Giacomo Lenzi

The Variable Hierarchy of the μ-Calculus Is Strict

Most of the logics commonly used in verification, such as LTL, CTL, CTL*, and PDL can be embedded into the two-variable fragment of the μ-calculus. It is also known that properties occurring at arbitrarily high levels of the alternation hierarchy can be formalised using only two variables. This raises the question of whether the number of fixed-point variables in μ-formulae can be bounded in general. We answer this question negatively and prove that the variable-hierarchy of the μ-calculus is semantically strict. For any k, we provide examples of formulae with k variables that are not equivalent to any formula with fewer variables. In particular, this implies that Parikhs Game Logic is less expressive than the μ-calculus, thus resolving an open issue raised by Parikh in~1983.

On modal μ-calculus with explicit interpolants

Abstract This paper deals with the extension of Kozens μ -calculus with the so-called “existential bisimulation quantifier”. By using this quantifier one can express the uniform interpolant of any formula of the μ -calculus. In this work we provide an explicit form for the uniform interpolant of a disjunctive formula and see that it belongs to the same level of the fixpoint alternation hierarchy of the μ -calculus than the original formula. We show that this result cannot be generalized to the whole logic, because the closure of the third level of the hierarchy under the existential bisimulation quantifier is the whole μ -calculus. However, we prove that the first two levels of the hierarchy are closed. We also provide the μ -logic extended with the bisimulation quantifier with a complete calculus.

A Note on Bisimulation Quantifiers and Fixed Points over Transitive Frames

We consider three basic questions regarding the extension of modal logic with a special kind of propositional quantifiers, known as bisimulation quantifiers, over arbitrary classes of frames: bisimulation invariance, uniform interpolation, and expressive power. In particular: – we discuss the relation between bisimulation invariance of bisimulation quantifiers and the semantical notion of amalgamation of the class of frames; – we consider a strong form of interpolation, uniform interpolation, and its relation with the closure under bisimulation quantifiers; – we compare bisimulation quantifiers logic with the better known extension of modal logic with extremal fixed points. In this article we show that the answers to these questions that are valid for the class of all frames do not generalize to arbitrary classes, but they do generalize if we restrict to classes of (finite) transitive or (finite) transitive and reflexive frames.

Relating levels of the mu-calculus hierarchy and levels of the monadic hierarchy

As is already known from the work of D. Janin & I. Walukiewicz (1996), the mu-calculus is as expressive as the bisimulation-invariant fragment of monadic second-order logic. In this paper, we relate the expressiveness of levels of the fixpoint alternation depth hierarchy of the mu-calculus (the mu-calculus hierarchy) with the expressiveness of the bisimulation-invariant fragment of levels of the monadic quantifiers alternation-depth hierarchy (the monadic hierarchy). From J. van Benthems (1976) results, we know already that the fixpoint free fragment of the mu-calculus (i.e. polymodal logic) is as expressive as the bisimulation-invariant fragment of monadic /spl Sigma//sub 0/ (i.e. first-order logic). We show that the /spl nu/-level of the mu-calculus hierarchy is as expressive as the bisimulation-invariant fragment of monadic /spl Sigma//sub 1/ and that the /spl nu//spl mu/-level of the mu-calculus hierarchy is as expressive as the bisimulation-invariant fragment of monadic /spl Sigma//sub 2/, and we show that no other level /spl Sigma//sub k/ (for k>2) of the monadic hierarchy can be related similarly with any other level of the mu-calculus hierarchy. The possible inclusion of all the mu-calculus in some level /spl Sigma//sub k/ of the monadic hierarchy, for some k>2, is also discussed.

The variable hierarchy of the µ-calculus is strict

The modal μ-calculus Lμ attains high expressive power by extending basic modal logic with monadic variables and binding them to extremal fixed points of definable operators. The number of variables occurring in a formula provides a relevant measure of its conceptual complexity. In a previous paper with Erich Gradel we have shown, for the existential fragment of Lμ , that this conceptual complexity is also reflected in an increase of semantic complexity, by providing examples of existential formulae with k variables that are not equivalent to any existential formula with fewer than k variables. In this paper, we prove an existential preservation theorem for the family of Lμ-formulae over at most k variables that define simulation closures of finite strongly connected structures. Since hard formulae for the level k of the existential hierarchy belong to this family, it follows that the bounded variable fragments of the full modal μ-calculus form a hierarchy of strictly increasing expressive power.

On the Variable Hierarchy of the Modal µ-Calculus

We investigate the structure of the modal µ-calculus Lµ with respect to the question of how many different fixed point variables are necessary to define a given property. Most of the logics commonly used in verification, such as CTL, LTL, CTL*, PDL, etc. can in fact be embedded into the two-variable fragment of the µ-calculus. It is also known that the two-variable fragment can express properties that occur at arbitrarily high levels of the alternation hierarchy. However, it is an open problem whether the variable hierarchy is strict.Here we study this problem with a game-based approach and establish the strictness of the hierarchy for the case of existential (i.e., ?-free) formulae. It is known that these characterize precisely the Lµ-definable properties that are closed under extensions. We also relate the strictness of the variable hierarchy to the question whether the finite variable fragments satisfy the existential preservation theorem.

A New Logical Characterization of Büchi Automata

We consider the monadic second order logic with two successor functions and equality, interpreted on the binary tree. We show that a set of assignments is definable in the fragment Σ2 of this logic if and only if it is definable by a Buchi automaton. Moreover we show that every set of second order assignments definable in Σ2 with equality is definable in Σ2 without equality as well. The present paper is sketchy due to space constraints; for more details and proofs see [7].

On the Structure of the Monadic Logic of the Binary Tree

Since the work of Rabin [9], it has been known that any monadic second order property of the (labeled) binary tree with successor functions (and not the prefix ordering) is a monadic Δ3 property. In this paper, we show this upper bound is optimal in the sense that there is a monadic Σ2 formula, stating the existence of a path where a given predicate holds infinitely often, which is not equivalent to any monadic II2 formula. We even show that some monadic second order definable properties of the binary tree are not definable by any boolean combination of monadic Σ2 and II2 formulas. These results rely in particular on applications of Ehrenfeucht-FraissE like game techniques to the case of monadic Σ2 formulas.

On modal μ-calculus over reflexive symmetric graphs

In this article, we consider the hierarchy of the modal μ-calculus over reflexive and symmetric graphs and show that in this class the modal μ-calculus hierarchy is infinite. In the proof, a parity game over a tree is transformed into a equivalent parity game where Duplicator, when playing over the reflexive and symmetric closure of the tree, will never use loops or back edges.

On fixpoint arithmetic and infinite time turing machines

In this paper we deal with Fixpoint Arithmetic and Infinite Time Turing Machines. We show that every set of first order assignments definable by a formula of fixpoint arithmetic can be recognized by an Infinite Time Turing Machine, and that the resulting inclusion is proper.Our result, combined with other known results, gives a chain of proper inclusions from Π11 to Fixpoint Arithmetic to Infinite Time Turing Machines to Δ21.