Notes on Model Checking and Abstraction in Rewriting Logic (original) (raw)

From Linear Temporal Logic Properties to Rewrite Propositions

Lecture Notes in Computer Science, 2012

In the regular model-checking framework, reachability analysis can be guided by temporal logic properties, for instance to achieve the counter example guided abstraction refinement (CEGAR) objectives. A way to perform this analysis is to translate a temporal logic formula expressed on maximal rewriting words into a "rewrite proposition" -a propositional formula whose atoms are language comparisons, and then to generate semidecision procedures based on (approximations of) the rewrite proposition. This approach has recently been studied using a non-automatic translation method. The extent to which such a translation can be systematised needs to be investigated, as well as the applicability of approximated methods wherever no exact translation can be effected. This paper presents contributions to that effect: (1) we investigate suitable semantics for LTL on maximal rewriting words and their influence on the feasibility of a translation, and (2) we propose a general scheme providing exact results on a fragment of LTL corresponding mainly to safety formulae, and approximations on a larger fragment.

Kripke models and intermediate logics

Publications of the Research Institute for Mathematical Sciences, 1970

In [10], Kripke gave a definition of the semantics of the intuitionistic logic. Fitting [2] showed that Kripke's models are equivalent to algebraic models (i.e., pseudo-Boolean models) in a certain sense. As a corollary of this result, we can show that any partially ordered set is regarded as a (characteristic) model of a intermediate logic ^ We shall study the relations between intermediate logics and partially ordered sets as models of them, in this paper. We call a partially ordered set, a Kripke model. 2^ At present we don't know whether any intermediate logic 'has a Kripke model. But Kripke models have some interesting properties and are useful when we study the models of intermediate logics. In §2, we shall study general properties of Kripke models. In §3, we shall define the height of a Kripke model and show the close connection between the height and the slice, which is introduced in [7]. In §4, we shall give a model of LP» which is the least element in n-ih slice S n (see [7]). §1. Preliminaries We use the terminologies of [2] on algebraic models, except the use of 1 and 0 instead of V and /\, respectively. But on Kripke models, we give another definition, following Schiitte [13]. 3) Definition 1.1. If M is a non-empty partially ordered set, then

On model checking durational Kripke structures (Extended abstract)

2002

We consider quantitative model checking in durational Kripke structures (Kripke structures where transitions have integer durations) with timed temporal logics where subscripts put quantitative constraints on the time it takes before a property is satisfied. We investigate the conditions that allow polynomial-time model checking algorithms for timed versions of CTL and exhibit an important gap between logics where subscripts of the form "= c" (exact duration) are allowed, and simpler logics that only allow subscripts of the form "≤ c" or "≥ c" (bounded duration). A surprising outcome of this study is that it provides the second example of a ∆ p 2 -complete model checking problem.

Substructure Temporal Logic

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science, 2013

In formal verification and design, reasoning about substructures is a crucial aspect for several fundamental problems, whose solution often requires to select a portion of the model of interest on which to verify a specific property.

Temporal Logics with Language Parameters

Lecture Notes in Computer Science, 2021

Computation Tree Logic (CTL) and its extensions CTL * and CTL + are widely used in automated verification as a basis for common model checking tools. But while they can express many properties of interest like reachability, even simple regular properties like "Every other index is labelled a" cannot be expressed in these logics. While many extensions were developed to include regular or even non-regular (e.g. visibly pushdown) languages, the first generic framework, Extended CTL, for CTL with arbitrary language classes was given by Axelsson et. al. and applied to regular, visibly pushdown and (deterministic) context-free languages. We extend this framework to CTL * and CTL + and analyse it with regard to decidability, complexity, expressivity and satisfiability.

On Model Checking Durational Kripke Structures

Lecture Notes in Computer Science, 2002

We consider quantitative model checking in durational Kripke structures (Kripke structures where transitions have integer durations) with timed temporal logics where subscripts put quantitative constraints on the time it takes before a property is satisfied. We investigate the conditions that allow polynomial-time model checking algorithms for timed versions of CTL and exhibit an important gap between logics where subscripts of the form "= c" (exact duration) are allowed, and simpler logics that only allow subscripts of the form "≤ c" or "≥ c" (bounded duration). A surprising outcome of this study is that it provides the second example of a ∆ p 2 -complete model checking problem.

Branching-Time Temporal Logics with Minimal Model Quantifiers

Lecture Notes in Computer Science, 2009

Temporal logics are a well investigated formalism for the specification and verification of reactive systems. Using formal verification techniques, we can ensure the correctness of a system with respect to its desired behavior (specification), by verifying whether a model of the system satisfies a temporal logic formula modeling the specification. From a practical point of view, a very challenging issue in using temporal logic in formal verification is to come out with techniques that automatically allow to select small critical parts of the system to be successively verified. Another challenging issue is to extend the expressiveness of classical temporal logics, in order to model more complex specifications. In this paper, we address both issues by extending the classical branching-time temporal logic CTL * with minimal model quantifiers (MCTL * ). These quantifiers allow to extract, from a model, minimal submodels on which we check the specification (also given by an MCTL * formula). We show that MCTL * is strictly more expressive than CTL * . Nevertheless, we prove that the model checking problem for MCTL * remains decidable and in particular in PSPACE. Moreover, differently from CTL * , we show that MCTL * does not have the tree model property, is not bisimulation-invariant and is sensible to unwinding. As far as the satisfiability concerns, we prove that MCTL * is highly undecidable. We further investigate the model checking and satisfiability problems for MCTL * sublogics, such as MPML, MCTL, and MCTL + , for which we obtain interesting results. Among the others, we show that MPML retains the finite model property and the decidability of the satisfiability problem. 6 6 J J J J J J i 6 6 J J J J J J

Logics of Kripke meta-models

Logic Journal of the IGPL, 2009

This paper introduces and studies a new type of logical construction, which allows to combine various non-classical propositional logics with the temporal or modal background. The possible candidates include (but are not restricted to) a number of epistemic, multi-agent, deontological and other well-studied logics. In this construction, that we call refinement, the Kripke structure of a chosen Kripke complete logic is imposed on clusters of the background transitive frame. Refinements fit in a wider framework of fibred logics, while having some unique features. First of all, when applied to classes of frames of Kripke complete logics, refinement preserves good meta-logical properties of constituent logics, in contrast with the well-known products of logics. Another advantage of refinements is that they allow for augmented languages of considerable expressive power, while preserving good meta-logical and semantical properties. In particular we show that refinement of logics preserves the effective finite model property and decidability for a wide class of constituent logics.

Temporal Logics over Transitive States

2005

We investigate the computational behaviour of ‘two-dimensional’ propositional temporal logics over (ℕ, \(\Pi^{\rm 1}_{\rm 1}\) -complete) if the domains of states with those relations are assumed to be constant. Motivated by applications in the areas of temporal description logic and specification & verification of hybrid systems, in this paper we analyse the computational impact of allowing the domains of states to expand. We show that over finite expanding domains (with an arbitrary, tree-like, quasi-order, or linear transitive relation) the logics are recursively enumerable, but undecidable. If these finite domains eventually become constant then the resulting O-free logics are decidable (but not in primitive recursive time); on the other hand, when equipped with O they are not even recursively enumerable. Finally, we show that temporal logics over infinite expanding domains as above are undecidable even for the language with the sole temporal operator ‘eventually.’ The proofs are based on Kruskal’s tree theorem and reductions of reachability problems for lossy channel systems.