Let $T(G;X,Y)$ be the Tutte polynomial for graphs. We study the sequence $t_{a,b}(n) = T(K_n;a,b)$ where $a,b$ are non-negative integers, and show that for every $\mu \in \N$ the sequence $t_{a,b}(n)$ is ultimately periodic modulo $\mu$ provided $a \neq 1 \mod{\mu}$ and $b \neq 1 \mod{\mu}$. This result is related to a conjecture by A. Mani and R. Stones from 2016. The theorem is a consequence of a more general theorem which holds for a wide class of graph polynomials definable in Monadic Second Order Logic and some of its extensions, such as the the independence polynomial, the clique polynomial, etc. We also show similar results for the various substitution instances of the bivariate matching polynomial and the trivariate edge elimination polynomial $\xi(G;X,Y,Z)$ introduced by I. Averbouch, B. Godlin and the second author in 2008. All our results depend on the Specker-Blatter Theorem from 1981, which studies modular recurrence relations of combinatorial sequences which count the number of labeled graphs. Comment: Accepted for publication in Fundamenta Informaticae 186(3): 1-19 (2022), the special volume celebrating B.A. Trakhtenbrot's centenary

Parameterized model checking is the problem of deciding if a given formula holds irrespective of the number of participating processes. A standard approach for solving the parameterized model checking problem is to reduce it to model checking finitely many finite-state systems. This work considers the theoretical power and limitations of this technique. We focus on concurrent systems in which processes communicate via pairwise rendezvous, as well as the special cases of disjunctive guards and token passing; specifications are expressed in indexed temporal logic without the next operator; and the underlying network topologies are generated by suitable formulas and graph operations. First, we settle the exact computational complexity of the parameterized model checking problem for some of our concurrent systems, and establish new decidability results for others. Second, we consider the cases where model checking the parameterized system can be reduced to model checking some fixed number of processes, the number is known as a cutoff. We provide many cases for when such cutoffs can be computed, establish lower bounds on the size of such cutoffs, and identify cases where no cutoff exists. Third, we consider cases for which the parameterized system is equivalent to a single finite-state system (more precisely a Büchi word automaton), and establish tight bounds on the sizes of such automata.

A large number of web applications is based on a relational database together with a program, typically a script, that enables the user to interact with the database through embedded SQL queries and commands. In this paper, we introduce a method for formal automated verification of such systems which connects database theory to mainstream program analysis. We identify a fragment of SQL which captures the behavior of the queries in our case studies, is algorithmically decidable, and facilitates the construction of weakest preconditions. Thus, we can integrate the analysis of SQL queries into a program analysis tool chain. To this end, we implement a new decision procedure for the SQL fragment that we introduce. We demonstrate practical applicability of our results with three case studies, a web administrator, a simple firewall, and a conference management system.

BIP is a component-based framework for system design that has important industrial applications. BIP is built on three pillars: behavior, interaction, and priority. In this paper, we introduce first-order interaction logic (FOIL) that extends BIP to systems parameterized in the number of components. We show that FOIL captures classical parameterized architectures such as token-passing rings, cliques of identical components communicating with rendezvous or broadcast, and client-server systems. Although the BIP framework includes efficient verification tools for statically-defined systems, none are available for parameterized systems with an unbounded number of components. The parameterized model checking literature contains a wealth of techniques for systems of classical architectures. However, application of these results requires a deep understanding of parameterized model checking techniques and their underlying mathematical models. To overcome these difficulties, we introduce a framework that automatically identifies parameterized model checking techniques applicable to a BIP design. To our knowledge, it is the first framework that allows one to apply prominent parameterized model checking results in a systematic way.

The finite satisfiability problem of monadic second order logic is decidable only on classes of structures of bounded tree-width by the classic result of Seese (1991). We prove the following problem is decidable: Input: (i) A monadic second order logic sentence $α$, and (ii) a sentence $β$ in the two-variable fragment of first order logic extended with counting quantifiers. The vocabularies of $α$ and $β$ may intersect. Output: Is there a finite structure which satisfies $α\landβ$ such that the restriction of the structure to the vocabulary of $α$ has bounded tree-width? (The tree-width of the desired structure is not bounded.) As a consequence, we prove the decidability of the satisfiability problem by a finite structure of bounded tree-width of a logic extending monadic second order logic with linear cardinality constraints of the form $|X_{1}|+\cdots+|X_{r}|

We study the description logic ALCQIO, which extends the standard description logic ALC with nominals, inverses and counting quantifiers. ALCQIO is a fragment of first order logic and thus cannot define trees. We consider the satisfiability problem of ALCQIO over finite structures in which k relations are interpreted as forests of directed trees with unbounded out degrees. We show that the finite satisfiability problem of ALCQIO with forests is polynomial-time reducible to finite satisfiability of ALCQIO. As a consequence, we get that finite satisfiability is NEXPTIME-complete. Description logics with transitive closure constructors or fixed points have been studied before, but we give the first decidability result of the finite satisfiability problem for a description logic that contains nominals, inverse roles, and counting quantifiers and can define trees.

Using a theorem of N. Chomsky and M. Schützenberger one can characterize sequences of integers which satisfy linear recurrence relations with constant coefficients (C-finite sequences) as differences of two sequences counting words in regular languag