Branching-time logic ECTL# and its tree-style one-pass tableau: Extending fairness expressibility of ECTL+

Temporal logic has become essential for various areas in computer science, most notably for the specification and verification of hardware and software systems. For the specification purposes rich temporal languages are required that, in particular, can express fairness constraints. For linear-time logics which deal with fairness in the linear-time setting, one-pass and two-pass tableau methods have been developed. In the repository of the CTL-type branching-time setting, the well-known logics ECTL and ECTL+ were developed to explicitly deal with fairness. However, due to the syntactical restrictions, these logics can only express restricted versions of fairness. The logic CTL*, often considered as ‘the full branching-time logic’ overcomes these restrictions on expressing fairness. However, CTL* is extremely challenging for the application of verification techniques, and the tableau technique, in particular. For example, there is no one-pass tableau construction for CTL*, while one-pass tableau has an additional benefit enabling the formulation of dual sequent calculi that are often treated as more ‘natural’ being more friendly for human understanding. These two considerations lead to the following problem - are there logics that have richer expressiveness than ECTL+, allowing the formulation of a new range of fairness constraints with ‘until’ operator, yet ‘simpler’ than CTL?, and for which a one-pass tableau can be developed? Here we give a positive answer to this question, introducing a sub-logic of CTL* called ECTL#, its tree-style one-pass tableau, and an algorithm for obtaining a systematic tableau, for any given admissible branching-time formulae. We prove the termination, soundness and completeness of the method. As tree-shaped one-pass tableaux are well suited for the automation and are amenable for the implementation and for the formulation of sequent calculi. Our results also open a prospect of relevant developments of the automation and implementation of the tableau method for ECTL#, and of a dual sequent calculi.Authors have been partially supported by Spanish Project TIN2017-86727-C2-2-R, and by the University of the Basque Country under Project LoRea GIU18/182

One-Pass Context-Based Tableaux Systems for CTL and ECTL

When building tableau for temporal logic formulae, applying a two-pass construction, we first check the validity of the given tableaux input by creating a tableau graph, and then, in the second “pass”, we check if all the eventualities are satisfied. In one-pass tableaux checking the validity of the input does not require these auxiliary constructions. This paper continues the development of one-pass tableau method for temporal logics introducing tree-style one-pass tableau systems for Computation Tree Logic (CTL) and shows how this can be extended to capture Extended CTL (ECTL). A distinctive feature here is the utilisation, for the core tableau construction, of the concept of a context of an eventuality which forces its earliest fulfilment. Relevant algorithms for obtaining a systematic tableau for these branching-time logics are also defined. We prove the soundness and completeness of the method. With these developments of a tree-shaped one-pass tableau for CTL and ECTL, we have formalisms which are well suited for the automation and are amenable for the implementation, and for the formulation of dual sequent calculi. This brings us one step closer to the application of one-pass context-based tableaux in certified model checking for a variety of CTL-type branching-time logics.Authors have been supported by the European Union (FEDER funds) under grant TIN2017-86727-C2-2-R, and by the University of the Basque Country under Project LoRea GIU18-182

The Structure of logarithmic advice complexity classes

A nonuniform class called here Full-P/log, due to Ko, is studied. It corresponds to polynomial time with logarithmically long advice. Its importance lies in the structural properties it enjoys, more interesting than those of the alternative class P/log; specifically, its introduction was motivated by the need of a logarithmic advice class closed under polynomial-time deterministic reductions. Several characterizations of Full-P/log are shown, formulated in terms of various sorts of tally sets with very small information content. A study of its inner structure is presented, by considering the most usual reducibilities and looking for the relationships among the corresponding reduction and equivalence classes defined from these special tally sets.Postprint (published version

Verified Model Checking for Conjunctive Positive Logic

We formalize, in the Dafny language and verifier, a proof system PS for deciding the model checking problem of the fragment of first-order logic, denoted FOAE/\ , known as conjunctive positive logic (CPL). We mechanize the proofs of soundness and completeness of PS ensuring its correctness. Our formalization is representative of how various popular verification systems can be used to verify the correctness of rule-based formal systems on the basis of the least fixpoint semantics. Further, exploiting Dafny’s automatic code generation, from the completeness proof we achieve a mechanically verified prototype implementation of a proof search mechanism that is a model checker for CPL. The model checking problem of FOAE/\ is equivalent to the quantified constraint satisfaction problem (QCSP), and it is PSPACE-complete. The formalized proof system decides the general QCSP and it can be applied to arbitrary formulae of CPL.This research has been supported by the European Union (FEDER funds) under grant TIN2017-86727-C2-2-R, and by the University of the Basque Country under Project LoRea GIU18-182

On Kobayashi's compressibility of infinite sequences

Kojiro Kobayashi's concept of compressibility of infinite sequences is presented. We point out here that there is a greater relationship between this concept of compressibility and the Kolmogorov complexity than we expected. In addition and with the same ideas used for comparing Kobayashi and Kolmogorov approaches, we show that two different ways of defining advice function classes coincide

On Kobayashi's compressibility of infinite sequences

Kojiro Kobayashi's concept of compressibility of infinite sequences is presented. We point out here that there is a greater relationship between this concept of compressibility and the Kolmogorov complexity than we expected. In addition and with the same ideas used for comparing Kobayashi and Kolmogorov approaches, we show that two different ways of defining advice function classes coincide.Postprint (published version

Comprenssibility and uniform complexity

We focus on notions of resource-bounded complexity for infinite binary sequences, and compare both, a definition based on Kobayashi's concept of compressibility, and the uniform approach studied by Loveland. It is known that for constant bounds on the complexity these definitions exactly coincide, and characterize the polynomial-time computable sequences when the running time is bounded by a polynomial, together with the recursive sequences when there is no time bound. We show here how for complexity functions that are monotonic, and recursive, the Kobayashi and Loveland complexity concepts are equivalent under a small constant factor. This also works under time bounds if instead of bounding functions that are recursive, those that are computed within the allowed time are considered.Preprin

On Kobayashi's compressibility of infinite sequences

Kojiro Kobayashi's concept of compressibility of infinite sequences is presented. We point out here that there is a greater relationship between this concept of compressibility and the Kolmogorov complexity than we expected. In addition and with the same ideas used for comparing Kobayashi and Kolmogorov approaches, we show that two different ways of defining advice function classes coincide

The structure of a logarithmic advice class

The complexity class Full-P/log, corresponding to a form of logarithmic advice for polynomial time, is studied. In order to understand the inner structure of this class, we characterize Full-P/log in terms of Turing reducibility to a special family of sparse sets. Other characterizations of Full-P/log, relating it to sets with small information content, were already known. These used tally sets whose words follow a given regular pattern and tally sets that are regular in a resource-bounded Kolmogorov complexity sense. We obtain here relationships between the equivalence classes of the mentioned tally and sparse sets under various reducibilities, which provide new knowledge about the logarithmic advice class. Another way to measure the amount of information encoded in a language in a nonuniform class, is to study the relative complexity of computing advice functions for this language. We prove bounds on the complexity of advice functions for Full-P/log and for other subclasses of it. As a consequence, Full-P/log is located in the Extended Low Hierarchy.Preprin

On Kobayashi's compressibility of infinite sequences

Kojiro Kobayashi's concept of compressibility of infinite sequences is presented. We point out here that there is a greater relationship between this concept of compressibility and the Kolmogorov complexity than we expected. In addition and with the same ideas used for comparing Kobayashi and Kolmogorov approaches, we show that two different ways of defining advice function classes coincide