Modal Logic and Non-Well-Founded Set Theory: Translation, Bisimulation, Interpolation
Giovanna d'Agostino
Abstract:
%Nr: DS-1998-04
%Title: Modal Logic and Non-Well-Founded Set Theory: Translation,
% Bisimulation, Interpolation
%Author: Giovanna d'Agostino
The notion of bisimulation is simple, natural, and central for many research
fields in Logic and Computer Science. It is the link that connects all the
topics discussed in this dissertation.
Bisimulation was proposed independently in many areas in the seventies.
Probably the first one was Modal Logic where, with the name of p-relation (see
[12]) it applies to Kripke models and it is an elegant and useful tool to prove
a large number of results (interpolation theorems, preservation theorems, etc.).
In Theoretical Computer Science the bisimulation relation applies to labeled
transition systems (see [54], [52]), which is another word for Kripke models.
Labeled transition systems are used to represent processes: the basic idea is
just to interpret nodes as possible states of the processes, unary relations as
properties of states, and binary relations as atomic actions that the processes
may undertake. Bisimulation (and its variations) can be considered as
equivalence relations on labeled transition systems: two bisimilar transition
systems represent the same process.
In this area, logics are used as languages that express behavioral properties
of processes. Extended modal logics, being invariant under various forms of
bisimulation, turn out to be of particular interest, because we can consider
properties expressed by these logics as process-invariants, instead of mere
transition system properties.
The notion of bisimulation is a central theme also in another area: the Theory
of non-well-founded Sets. It was first introduced as an axiom characterizing
non-well-founded sets in [31] and since then is a fruitful notion in this
field. Working with non-well-founded sets, the usual criterion for equality,
the so-called extensionality axiom, cannot be applied: the argument becomes
circular (!); on the contrary, the notion of bisimulation applies to
non-well-founded sets and provides a criterion for equality. Different notions
of bisimulation give rise to different theories of non-well-founded sets. In
[2], Aczel compares many such
theories, obtaining in this way a deep insight in the possible structure of
nonwell-founded sets.
Altogether bisimulation can be seen as a bridge between Modal Logic,
nonwell-founded Set Theory, and Process Theory, and one can fruitfully use this
bridge to transfer results and techniques from one area to the others. In this
dissertation we cross this bridge a number of times.
* We prove results in extended modal logics that can be used in Process Theory.
* We use extended modal formulae to describe different non-well-founded
universes. This allows us to provide alternative definitions of classes of
extended modal logics well-known by the modal logic community.
* We use non-well-founded set theories to study derivability in Modal Logic.
As one naturally expects, the choice of representing a process as an
equivalence class of transition systems modulo bisimulation has a strong
influence on the choice of the logics used to express properties of such
processes. Essentially, we want to restrict our attention to formulae OE which
are bisimulation invariant in the following sense: if two transition systems
are in the same bisimulation class, then they must agree on OE. As we shall
see, even though not all formulae of First Order or Monadic Second Order Logic
have this property, in these environments one can try to isolate those formulae
which are bisimulation invariant: the Van Benthem Theorem [12] and the
Janin-Walukiewicz Theorem [41] do the job. The first one characterizes the
formulae of Basic Modal Logic as those first-order properties of states that
are bisimulation invariant; the second one proves a similar results in the
second-order setting: the formulae of the Modal _-Calculus are exactly the
monadic second-order properties of states which are bisimulation invariant.
One of the most important aspects of these theorems lies in the fact that Basic
Modal Logic and Modal _-Calculus have desirable properties, such as a complete
calculus, decidability, and the finite model property; van Benthem's and
JaninWalukiewicz's theorems guarantee that, by restricting to bisimulation
invariant properties, we obtain this nice behavior without losing the
expressive power of the logic we started with, First Order or Monadic Second
Order. In particular, as far as Process Theory is concerned, the second-order
setting is especially interesting, because well-known and much used properties
of processes such as fairness and termination are expressible in it and hence,
by Janin-Walukiewicz Theorem, in the Modal _-Calculus.
Our contribution in this dissertation with respect to this area is an
interpolation theorem for the Modal _-Calculus. Interpolation is another
desirable property that a well-behaved logic is supposed to have. Since
interpolation can be proved whenever a Gentzen-style sequent calculus without
cut is available, a failure of interpolation can be seen as a signal that the
logic cannot have such an elegant calculus. Moreover, interpolation has a nice
consequence known as the Beth property, which says that implicit and explicit
definitions in the logic coincide.
Interpolation was also re-discovered in recent years by computer scientists
from a more practical point of view, as a useful property in the context of
modular databases. A formula OE can be seen as a description of a database and
interpolation, in its uniform version, says that the database can be split into
modules: if we submit a query dealing with a specific aspect of the database,
we can restrict ourselves to querying the corresponding module.
Summarizing, the Modal _-Calculus is a useful logic for application in Computer
Science and (uniform) interpolation is a useful property of a logic. In Chapter
3 we show that Modal _-Calculus enjoys (uniform) interpolation by using
automata techniques and the so-called bisimulation quantifiers.
Let us now cross another bridge and go into the realm of non-well-founded sets.
As we already said, bisimulation has a well-established role in this theory and
hence it is quite surprising that a modal-logic perspective acquired a place in
this context only in recent times (see [7]). Using bisimulation, Modal Logic
can be used to describe sets: in the most well-known axiomatization of
non-well-founded sets, the theory ZF C\Gamma + AF A, any set has a precise
description in terms of an infinitary modal formula. Moreover, by means of this
description sets can be seen as formulae and a model of ZF C\Gamma + AF A
consisting of infinitary modal formulae can be built.
In addition to its fairly recent discovery, notice also that the role of Modal
Logic in non-well-founded Set Theory has been considered only in connection
with the theory ZF C\Gamma + AF A. However important, this theory does not
certainly exhaust all possible descriptions of non-well-founded sets. Other
reasonable theories are known, whose study helped to understand the possible
structure of such sets in more depth. Can (extended) Modal Logic be used to
describe sets in other non-well-founded theories? In Chapter 4 we show that the
Scott anti-foundation theory ([62], [2]) admits such a description in terms of
a natural infinitary extension of the so-called graded modal logics. We also
give some natural variations on this theme, describing other non-well-founded
universes having a similar description. A central role in this respect is
played by the expansion operations, which transform structures into bisimilar
trees. The simplest expansion, the unraveling, can be used to describe the
Scott axiom; more elaborate expansions are used to described different kinds of
non-well-founded sets.
Crossing the bridge once more and re-entering the realm of Modal Logic, in
Chapter 5 we study interpolation for the class of logics suggested by non
well-founded theories. This is the well-known class of graded modal logics (in
its infinitary version) and the problem of interpolation for this class has
already been considered by Andr'eka in [1]. She showed that the behavior of the
logics in the class is not uniform: some logics in the class enjoy
interpolation and some other do not. In this setting we prove a weak form of
interpolation for the class of graded modal logics, the so-called elementary
interpolation and prove full interpolation whenever possible. The main idea
used in this context is that of a consistency property, which is often used in
the context of infinitary logic. Following an idea proposed by van Benthem
([11]), we tune this notion over bisimulation and introduce consistency
property modulo bisimulation to prove elementary and Craig interpolation in the
class of graded modal logics.
Finally, in Chapter 6 we use sets to describe derivability in Modal Logic. As
we shall see, in non-well-founded Set Theory Kripke frames can be used to
represent sets, with the accessibility relation playing the role of the inverse
membership relation. From this point of view, we can define a semantics for
Modal Logic in which the role of a Kripke frame is taken by the simpler concept
of set. Formulae of Modal Logic are then naturally translated into set-terms
representing the set of worlds in the Kripke frame which make the formula true.
We obtain in this way a natural generalization of the interpretation of
propositional connectives as set operations: together with the set
interpretation of disjunction as a union, conjunction as intersection, and the
other set interpretations of propositional connectives, we prove that we can
consider the 2-operator as the powerset operator (to be applied to
non-well-founded sets). This leads us to a translation from Basic Modal Logic
to a simple theory of non-well-founded sets, the theory \Omega .
The above mentioned translation was originally proposed in the context of
Automated Theorem Proving for Modal Logic. In this area, translations from
modal logics into First Order Logic are often used, since they allow the use of
very sophisticated and well performing theorem provers to automatically derive
modal logic formulae. From this point of view, the larger the class of
translatable logics, the better. However, the most used translations in the
field are the standard one and variations of it, which essentially allow one to
translate only the class of firstorder complete logics. The above mentioned
set-translation is, in this sense, a better choice: it works for all complete
logics, not necessarily first-order complete. In general, we only claim that
the technique for automated deduction in modal logic that opens up with the
introduction of the set translation is widely applicable and genuinely new. As
far as computationally related issues are concerned, we simply mention the fact
that new techniques from Computable Set Theory can be used to produce theorem
provers for the set theory \Omega (see [16], [58]).
The completeness and soundness of the set translation was first proved in [20]
by using tools of non-well-founded Set Theory. In this dissertation, we give a
simpler proof based on a comparison between the standard translation and the
set translation.
One of the advantage of the theory \Omega lies in its simplicity: we just have
axioms describing the relationship of 2 with the union operator, the
set-difference operator, and the powerset operator. However, even though \Omega
is strong enough to deal with Basic Modal Logic, modern Modal Logic goes
toward extensions of the basic formalism and more complex logics arise and are
studied. A natural question arises: can we tune the theory \Omega to deal with
extended modal logics? Is the new theory still a set theory, or do we need some
artificial axioms that have nothing to do with sets? What kind of extensions of
Basic Modal Logic are we able to cope with?
Extended modal logics may be obtained introducing new operators and, in most
cases, they have a first-order definable semantics in the language of Kripke
models. Example of such operators are the difference operator, the past
operator, the graded operators, and so forth. In this dissertation we consider
a (secondorder) logic L2 (see [8], [10]) within which all operators with such a
first-order definable semantics can be embedded. We show that we can strengthen
the theory \Omega in order to capture modal derivability in L2 and that the
new theory is still a genuine set theory: it is obtained by adding to \Omega
the operators that were introduced by G"odel to describe the universe of
constructible sets used to prove the consistency of the continuum hypothesis.
Organization and origin of the chapters:
Chapter 2 gives a general introduction to the topics of this dissertation.
Besides notations and basic definitions, it contains five more elaborate
sections on the main themes. We have a section on bisimulation and two sections
introducing the logics we are mainly interested in: the Modal _-Calculus and
the family of graded modal logics. Then we have a section on the different
forms of interpolation that we shall encounter and another section discussing
the role played by bisimulation in non-well-founded set theories.
Chapter 3 deals with uniform interpolation for the Modal _-Calculus. This work
is the result of a fruitful collaboration with Marco Hollenberg and, together
with other results about the Modal _-Calculus, it has been accepted for
publication in the Journal of Symbolic Logic. It is now available as a preprint
([21]).
Chapter 4 discusses the relationship between non-well-founded sets and Modal
Logic. It is still unpublished material.
Chapter 5 considers the problem of interpolation for graded modal logics. As
the material in the preceding chapter, it is still unpublished.
Chapter 6 is about the set translation of Modal Logic and extended modal
logics. This is again the result of a fruitful collaboration, this time with
Johan van Benthem, Angelo Montanari, and Alberto Policriti. The various stages
of the work have been published in [20], [14], [15].
Each chapter ends with brief concluding remarks and open questions. The further
directions of investigation listed there are by no means exhaustive, but I hope
that they do show that our intellectual bridges can support a good deal of
traffic.
Finally, I wish to ask the reader a little patience with my roman-english.