Definite Clause Grammars
[email protected]
http://www.knoesis.org/tkprasad/
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Review : Difference Lists
• Represent list L as a difference of two lists
L1 and L2
– E.g., consider L = [a,b,c] and various L1-L2
combinations given below.
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L1
L2
[a,b,c|T]
T
[a,b,c,d|T]
[d|T]
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Review: Append using Difference Lists
append(X-Y, Y-Z, X-Z).
• Ordinary append complexity = O(length of first list)
• Difference list append complexity = O(1)
X-Z
X
X-Y
Y
Y
Y-Z
Z
Z
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DCGs
• Mechanize attribute grammar formalism (a
generalization of CFG)
– Executable specification
• Use difference lists for efficiency
• Translation from DCGs to Prolog clauses is
automatic
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Sample Applications of DCGs
• Coding recursive descent backtracking
parser
• Encoding and checking context-sensitive
constraints
• Simple NLP
• In general, enabling syntax directed
translation
• E.g., VHDL Parser-Pretty Printer
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DCG Example : Syntax
sentence --> noun_phrase, verb_phrase.
noun_phrase --> determiner, noun.
verb_phrase --> verb, noun_phrase.
determiner --> [a].
determiner --> [the].
determiner --> [many].
noun --> [president].
noun --> [cat].
noun --> [cats].
verb --> [has].
verb --> [have].
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DCG to Ordinary Prolog Syntax
sentence(S,R) :noun_phrase(S,T), verb_phrase(T,R).
noun_phrase(S,T) :- determiner(S,N), noun(N,T).
verb_phrase(T,R) :- verb(T,N), noun_phrase(N,R).
determiner([a|R],R).
determiner([the|R],R).
determiner([many|R],R).
noun([president|R],R).
noun([cat|R],R).
noun([cats|R],R).
verb([has|R],R).
verb([have|R],R).
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Queries
?- sentence([the, president, has, a, cat], []).
?- sentence([the, cats, have, a, president], []).
?- sentence([a, cats, has, the, cat, president], [president]).
?- sentence([a, cats, has, the, cat, President], [President]).
• Each non-terminal takes two lists as arguments.
• In difference list representation, they together
stand for a single list.
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DCG Example: Number Agreement
sentence --> noun_phrase(N),verb_phrase(N).
noun_phrase(N) --> determiner(N), noun(N).
verb_phrase(N) --> verb(N), noun_phrase(_).
determiner(sgular) --> [a].
determiner(_) --> [the].
determiner(plural) --> [many].
noun(sgular)
noun(sgular)
noun(plural)
verb(sgular)
verb(plural)
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-->
-->
-->
-->
-->
[president].
[cat].
[cats].
[has].
[have].
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Extension: AST plus Number agreement
sentence(s(NP,VP)) --> noun_phrase(N, NP),verb_phrase(N, VP).
noun_phrase(N, np(D,NT)) --> determiner(N, D), noun(N, NT).
verb_phrase(N, vp(V,NP)) --> verb(N, V), noun_phrase(_, NP).
determiner(sgular, dt(a)) --> [a].
determiner(_, dt(the)) --> [the].
determiner(plural, dt(many)) --> [many].
noun(sgular, n(president)) --> [president].
noun(sgular, n(cat)) --> [cat].
noun(plural, n(cats)) --> [cats].
verb(sgular, v(has)) --> [has].
verb(plural, v(have)) --> [have].
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Queries
?- sentence(T,[the, president, has, a, cat], []).
T = s(np(dt(the), n(president)), vp(v(has), np(dt(a), n(cat)))) ;
?- sentence(T,[the, cats, have, a, president|X], X).
?- sentence(T,[a, cats, has, the, cat, preside], [preside]).
• Each non-terminal takes two lists as arguments for
input sentences, and additional arguments for the
static semantics (e.g., number, AST, etc).
• Number disagreement causes the last query to fail.
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Prefix Expression DCG
expr --> [if], expr, [then], expr,
[else], expr.
expr --> [’+’], expr, expr.
expr --> [’*’], expr, expr.
expr --> [m].
expr --> [n].
expr --> [a].
expr --> [b].
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Queries
?-expr([’*’, m, n], []).
?-expr([m, ’*’, n], []).
?-expr([’*’, m, ’+’, ’a’, n, n], [n]).
?-expr([if, a, then, m, else, n], []).
?-expr([if, a, then, a, else, ’*’, m, n], []).
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Prefix Expression DCG : Type Checking Version
tExpr(T) --> [if], tExpr(bool), [then], tExpr(T),
[else], tExpr(T).
tExpr(T) --> [’+’], tExpr(T), tExpr(T).
tExpr(T) --> [’*’], tExpr(T), tExpr(T).
tExpr(int)
--> [m].
tExpr(int)
--> [n].
tExpr(bool) --> [a].
tExpr(bool) --> [b].
• Assume that + and * are overloaded for int and bool.
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Queries
?-tExpr(T,[’*’, m, n], []).
?-tExpr(T,[m, ’*’, n], []).
?-tExpr(T,[’*’, m, ’+’, ’a’, n, n], [n]).
?-tExpr(T,[if, a, then, m, else, n], []).
T = int ;
?-tExpr(T,[if, a, then, b, else, ’*’, m, n], []).
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Prefix Expression DCG :
Type Checking and Evaluation Version
evalExpr(V) --> etExpr(V,_).
etExpr(V,T) --> [if], etExpr(B,bool), [then],
etExpr(V1,T), [else], etExpr(V2,T),
{B==true -> V = V1 ; V = V2}.
etExpr(V,bool) --> [’+’],
etExpr(V1,bool), etExpr(V2,bool),
{or(V1,V2,V)}.
etExpr(V,int) --> [’+’],
etExpr(V1,int), etExpr(V2,int),
{V is V1 + V2}.
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(cont’d)
etExpr(V,bool) --> [’*’],
etExpr(V1,bool), etExpr(V2,bool),
{and(V1,V2,V)}.
etExpr(V,bool) --> [’*’],
etExpr(V1,int), etExpr(V2,int),
{V is V1 * V2}.
etExpr(V,int)
--> [m], {value(m,V)}.
etExpr(V,int)
--> [n], {value(n,V)}.
etExpr(V,bool) --> [a], {value(a,V)}.
etExpr(V,bool) --> [b], {value(b,V)}.
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(cont’d)
value(m,10).
value(a,true).
value(n,5).
value(b,false).
and(true,true,true).
and(false,true,false).
and(true,false,false).
and(false,false,false).
or(true,true,true).
or(false,true,true).
or(true,false,true).
or(false,false,false).
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Prefix Expression DCG : AST Version
treeExpr(V) --> trExpr(V,_).
trExpr(cond(B,V1,V2),T) -->
[if], trExpr(B,bool), [then],
trExpr(V1,T), [else], trExpr(V2,T).
trExpr(or(V1,V2),bool) --> [’+’],
trExpr(V1,bool), trExpr(V2,bool).
trExpr(plus(V1,V2),int) --> [’+’],
trExpr(V1,int), trExpr(V2,int).
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(cont’d)
trExpr(and(V1,V2),bool) --> [’*’],
trExpr(V1,bool), trExpr(V2,bool).
trExpr(mul(V1,V2),int) --> [’*’],
trExpr(V1,int), trExpr(V2,int).
trExpr(m,int)
--> [m].
trExpr(n,int)
--> [n].
trExpr(a,bool) --> [a].
trExpr(b,bool) --> [b].
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Other Compiler Operations
• From parse tree and type information, one
can:
– compute (stack) storage requirements for
variables and for expression evaluation
– generate assembly code (with coercion
instructions if necessary)
– transform/simplify expression
•
Ref: http://www.cs.wright.edu/~tkprasad/papers/Attribute-Grammars.pdf
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Variation on Expression Grammars
Inefficient
Backtracking Parser Exists
E -> T + E | T
T -> F * T | F
F -> (E)
| x
| y
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Unsuitable Grammar
E -> E + E
| E * E
| x
| y
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RELATIONSHIP TO
ATTRIBUTE GRAMMARS
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Attribute Grammars
•
Formalism for specifying semantics based on
context-free grammars (BNF)
 Static semantics (context-sensitive aspects)
 Type checking and type inference
 Compatibility between procedure definition and call
 Dynamic semantics
•
•
Associate attributes with terminals and nonterminals
Associate attribute computation rules with
productions
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• Attributes
– Synthesized
– Inherited
A(X)
S(X)
I(X)
• Attribute computation rules (Semantic functions)
X0
->
X1
X2
…
Xn
S(X0) = f( I(X0), A(X1), A(X2), …, A(Xn) )
I(Xj) = Gj( I(X0), A(X1), A(X2), …, A(Xj-1))
for all j in 1..n
P( A(X0), A(X1), A(X2), …, A(Xn) )
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Information Flow
inherited
computed
available
synthesized
...
...
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• Synthesized Attributes
Pass information up the parse tree
• Inherited Attributes
Pass information down the parse tree or
from left siblings to the right siblings
• Attribute values assumed to be available from the
context.
• Attribute values computed using the semantic rules
provided.
The constraints on the attribute evaluation rules
permit top-down left-to-right (one-pass)
traversal of the parse tree to compute the meaning.
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An Extended Example
• Distinct identifiers in a straight-line program.
BNF
<exp> ::=
<var>
| <exp> + <exp>
<stm> ::= <var> := <exp> | <stm> ; <stm>
Attributes
<var>
<exp>
<stm>
 id
 ids
 ids
 num
• Semantics specified in terms of sets (of identifiers).
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<exp>
::= <var>
<exp>.ids = { <var>.id }
<exp>
::= <exp1> + <exp2>
<exp>.ids = <exp>.ids U <exp>.ids
<stm>
::=
<var> := <exp>
<stm>.ids ={ <var>.id } U <exp>.ids
<stm>.num = | <stm>.ids |
<stm>
<stm>.ids =
::=
<stm1> ; <stm2>
<stm1>.ids
U <stm2>.ids
<stm>.num = | <stm>.ids |
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Alternative Semantics using lists
• Attributes



envi : list of vars in preceding context
envo : list of vars for following context
dnum : number of new variables
<exp>
::=
<var>
<exp>.envo =
if member(<var>.id,<exp>.envi)
then
else
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<exp>.envi
cons(<var>.id,<exp>.envi)
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Attribute Computation Rules
<exp>



::=
envi
envo
dnum
<exp1>.envi
<exp2>.envi
<exp>.envo
<exp>.dnum
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<exp1> + <exp2>



=
=
=
=
envi
envo
dnum



envi
envo
dnum
<exp>.envi
<exp1>.envo
<exp2>.envo
length(<exp>.envo)
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