BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LX (LXIV), Fasc. 1, 2014
SecŃia
AUTOMATICĂ şi CALCULATOARE
ON THE DEVELOPMENT OF THE FRAENKEL-MOSTOWSKI
SET THEORY
BY
ANDREI ALEXANDRU∗ and GABRIEL CIOBANU
Romanian Academy, Institute of Computer Science, Iaşi
Received: February 17, 2014
Accepted for publication: March 24, 2014
Abstract. The Fraenkel-Mostowski set theory is an axiomatic set theory
which provides a mathematical model for names in syntax. We present an
overview of applications of the Fraenkel-Mostowski set theory in computer science
emphasizing our personal contributions in this topic. We also present the
possibilities of rephrasing mathematics according to the Fraenkel-Mostowski
axioms, and the development of another set theory (the Extended FraenkelMostowski set theory) obtained by replacing a strong axiom in the FraenkelMostowski set theory with a weaker one.
Key words: nominal set, finitely supported structure, Fraenkel-Mostowski
set theory.
2010 Mathematics Subject Classification: 03E25, 03E30, 08A70.
1. Introduction
The act of choosing a fresh name often arises when manipulating
syntactic expressions, and usually it is necessary to indicate some constraints
whenever describing such a syntactic manipulation. Sometimes it is just said
that a name is fresh, without specifying any restrictions. In such a case, we
∗
Corresponding author; e-mail: [email protected]
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Andrei Alexandru and Gabriel Ciobanu
mean that the fresh name must be different from any name occurring anywhere
else in the expression/program. Some programming systems have mechanisms
for renaming, for binding a name with a value and for managing sets of such
bindings. Modern programming languages are designed to manage bindings and
fresh names by using the notions of scopes, workspaces, or environments. Since
renaming, binding and fresh names appear in several approaches, it became
evident that they deserve to be studied in their own terms.
The Fraenkel-Mostowski (FM) set theory over an infinite set of atoms
represents the mathematical model for names in syntax. An atom is an entity
that contains no elements; the set of all atoms is generally denoted by A. Atoms
have the same properties as names in computer science. Most often the precise
nature of names is unimportant; what matters is their ability to identify and their
distinctness.
Let us consider X a set endowed with a group action of the group of all
finitary permutations of A . A set S of atoms supports an element x ∈ X if x
is invariant under every finitary permutation of atoms which is the identity on
S .The finite support axiom in the FM set theory claims that claims that for any
element in the theory there exists a finite set supporting it. This axiom is
motivated by the fact that syntax can only ever involve finitely many names.
The applications of the FM set theory have roots in various areas of
computer science as semantics, database theory, verification, programming,
proof theory, game theory, algebra, logic, topology, or automata theory. A
collection of applications of the FM set theory in computer science in presented
Section 2. This paper is a survey paper which presents the existing development
in the topic of the FM set theory, emphasizing our personal contributions in this
topic. We start reviewing our original contribution from the original motivation
of Franekel and Mostowski for defining a set theory with atoms. In Section 3
we present the relationship between the axioms of the Fraenkel-Mostowski set
theory and various forms of choice. In Section 4 we emphasize the benefits of
using the new set theory in the study of various process calculi. The main
benefit is that we are able to provide transition rules for various process calculi
using a mixing of binding operators instead of side conditions. In Section 5 we
present the existing development in the study of the so-called nominal algebraic
structures, and some new research directions. The nominal algebraic structures
are defined according to the finite support axiom in the FM set theory.
Therefore they are expressed in terms of finitely supported objects. These
nominal algebraic structures can be related to the usual algebraic structures as it
is emphasized in Section 5. Finally, in Section 6 we present the attempt of
extending the FM set theory by replacing the finite support axiom with a weaker
axiom. The impact of this action in algebra, topology and logic is analyzed in a
series of papers presented in Section 6.
Bul. Inst. Polit. Iaşi, t. LX (LXIV), f. 1, 2014
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2. Motivation and Related Work
Nominal sets represent a different set theory which allows a more
relaxed interpretation for the notion of finiteness. A nominal set X is defined as
a set endowed with a group action of the group of all finitary permutations of A
with the additional property that any element of X is finitely supported. An
empty supported element in a nominal set is called equivariant. Nominal sets
offer an elegant formalism for describing λ-terms modulo α-conversion, or
automata on data words. They also serve as a good framework for database
theory since atoms can be used as an abstraction for data values, which can
appear in a relational database or in an XML document. Atoms can also be used
to model sources of infinite data in other applications, such as software
verification, where an atom can represent a pointer or the contents of an array
cell. The theory of nominal sets was developed initially by Fraenkel and
Mowstowski in 1930s, and it is also known as the FM set theory. At that time,
nominal sets were used to prove independence of the axiom of choice and other
axioms in the classical Zermelo-Fraenkel (ZF) set theory. In computer science,
nominal sets have been rediscovered (Gabbay & Pitts, 2001) to properly model
the syntax of formal systems involving variable binding operations. Since then,
nominal sets have become a lively topic in semantics. An advantage of
modeling syntax in a model of FM set theory is that datatypes of syntax modulo
α-equivalence can be modeled inductively because FM set theory provides a
model of variable symbols and α-abstraction. Nominal sets are used to define
new semantics for various process algebras; these semantics have the property
that the transition rules are expressed by using binding operators instead of side
conditions (Alexandru & Ciobanu, 2013a; Alexandru & Ciobanu, 2012c;
Alexandru & Ciobanu, 2012d). In (Bengtson & Parrow, 2009) the π-calculus is
also formalized in Isabelle using the nominal datatype package (Urban, 2008).
Nominal sets were also independently rediscovered by the concurrency
community, as a basis for syntax-free models of name-passing process algebras
(Montanari & Pistore, 2005), and used in automata theory as a framework for
describing automata on data words (Bojanczyk et al., 2011). History-dependent
automata (HD-automata) descibed in (Montanari & Pistore, 2005) and (Pistore,
1999) have been developed in order to check π-calculus expressions for
bisimilarity. HD-automata are internal in the sense of (Arbib & Manes, 1974) in
the category of named sets (Ciancia, 2008) which is equivalent with the
category of nominal sets according to (Fiore & Staton, 2006; Gadducci et al.,
2006). In (Kurz et al., 2012) formal languages over infinite alphabets where
words may contain binders are introduced. HD-automata is extended by adding
stack, and study the recognisability of nominal languages. In (Bojanczyk, 2011)
the monoids defined in the category of nominal sets (also called nominal
monoids) are used in the study of languages over infinite alphabets. The theory
of syntactic monoids for languages of data words represents the same theory as
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Andrei Alexandru and Gabriel Ciobanu
the theory of finite monoids in the category of nominal sets, and under certain
conditions, a language of data words is definable in first-order logic if and only
if its syntactic monoid is aperiodic (Bojanczyk, 2013b). Nominal techniques are
used in (Shinwell & Pitts, 2005) in order to implement a functional
programming language incorporating facilities for manipulating syntax
involving names and binding operations. Computation in nominal sets has
also been defined in (Bojanczyk et al., 2012) by presenting a basic functional
programming language called Nλ. An imperative programming language
which extends while programs and works with nominal sets is introduced in
(Bojanczyk & Torunczyk, 2012). Unlike in Nλ, in the programming language
presented in (Bojanczyk & Torunczyk, 2012) the author was able to correctly
type a program for minimising the deterministic orbit-finite automata
introduced in (Bojanczyk, Klin & Lasota, 2011). A more recent paper
(Bojanczyk, 2013a) studies a variant of the first-order logic in the framework
of nominal sets and presents a notion of model for this logic (the so-called
‘stratified models’) which is well founded according to the FM axioms and,
furthermore, admits compactness. Nominal algebraic structures are presented
in terms of finitely supported objects in (Alexandru & Ciobanu, submitted_b;
Alexandru & Ciobanu, 2014c; Alexandru & Ciobanu, 2013b; Alexandru &
Ciobanu, 2012b). The solution of the Scott recursive domain equation
D ≅ ( D → D ) in the FM framework is presented in (Shinwell, 2005).
Nominal techniques have also been used in game theory (Abramski et al.,
2004), in logic (Gabbay & Mathijssen, 2008; Pitts, 2003), in topology
(Petrisan, 2011), in domain theory (Shinwell, 2005; Turner, 2009), and in
proof theory (Urban, 2008).
The FM axioms are precisely the Zermelo-Fraenkel with atoms (ZFA)
axioms over an infinite set of atoms (Gabbay & Pitts, 2001), together with the
special property of finite support which claims that for each element x in an
arbitrary set we can find a finite set supporting x. Hence in the FM universe
only finitely supported objects are allowed. The FM set theory represents the
mathematical model for names in syntax; the finite support property is
motivated by the fact that syntax can only ever involve finitely many names.
A motivating example is that in λ-calculus, the set of α-equivalence classes of
λ-terms t can be modelled as a nominal set; if t is chosen to be a representative
of its α-equivalence class, then the least set supporting t (i.e. the support of t)
coincides with the set of free variables of t.
3. The Axiom of Choice
The Axiom of Choice (AC) is probably the most discussed axiom in
mathematics after Euclid’s Axiom of Parallels. The first formulation of AC is
due to Zermelo (Zermelo, 1904). It claims that given any family of nonempty
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set F , it is possible to select a single element from each member of F . Various
weaker forms of the axiom of choice, also called choice principles, are
presented in Chapter 2 from (Herrlich, 2006). Many areas in mathematics,
especially in functional analysis, real analysis, topology, the theory of
differential equations, the theory of algebraic structures, measure theory, firstorder logic, and category theory heavily depends on choice principles. The
axiom of choice has a lot of elegant consequences, but that is an argument for
its mathematical interest, not for its truth. Since its first postulation the axiom of
choice became the subject of many controversies. The first controversy is about
the meaning of the word “exists” since this term is very vague. One group of
mathematicians (called intuitionists) believes that a set exists only if each of its
elements can be designated specifically or at least if there is a law by which
each of its elements can be constructed. Another controversy is represented by a
geometrical consequence of AC known as the Banach and Tarski’s paradoxical
decomposition of the sphere. In (Banach & Tarski, 1924) they showed that any
solid sphere can be split into finitely many subsets which can themselves be
reassembled to form two solid spheres, each of the same size as the original;
and any solid sphere can be split into finitely many subsets in such a way as to
enable them to be reassembled to form a solid sphere of arbitrary size.
Questions about the AC’s independence of the systems of set-theoretic axioms
appeared naturally. In 1922 Fraenkel introduces the permutation method to
establish the independence of AC from a system of set theory with atoms
(Fraenkel, 1922). That is, Fraenkel constructed a model in which the axioms of
set theory excluding the axiom of choice are satisfied but this model contains a
set which does not satisfy the axiom of choice. Fraenkel’s model was refined
and extended by Lindenbaum and Mostowski (1938). Inspired by the ZF
models of Fraenkel and Mostowski, in 2000s the FM set theory was described
axiomatically as an independent set theory, named the Fraenkel-Mostowski set
theory, by two reserchers from Cambridge University: M. Gabbay and A. Pitts
(2001). In (Gödel, 1940) Gödel proved that the axiom of choice is consistent
with the other axioms of set theory (von Neumann-Bernays-Gödel set theory).
He proved that given a model for set theory in which there are no atoms and the
axiom of foundation is true, there exists a model in which, in addition, the
axiom of choice is true. Moreover, if Gödel’s model is modified so that either
atoms exist or the axiom of foundation is false, the validity of the axiom of
choice is not disturbed. Therefore, the axiom of choice is consistent with the
other axioms of set theory whether there exist or not atoms, and whether the
axiom of foundation is valid or not. Fraenkel shows that the collections of sets
of atoms need not necessarily have choice functions (Fraenkel, 1922). However,
at that time it couldn’t establish the same fact for the usual sets of mathematics,
for example the set of real numbers. This problem remains unsolved until 1963
when Cohen proved the independence of AC (and of the axiom of countable
choice) from the standard axioms of set (Cohen, 1963a; Cohen, 1963b; Cohen,
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1964). Cohen’s independence proof (caled the method of forcing) also made use
of permutation in essentially the form in which Fraenkel had originally
employed them.
According to (Gabbay & Pitts, 2001) the full axiom of choice (AC) is
inconsistent with the axioms of the FM set theory. An original result of the
authors is stronger and provides a comparison between the finite support axiom
in the FM set theory and various weaker forms of choice. In (Alexandru &
Ciobanu, submitted_a) the choice principles presented in Chapter 2 from
(Herrlich, 2006) are rephrased and compared in the FM framework by
considering that each object appearing in their statement is finitely supported.
For example, AC has in the FM framework the form “Given any nominal set X,
and any finitely supported family F of nonempty finitely supported subsets of
X, there exists a finitely supported choice function on F ”, DC has in the FM
framework the form “Let R be a nonempty finitely supported relation on a
nominal set X with the property that for each x ∈ X there exists y ∈ X with xRy.
Then there exists a finitely supported function f :ω → X such that
f ( n ) Rf ( n + 1) , ∀n ∈ ω ”, and so on (Alexandru & Ciobanu, submitted_a).
According to (Alexandru & Ciobanu, submitted_a), the choice principles AC,
DC, CC, PCC, AC(fin), Fin, PIT, UFT, OP, KW and OEP presented in
(Herrlich, 2006) (and rephrased in (Alexandru & Ciobanu, submitted_a) in the
FM framework, in terms of finitely supported objects) are inconsistent with the
axioms of the FM set theory for all possible sorts of atoms. In the particular
case when the set of atoms is countable, the choice principle CC(fin) is also
inconsistent with the axioms of the FM set theory.
4. Nominal Sets in Process Algebras
Process algebras are used as a formal framework for the study of
concurrent computation. In (Alexandru & Ciobanu, 2013a; Alexandru &
Ciobanu, 2012c; Alexandru & Ciobanu, 2012d) we provide new (nominal)
transition rules for several process algebras. In order to reach our goal we make
use of the nominal quantifier “new” introduced in (Gabbay & Pitts, 2001). The
meaning of the quantifier “new” is as follows: for a predicate P on A , we have
newa.P (a) if P (a) is true for all but finitely many atoms in A . We use “new”
to encode the freshness conditions in the hypothesis of the transition rules of
these process algebras, and we obtain some sets of compact transition rules
which defines the nominal semantics of the related process algebras. The central
idea is to use atoms to represent variable symbols, and the nominal abstraction
defined in (Gabbay & Pitts, 2001) to represent the binding operators in various
process algebras. A mixing of the quantifiers ∀ and “new” is used to replace
the side conditions in the transition rules of these process algebras. Finally, we
are able to make a comparison between the expressive prower of the usual
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semantics and of the nominal semantics of the related process algebras.
According to Theorem 4.1. in (Alexandru & Ciobanu, 2012c) the new nominal
semantics of the monadic fusion calculus and the original semantics of the
monadic fusion calculus presented in (Parrow & Victor, 1997) have the same
expressive power. The nominal transition rules of the πI-calculus defined in
(Alexandru & Ciobanu, 2013a) and the original transition rules of the πIcalculus presented in (Sangiorgi, 1995) provide the same transitions; this is
proved as Theorem 4.11 in (Alexandru & Ciobanu, 2013a). Also, according to
Theorem 6.4 in (Alexandru & Ciobanu, 2012d), the nominal semantics of the πcalculus defined in (Alexandru & Ciobanu, 2012d) and the late semantics of the
π-calculus presented in (Sangiorgi & Walker, 2001) are equivalent.
There exists many papers where process algebras are modeled by using
different techniques for binding. In (Hirschkoff, 1997), D. Hirschkoff
formalised a subset of the π-calculus excluding match, mismatch and sum in
Coq by using de Bruijn indices. Higher order abstract syntax (HOAS) was used
to model the π-calculus in both Isabelle (Rockl & Hirschkoff, 2003) and in Coq
(Honsell et al., 2001). However, approaches based on HOAS can suffer by
some problems. For example the function spaces can be too large. Also,
function spaces can destroy the inductive structure. When using HOAS terms,
binders are represented as functions of type name → term. If these functions
range over the entire function space they may produce exotic terms, so the
formalisations have to ensure that those terms are avoided. The nominal logic is
a first order logic (Pitts, 2003) and, hence, exotic terms can not appear.
Moreover, in the nominal framework the existence of fresh names is
axiomatically assumed (each time when a such a name is requested) whilst in
HOAS, if we need to generate names, we may need to index all predicates and
relations by an explicit set of known names (Honsell et al., 2001).
Nominal techniques have also been used in order to formalise the πcalculus. In (Bengtson & Parrow, 2009) the π-calculus is formalised in Isabelle
using the nominal datatype package (Urban, 2008). The nominal techniques
used in (Bengtson & Parrow, 2009) are quite different from those proposed by
us because the nominal quantifier is not used in (Bengtson & Parrow, 2009).
Our nominal semantics of process algebras are based on the nominal quantifier
which is used in order to replace the side conditions in the hypothesis of the
transition rules. In paper (Gabbay, 2003), M.Gabbay also uses nominal
techniques to obtain new transition rules for the π-calculus. In (Alexandru &
Ciobanu, 2012d) we made a refinement of the transition rule L − CLOSEL such
that we assumed a similar rule to be valid only when some freshness conditions
are satisfied. These freshness conditions are motivated by the manner in which
the links are moved in a virtual space of linked processes. In the new semantics
of the π-calculus defined in (Alexandru & Ciobanu, 2012d), which is different
from the related nominal semantics of the π-calculus defined in (Gabbay, 2003),
we can explain mobility using a transition rule which is assumed to be valid
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only in a more restrictive hypothesis than its related rule in the late semantics of
the π-calculus. Moreover, in (Alexandru & Ciobanu, 2012d) we presented a
nominal semantics of the π-calculus without using a structural congruence, and
we proved that, even one transition rule in the nominal semantics of the πcalculus is assumed to be valid only in a more restricted hypothesis than its
related rule in the late semantics of the π-calculus, we are able to establish a
complete equivalence between the nominal semantics and the late semantics of
the π-calculus. This equivalence was proved using the finite support axiom in
the FM framework.
5. Algebraic Structures in the Framework
of Nominal Sets
Various algebraic structures involved in computer science are presented
in the FM framework emphasizing a strong connection between the FM
properties of these algebraic structures and their related ZF properties
(Alexandru & Ciobanu, submitted_b; Alexandru & Ciobanu, 2014c; Alexandru
& Ciobanu, 2013b; Alexandru & Ciobanu, 2012b). The general idea in these
papers is to rephrase the classical ZF properties of some algebraic structures by
replacing ‘object’ with ‘finitely supported object’, and to prove that the validity
of the ZF properties of these algebraic structures is preserved in the new
framework. Informally, this approach allow us to replace ‘finite’ with ‘infinite
but with finite support’. Obviously, any ZF set is a particular nominal set
equipped with a trivial permutation action. Therefore the general properties of
nominal sets lead to valid properties of ZF sets. However, the converse is not
always valid, that is, not every ZF result can be directly rephrased in the world
of nominal sets, in terms of finitely supported objects according to arbitrary
permutation actions. This is because, given a nominal set X, there could exist
some subsets of X which fail to be finitely supported. This is the reason why
the translation of a ZF result to the framework of nominal sets deserves a
special attention.
In (Alexandru & Ciobanu, submitted_b) we extend the notion of
multiset over a finite alphabet by considering the notion of extended nominal
multiset. A multiset on an alphabet Σ is a function from Σ to the set of all
natural numbers, where each element in Σ has associated its multiplicity. An
extended nominal multiset is actually an algebraically finitely supported
multiset over a possibly infinite alphabet. We study the correspondence between
some properties of multisets obtained in the Fraenkel-Mostowski framework
where only finitely supported objects are allowed, and those obtained in the
classical Zermelo-Fraenkel framework. Analogue the generalized multisets, i.e.,
the multisets with possibly negative multiplicities are translated in (Alexandru
& Ciobanu, 2013b) to the FM framework, and presented in terms of finitely
supported objects. Using the finite support property, many ZF properties of a
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85
generalized multisets on a finite alphabet still hold when the finite alphabet is
replaced by an infinite alphabet with the mention that the generalized multisets
have to be algebraically finitely supported.
A nominal theory for partially ordered sets was first developed by
Pitts (2013) and Shinwell (2005). In this topic we also proved some
important results. An example of such a result is represented by the Tarskilike theorem for nominal complete lattices (we mention that a nominal
complete lattice nominal set endowed with an equivariant lattice order and
with the additional property that any finitely supported subset has a least
upper bound according to this order), i.e., Theorem 3.3. in (Alexandru &
Ciobanu, 2012b) which was originally presented in terms of event structures.
Rephrased in terms of latices which states that given any nominal complete
lattice L and any equivariant (i.e. empty supported) monotone function f over
L, the set of fixed points of f form another nominal complete lattice. Using
this fixpoint theorem and some equivariant Galois connections we are able to
develop a nominal theory of abstract interpretation of programming
languages and to approximate finitely supported subsets of a possibly infinite
nominal set in terms of nominal complete Boolean lattices. Presenting this in
detail represents our future work.
Nominal groups are defined as groups which are also nominal sets and
whose internal laws are equivariant. Several algebraic properties of nominal
groups are presented in Section 3 from (Alexandru & Ciobanu, 2014c). The
classical correspondence and isomorphism theorems from the ZF groups theory
are translated in the FM framework in terms of equivariant (empty supported)
homomorphisms in Section 4 from (Alexandru & Ciobanu, 2014c). Also,
uniform nominal groups (a uniform nominal group is a group with the property
that there exist a finite set which supports all the elements in the group) are
defined in Section 5 from (Alexandru & Ciobanu, 2014c), and several nominal
embedding theorems valid for this class of groups are presented. Nominal
groups are, actually, the natural extension of nominal monoids defined in
(Bojanczyk, 2013b). However, the presence of the inverse elements in a group
allows us to think to the study of reversibility in nominal terms. In (Alexandru
& Ciobanu, 2014c) we present an algebraic framework from such a future
development. The wreath product and one of its generalizations are used in the
algebraic theory of automata in order to prove the Krohn-Rhodes theorem
which states that any deterministic automaton is a homomorphic image of a
cascade of some simple automata which realize either resets or permutations
(Krohn & Rhodes, 1965). According to Theorem 3.20. in (Alexandru &
Ciobanu, 2014c), the regular wreath product of two nominal groups is also
nominal. The cascade product of two (nominal) automata can be studied in a
similar way. We conjecture that a similar decomposition theorem for a class of
nominal automata can be presented in the nominal framework in terms of
nominal wreath products. Presenting this also represents future work.
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The classical theory of nominal sets over a fixed set A of atoms is
generalized in (Bojanczyk et al., 2011) to a new theory of nominal sets over
arbitrary unfixed sets of data values. In the standard theory of nominal sets, S A
represents the group of all finitary permutations on A , and an S A -set is a ZF set
X together with a group actions of S A on X . In (Bojanczyk et al., 2011), the
notion of ‘SA-set’ is replaced by the notion of ‘set endowed with an action of a
subgroup of the symmetric group of D’ for an arbitrary set of data values D, and
the notion of ‘finite set’ is replaced by the notion of ‘set with a finite number of
orbits according to the previous group action (orbit-finite set)’. The theory of
automata have been studied in this framework (Bojanczyk et al., 2011).
According to definitions in (Bojanczyk et al., 2011) the set A of atoms is a
single-orbit set. However the set of all extended nominal multisets on A
(denoted by Next(A)) has an infinite number of orbits because if two functions
from Next(A) are in the same orbit, then their correspondent algebraic supports
must have the same cardinality. In (Alexandru & Ciobanu, submitted_b) we are
able to develop a nominal theory of multisets even when the alphabet is
possibly infinite and the set of all multisets over the related alphabet is not finite
nor orbit finite. Informally, our approach extends the framework from
‘finite/orbit-finite’ to ‘infinite/orbit-infinite but with finite algebraic support’.
Bojanczyk introduces a notion of nominal monoid over arbtirary data
symmetries (Bojanczyk, 2013b) in order to prove that a language of data words
is definable in first-order logic if and only if its syntactic monoid is aperiodic.
We have a little different perspective. The notion of nominal monoid (nominal
set together with an equivariant monoid internal law) introduced in (Alexandru
& Ciobanu, submitted_b), which is similar to the notion of nominal group
introduced in (Alexandru & Ciobanu, 2014c), makes sense only when the set of
atoms is fixed, and nominal monoids are used in order to obtain nominal
properties of multisets and to connect several such nominal properties in terms
of finitely supported homomorphisms. For example in (Alexandru & Ciobanu,
submitted_b) we proved that the set of all extended multisets over a (possibly
infinite) nominal set Σ is a free abelian nominal monoid, and the free monoid
over Σ is also a nominal monoid. Moreover, these monoids satisfy some
universality properties, and they can be connected by using nominal
isomorphism theorems similar to those presented in (Alexandru & Ciobanu,
2014c). Analogue results are obtained in (Alexandru & Ciobanu, 2013b) for the
set of all generalized multisets over a possibly infinite alphabet. Bojanczyk uses
alternative definitions for nominal algebraic structures in the generalized
framework of G-sets described (Bojanczyk, Klin & Lasota, 2011), in order to
study languages of data words. In (Alexandru & Ciobanu, submitted_b;
Alexandru & Ciobanu, 2014c; Alexandru & Ciobanu, 2013b; Alexandru &
Ciobanu, 2012b) we present the nominal algebraic structures in the classical
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87
framework of nominal sets in order to prove that several classical ZF properties
of algebraic structures have related FM correspondents. Therefore we focus on
an algebraic study of nominal structures. Presenting isomorphism, embedding,
fixed point, correspondence, decomposing, nominality-preserving theorems was
not the object of Bojanczyk’s papers, and represent the original part of our
papers (Alexandru & Ciobanu, submitted_b; Alexandru & Ciobanu, 2014c;
Alexandru & Ciobanu, 2013b; Alexandru & Ciobanu, 2012b).
Note that extending a ZF algebraic structure to an FM algebraic
structure is not trivial, because we can not always obtain an FM result starting
from a ZF result only by replacing ‘structure’ with ‘nominal structure’ in a
natural way. For example, the nominal embedding theorems for groups, of
Cayley-type and core-type, presented in Section 5 from (Alexandru & Ciobanu,
2014c) can be proved only for a particular class of nominal groups namely the
uniform nominal groups, and not for all nominal groups. Another example of
why extending algebraic structures to nominal algebraic structures is not trivial
and deserves a special attention is presented in (Alexandru & Ciobanu, 2012b).
Theorem 3.3. from (Alexandru & Ciobanu, 2012b) shows that the Tarski
Theorem for complete lattices cannot be translated in FM using a standard
algorithm. The FM Tarski-like theorem is not valid for all finitely supported
functions defined on an FM complete lattice, but only for those functions which
are equivariant. Also, we cannot prove an FM result only involving a ZF result
(without an additional proof presented according to the FM axioms). There exist
a lot of results which are valid in the ZF framework, but fail in the FM
framework. This thing could happen because, in our approach, the FM set
theory is not a model of the ZF set theory, but an independent axiomatic set
theory. The classical example is represented by the statement “The axiom of
choice is consistent”. This statement is a valid theorem in the ZF framework,
but fails in the FM framework. Moreover, as proved in (Alexandru & Ciobanu,
submitted_a), all the ZF choice principles presented in (Herrlich, 2006) are
inconsistent with the axioms of the FM set theory. Another example of a
mathematical result which fails in the FM framework is the Stone representation
theorem for Boolean lattices (claiming that every Boolean lattice is isomorphic to
the dual algebra of its associated Stone space). According to Section 5.2. from
(Petrisan, 2011), Stone duality fails in the framework of nominal sets because its
proof would require a choice principle, namely the ultrafilter theorem, and this
theorem is proved to be inconsistent with the FM axioms according to Proposition
5.2.2. from (Petrisan, 2011). Other results which fail in the FM settings, such as
determinization of finite automata, or equivalence of two-way and one-way finite
automata, are presented in (Bojanczyk et al., 2011).
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Andrei Alexandru and Gabriel Ciobanu
6. Extensions of the Fraenkel-Mostowski Set Theory
The finite support axiom of the FM set theory is very strong. In
(Alexandru & Ciobanu, 2012a) we study the consequences of replacing this
strong axiom with a weaker one. We defined the Extended Fraenkel-Mostowski
(EFM) set theory by replacing the finite support axiom with an implication of it
which states that each subset of A is either finite or cofinite. The main goal of
describing the EFM set theory was to prove that some properties of sets which
are valid in the FM framework remain also valid if instead of the finite support
axiom in the description of FM set theory we introduce a related weaker axiom.
According to (Alexandru & Ciobanu, 2014b; Alexandru & Ciobanu, 2012a) we
do not necessary need to assume that each element in each set is finitely
supported in order to prove some algebraic, order and topological properties of
sets; it is enough to require only that the set of atoms has an amorphous
structure. The main result proved in (Alexandru & Ciobanu, 2012a) states that
the group of all bijections on the set of atoms is locally finite if we work in the
EFM framework. The notion of renaming can be generalized in the EFM set
theory. According to (Alexandru & Ciobanu, 2014a) the permutative renamings
defined in the EFM framework have similar combinatorial properties as the
clasical permutative renamings defined in (Gabbay & Hofmann, 2008).
According to (Alexandru & Ciobanu, submitted_a) the choice principles AC, DC,
CC, PCC, AC(fin), Fin, PIT, UFT, OP, KW and OEP presented in (Herrlich,
2006) are inconsistent with the axioms of the EFM set theory for all set of atoms.
In the particular case when the set of atoms is countable, the choice principle
CC(fin) is also inconsistent with the axioms of the EFM set theory.
REFERENCES
Abramsky S., Ghica D.R., Murawski A.S., Luke Ong C.H., Stark I.D.B., Nominal
Games and Full Abstraction for the Nu-Calculus. In Proc. of 19th IEEE
Symposium on Logic in Computer Science, IEEE Computer Society Press,
150−159, 2004.
Alexandru A., Ciobanu G., Choice Principles in Finitely Supported Mathematics.
submitted_a.
Alexandru A., Ciobanu G., Mathematics of Multisets in the Fraenkel-Mostowski
Framework. submitted_b.
Alexandru A., Ciobanu G., Permutative Renamings in Extended Fraenkel-Mostowski
Model. Scientific Annals of “Al. I. Cuza” University (in press, 2014a).
Alexandru A., Ciobanu G., A Topological Approach in the Extended FraenkelMostowski Model. Scientific Annals of “Al. I. Cuza” University (online
available) (in press, 2014b).
Alexandru A., Ciobanu G., Nominal Groups and their Homomorphism Theorems.
Fundamenta Informaticae, 131, 3-4, 279−298, 2014c.
Bul. Inst. Polit. Iaşi, t. LX (LXIV), f. 1, 2014
89
Alexandru A., Ciobanu G., Nominal Techniques for πI-Calculus. Romanian Journal of
Information Science and Technology, 16, 4, 261−286, 2013a.
Alexandru A., Ciobanu G., Algebraic Properties of Generalized Multisets. In Proc. of
15th International Symposium on Symbolic and Numeric Algorithms for
Scientific Computing, IEEE Computer Society Press, 369−377, 2013b.
Alexandru A., Ciobanu G., An Extension of a Permutative Model of Set Theory.
Scientific Annals of “Al. I. Cuza” University, LVIII, s.1, 1−18, 2012a.
Alexandru A., Ciobanu G., Nominal Event Structures. Romanian Journal of Information
Science and Technology, 15, 2, 79−90, 2012b.
Alexandru A., Ciobanu G., Nominal Fusion Calculus. In Proc. of 14th International
Symposium on Symbolic and Numeric Algorithms for Scientific Computing,
IEEE Computer Society Press, 376−384, 2012c.
Alexandru A., Ciobanu G., Nominal Semantics of Mobility. Romanian Journal of
Information Science and Technology, 15, 4, 171−214, 2012d.
Arbib M.A., Manes E.G., Machines in a Category: An Expository Introduction. SIAM
Review, 16, 285−302, 1974.
Banach S., Tarski A., Sur la decomposition des ensembles de points en parties
respectivement congruentes. Fundamenta Mathematicae, 6, 244−277, 1924.
Bengtson J., Parrow J., Formalising the π-Calculus Using Nominal Logic. Logical
Methods in Computer Science, 5, 1−36, 2009.
Bojanczyk M., Modelling Infinite Structures with Atoms. In Proc. of 20th Workshop on
Logic, Language, Information and Computation, Lecture Notes in Computer
Science, 8071, 13−28, 2013a.
Bojanczyk M., Nominal Monoids. Theory of Computing Systems, 53, 194−222, 2013b.
Bojanczyk M., Data Monoids. In Proc. of 28th Symposium on Theoretical Aspects of
Computer Science, 105−116, 2011.
Bojanczyk M., Braud L., Klin B., Lasota S., Towards Nominal Computation. In Proc. of
39th ACM SIGPLAN-SIGACT Symposium on Principles of Programming
Languages, 401−412, 2012.
Bojanczyk M., Klin B., Lasota S., Automata with Group Actions. In Proc. of 26th IEEE
Symposium on Logic in Computer Science, IEEE Computer Society Press,
355−364, 2011.
Bojanczyk M., Torunczyk S., Imperative Programming in Sets with Atoms. In Proc. of
32nd Foundations of Software Technology and Theoretical Computer Science
Conference, LIPIcs, 18, 4−15, 2012.
Ciancia V., Nominal Sets. Accessible Functors and Final Coalgebras for Named Sets,
Ph. D. Thesis, Dipartimento di Informatica, Universita di Pisa, 2008.
Cohen P.J., The Independence of the Axiom of Choice. Mimeographed, 1963a.
Cohen P.J., The Independence of the Continuum Hypothesis. I. Proc. Nat. Acad. Sci.
USA, 50, 1143−1148, 1963b.
Cohen P.J., The Independence of the Continuum Hypothesis. II. Proc. Nat. Acad. Sci.
USA, 51, 105−110, 1964.
Fiore M., Staton S., Comparing Operational Models of Name-Passing Process Calculi.
Information and Computation, 204, 4, 524−560, 2006.
Fraenkel A., Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre.
Mathematische Annalen, 86, 230−237, 1922.
90
Andrei Alexandru and Gabriel Ciobanu
Gabbay M.J., The π-Calculus in FM. Thirty Five Years of Automating Mathematics,
Kluwer Applied Logic, 28, 247−269, 2003.
Gabbay M.J., Hofmann M., Nominal Renaming Sets. Lecture Notes in Computer
Science, 5330, 158−173, 2008.
Gabbay M.J., Mathijssen A., One-and-a-Halfth-Order Logic. Journal of Logic and
Computation, 18, 521−562, 2008.
Gabbay M.J., Pitts A., A New Approach to Abstract Syntax with Variable Binding.
Formal Aspects of Computing, 13, 341−363, 2001.
Gadducci F., Miculan M., Montanari U., About Permutation Algebras, (Pre)Sheaves and
Named Sets. Higher-Order and Symbolic Computation, 19, 2-3, 283−304, 2006.
Godel K., The Consistency of the Axiom of Choice and of the Generalized ContinuumHypothesis with the Axioms of Set Theory. Annals of Mathematics Studies, No.
3, Princeton University Press, 1940.
Herrlich H., Axiom of Choice. Lecture Notes in Mathematics, Springer, Berlin, 2006.
Hirschkoff D., A Full Formalisation of π-Calculus Theory in the Calculus of
Constructions. In Proc. of 10th International Conference on Theorem Proving
in Higher Order Logics, 153–169, 1997.
Honsell F., Miculan M., Scagnetto I., π-Calculus in (Co)Inductive Type Theory.
Theoretical Computer Science, 253, 239−285, 2001.
Krohn K., Rhodes J., Algebraic Theory of Machines. Prime Decomposition Theorem for
Finite Semigroups and Machines. Transactions of the American Mathematical
Society, 116, 450−464, 1965.
Kurz A., Suzuki T., Tuosto E., Towards Nominal Formal Languages. In Proc. of 15th
Conference of Foundations of Software Science and Computational Structures,
Lecture Notes in Computer Science, 7213, 255−269, 2012.
Lindenbaum A., Mostowski A., Uber die Unabhangigkeit des Auswahlsaxioms und
einiger seiner Folgerungen. Comptes Rendus des Seances de la Societe des
Sciences et des Lettres de Varsovie, 31, 27−32, 1938.
Montanari U., Pistore M., History-Dependent Automata: an Introduction. In Proc. of
5th International Conference on Formal Methods for the Design of Computer,
Communication, and Software Systems: Mobile Computing, 1−28, 2005.
Parrow J., Victor B., The Update Calculus. In Proc. of 6th International Conference on
Algebraic Methodology and Software Technology, Lecture Notes in Computer
Science, 1349, 409−423, 1997.
Petrisan D., Investigations into Algebra and Topology over Nominal Sets. Ph. D. Thesis,
University of Leicester, 2011.
Pistore M., History Dependent Automata. Ph. D. Thesis, Dip. di Informatica, Pisa, 1999.
Pitts A., Nominal Sets Names and Symmetry in Computer Science. Cambridge
University Press, 2013.
Pitts A., Nominal Logic, a First Order Theory of Names and Binding. Information and
Computation,186, 165−193, 2003.
Rockl C., Hirschkoff D., A Fully Adequate Shallow Embedding of the π-Calculus in
Isabelle/HOL with Mechanized Syntax Analysis. Journal of Functional
Programming, 13, 415-451, 2003.
Sangiorgi D., π-Calculus, Internal Mobility, and Agent-Passing Calculi. Rapport INRIA
No. 2539, 1995.
Bul. Inst. Polit. Iaşi, t. LX (LXIV), f. 1, 2014
91
Sangiorgi D., Walker D., The π-Calculus: A Theory of Mobile Processes. Cambridge
University Press, 2001.
Shinwell M.R., The Fresh Approach: Functional Programming with Names and
Binders. Ph. D. Thesis, University of Cambridge, Computer Laboratory,
February 2005.
Shinwell M.R., Pitts A., Fresh Objective Caml User Manual. Tech. Report UCAM-CLTR-621, University of Cambridge, 2005.
Turner D., Nominal Domain Theory for Concurrency. Tech. Report, 751, University of
Cambridge, 2009.
Urban C., Nominal Techniques in Isabelle/HOL. Journal of Automated Reasoning, 40,
327−356, 2008.
Zermelo E., Beweis, daβ jede Menge wohlgeordnet werden kann. Mathematische
Annalen, 59, 514−516, 1904.
DEZVOLTAREA TEORIEI FRAENKEL-MOSTOWSKI
A MULłIMILOR
(Rezumat)
Teoria Fraenkel-Mostowski (FM) a mulŃimilor îşi are originile în abordarea
oferită de Fraenkel şi Mostowski în anii 1930 pentru a proba independenŃa axiomei
alegerii faŃă de celelalte axiome ale teoriei Zermelo-Fraenkel (ZF) a mulŃimilor. Pornind
de la modelul permutativ al teoriei ZF a mulŃimilor definit de Fraenkel în 1922 şi extins
de Mostowski şi Lindenbaum în 1938, în anii 2000 Gabbay şi Pitts definesc o teorie
axiomatică a mulŃimilor, independentă de teoria axiomatică ZF, care reprezintă modelul
matematic pentru nume în sintaxă. Axioma de suport finit din această nouă teorie este
motivată de faptul că sintaxa utilizează în fapt doar un numar finit de nume (acele nume
libere). În această lucrare de sinteză prezentăm dezvoltarea teoriei Fraenkel-Mostowski
a mulŃimilor de la origini până în prezent, evidenŃiind aplicaŃiile sale în matematică şi
informatica teoretică. Deasemenea punctăm în special contribuŃiile personale avute în
ultimii ani în acest topic. Acestea s-ar rezuma prin: studierea consistenŃei unor principii
de alegere cu axiomele FM, definirea unor semantici în acord cu axiomatica FM pentru
diferite algebre de procese, rescrierea algebrei în universul FM şi extinderea teoriei FM
prin relaxarea unei axiome.