First-Order Model Checking Problems Parameterized by the Model

In this paper, we investigate the model checking (MC) problem for Halpern and Shoham's interval temporal logic HS. In the last years, interval temporal logic MC has received an increasing attention as a viable alternative to the traditional (point-based) temporal logic MC, which can be recovered as a special case. Most results have been obtained under the homogeneity assumption, that constrains a proposition letter to hold over an interval if and only if it holds over each component state. Recently, Lomuscio and Michaliszyn proposed a way to relax such an assumption by exploiting regular expressions to define the behaviour of proposition letters over intervals in terms of their component states. When homogeneity is assumed, the exact complexity of MC is a difficult open question for full HS and for its two syntactically maximal fragments AA'BB'E' and AA'EB'E'. In this paper, we provide an asymptotically optimal bound to the complexity of these two fragments under the more expressive semantic variant based on regular expressions by showing that their MC problem is AEXP_pol-complete, where AEXP_pol denotes the complexity class of problems decided by exponential-time bounded alternating Turing Machines making a polynomially bounded number of alternations.

A successor-invariant first-order formula is a formula that has access to an\nauxiliary successor relation on a structure's universe, but the model relation\nis independent of the particular interpretation of this relation. It is well\nknown that successor-invariant formulas are more expressive on finite\nstructures than plain first-order formulas without a successor relation. This\nnaturally raises the question whether this increase in expressive power comes\nat an extra cost to solve the model-checking problem, that is, the problem to\ndecide whether a given structure together with some (and hence every) successor\nrelation is a model of a given formula. It was shown earlier that adding\nsuccessor-invariance to first-order logic essentially comes at no extra cost\nfor the model-checking problem on classes of finite structures whose underlying\nGaifman graph is planar [Engelmann et al., 2012], excludes a fixed minor\n[Eickmeyer et al., 2013] or a fixed topological minor [Eickmeyer and\nKawarabayashi, 2016; Kreutzer et al., 2016]. In this work we show that the\nmodel-checking problem for successor-invariant formulas is fixed-parameter\ntractable on any class of finite structures whose underlying Gaifman graphs\nform a class of bounded expansion. Our result generalises all earlier results\nand comes close to the best tractability results on nowhere dense classes of\ngraphs currently known for plain first-order logic.\n

A successor-invariant first-order formula is a formula that has access to an auxiliary successor relation on a structure's universe, but the model relation is independent of the particular interpretation of this relation. It is well known that successor-invariant formulas are more expressive on finite structures than plain first-order formulas without a successor relation. This naturally raises the question whether this increase in expressive power comes at an extra cost to solve the model-checking problem, that is, the problem to decide whether a given structure together with some (and hence every) successor relation is a model of a given formula. It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph is planar [1], excludes a fixed minor [2] or a fixed topological minor [3], [4]. In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded expansion. Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic.

The paper proposes a new technique to mitigate the state explosion in model checking. The technique is called a divide & conquer approach to leads-to model checking. As indicated by the name, the technique is dedicated to leads-to properties. It is known that many important systems requirements can be expressed as leads-to properties, thus it is worth focusing on leads-to properties. The technique divides an original leads-to model checking problem into multiple smaller model checking problems and tackles each smaller one. We prove a theorem that the multiple smaller model checking problems are equivalent to the original leads-to model checking problem. We conduct two case studies demonstrating the power of the proposed technique.

We study the definability problem for first-order logic, denoted by FO-D ef . The input of FO-D ef is a relational database instance I and a relation R ; the question to answer is whether there exists a first-order query Q (or, equivalently, a relational algebra expression Q ) such that Q evaluated on I gives R as an answer. Although the study of FO-D ef dates back to 1978, when the decidability of this problem was shown, the exact complexity of FO-D ef remains as a fundamental open problem. In this article, we provide a polynomial-time algorithm for solving FO-D ef that uses calls to a graph-isomorphism subroutine (or oracle). As a consequence, the first-order definability problem is found to be complete for the class GI of all problems that are polynomial-time Turing reducible to the graph isomorphism problem, thus closing the open question about the exact complexity of this problem. The technique used is also applied to a generalized version of the problem that accepts a finite set of relation pairs, and whose exact complexity was also open; this version is also found to be GI -complete.

In the last years, the model checking (MC) problem for interval temporal logic (ITL) has received an increasing attention as a viable alternative to the traditional (point-based) temporal logic MC, which can be recovered as a special case. Most results have been obtained by imposing suitable restrictions on interval labeling. In this paper, we overcome such limitations by using regular expressions to define the behavior of proposition letters over intervals in terms of the component states. We first prove that MC for Halpern and Shoham’s ITL (HS), extended with regular expressions, is decidable. Then, we show that formulas of a large class of HS fragments, namely, all fragments featuring (a subset of) HS modalities for Allen’s relations meets, met-by, starts, and started-by, can be model checked in polynomial working space (MC for all these fragments turns out to be PSPACE-complete).

In the author's recent paper (ibid., vol.20, no.8 (1998)), it was shown that partial classification, which allows for indecision in certain regions of the data space, can increase a benefit function. It was also claimed that in the case of ideal separation, corresponding to the maximum likelihood criterion, partial classification cannot be more beneficial than full classification. The latter claim is false, and a counterexample is given to support the argument. The error in the original statement was caused by the omission of certain terms in the calculation of the decision probability. The erroneous result may be omitted without changing the rest of the paper. In fact, the present observation strengthens the main claim of the paper, that partial classification can be beneficial.

At present, most of the first-order logic theorem provers use a binary-resolution method, which can effectively solve the general first-order logic problems to a certain extent. However, the cooperative processing ability of this method for multiple clauses is insufficient, and it is easy to cause rapid expansion of clause set in the deductive process. In this paper, we propose a novel first-order logic theorem prover based on the standard contradiction separation (S-CS) rule. This prover can realize the dynamic cooperative deduction of multiple clauses in each deduction process, as well as it can effectively learn and control the deduction process. This paper incorporates the S-CS rule with the well-known prover Prover9, to build a combined system, which effectively integrates the advantages of the two methods, not only improves the binary-resolution prover’s ability, but also solves more than 100 problems that cannot be solved by other provers.

As a recently proposed membrane computing model, cP systems are capable of solving computational hard and distributed problems. Although several membrane system variants were formally verified in previous research, none of their approaches was applicable to cP systems. To formally verify the safety and liveness properties of cP systems, we solve a famous NP-Complete problem—the subset sum problem—in cP systems and use the PAT3 and ProB model checkers to verify the cP solution. Our cP solution outperforms previous work in time complexity and uses fewer rules. To perform model checking in cP systems, we define several mapping rules from cP notation to formal verification languages CSP# and B. We show that we can use model checkers to effectively detect design errors in cP systems. This work is the first study on formal verification of cP systems, and it showed that cP models can be effectively transformed into model checking problems and verified automatically.

Satisfiability Modulo Theories (SMT) is an extension of SAT towards FOL. SMT solvers have proven highly scalable and efficient for problems based on some ground theorems. However, SMT problems involving quantifiers and combination of theorems is a long-standing challenge, which has been a major bottleneck of practical application of SMT solvers in some fields. We reveal a decidable fragment of FOL involving quantifiers, which could not be solved by SMT solvers such as Z3, CVC3, etc., and show how to convert them into model checking problems.

Model-Checking of Correctness Conditions for Concurrent Objects

The process of completing, correcting and prioritising specifications is an essential but very complex task for the maintenance and improvement of software systems. The preservation of functionalities and the ability to accommodate changes are main objectives of the software development cycle to guarantee system reliability. Logical theories able to fully model such processes are still insufficient. In this paper we propose a full formalisation of such operations on software systems inspired by the Alchourron–Gardenfors–Makinson (AGM) paradigm for belief revision of human epistemic states. We represent specifications as finite sets of formulas equipped with a priority relation that models functional entrenchment of properties. We propose to handle specification incompleteness through ordered expansion, inconsistency through ordered safe contraction and prioritisation through revision with reordering, and model all three in an algorithmic fashion. We show how the system satisfies basic properties of the AGM paradigm, including Levi’s and Harper’s identities. We offer a concrete example and complexity results for the inference and model checking problems on revision. We conclude by describing resilience and evolvability of software systems based on such revision operators.

Explicit Versus Symbolic Algorithms for Solving ALFP Constraints

We develop a generic framework to extend the logics LTL, CTL\(^+\) and CTL\(^*\) by automata-based connectives from formal language classes and analyse this framework with regard to regular languages, visibly pushdown languages and (deterministic) context-free languages. More precisely, we consider how the use of different automata classes changes the expressive power of the logics and provide algorithms for the satisfiability and model checking problems induced by the use of different automata. For the model checking problem, we treat not only finite Kripke transition systems, but also visibly pushdown systems and pushdown systems. We provide completeness or undecidability results in almost all cases and show that the extensions we consider can formulate properties not expressible in classical temporal logics or regular extensions thereof.

Many systems, both natural and artificial, exhibit a mixed discrete-continuous behavior that cannot be fully captured by either continuous nor discrete models: they evolve in accordance to continuous laws, but these laws are controlled by a finite set of modes. Hybrid automata were proposed to represent such kind of behaviors and they have been used to model numerous natural phenomena in the last decades. Unfortunately, the Model Checking problem over them was proved undecidable and, because of that, many techniques were suggested so far to both approximate the original models and reduce the analysis complexity. This paper surveys some of such techniques and reports some open questions.