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.

The past decade has witnessed substantial developments in string solving. Motivated by the complexity of string solving strategies adopted in existing string solvers, we investigate a simple and generic method for solving string constraints: regular constraint propagation. The method repeatedly computes pre- or postimages of regular languages under the string functions present in a string formula, inferring more and more knowledge about the possible values of string variables, until either a conflict is found or satisfiability of the string formula can be concluded. Such a propagation strategy is applicable to string constraints with multiple operations like concatenation, replace, and almost all flavors of string transductions. We demonstrate the generality and effectiveness of this method theoretically and experimentally. On the theoretical side, we show that RCP is sound and complete for a large fragment of string constraints, subsuming both straight-line and chain-free constraints, two of the most expressive decidable fragments for which some modern string solvers provide formal completeness guarantees. On the practical side, we implement regular constraint propagation within the open-source string solver OSTRICH. Our experimental evaluation shows that this addition significantly improves OSTRICH’s performance and makes it competitive with existing solvers. In fact, it substantially outperforms other solvers on random PCP and bioinformatics benchmarks. The results also suggest that incorporating regular constraint propagation alongside other techniques could lead to substantial performance gains for existing solvers.

Abstract Sequence theories are an extension of theories of strings with an infinite alphabet of letters, together with a corresponding alphabet theory (e.g. linear integer arithmetic). Sequences are natural abstractions of extendable arrays, which permit a wealth of operations including append, map, split, and concatenation. In spite of the growing amount of tool support for theories of sequences by leading SMT-solvers, little is known about the decidability of sequence theories, which is in stark contrast to the state of the theories of strings. We show that the decidable theory of strings with concatenation and regular constraints can be extended to the world of sequences over an alphabet theory that forms a Boolean algebra, while preserving decidability. In particular, decidability holds when regular constraints are interpreted as parametric automata (which extend both symbolic automata and variable automata), but fails when interpreted as register automata (even over the alphabet theory of equality). When length constraints are added, the problem is Turing-equivalent to word equations with length (and regular) constraints. Similar investigations are conducted in the presence of symbolic transducers, which naturally model sequence functions like map, split, filter, etc. We have developed a new sequence solver, SeCo, based on parametric automata, and show its efficacy on two classes of benchmarks: (i) invariant checking on array-manipulating programs and parameterized systems, and (ii) benchmarks on symbolic register automata.

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 propose a machine learning framework to synthesize reactive controllers for systems whose interactions with their adversarial environment are modeled by infinite-duration, two-player games over (potentially) infinite graphs. Our frame-work targets safety games with infinitely many vertices, but it is also applicable to safety games over finite graphs whose size is too prohibitive for conventional synthesis techniques. The learning takes place in a feedback loop between a teacher component, which can reason symbolically about the safety game, and a learning algorithm, which successively learns an approximation of the winning region from various kinds of examples provided by the teacher. We develop a novel decision tree learning algorithm for this setting and show that our algorithm is guaranteed to converge to a reactive safety controller if a suitable approximation of the winning region can be expressed as a decision tree. Finally, we empirically compare the performance of a prototype implementation to existing approaches, which are based on constraint solving and automata learning, respectively.