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 framework 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.
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 modulos constraints). Our experiments suggest that our learning algorithms substantially outperform the existing algorithms.
Parameterized synthesis offers a solution to the problem of constructing correct and verified controllers for parameterized systems. Such systems occur naturally in practice (e.g., in the form of distributed protocols where the amount of processes is often unknown at design time and the protocol must work regardless of the number of processes). In this paper, we present a novel learningbased approach to the synthesis of reactive controllers for parameterized systems from safety specifications. We use the framework of regular model checking to model the synthesis problem as an infinite-duration two-player game and show how one can utilize Angluin's well-known L * algorithm to learn correctby-design controllers. This approach results in a synthesis procedure that is conceptually simpler than existing synthesis methods with a completeness guarantee, whenever a winning strategy can be expressed by a regular set. We have implemented our algorithm in a tool called L * -PSynth and have demonstrated its performance on a range of benchmarks, including robotic motion planning and distributed protocols. Despite the simplicity of L * -PSynth it competes well against (and in many cases even outperforms) the state-of-the-art tools for synthesizing parameterized systems.
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.
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