A logic with temporally accessible iteration

Deficiency in expressive power of the first-order logic has led to developing its numerous extensions by fixed point operators, such as Least Fixed-Point (LFP), inflationary fixed-point (IFP), partial fixed-point (PFP), etc. These logics have been extensively studied in finite model theory, database theory, descriptive complexity. In this paper we introduce unifying framework, the logic with iteration operator, in which iteration steps may be accessed by temporal logic formulae. We show that proposed logic FO+TAI subsumes all mentioned fixed point extensions as well as many other fixed point logics as natural fragments. On the other hand we show that over finite structures FO+TAI is no more expressive than FO+PFP. Further we show that adding the same machinery to the logic of monotone inductions (FO+LFP) does not increase its expressive power either

Finite Model Finding for Parameterized Verification

In this paper we investigate to which extent a very simple and natural "reachability as deducibility" approach, originated in the research in formal methods in security, is applicable to the automated verification of large classes of infinite state and parameterized systems. The approach is based on modeling the reachability between (parameterized) states as deducibility between suitable encodings of states by formulas of first-order predicate logic. The verification of a safety property is reduced to a pure logical problem of finding a countermodel for a first-order formula. The later task is delegated then to the generic automated finite model building procedures. In this paper we first establish the relative completeness of the finite countermodel finding method (FCM) for a class of parameterized linear arrays of finite automata. The method is shown to be at least as powerful as known methods based on monotonic abstraction and symbolic backward reachability. Further, we extend the relative completeness of the approach and show that it can solve all safety verification problems which can be solved by the traditional regular model checking.Comment: 17 pages, slightly different version of the paper is submitted to TACAS 201

Temporal logic with predicate abstraction

A predicate linear temporal logic LTL_{\lambda,=} without quantifiers but with predicate abstraction mechanism and equality is considered. The models of LTL_{\lambda,=} can be naturally seen as the systems of pebbles (flexible constants) moving over the elements of some (possibly infinite) domain. This allows to use LTL_{\lambda,=} for the specification of dynamic systems using some resources, such as processes using memory locations, mobile agents occupying some sites, etc. On the other hand we show that LTL_{\lambda,=} is not recursively axiomatizable and, therefore, fully automated verification of LTL_{\lambda,=} specifications is not, in general, possible.Comment: 14 pages, 4 figure

Finite Models vs Tree Automata in Safety Verification

In this paper we deal with verification of safety properties of term-rewriting systems. The verification problem is translated to a purely logical problem of finding a finite countermodel for a first-order formula, which is further resolved by a generic finite model finding procedure. A finite countermodel produced during successful verification provides with a concise description of the system invariant sufficient to demonstrate a specific safety property. We show the relative completeness of this approach with respect to the tree automata completion technique. On a set of examples taken from the literature we demonstrate the efficiency of finite model finding approach as well as its explanatory power

First-order logic for safety verification of hedge rewriting systems

In this paper we deal with verification of safety properties of hedge rewriting systems and their generalizations. The verification problem is translated to a purely logical problem of finding a finite countermodel for a first-order formula, which is further tackled by a generic finite model finding procedure. We show that the proposed approach is at least as powerful as the methods using regular invariants. At the same time the finite countermodel method is shown to be efficient and applicable to the wide range of systems, including the protocols operating on unranked trees

Finite Countermodel Based Verification for Program Transformation (A Case Study)

Both automatic program verification and program transformation are based on program analysis. In the past decade a number of approaches using various automatic general-purpose program transformation techniques (partial deduction, specialization, supercompilation) for verification of unreachability properties of computing systems were introduced and demonstrated. On the other hand, the semantics based unfold-fold program transformation methods pose themselves diverse kinds of reachability tasks and try to solve them, aiming at improving the semantics tree of the program being transformed. That means some general-purpose verification methods may be used for strengthening program transformation techniques. This paper considers the question how finite countermodels for safety verification method might be used in Turchin's supercompilation method. We extract a number of supercompilation sub-algorithms trying to solve reachability problems and demonstrate use of an external countermodel finder for solving some of the problems.Comment: In Proceedings VPT 2015, arXiv:1512.0221

Machine learning discovers invariants of braids and flat braids

We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our ML takes form of supervised learning using neural networks (multilayer perceptrons). When they achieve good results in classification, we are able to interpret their structure as mathematical conjectures and then prove these conjectures as theorems. As a result, we find new convenient invariants of braids, including a complete invariant of flat braids.Comment: 24 page