pruned ' are consistent. This pruning should work on all denotations
of intensional type (IT ). We shall call this function ". This function also needs to know
s, which we write as a superscript to ".
Denition 4.5 (Consistency Filter). We dene the consistency lter function "
thus6 :
"s (') =f(X; "s('(X )))g
where "s (A)= hVAR(A); fr 2 FUN (A) j r k sgi; if A 2 Dt
= A; if A 2 D ; where 2 f(s; e); (s; o)g
= fa ! "s (A(a)) j a 2 D g; if A 2 D(; )
Notation 4.6. We shall omit the superscript, if it is clear which superscript we need to
pass to "; for example, in denition 4.9 below I"t ' should really write I"tt ' .
( )
( )
We shall further abbreviate "t ("s (')) by "s;t (').
The denition of the interpretation function I's that we are about to dene uses the
idea of an assignment continuation s[X ]t which the reader may know from the literature
such as [GS91]. We shall dene a variant of the common notion, however, which takes into
consideration the partiality of the assignment functions and the need to foresee values, as
motivated in the introduction to this section.
We shall later (cf. example 12) work through this example in more detail.
We say that two partial functions t1 and t2 agree, t1 k t2 , if they assign the same value to all members
of Dom(t1 ) \ Dom(t2 ).
Furthermore, s + ' denotes the (simple) addition of the two partial functions s and '; note that their
domains are disjunct (cf. denition of well-formed formulae, 2.8).
6
Note that the function "s is overloaded, being dened on both, assignment functions ' whose domain
is restricted to variables with intensional type, and denotations A of intensional-type-objects.
4
5
17
4 SEMANTICS FOR -DRT
4.1 A Model for -DRT
Denition 4.7 (Continuation of an Assignment Function). Let s and t be partial
assignment functions. We call t a continuation of s by X , denoted s[X ]t, i it is dened
at least on Dom(s) [ X and we have t(X ) = s(X ) for all X 2 Dom(s) ? X .
A continuation s[X ]t can thus alter the value for all X 2 Dom(s) \X and adds values
for all other X 2 X . t may also be dened on other variables Y 62 Dom(s) [ X ; these Y
are assigned any values. Because the cardinality of t's domain can not be pre-determined,
we speak of a variable continuation.
Denition 4.8 (Model Structure). Let
D be a carrier and
F be a typed function F : ?! D
We call M = hD; F i a model structure, if all operators in F are strict, and in particular
F (:)@A = T, if FUN (A) = ;, else = F.
F (_)@A@B = T, if FUN (A) 6= ; or FUN (B) 6= ;,
else = F.
F (!)@A@B = T, if for all t 2 FUN (A) we have
9r 2 FUN (B) : t[VAR(B)]r,
else = F.
F (^) @ a @ b= T, if a = T and b = T,
= ?, if at least one of a = ? or b = ?
= F, else.
F (
)@A@B= h VAR(A) [ VAR(B) ; FUN (A) \ FUN (B) i
= @A@B
Denition 4.9 (Denotation I's (A)). The interpretation of a safe formula A with respect to M and the assignment function s and ' with (s; '), the denotation I's (A), is
dened inductively as follows7 :
1.
2.
3.
4.
5.
6.
7.
I's (c) = F (c) for any c 2 I's (V ) = '(V ) if V 2 Dom(')
= s(V ) if V 2 Dom(s)
= ? else.
I's (X :B) = hX ; ft j s[X ]t ; (t; "t(')) ; I"t ' (B) = Tgi
I's (X:D) = A:I"s '; A=X (D)
I's (A(B)) = I's (A)@I's (B)
I's ( ^A) = t:I't (A)
I's ( _U ) = I's (U )(s)
( )
(
[
])
8
Using partial assignment functions always mean that we must take caution that no
undened value may leap in when we interpret representations of linguistically correct expressions. It turns out that we are able to prove that if we restrict ourselves to functionally
closed, contextually closed formulae (cf. denitions 2.12 and 2.34) no undened value will
leap in.
7
8
We use for lambda-abstraction in the meta-language with the intuitive meaning.
Here, t is a state.
18
4 SEMANTICS FOR -DRT
4.1 A Model for -DRT
Lemma 4.10. Each variable X 2 V occurring in any functionally closed expression A 2
ccf will be assigned a proper (i.e. non-?) value in the course of I's (A).
Proof: We have to look at two sub-cases according to whether X 2 VnVD is a functional
variable or X 2 VD is a dynamic variable.
At the interpretation of any occurrence of X , where X 2 V n VD, X will be dened
by ' since A is functionally closed. This is just as in standard interpretion of type
theoretic expressions.
If A 2 VD, then we can use the fact that since A 2 ccf there must be a contextually
closed sub-expression of A of form either X :B or ^B, such that the occurrence of
X is in this sub-expression. Here, we need not have X 2 X .
If the sub-expression is of form A0 = X :B, then I's (A0) = hX ; ft j s[X ]t ; (t; "('));
I"t(')(B) = Tgi ensures that only those assignments t are chosen which assign T to
the condition of the DRS. Thus, since all operators in F are dened to be strict
(cf. 4.8), all free and -bound variables occurring in A, i.e. all variable occurrences
X 2 VD, are assigned a value by the functions IF's (A).
If the sub-expression is of form A0 = ^B, then the value of B is abstracted over
by states. Thus the interpretation of the occurrence of X in B is delayed and not
undened.
This proof demonstrates how -abstraction and the intensionalizer complement each
other in the interpretation of -DRT expressions. Note that it is the -free variables which
are critical in this proof; in case of a DRS, a -free variable will be assigned a value on
the way of making the conditions of the DRS true. On the other hand, if a -free variable
occurs in the scope of an ^-operator, the interpretation does not evaluate the variable
with respect to one particular assignment function but abstracts over such a look-up.
This complement role of -abstraction and intensionalisation in semantics will be further
demonstrated in the comment of example 11.
Based on this observation we will now dene the notion of a model for -DRT.
Denition 4.11 (-DRT Model). Let M = hD; F i be a model structure such that I's
is dened for all safe expressions A 2 sf, for all s and ' with (s; '). Then we call M a
-DRT model.
To understand how the construction of the denotations works let us look at some examples. We will trace three things here: rst, how the interpretation of DRSes facilitates
the simple denition of the (symmetric) merge operator and second, how the intensionality operator allows correct coordination of values for variables in partially instantiated
expressions. Lastly, after having discussed it at such great length, it shall be interesting
how the variable capture works on the level of denotations.
In the introduction to this section we already motivated the partiality of the assignment
function s and the concept of assignment continuation which play an important role in the
interpretation of a DRS in the present approach. We observed in the proof of lemma 4.10,
case (3), that the variable continuation ensures that no undened values occur within a
DRS denotation by being able to assign a value to all -abstracted and all free variables of
the DRS. However, the continuation also allows for any other variable 2 VD to be assigned
a value, that is, variables other than those occurring in the DRS in focus. This implies
the following denition.
19
4 SEMANTICS FOR -DRT
4.1 A Model for -DRT
Denition 4.12 (Minimal Assignment Function). Those assignment functions t of
the second component of the denotation of a DRS A, t 2 IF's (A), which are only dened
on those variables which occur in A (i.e. free variables and discourse referents), are
called minimal assignment functions. All other functions t in IF's (A) are themselves
continuations of the minimal assignment functions by the empty set of variables.
It is the occurrence of non-minimal assignment functions in IF's (A) that allows for using
set intersection to dene the merge operator. Through the intersection we get exactly those
functions which also assign a value to variables occurring in other arguments of the .
To illustrate this, look at the following example taken from example (6) in section 3; the
three parts in (10) are the interpretations of the arguments of the resulting expression of
step 2 there9 .
(10) (a)
(b)
(c)
I's (fXj g: Xj = j ) = hfXj g; ft j s[fXj g]t ; I"t ' (Xj = j ) = Tgi
I's (fEag:walk(Ea; Xj ))) = hfEag; ft j s[fEag]t ; I"t ' (walk(Ea; Xj )) = Tgi
I's (fg:loc(Ea) = in-park) = hfg; ft j s[fg]t ; I"t ' (loc(Ea) = in-park) = Tgi
1
1
2
3
1
( )
2
3
2
( )
3
( )
Each of the functions t1 in case (a) is at least dened on Xj but some of the t1 may
also be dened on any other variables. Indeed, there will be functions t1 which are dened
on the variable Ea. In case (b), all of the functions t2 must be dened on both Xj and
Ea, in order to attain the value T in the condition; again, some of the t2 will be dened
on other variables, too. Similarly for case (c): each one of the t3 will be dened on Ea
and some of them on Xj or other variables. The set intersection of the three sets will not
only match the values of the variables but also the domains of the functions; thus in this
example, all functions t of the result are dened at least on both Xj and Ea. Therefore, by
means of the variable continuations, DRS-interpretations are able to foresee or be ready
for the assignments of yet unknown variables.
Notice what happens if the starting s already assigns a value to Xj , say. Then if s(Xj )
is not an individuum that walks, the second component of (b) is empty and thus the
second component of the merge of (a) and (b) is, too. Indeed, only if s(Xj ) is the walking
John do we get a non-empty second component of the merge of (a) and (b). We have
already discussed that if free and bound variables are to behave the same, the starting
assignment should not be dened on that variable (cf. the introduction to this section).
We therefore note the following10 .
Remark 4.13. We shall always interpret an expression with respect to the empty assignment s.
Here is our second example. It is in fact a variation of example (6) of section 3, and
we look at the interpretation of the expression which corresponds to step 2 there11 . The
interpretation function for constants, F , will be dened so that F (man) = man', etc.
(11) (a)
A man walks in the park.
For reasons of space and to make the formulae easier to read, we will not write the consistency relation
here and in the following example; the reader should consider it as being implicit.
10
We will also come back to this point extensively in section 5.1
11
As just noted in remark 4.13, we start with an empty assignment s here and therefore simplify "s;t to
t
" , without loss of generality.
Also note that in line 2 g is an object of type (d; t), so "s (g)@s:s(X ) = "s(g@s:s(X ), from denition 4.5, and in the same way in line 4 we have "r (a)(r) = a(r), since a 2 D(s;e).
9
20
4 SEMANTICS FOR -DRT
4.1 A Model for -DRT
I's ((a man) @ (walk))
= I's (G:( fX g:man(X ) G( ^X ))@(Zd: fE g:walk(E; _Z )))
= g:I"s '; g=G ( fX g:man(X ) G( ^X ))@a:I"s '; a=Z ( fE g:walk(E; _Z ))
= g:(hfxg; ft j s[fX g]t; I"t '; g=G (man(X )) = Tgi "s (g )@s:s(X ))
@a:hfE g; fr j s[fE g]r; I"r '; a=Z (walk(E; _Z )) = Tgi
4. = g:(hfX g; ft j s[fX g]t; man(t(X )) = Tgi "s (g @s:s(X )))
@a:hfE g; fr j s[fE g]r; walk(r(E ); "r(a)(r)) = Tgi
5. = hfX g; ft j s[fX g]t; man(t(X )) = Tgi
"s (a:hfE g; fr j s[fE g]r; walk(r(E ); a(r)) = Tgi@s:s(X ))
6. = hfX g; ft j s[fX g]t; man(t(X )) = Tgi
"s (hfE g; fr j s[fE g]r; walk(r(E ); s:s(X )@r) = Tgi)
7. = hfX g; ft j s[fX g]t; man(t(X )) = Tgi
"s (hfE g; fr j s[fE g]r; walk(r(E ); r(X )) = Tgi)
8. = hfX; E g; fq j s[fX g]q; s[fE g]q; man(q (X )) = T ^ walk(q (E ); q (X )) = Tgi
the latter being the intended value since s[X ]q and s[Y ]q implies s[X [ Y ]q .
(b)
1.
2.
3.
(
[
])
(
(
[
(
[
])
])
[
])
In this example we focus on how the look-up of X in G( ^X ) is delayed through the
interpretation of the ^-operator until X has moved into the dominance of the second abstraction operator (i.e. the DRS that represents walk) and can then be assigned a value
by the right states, in this case state r in clause 7. Until there, the ( ^X ) of clauses 1 and 2
becomes s:s(X ), which is nally applied to r in step 6. If s is empty from the start then
"s (a: : : :) = a: : : : in lines 5 . The coordination of values for the X 's in the rst and
second argument for the merge is therefore eected by the interpretation of , not by the
". If s is not empty, then "s (a: : : :) may eect some pruning for coordination, but note
that this pruning only preempts some of the coordination work.
Finally we look at the interpretation of an expression in which variable capture would
occur if we had reduced the expression prior to interpretation. Here, the occurrence of
the variable X in the argument is bound through the -abstraction in the functor. This
example shows how the interpretation of function application forces the assignment of the
occurrence of X in the argument to be the same as that in the functor. We will reuse
example (9) here and extend it.
(12) (a) I's (P:( fX g:man(X ) ^ :P )@( fg:dogowner(X )))
1. = p:I"s(';[p=P ])( fX g:man(X ) ^ :P )@I's ( fg:dogowner(X ))
2. = p:(hfX g; ft j s[fX g]t; I"ts;t(';[p=P ])(man(X ) ^ :P ) = T; (t; "s;t('; [p=P ]))gi
@hfg; fr j s[fg]r; I"r(')(dogowner(X )) = T; (r; "r('))gi
3. = hfX g; ft j s[fX g]t; I"ts;t(';[hfg;fr j s[fg]r;:::gi=P ])(man(X )) = T; (t; "(hfg; fr j : : :gi)g
^ there exists no r in "s;tfr j s[fg]r; I"r(')(dogowner(X )) = T; (r; "r('))g)i
The coordination of the free occurrence of X in the argument and its dynamically
bound occurrence in the functor is forced by the consistency condition in the interpretation
of the functor's dynamic body. To illustrate this consider the following situation. If we
have a group of men, among them John, and women, but Mary is the only dogowner
in the world, the resultant denotation should contain a state that assigns John to X .
21
4 SEMANTICS FOR -DRT
4.1 A Model for -DRT
However, the second component of the argument's denotation is not empty since Mary
owns a dog. This would mean, however, that X is assigned John in the functor and Mary
in the argument, an inconsistency. The " now prunes the argument's denotation to contain
no more inconsistencies, thus pruning the (only) state that assigns X to Mary. Now the
second component is empty, and applied to the negation operator we get the value T, as
required.
It is routine to prove, as we shall do now, that during the interpretation process, the
two assignment functions do indeed stay in the consistency relation.
Theorem 4.14 (Conservation of Consistency). If (s; '), then we have (t; ) for
every recursion step I t of I's interpreting contextually closed (safe) expressions.
Proof: We only need to consider two cases, namely those cases in the semantic's denition in which an assignment function is extended or changed; these are cases (3) and (4)
of I's 's denition, the former extending s, the latter '. Note that the argument A in (4)
is from D where 2 IT .
3.
4.
I's (X :B): we have (t; "t(')) by denition.
I's (X:D)@A: if A 2 Dt: By denition we have r k s for all r 2 "(A), and thus
(s; ("('; [A=X ]))).
if A 2 D , where 2 f(s; e); (s; o)g, we have "s (A) = A. Also,
since, by denition 2.8, case (4), X 62 VD, X is not
assigned a value by a state and therefore we nd that no
value for a type-t variable assigned by ' needs to be
pruned.
if A 2 D(; ): Here, the functor is pruned if necessary (cf. previous
cases). We need not look at the arguments since only
pruned arguments will be considered (cf. denition 4.9,
case 4.).
To sum up on denition 4.9 of the denotational semantics of -DRT we note that
those parts of the denition which originate from the -calculus resemble very much their
counterparts in pure typed -calculus. However, the consequences of the interference with
dynamicity in -DRT can be clearly pin-pointed in 4.9. Firstly, in clause (2), we remark
that the interpretation of variables considers two sources for the denotations. We not
only have two kinds of abstraction operators, but also two assignment functions with clear
responsibilities for the variables bound by each of these operators. It is interesting to note
that using two assignment functions is not only a design choice to underline the presence
of two concepts but will indeed be necessary to be able to prove -reduction correct. Thus
it is something the calculus itself asks for. See remark 5.6 at the end of the proof of
theorem 5.5 for what eect the separation has.
Remark 4.15. Secondly, the need to prune the denotation of the argument when extending ' in clause (4) is striking. We have just studied how this allows the | from a
purely functional point of view inconceivable | capturing of variables by correcting the
non-local interpretation to t with the local information. It became clear in the proof of
lemma 4.10 that what lies behind the lter function " is the consistency relation. The
consistency relation was necessary since values of DRSes contain assignments to variables
themselves (and these assignments can clash with other information). We observe here
22
4 SEMANTICS FOR -DRT
4.2 Validity and Equivalences
that it is the availability of the variable name within the DRS denotation that allows
the conservation of consistency despite variable capture. Indeed, the presence of variable
names in the semantic object, which makes the discourse referents globally accessible, is
the key to discourse semantics. This feature is in contrast with pure -calculus, or classical
logic in general, where variables are merely place holders and names can be abstracted
away (cf. [Bru72]). To stress the global accessibility of variable names we suggest to call
-abstracted variables also dynamic declarations.
It is a much debated question how actually to view discourse referents. They are
neither variables nor constants in the classical sense. Through the above interpretation
we have given them a status in between: the variable-like character, underlined by the fact
that free variables can be bound by the -operator, is prominent. Yet the accessibility of
the variable's name makes the discourse referents resemble constants very much.
Remark 4.16. Finally, we remark that the asymmetric dynamic conjunction operators
used elsewhere in the literature (the ^ in [GS91] and the ; of [Mus94] and [EK95]) is a
specialization of the symmetric of -DRT. Syntactically, ; may be dened just as was,
restricted by the constraint that no -abstraced variable of the second argument may be
used in the rst argument. ;'s relational semantics is just like the one for , i.e. we take
the union of the abstracted variables and conjoin the conditions of the two arguments.
The asymmetry constraint on ; suggests left-to-right construction and use of contexts.
We will try to mirror this notion in the semantics by interpreting ; as follows:
I's (A;B) = h IV's (A) [ IV's (B) ;
fr j r 2 IF's (B) und 9t 2 IF's (A) : rj?IV's (B) = tgi12
We remark that with this semantics if -DRT used ; instead of or in addition to , all
technical results would hold just as we will present them in the later sections. The proofs
have been worked through for ; but we will not include this case in this paper.
Note that the above interpretation of ; resembles very much the denition given in
approaches such as the following from [EK95]:
M; s; r j= A; B i there exists an assignment function t with M; s; t j= A and
M; t; r j= B.
In the same way as the t in [EK95]'s denition is the output assignment of the rst
argument DRS, so is the t in the -DRT denition. Since rj?VAR(B ) = t merely requires
that r extends t by values for the VAR(B )13 , t acts as input assignment of the second DRS.
4.2 Validity and Equivalences
The focus of attention so far has mainly been the context properties in syntactic processing
and the semantic representations. In the understanding task we may want to abstract
away from these matters of context at particular stages of processing in order to nd out
whether the utterances are true in the rst place. For the denition of validity in -DRT
we shall revert to the respective notion in DRT. There, a DRS is assigned a truth value
T or F depending on whether it has an embedding or not. Applied to -DRT, having an
embedding parallels having at least one assignment in the second component of the DRS's
denotation. Thus, the denition of validity of a DRS is a straightforward check on the
non-emptyness of the second component of its denotation. It has, in fact, already been
pre-empted in the denitions of : and _, which include a test on the empty set.
Where rj?Z := f(a; r(a)) j a 62 Zg.
Note that we can neglect the cases in which the value for some variable in VAR(B) is changed since we
assume that none of the VAR(B) occurs in A.
12
13
23
4 SEMANTICS FOR -DRT
4.2 Validity and Equivalences
Denition 4.17 (Validity). A DRS A is valid in a model M with respect to s and ',
if IF's (A) =
6 ;.
If we want to compare two -DRT expressions, we may again "`switch o"' context
aspects, or consider the complete range of contextual and truth-conditional properties. We
shall thus be interested in static and dynamic equivalence14 respectively. The denition
of static equivalence, or s-equivalence, is simple; it straightforwardly builds on the above
denition of validity in -DRT.
Denition 4.18 (s-Equivalence). Two DRSes A and B are s-equivalent, denoted
A 's B, if IF't (A) 6= ; , IF't (B) 6= ; for all states t and assignments '.
By way of example, the two DRSes A = fX g:p(X ) and B = fY g:p(Y ) are sequivalent, and it is clear that this notion neglects the dierent binding properties of A
and B. The binding properties are important, however, if we carry on in the utterance
and thus merge more representations onto A and B. Imagine a text carries on with an
utterance which is represented by C = fZ g:q (X; Z ). Then A C and B C are not
s-equivalent.
Let us introduce the notion of replacement thus
Denition 4.19 (Replacement of B at Some Position p). We write Cjp = A if the
sub-expression of C at position p is A. The replacement of B at position p [B=p]C
is dened as ([B=p]C)jp = B, if no -bound variable occurrence in B gets bound through
this, if Cjp is not in the scope of an ^ in C and if [B=p]C is a safe expression. Else
[B=p]C = C.
In this notation the above negative result is, in the general case, stated by saying,
if A 's B and Bjp = A then we do not have C 's [B=p]C for all C. However, we
would like to coin some notion of equivalence for which the principle of extensionality can
be stated. For this we note that just as in intensional type theory expressions with the
same intension may be substituted for one another without changing the truth value, i.e.
^ = ^ j= = [= ] , so the principle holds if we require full (contextual and truthconditional) equivalence, i.e. equivalence of the whole denotation. We shall call this kind
of equivalence c-equivalence.
Denition 4.20 (c-Equivalence). Two DRSes A and B are c-equivalent, A 'c B, if
I's (A) = I's (B) for all states s and assignments ', i.e. if the denotations are exactly alike.
Thus, if A and B are c-equivalent, then they may be substituted for one another in
any -DRT expression (while making sure that the resulting expressions stay well-formed)
without a change in the truth value, i.e. if Cjp = A then A 'c B j= C 's [B=p]C for all
C. This observation comes with no surprise if we remember what has been said about the
relation between the intension operator and -abstraction.
Before we set o to prove the extensionality principle for dynamic equivalence, we
introduce a slightly weaker notion than c-equivalence, to be called d-equivalence. We will
then show the extensionality principle using d-equivalent A and B; extensionality for
c-equivalence will be a trivial matter then.
It turns out that to capture the equality of dynamic behaviour we need not require
the second components to be exactly alike. They may as well relate by Z , dened thus:
Denition 4.21 (Contextual Detail). A set of states A is contextually less detailed (by a set of variables Z ) than the set of states B , A Z B , if
14
These terms have been coined in [GS91].
24
4 SEMANTICS FOR -DRT
4.2 Validity and Equivalences
1. either ; =
6 A B or ; = A = B , and
2. for all s 2 B n A there exists a t 2 A and some subset Z 0 Z with s = tj?Z 0 .
We may extend this notion on all kinds of semantic objects thus: for two type-t
denotations we have At Z Bt , i VAR(A) = VAR(B ) and FUN (A) Z FUN (B ). If
A; B 2 D(;) then A Z B, if for all C 2 D A@C Z B@C , provided these applications
go back to safe formulae. For all other semantic objects we write A Z B , if the two
objects A and B are equal.
We may also write simply , if Z is not in focus.
If the set of states B is contextually more detailed than A , then it is of larger
cardinality; the states it adds to A all result from states in B by taking away the values
for some or all members of Z . By including the smaller states in addition to the broader
dened states, one can think of adding some detail, hence the name. Note that if one of
the sets is empty, the other one must be, too.
Using the notion of contextual detail we dene d-equivalence thus:
Denition 4.22 (d-Equivalence). Two DRSes A and B are d-equivalent, denoted
A 'd B, if for all states s and assignments ' there exists some Z such that
I's (A) Z I's (B) or I's (B) Z I's (A).
How do two DRSes, A and B, relate, if they are d-, but not c-equivalent? The rst
guess is that their sets of discourse referents is the same but that one of them, say B,
has at least one free variable, say X 2 Z , which does not occur in the other. Then,
of course, all states in the interpretation of B must assign a value to X , whereas the
states in the interpretation of A need not. The extra states in IF's (B) are those which
do this. Such A and B are only d-equivalent, however, if the predication on X is a
tautology, i.e. if X can be assigned any value from the domain. Take A = fX g:r(X )
vs. B = fX g:r(X ) ^ (s(Y ) _ :s(Y )) as an example. If, on the other hand, the extra
predicate cuts down the number of possibilities of values for Y , then IF's (A) would include
15
states that assign one of the impossible values to Y , and thus no longer be a subset of
IF's (B).
We shall need the notion of contextually less detailed sets of states when we compare
the interpretation of an expression A with respect to a ' extended by the assignment of
the denotation of A0 to P and the denotation of another expression B which is like A with
a sub-expression A0 of A replaced by a variable P . This is, of course, the situation we
are faced with at -reduction. In such a case we have indeed less contextual detail in A
than we have in B and no clash like in the case above, since the states in the denotation
in B have to stay consist with the assignments in the denotation of P assigned by '. The
exact working of this will become clear when we look at the proof of the substitution-value
lemma 5.5.
Before we prove the extensionality principle for d-equivalence, let us rst prove two
auxiliary lemmas. First, we will check that reducing the amount of contextual detail (by
the same set of variables) in the arguments of the logical operators that lead to type-o
denotations does not destroy the truth value.
Lemma 4.23. If A0t Z At and Bt0 Z Bt for some Z , then
Note that the representation of a text like It rains. Mary sings or does not sing. is not an
example to be considered in place of B, since Mary introduces a new discourse referent while we are talking
about free variables here.
15
25
4 SEMANTICS FOR -DRT
4.2 Validity and Equivalences
I (:)@A = I (:)@A0.
I (^)@A@B = I (^)@A0@B0 .
I (_)@A@B = I (_)@A0@B0 .
I (!)@A@B = I (!)@A0@B0 .
Proof: 1. to 3. are easy to prove, since I (:), I (wedge) and I (_) check on the second
component of the denotations to be empty. However, denotations in the -relation are
1.
2.
3.
4.
either both empty or not empty.
4. is more dicult. We need to consider both directions of
I (!)@A@B = T , I (!)@A0@B0 = T
) For this we will contradict the property that for some s 2 A0 there is a t 2 B n B0
such that s[XB ]t but no t0 2 B 0 such that s[XB ]t0.
Remember that the minimal assignment of B is dened on Z , whereas the minimal
assignment of B 0 is not; otherwise the two sets are equal. But if for some s 2 A0
there is a t 2 B n B 0 with s[XB ]t, then due to the denition of a continuation there
also is a t0 2 B 0 such that s[XB ]t0 .
( We have to prove that for every S 2 A n A0 there exists a t 2 B with s[XB ]t. We
know that there exists an s0 such that s = s0 j?Z and for this s0 there exists a t0 with
s0[XB ]t0. Therefore take t = t0j?Z .
Second, we want to prove that d-equivalence is conserved upon merging with another
expression.
Lemma 4.24. Let A 'd B. Then I's (A) I's (C) 'd I's (B) I's (C).
Proof: Let us assume that IF's (A) Z IF's (B) for some Z . We will use the following
abbreviations: IF's (A) = A , IF's (B) = B , IF's (C) = C and B n A = BA , i.e.
IF's (B) = A [ BA. We will show that A \ C Z B \ C , and do this by showing the
two conditions of denition 4.21 in turns.
1. We need to show that A \ C = ; i B \ C = ;
Since IF's (A) is a subset of IF's (B), the (-direction is trivial. This leaves us to show
that if the merging A with C results in an empty set of states, then so does merging
B with C. We will show this by contradiction, i.e. try to prove that at least one
member of BA survives but none of A .
Let us rst note that C cannot equal just BA because it will also include all
continuations of members of BA, and these are also in A , by denition of .
Since is translated by simple set intersection, we need to nd a set C which
includes some state s 2 BA, but no state t 2 A . Again, if s is in C , then the
state s = s0 j?Z 0 is in C , too, by way of the variable continuation; this means in
particular, that s0 (Z ) for some Z 2 Z can take all sorts of values, since Z apparently
it does not occur in C, for otherwise C would not contain states that do not assign
a value to Z . So, this state s0 must also be a member of A .
2. We need to show that if for all s 2 BA there exists a t 2 A with s = tj?Z 0 , then
also for all s0 2 BA \ C there exists a t0 2 A \ C with s0 = t0 j?Z 0 .
This is true since for every element in C there are also all continuations of that
element in C .
26
5 CORRECTNESS OF REDUCTION RULES
We now prove the existensionality principle for d-equivalence in -DRT.
Lemma 4.25. If A 'd B and Cjp = A, then C 'd [B=p]C for all C.
Proof: Proof by induction on the structure of C.
1. C is a constant: Cjp = A means that C = A. Hence trivially C = A 'd [B=p]C =
B.
2. C is a variable: same argument as in the previous case.
3. C = X :D: Then [B=p]C = X :[B=p0]D where Cjp = Djp0 . We must verify that
each condition has the same truth value upon the replacement. This was shown
in 4.23.
4. X:D: If X occurs in B, then [B=p]C = C and the above is given trivially. Else use
induction hypothesis.
5. C = C1 (C2): If C1 is one of :; ^; _; ! or , then use lemmas 4.23 or 4.24. Else
use induction hypothesis.
6. C = ^D: Then [B=p]C = C by denition of replacement.
7. C = _D: Use induction hypothesis on D.
Corollary 4.26. If A 'c B and Cjp = A, then C 'c [B=p]C for all C.
5 Correctness of Reduction Rules
Having dened the operational and denotational semantics of -DRT so far, we are now
eager to examine the properties of their relation, most notably we shall strive to prove
correctness and completeness of the operational semantics with respect to the denotational
semantics. In this section we shall start with correctness.
The major challenge of -DRT's semantics has been the capturing of free variables.
In the previous section we looked at a couple of examples to understand how the given
semantics mirrors this unorthodox property. We recall that we employed two complementing techniques to model variable capturing, the pruning of local interpretations and the
delaying of interpretation.
The substitution value lemma of section 5.2 will be the central result of this chapter.
It can be proven in a fairly straightforward way except | not surprisingly | for the case
where variable capture is modelled by pruning local information; we will have to reason
precisely on the behaviour of the pruning.
However, we need to remember that variable capturing not only requires the above
techniques to ensure correlation of values but also demands that free and bound occurrences of a variable behave in the same way. For this purpose we used partial assignments
and built on the observation that a free and a bound occurrence of a variable X in A
have the same eect only if the assignment with respect to which A is interpreted does
not assign a value to X . Therefore, before we embark on the proof of correctness, we shall
make this requirement explicit in an alternative semantics and prove that this semantics
can be used in place of the original.
27
5 CORRECTNESS OF REDUCTION RULES
5.1 Alternative Semantic Denition
5.1 Alternative Semantic Denition
In the analysis of example (10) of last section, we observed that if we interpret an unreduced merge of two DRSes A = A1 A2 with respect to an assignment s such that one
of the variables which get captured through -reduction is in Dom(s), then we get only a
(singleton) subset of the full denotation for A. This observation is due to the fact that if
a free variable has not yet been given a value, then upon interpretation it can be assigned
any value of the appropriate carrier set and thus behaves just like a bound variable.
Further, we note that even if we start interpretation of A with an assignment which
does not assign a value to any discourse referents in A, or even the empty assignment, we
may get to such a "troublemaking" assignment at a lower interpretation step due to the
variable continuation which may assign values to any new variable. However, it is through
the continuation that free and bound variables may get the same treatment. So we will
dene an alternative semantics, I's , which cuts down on the assignment function before it
is passed to the continuation operation. Parallel to the denitions derived from I's we use
IV's and IF's .
In the proofs of this section we will often fall back on this alternative semantics and
therefore we need to prove that the two denitions are equal. The reader may ask why the
alternative semantics was not introduced as the semantics denition proper. We believe
that the semantics dened in 4.9 is not only more smooth and regular but also more in
the spririt of interpretation with respect to some assignments.
Denition 5.1 (Alternative Semantics Denition). We dene I's , the alternative
semantics of -DRT, on the basis of the original denition: I's (A) = I's (A) if A is not
of the form X :B; else, using Y = DR(A), the set of all discourse referents in A, we dene
I's (X :B) = hX ; ft j sj?Y [X ]t ; (t; "t(')) ; I"t(')(B) = Tgi.
We shall now verify that the alternative semantics is equivalent to the original semantics. The proof is based on two observations:
1. If an assignment does not dene any of the subordinate discourse referents, then its
continuation by the actual discourse referents does always also contain assignments
which are not dened on any of the subordinate discourse referents, due to the fact
that any two discourse referents may not have the same name. Therefore, when the
further interpretation process hits a DRS, the above modied interpretation function
does not eect a cutting down on the incoming assignment s. This means that the
result of the alternative semantics will be part of the result of the original semantic
denition.
2. The interpretation with respect to an assignment which does not dene variables
with the same name as subordinate discourse referents subsumes the interpretations
with respect to assignments that do not. This means that the alternative semantics
does not cut away any results of the original interpretation function would have
achieved.
These two observations will be shown in the two lemmas below respectively; following
that we then argue that I;; (A) = I;;(A).
Lemma 5.2. Given a well-formed expression A = X :D, and let s not be dened on
DR(A), and let there be any '. Then there exists an assignment t 2 IF's (A) such that
Dom(t) \ (DR(A) n X ) = ;.
28
5 CORRECTNESS OF REDUCTION RULES
5.2 Substitution-value lemma
Proof: By denition of continuation, denition 4.7: Dom(t) may be Dom(s) [ X . We
are left with the question whether it may be possible that I't (D) = F for all assignments
t which do not dene DR(A) n X while I't0 (D) = T for an assignment t0 which denes
DR(A) n X (and therefore is in IF's (A)). This is not possible since in a well-formed
expression any Z 2 DR(A) n X may not be used above where Z is introduced.
Lemma 5.3. Given a well-formed expression A = X :D and Y = DR(A). For any state
s we have IF's (A) IF's (A).
Proof: All IF's (A) are in IF's (A), since the continuation s[X ]t may assign values to
members of Y also; the other conditions of the denition of the interpretation function
for DRSes stayed the same. However, there may be some t 2 IF's (A) but not in IF's (A),
namely those for which for some X 2 Y n X we have X 62 Dom(t).
We can now show that both semantics denitions are equivalent.
Theorem 5.4. For all A 2 w we have
I;;(A) = I;;(A).
Proof: In lemma 5.2 we showed that at any step interpreting a DRS, if the state does not
dene the discourse referents of subordinate DRSes then among the continuations of this
interpretation there is an assignment which also does not dene the discourse referents of
subordinate DRSes. If we start with an empty assignment, we can assume this for every
interpretation step which interprets a DRS. In lemma 5.3 we then veried that all other
assignments of the continuation do not contribute anything new; so we conclude the above
statement.
5.2 Substitution-value lemma
Theorem 5.5. Let A; B 2 sf and Y not bound in A. Let (Y ) = (B) 2 IT . Then
I's ([B=Y ]A) Z I"s '; I's B =Y (A)
(
[
(
)
])
where Z is the set of variables occurring in B but not in A.
Having proved this, we then argue that, by denition of d-equivalence, denition 4.22,
I's ([B=Y ]At) and I"s '; I's B =Y (At) reect expressions which are d-equivalent.
(
[
(
)
])
Note that we need not consider the case that B is not substitutable for Y in A, because
of the denition of substitution, Def. 2.14, and therefore free variables are not captured
by -bound variables.
Proof: According to the results regarding the alternative semantic denition 5.4, we can
assume that X 2= Dom(s) for all X introduced as discourse referents at some lower level.
We will conduct the proof by induction on the structure of A.
1. If A = c a constant:
I's ([B=Y ]c) = I's (c) = F (c) = I"s '; I's B =Y (c).
(
[
(
)
])
Due to the denition of interpretation of a constant.
29
5 CORRECTNESS OF REDUCTION RULES
5.2 Substitution-value lemma
2. If A = V a variable, then if V 6= Y , we have the same argument as in the case of a
constant. If V = Y , then we have to prove that
I's ([B=Y ]V ) = I's (B) I"s(';[I's (B)=Y ])(V ) = "s(I's (B)).
This shall be done by induction on the type of B (i.e. (B)). If B 2 Dt , then "s
has no eect since we can assume that all discourse referents X in B are not in
Dom(s) due to the alternative semantics. If B 2 D(s;e) [D(s;o) then "s has no eect
by denition, and if B 2 D(; ), then by denition we can reuse the rst two cases
of this argument.
3. Let A = X :D:
I's ([B=Y ](X :D))= I's (X :[B=Y ]D)
= hX ; ft j s[X ]t ; (t ; "t (')) ; I"t ' ([B=Y ]D) = Tgi
= hX ; T i
1
1
1
1
1
( )
1
Because of induction hypothesis.
We also have:
I"s '; I's B =Y (X :D)= hX ; ft j s[X ]t ; (t ; "s;t ('; [I's (B)=Y ])) ;
I"ts;t '; I's B =Y (D) = Tgi
= hX ; T i
(
[
(
)
2
])
2
2
2
2
2(
[
(
)
])
2
Since the rst components of these two resulting denotations are equal, we are left
with proving T1 Z T2 for Z the set of variables occurring in B but not in A. We
do this by structural induction over B with (B) 2 IT .
(a) B is a constant. T1 and T2 are equal, since the interpretation of B does not
depend on the assignment functions.
(b) B is a variable. Note that B 62 X since (X ) = e 62 IT for all X 2 X . We
are done if I"tt (') (B) = "s;t (I's (B)). Here, the critical case is (B) = t; the
reader is referred to the argument in the following case (B a DRS) which is
quite similar.
(c) B = X :E is a DRS. We will consider two cases.
i. No free variables of B are captured by a discourse referent in A, i.e. in
particular X \FV (B) = ; (we should really look at a larger set of discourse
referents than X , namely the set of all discourse referents Y is in scope of.
However, the argument may be easily extended.). Remember that we may
assume that Dom(s) does not contain X or any other discourse referents
introduced in A. We shall look at what constraining eects the conditions
I"t(')([B=Y ]D) = T and I"ts;t (';[I's (B)=Y ])(D) = T have. The reader may
convince himself that for every t2 of T2 there is a t1 of T1 which agrees
with t2 on the variables that occur in D but not in B, since due to the just
mentioned assumption t1 and t2 may assign any value to all free variables
with the same name as a discourse referent, with the same propositions (the
conditions in D and B) as constraints. Note again that variables that occur
both in B and D trivially are assigned the same value in the rst expression,
whereas this is assured by the "-relation in the second expression.
We therefore note for the two conditions of :
1
1
1
2
2
2
30
5 CORRECTNESS OF REDUCTION RULES
5.2 Substitution-value lemma
A. If one of T1 and T2 is empty, then so is the other. This is a direct
consequence of the above observation.
B. To check on condition 2 of denition 4.21, we will prove that for every
t2 2 T2 n T1 we have a t1 2 T1 such that t2 = t1j?Z , where Z is the set
of variables occurring in B but not in D. Take any t1 2 T1 . There may
indeed be a t2 2 T2 n T1 with t2 = t1 j?Z , since in I"ts;t (';[I's (B)=Y ])(D)
we note that t2 need not assign a value to any variable occurring in B
but not in D, since I's (B) is doing that.
ii. If X \ FV (B) 6= ;, i.e. at least one free variable of B will be captured
by a discourse referent in X . Here again, the "-relation ensures that this
variable will not be assigned a dierent value by s than by t1 .
Let B = X:D. Here ' is extended by a value for X , and the induction
hypothesis can be used for D.
Let B = CD. Using induction hypothesis on C and D.
Let B = ^X. Here I's (B) = I't (B), since the interpretation of B does not use
any of the assignment functions.
Let B = _U. The restriction (B) 2 IT restricts U to be an intension of an
expression of intensional type. We can therefore use induction hypothesis.
2
(d)
(e)
(f)
(g)
2
1
4. A = X:D: Let = '; ["(a)=X ]. Note that we have X 62 FV (B), since otherwise
B is not substitutable for Y in A.
I's ([B=Y ]A)@a= I's (X:[B=Y ]D)@a since Y free in X:D, i.e. X 6= Y
= I"s '; a=X ([B=Y ]D)
Z I"ss "s '; a=X ; I"s '; a=X B =Y (D)
= I"s '; a=X ; I's B =Y (D) since "s ('; [a=X ]) = ' on FV (B)
= I"s '; I's B =Y (X:D)@a.
(
[
(
(
[
(
[
])
(
[
][
(
)
]) [
(
(
)
[
])
(
)
]
]
])
5. Let A = CD: By induction hypothesis we know that
I's ([B=Y ]C) = C 0 I"s(';[I's (B)=Y ])(C) = C and
I's ([B=Y ]D) = D0 I"s(';[I's (B)=Y ])(D) = D.
and we also know that both C and D are of intensional type.
If D0 is equal to D, then by denition of we know that I's ([B=Y ]A) = A0 I"s(';[I's (B)=Y ])(A) = A. Now assume that D0 and D are not equal. If D0; D 2 D,
2 IT n ftg, then we have either D0 = D or, in case of a functional object, can
argue back on the non-functional case. Therefore assume D0; D 2 Dt .
If we have D0 D because the second components of both D0 and D are empty,
then we have A0 A trivially by denition again.
If we assume that the second components of D0 and D are not empty, then it is not
possible for the second component of A0 to be empty while the second component of
A is non-empty; since if this were so, then C 0 had to be such that it was successful
only with smaller states but not with bigger states which subsume the smaller states.
This would mean that there must be a proposition that switches its truth value if
variables which do not occur in the proposition are assigned a value.
6. Let A = ^X. Then I's ([B=Y ] ^X) = I's ( ^X) = t:t(X) = I"s(';[I's (B)=Y ])( ^X).
31
5 CORRECTNESS OF REDUCTION RULES
5.3 Correctness of -conversion
7. Let A = _U. We may use induction hypothesis here.
Remark 5.6. Since the substitution-value lemma is the very basis for the correctness
proofs for the -, - and -reductions to follow, it very much helps us understand what
happens during these reductions at the presence of dynamic abstraction. In particular,
one design decision is a direct consequence of this understanding, namely the separation
of the assignment function in s and '.
Consider, in the proof of 5.5, case of A being a DRS, what would happen if the interpretation would be with respect to one function, responsible for both - and -abstracted
variables. We would need to compare16
()
Is (X :[B=Y ]D) = hX ; ft j s[X ]t ; (t) ; It([B=Y ]D) = Tgi
and
() I"(s;[Is (B)=Y ]) ( X :D) = hX ; ft j "(s; [Is(B)=Y ])[X ]t ; (t) ; It(D) = Tgi
Here, if s(Y ) = ? at the beginning, the t in () would be such that t(Y ) = Is (B),
whereas in (), t could assign anything to Y , including t(Y ) = ?, for It ([B=Y ]D) = T. If,
on the other hand, s(Y ) = a for some a 6= Is (B) at the beginning, then t(Y ) is dierent
for the two cases.
Note that this proof also shows that the pruning only works on type-t denotations
since these provide the name of the variables within the denotation. In denotations of
expressions of type o and e we lose track of the information what value was assigned to
the variables occurring therein and therefore could not do any pruning. This is just like
in classical logic. As a result, type-t expression complement with types of pattern (s; )
in -DRT.
5.3 Correctness of -conversion
Theorem 5.7. If Y 2 V n VD is new in A, then for all assignments s and ':
I's (X:A) 'd I's (Y:[Y=X ]A).
Proof: We prove that I's (Y:[Y=X ]A)@a I's (X:A)@a and therefore, by denition 4.22 I's (X:A) and I's (Y:[Y=X ]A) represent expressions that are d-equivalent.
I's (Y:[Y=X ]A)@a= I"s '; a=Y ([Y=X ]A)
I"s '; a=Y ; Y=X (A)
= I"s '; a=X (A)
= I's (X:A)@a
(
[
])
(
[
][
(
[
])
])
Due to denition of -conversion and the substitution-value lemma.
As a consequence we can now dene -congruence.
Denition 5.8 (alphabetical variants, -congruence). If B results from A by a sequence of -reductions, A ?! B, then A and B are called alphabetical variants. A
and B are -congruent.
16
Here we assume the obvious one-place equivalences to and ", e.g. checking on consistency within.
32
5.4 Correctness of -reduction
5 CORRECTNESS OF REDUCTION RULES
Remark 5.9. Variable convention: We assume a built-in -equality, i.e. that expres-
sions are syntactically equal, if they are alphabetical variants of each other. Therefore,
we can safely assume that all functionally bound variables are distinct to all free and all
dynamically bound variables, without loss of generality, and strictly speaking, the use of
a distinguished set VD, used in denition 2.8, is not necessary.
Also, restricting substitution to substitutable cases really is no restriction, since we
can assume substitutablity to be given always, through -conversion. Thus, -reduction,
which builds on substitution, is executable for all safe expressions.
5.4 Correctness of -reduction
Theorem 5.10. If X is not bound in A, then for all assignments s and ':
I's ((X:A)B) 'd I's ([B=X ]A).
Proof: We shall prove that I's ([B=X ]A) I's ((X:A)B). Therefore, by denition 4.22,
the two expressions are d-equivalent.
I's ([B=X ]A) I"s '; I's B =X (A) = I's (X:A)@I's (B) = I's ((X:A)B)
(
[
(
)
])
Due to the denition for -reduction and the substitution-value lemma.
5.5 Correctness of -reduction
Theorem 5.11. If X 62 FV (A) and X 2 V n VD, then for all assignments s and ':
I's (X:AX ) 'd I's (A).
Proof: We shall prove that I's (A)@a I's (X:AX )@a for the appropriate X . Therefore
by denition 4.22 I's (X:AX ) and I's (A) represent expressions that are d-equivalent.
I's (A)@a= I's ([a=X ]A)@a
I"s '; a=X (A)@a
I"s '; a=X (A)@I"s '; a=X (X )
= I"s '; a=X (AX ) = I's (X:AX )@a
Since X 62 FV (A) and due to the substitution-value lemma and the denition of I .
(
[
])
(
[
])
(
[
])
(
[
])
5.6 Correctness of - and -reduction
To prove the correctness of -reduction, we shall use the alternative semantics (cf. denition 5.1), together with the fact that it is equal to the original semantics denition.
Theorem 5.12. Let s be undened on X [ Y . Then
I's (X :A) I's (Y :B) = I's (X [ Y :A ^ B)
Proof:
I's (X :A) I's (Y :B)= hX ; ft j s[X ]t ; (t ; "t (')) ; I"t ' (A)gi
hY ; ft j s[Y ]t ; (t ; "t (')) ; I't (B)gi
= hX [ Y ; ft j s[X ]t and s[Y ]t ; (t; "t(')) ;
I"t ' (A) = T and I"t ' (B) = Tgi
1
1
2
1
2
( )
1
1
2
( )
2
2
( )
33
6 -NORMAL FORMS
On the other hand
I's (X [ Y :A ^ B) = hX [ Y ; fr j s[X [ Y ]r ; (r; "r(')) ; I"r ' (A ^ B) = Tgi
( )
We have to show, rst, that the assignments t and r have equal domains and second,
that the interpretations of the parts (in one case the interpretations of A and B, in the
other case the interpretations of A ^ B) have the same ltering eect.
1. Since s is undened on X [Y , we have s[X [Y ]t for all t where both s[X ]t and s[Y ]t.
Thus t and r have the same domain.
2. Due to the denition of ^ we have I't (A ^ B) = T i I't (A) = T and I't (B) = T,
since I't (A ^ B) = I't @T@T = T.
Without using the restriction on s we can have the following situation: Suppose a
Y 2 Y is dened in s, such that s(Y ) 6= t2 (Y ) for all t2 2 IF's (B). Since t1(Y ) = s(Y )
for all t1 2 IF's (A), set intersection results in an empty set. Here it is not possible for the
free variable to act for change like the discourse referent Y in B.
Theorem 5.13. Let A ?! B. Then I's (A) = I's (B).
Proof: By the observations of section 5.1, we may assume that the incoming assignment s
does not assign a value to discourse referents introduced in the expression to be interpreted.
With this, the theorem follows directly from 5.12.
Theorem 5.14. If A ?! B, then I's (A) = I's (B).
Proof: We have I's ( _^U) = I's ( ^U)(s) = (t:It (U))(s) = I's (U).
6
-Normal
Forms
Since the -, -, -, and reductions are the basic tools for the semantic construction in
-DRT, we have to ensure that the reduction system consisting of these reductions has
unique normal forms. We will thus show termination and conuence in this section.
Again, as in the previous section, we will look at safe expressions only (however, the
property of being contextually closed will not be used here). If a reduction cannot be
executed due to an ambiguity of the concept of boundedness, we may view the reduction
process as failed. Failure, however, dees comparison to check conuence.
Both properties, termination and conuence, can be shown independently for -,
and -reduction, because the three groups of reductions address dierent abstraction
operators and can therefore be separated. The following diagrams illustrate that the each
two reduction steps can be exchanged:
(X (A1 A2))D
[D=X ]A1 [D=X ]A2
(X (A))D
[D=X ](A)
34
6 -NORMAL FORMS
6.1 Termination of -Reduction
(X ( _^A))D
[D=X ]( _^A)
(X (A))D
_^(A1 A2)
A A
1
[D=X ](A)
_^(A)
2
A
The proofs for - and -reduction alone are trivial. First let us look at termination.
Regarding -reduction, every occurrence of a -operator is eliminated in one step, without
producing new occurrences of one of the merging operators. Regarding -reduction, each
pair _^ of intension operators is eliminated by a single reduction as well, and no other
reduction step produces new intension operators.
Conuence says that regardless of the order of reduction steps every reduction results
in the same result. For -reduction this is easy to show informally: since -reduction
introduces set union and logical conjunction and since both are associative operators,
the order of -reduction is unimportant. Since -reduction simply eliminates operators,
reduction order is unimportant here, too.
It thus remains to be shown that -reduction alone terminates and is conuent.
Termination of -reduction is trivial; the number of reduction steps is limited by the
number of -abstraction.
6.1 Termination of -Reduction
We will prove termination of -reduction following Tait's proof for -reduction in typed
-calculus [Tai67] by doing a combined induction over both the structure and the types
of the expressions. Cf. the version of this proof in Hindley und Seldin [HS86] which only
needs slight revision to t for -DRT.
To state the termination property we will use the notion of strong reducibility:
Denition 6.1 (Strong Reducibility). An expression A is called strongly reducible,
SR(A), i every sequence of -reductions terminates.
We will also need the notion of the logical relation, in a similar manner to the logical
relation in Tait's proof:
Denition 6.2 (Logical Relation). The logical relation, LR, is dened as:
1. LR(A) i SR(A) if 2 fe; d; og.
2. LR((X:Bo)t ) i LR(Bo ).
3. LR(A(; )) i for all B , LR(B ) implies LR(AB )
Our aim is to show that:
35
6 -NORMAL FORMS
6.1 Termination of -Reduction
Theorem 6.3 (Termination of -Reduction). For all well-formed expression of DRT A 2 w we have SR(A).
Proof: The proof is in two parts: rst, we have to prove that for all A which are in the
logical relation, i.e. LR(A), we have SR(A), that is, termination is a consequence of the
logical relation. Second, we will prove that every well-formed expression is in the logical
relation. Lemmas 6.4 and 6.6 state these two claims, followed by their respective proof.
Theorem 6.3 follows directly from 6.4 and 6.6.
Theorem 6.4 (LR SR). This lemma consists of two parts:
1. If h an atomic expression of type n ! and SR(Aii ), then LR(hAn).
2. LR(A) implies SR(A).
Proof: Joint induction on the type :
If of basic type:
1. Since SR(Aii ) for all i, we also have SR(hA : : : Ann ), since no Aii will
be applied to another Ajj . Therefore, due to the denition of LR, we have
LR((hAnn )), because is of basic type.
2. Follows directly from the denition of LR.
If functional type, = (; ):
1. Let LR(B ). Due to induction assumption (2) we have SR(B ) and due to
induction assumption (1) we have LR((hAnn B) ). It thus follows from denition 6.2, case (3) that LR((hAnn )).
2. Let LR(A) and given a new variable X , i.e. X 62 FV (A). Taking n = 0
in induction assumption (1), we have LR(X ) and thus LR((AX ) ) due to
denition 6.2. Due to induction assumption (2) it follows that SR((AX ) ).
Thus we also have SR(A), because if there was an innite sequence of reductions in A, then there also was one in (AX ), which is in contradiction with
SR((AX )).
1
1
We thus have LR(A) ) SR(A) for all A.
The following lemma prepares the proof of lemma 6.6.
Theorem 6.5 (Closure under -Head-Expansion). If LR([B=X ]A) and also LR(B )
if X 62 FV (A), then we have LR(((X:A)B)).
Proof: Let = (~; ), where is of basic type, and let LR(Cii ).
Since is of basic type, it suce to show that SR(((X:A)BC) ).
Due to induction hypothesis we have LR([B=X ]A) and LR(Ci i ). Thus because of the denition of LR: LR(([B=X ]AC) ). But since is of basic type,
we also have SR(([B=X ]AC) ).
Since we look at SR, i.e. the property that all reduction sequence terminates,
we have SR for all sub-terms of (([B=X ]AC) ), thus also SR([B=X ]A) and
SR(Cii ).
36
6 -NORMAL FORMS
6.1 Termination of -Reduction
Due to hypothesis LR(B ) and lemma 6.4 we have SR(B ).
So, an innite reduction of (X:A)BC does not only contain redexes of [B=X ]A
and the Ci , but must be of the form
(X:A)BC?! (X:A0)B0 C0
?! [B0=X ]A0C0
?! : : :
where A ?! A0, B ?! B0 and Ci ?! C0i. We therefore also have
[B=X ]A ?! [B0=X ]A0, and thus an innite reduction
[B=X ]AC?! [B0=X ]A0C0
?! : : :
This is in contrast to the assumption.
We thus showed that SR(((X:A)BC) ), and so we also have LR(((X:A)BC) ).
Since also LR(Ci i ), we have LR(((X:A)B)) due to the denition of LR.
Theorem 6.6. Let LR(((X))) for all X 2 Dom() and A 2 w(), then LR((A)),
i.e. every well-formed expression is LR.
Proof: Induction on the structure of A:
1. A 2 : Then (A) = A and thus LR(A) due to lemma (6.4) with n = 0.
2. A 2 V : We need to consider two cases:
A = X 2 Dom(). Then LR((A)) due to assumption.
A 62 Dom(). Then (A) = A and LR(A) due to lemma (6.4) with n = 0.
3. A = X :B: If LR( (B)o) and = [ is a splitting of , such that contains
exactly those substitutions which are not dened on the discourse referents, i.e.
Dom() \ X = ; and Dom( ) \ X = X . Then (X :B)t = (X :(B))t.
But since LR(B) due to induction hypothesis, we also have LR( X :(B)) due to
denition of LR.
4. A = B@C: Due to induction hypothesis we have LR(B ; ) and LR(C ), thus
LR(B@C) due to denition of LR.
5. A = (X :C ): Let LR(B ) and := ; [B=X ], then meets the conditions
of the induction hypothesis. Also ((X :C )B) ! ([B=X ]C) = (C). Since
LR((AB)) due to induction hypothesis, we have LR((C)) due to lemma (6.5).
Thus we have LR(A) due to denition of LR.
6. A = ^X : We have (A) = A and thus LR(Ad ) due to lemma (6.4) with n = 0.
7. A = _U : analogous to case of variables, case (2).
(
37
)
6 -NORMAL FORMS
6.2 Conuence of -Reduction
6.2 Conuence of -Reduction
The successful proof of termination for -reduction in the last section will greatly simplify
our proof for conuence in this section. By Newman's lemma (cf. [Bar81]) a reduction is
conuent, i it is terminating and locally conuent.
Denition 6.7 (Local Conuence). Let ?!R be the transitive closure of a relation
?!R.
A reduction R of a language is called locally conuent, if for all well-formed expressions A; B und C of this language with A ?!R B and A ?!R C we have a well-formed
expression D with B ?!R D and C ?!R D. It is usual to represent this graphically
thus:
A
B
C
D
Theorem 6.8 (Local Conuence for -Reduction in -DRT). -reduction in -
DRT is locally conuent.
Proof: Let A and D be any -DRT expressions, and let A ?! A0 and D ?! D0. We
prove this theorem by rst proving local conuence of -reduction in detail, and by then
referring to standard -calculus results for the proof of -reduction and the commutation
of both reductions.
To prove -reduction conuent, let us look at all paths of the following reduction graph.
(X:D)@A
(X:D0)@A
(X:D)@A0
[A0=X ]D0
[A=X ]D
[A0=X ]D0
[A0=X ]D
So, for every combination of -reductions of (X:D)@A (expressions of the second
line), a common -reduction can be found.
The proof for local conuence of -reduction turns out to be just like in ordinary
-calculus as well, and so does the proof that ?! and ?! commute17.
The termination and conuence of the -reduction has two important consequences:
rstly, the normal forms of this reduction system are unique, and secondly, we can dene
-equality using -normal forms. So, let us redene -equality (cf. 2.23) in the
following way.
Relations R and S commute, if from R(X;Y ) and S (X;Z ) it follows that there exists a W such that
S (Y; W ) and R(Z; W ).
17
38
8 CONCLUSION
Denition 6.9 (-Equality). Two -DRT expressions A and B are called equal, if there exists an expression C with A ?! C and B ?! C.
Theorem 6.10 (Decidability of -Equality). -equality is decidable.
Proof: Since -reduction terminates, we can verify A = B by checking, in a
nite number of -reduction steps, whether there exists a C with A ?! C and
B ?! C.
7 Completeness
In relating the operational and the denotational semantics of a logic, we are not only
interested in correctness. We also want to know whether the equality relation induced by
the reduction rules, in this case = , captures the full (semantic) equality relation of
that logic. In other words, can all terms which are mapped to the same denotation by the
interpretation function be related by = ?
In -DRT, the answer is No. Two sources of incompleteness are quite obvious and
inherent in the nature of two sub-parts of the language.
Firstly, = cannot capture equality of type-o expressions, e.g. the equality of p(X ) ^
(q (Y ) _ :q (Y )) = p(X ). This problem, the problem of coping with the logical structure,
is well-known from standard higher order logic. In order to ll this hole to completeness,
we need to add a prover for the equality of type-o expressions.
The second source of incompleteness has to do with the fact that we fully excluded
-conversion for dynamically bound variables. We thereby neglected the fact that if the
dynamicity of a dynamically bound variable is closed o by a negation, a disjunction or
an implication, then the name of this variable is no longer visible, just like a functionally
bound variable. Consider the two expressions
(13) fX g:( fY g:q (Y; X ) ! fg:r(Y ))
and
(14) fX g:( fZ g:q (Z; X ) ! fg:r(Z ))
These two expressions have the same interpretation, exactly because the discourse
referent of the antecedent DRS of the implication has no binding eect outside the implication. However, with the conversion rules dened as they are here, this denotational
equality cannot be captured by operational means.
8 Conclusion
-DRT was born by the idea to straightforwardly combine standard DRT and -reduction
of typed -calculus for the semantic processing core of actual NL projects. The operational
semantics immediately suggested itself from this very idea of -DRT. The work presented
here gives the formal background for systems using -DRT as their semantic formalism.
The most important result is the denition of a denotational semantics which can be
proved to be correct with respect to the operational semantics.
The interaction of s and s, which looks so simple in the reduction system, turned
out to be not quite so trivial in the denotational semantics. We are faced with the fact
that functionality and dynamics are based on quite contrary ideas and principles. Essentially, the interaction was enabled by the joint forces of pruning to consistent states and
39
REFERENCES
REFERENCES
intensionalisation. In all, the accessibility of the variable names was the key to success.
In fact, we believe that the dierent character of functionality and dynamicity justify the
clear separation of the two parts of -DRT as done in this paper. Furthermore, this study
identied the semantically problematic formulae of a straightforward interaction of functional and dynamic languages; this may serve as a list of caveats for any further work in
this eld.
Another focus of the denotational semantics was the symmetry of the -operator. To
interpret this operator correctly, we had to make the dynamics bi-directional. Besides
the original motivation for the -operator (cf. section 1), one might investigate whether
bi-directional dynamics is cognitively adequate, and if so, how do humans perform the
`backward binding'?
This work lays the ground for a number of directions of further research. For one, DRT has been and will be further extended. In section 3 we mentioned the assumption that
coindexation of anaphors with their antecedents is provided by the syntactical analysis. It
will be interesting to explore the alternatives of this approach in -DRT. We also already
mentioned the goal of building an inference system. This, indeed, has been one of the
major motivations for our enterprise of formalising -DRT.
In section 7 we mentioned that we need to be able to rename discourse referents if they
have no binding power outside the expression they occur in because certain operators close
o the scope of dynamic abstraction. It turns out that an -conversion rule which correctly
recognises the closing o of dynamic binding is non-trivial to dene at the presence of abstraction. Consider the following slight variation of (13)
(15) P: fX g:( fY g:q (Y; X ) ! P )
Here, the dynamic variable Y does still have binding power, namely on variables Y
occurring in the term to be inserted in P through -conversion. We suspect that by using
a richer type system we can dene these binding properties to a full extent, and by this
not only dene a richer -conversion but also lay the ground to a unication algorithm
for -DRT.
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