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.