Abstract We present $$\textsf{HornStr}$$ HornStr , the first solver for invariant synthesis for Regular Model Checking (RMC) with the specification provided in the SMT-LIB 2.6 theory of strings. It is well-known that invariant synthesis for RMC subsumes various important verification problems, including safety verification for parameterized systems. To achieve a simple and standardized file format, we treat the invariant synthesis problem as a problem of solving Constrained Horn Clauses (CHCs) over strings. Two strategies for synthesizing invariants in terms of regular constraints are supported: (1) L* automata learning, and (2) SAT-based automata learning. $$\textsf{HornStr}$$ HornStr implements these strategies with the help of existing SMT solvers for strings, which are interfaced through SMT-LIB. $$\textsf{HornStr}$$ HornStr provides an easy-to-use interface for string solver developers to apply their techniques to verification. At the same time, it allows verification researchers to painlessly tap into the wealth of modern string solving techniques. To assess the effectiveness of $$\textsf{HornStr}$$ HornStr , we conducted a comprehensive evaluation using benchmarks derived from applications including parameterized verification and string rewriting tasks. Our experiments highlight $$\textsf{HornStr}$$ HornStr ’s capacity to effectively handle these benchmarks, e.g., as the first solver to verify the challenging MU puzzle automatically. Finally, $$\textsf{HornStr}$$ HornStr can be used to automatically generate a new class of interesting SMT-LIB 2.6 string constraint benchmarks, which might in the future be used in the SMT-COMP strings track. In particular, our experiments on the above invariant synthesis benchmarks produce more than 30000 new constraints. We also detail the performance of various integrated string solvers, providing insights into their effectiveness on our new benchmarks.

Concurrent stochastic games are an important formalism for the rational verification of probabilistic multi-agent systems, which involves verifying whether a temporal logic property is satisfied in some or all game-theoretic equilibria of such systems. In this work, we study the rational verification of probabilistic multi-agent systems where agents can cooperate by communicating over unbounded lossy channels. To model such systems, we present concurrent stochastic lossy channel games (CSLCG) and employ an equilibrium concept from cooperative game theory known as the core, which is the most fundamental and widely studied cooperative equilibrium concept. Our main contribution is twofold. First, we show that the rational verification problem is undecidable for systems whose agents have almost-sure LTL objectives. Second, we provide a decidable fragment of such a class of objectives that subsumes almost-sure reachability and safety. Our techniques involve reductions to solving infinite-state zero-sum games with conjunctions of qualitative objectives. To the best of our knowledge, our result represents the first decidability result on the rational verification of stochastic multi-agent systems on infinite arenas.

Abstract The learning problem consists in finding for a given sample of positive and negative Kripke structures a distinguishing formula that is verified by the former but not by the latter. Further constraints may bound the size and shape of the desired formula or even ask for its minimality in terms of syntactic size. This synthesis problem is motivated by explanation generation for dissimilar models, e.g. comparing a faulty implementation with the original protocol. We devise a -based encoding for a fixed size formula, then provide an incremental approach that guarantees minimality. We further report on a prototype implementation whose contribution is twofold: first, it allows us to assess the efficiency of various output fragments and optimizations. Secondly, we can experimentally evaluate this tool by randomly mutating Kripke structures or syntactically introducing errors in higher-level models, then learning distinguishing formulas.

Abstract We study the problem of learning a finite union of integer (axis-aligned) hypercubes over the d-dimensional integer lattice, i.e., whose edges are parallel to the coordinate axes. This is a natural generalization of the classic problem in the computational learning theory of learning rectangles. We provide a learning algorithm with access to a minimally adequate teacher (i.e. membership and equivalence oracles) that solves this problem in polynomial-time, for any fixed dimension d. Over a non-fixed dimension, the problem subsumes the problem of learning DNF boolean formulas, a central open problem in the field. We have also provided extensions to handle infinite hypercubes in the union, as well as showing how subset queries could improve the performance of the learning algorithm in practice. Our problem has a natural application to the problem of monadic decomposition of quantifier-free integer linear arithmetic formulas, which has been actively studied in recent years. In particular, a finite union of integer hypercubes correspond to a finite disjunction of monadic predicates over integer linear arithmetic (without modulo constraints). Our experiments suggest that our learning algorithms substantially outperform the existing algorithms.

We study the almost-sure reachability problem in a distributed system obtained as the asynchronous composition of N copies (called processes) of the same automaton (called protocol), that can communicate via a shared register with finite domain. The automaton has two types of transitions: write-transitions update the value of the register, while read-transitions move to a new state depending on the content of the register. Non-determinism is resolved by a stochastic scheduler. Given a protocol, we focus on almost-sure reachability of a target state by one of the processes. The answer to this problem naturally depends on the number N of processes. However, we prove that our setting has a cut-off property: the answer to the almost-sure reachability problem is constant when N is large enough; we then develop an EXPSPACE algorithm deciding whether this constant answer is positive or negative.

We study mixed-strategy Nash equilibria in multiplayer deterministic concurrent games played on graphs, with terminal-reward payoffs (that is, absorbing states with a value for each player). We show undecidability of the existence of a constrained Nash equilibrium (the constraint requiring that one player should have maximal payoff), with only three players and 0/1-rewards (i.e., reachability objectives). This has to be compared with the undecidability result by Ummels and Wojtczak for turn-based games which requires 14 players and general rewards. Our proof has various interesting consequences: (i) the undecidability of the existence of a Nash equilibrium with a constraint on the social welfare; (ii) the undecidability of the existence of an (unconstrained) Nash equilibrium in concurrent games with terminal-reward payoffs.