Backward Modalities Research Articles

On-the-Fly Data Flow Analysis Based on Verification Technology

The combination of static and dynamic software analysis, such as data flow analysis (Dfa) and model checking, provides benefits for both disciplines. On the one hand, the information extracted by Dfas about program data may be utilized by model checkers to optimize the state space representation. On the other hand, the expressiveness of logic formulas allows us to consider model checkers as generic data flow analyzers. Following this second approach, we propose in this paper an algorithm to calculate Dfas using on-the-fly resolution of boolean equation systems (Bess). The overall framework includes the abstraction of the input program into an implicit labeled transition system (Lts), independent of the program specification language. Moreover, using Bess as an intermediate representation allowed us to reformulate classical Dfas encountered in the literature, which were previously encoded in terms of μ-calculus formulas with forward and backward modalities. Our work was implemented and integrated into the widespread verification platform Cadp, and experimented on real examples.

The finite graph problem for two-way alternating automata

Two-way alternating automata on infinite trees were introduced by Vardi (Reasoning about the part with two way automata, Lecture Notes in Computer Science, vol. 11, Springer, Berlin, 1998, pp. 628–641). Here we consider alternating two-way automata on graphs and show the decidability of the following problem: “does a given automaton with the Büchi condition accept any finite graph?” Using this result we demonstrate the decidability of the finite model problem for a certain fragment of the modal μ-calculus with backward modalities.

Substructural logics on display

Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponential-free linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a 'cyclic' counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic Bi-Lambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bi-linear, Bi-relevant, Bi-BCK and Bi-intuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and some even have a 'cyclic' counterpart. These (bi-intuitionistic and bi-classical) extensions of Bi-Lambek logic are not so well understood. Cut-elimination for Classical Bi-Lambek logic, for example, is not completely clear since some cut rules have side conditions requiring that certain constituents be empty or non-empty. The display logic of Nuel Belnap is a general Gentzen-style proof theoretical framework designed to capture many different logics in one uniform setting. The beauty of display logic is a general cut-elimination theorem, due to Belnap, which applies whenever the rules of the display calculus obey certain, easily checked, conditions. The original display logic, and its various incarnations, are not suitable for capturing bi-intuitionistic and bi-classical logics in a uniform way. We remedy this situation by giving a single (cut-free) Display calculus for the Bi-Lambek Calculus, from which all the well-known (bi-intuitionistic and bi-classical) extensions are obtained by the incremental addition of structural rules to a constant core of logical introduction rules. We highlight the inherent duality and symmetry within this framework obtaining 'four proofs for the price of one'. We give algebraic semantics for the Bi-Lambek logics and prove that our calculi are sound and complete with respect to these semantics. We show how to define an alternative display calculus for bi-classical substructural logics using negations, instead of implications, as primitives. Borrowing from other display calculi, we show how to extend our display calculus to handle bi-intuitionistic or bi-classical substructural logics containing the forward and backward modalities familiar from tense logic, the exponentials of linear logic, the converse operator familiar from relation algebra, four negations, and two unusual modalities corresponding to the non-classical analogues of Sheffer's 'dagger' and 'stroke', all in a modular way. Using the Gaggle Theory of Dunn we outline relational semantics for the binary and unary intensional connectives, but make no attempt to do so for the extensional connectives, or the exponentials. Finally, we flesh out a suggestion of Lambek to embed intuitionistic logic using two unusual 'exponentials', and show that these 'exponentials' are essentially tense logical modalities, quite at odds with the usual exponentials. Using a refinement of the display property, you can pick and choose from these possibilities to construct a display calculus for your needs. Keywords:display logic, substructural logics, linear logic, relevant logic, BCK-logic, bi-Heyting logic, intuitionistic substructural tense logics, intuitionistic modal logics, proof theory, gaggle theory

Translations between modal logics of reactive systems

We propose meaning-preserving translations between L B , L U and L sb (three modal logics in full agreement with branching bisimulation), thus proving that they all have the same expressivity. The translations can be implemented and have potential applications in the automated analysis of reactive systems. In this work the main difficulty is that L B uses both forward and backward modalities, whereas L U and L sb only have forward modalities. The technique we developed to cope with this, is an adaptation in a branching-time framework of the methods underlying Gabbay's separation theorem for PTL (Gabbay, 1987). This technique is powerful and has been applied successfully to related problems.

Three logics for branching bisimulation

Three temporal logics are introduced that induce on labeled transition systems the same identifications as branching bisimulation, a behavioral equivalence that aims at ignoring invisible transitions while preserving the branching structure of systems. The first logic is an extension of Hennessy-Milner Logic with an “until” operator. The second one is another extension of Hennessy-Milner Logic, which exploits the power of backward modalities. The third logic is CTL* without the next-time operator. A relevant side-effect of the last characterization is that it sets a bridge between the state- and action-based approaches to the semantics of concurrent systems.