A constraint-based approach to solving games on infinite graphs

Abstract

We present a constraint-based approach to computing winning strategies in two-player graph games over the state space of infinite-state programs. Such games have numerous applications in program verification and synthesis, including the synthesis of infinite-state reactive programs and branching-time verification of infinite-state programs. Our method handles games with winning conditions given by safety, reachability, and general Linear Temporal Logic (LTL) properties. For each property class, we give a deductive proof rule that --- provided a symbolic representation of the game players --- describes a winning strategy for a particular player. Our rules are sound and relatively complete. We show that these rules can be automated by using an off-the-shelf Horn constraint solver that supports existential quantification in clause heads. The practical promise of the rules is demonstrated through several case studies, including a challenging "Cinderella-Stepmother game" that allows infinite alternation of discrete and continuous choices by two players, as well as examples derived from prior work on program repair and synthesis.

Strategy synthesis for linear arithmetic games

“…There has been a great deal of decidability results and algorithms for solving finite state games of infinite duration [Emerson and Jutla 1991;Kupferman and Vardi 1999;Pnueli and Rosner 1989; Strategy Synthesis for Linear Arithmetic Games 61:3 Thomas 1995]. Linear reachability games are infinite state and infinite duration, a class of games that have received relatively less attention [Beyene et al 2014;De Alfaro et al 2001]. There is no algorithm that can synthesize a winning strategy for the entire class, so it is important to have a variety of different techniques with different strengths.…”

“…This paper contributes fully automated procedures for synthesizing winning strategies for both satisfiability and reachability games. In contrast to many (but not all) approaches to synthesis, our technique does not require user-supplied hints (such as a program grammar as in syntax guided synthesis (SyGuS) [Alur et al 2013], or solution hints in the form of templates in the style of Beyene et al [2014]; Srivastava et al [2013]). While such hints are sometimes natural or even desirable, fully automated strategy synthesis remains a fundamental algorithmic challenge.…”

“…It is a turn-based two-player game on an undirected graph, with players Cinderella and her Stepmother. The graph is a regular pentagon (same configuration as Beyene et al [2014]; Hurkens et al [2011]) for the purposes of this example. Each vertex of the pentagon contains a bucket (which represents a buffer) that can store water (which represents data).…”

Optimal Guard Synthesis for Memory Safety

“…Here, variable optind used at lines 16-17 is initialized by getopt to be the index of the next element to be processed in argv. Looking at lines 16-23, the programmer expects the user to pass some required arguments and accesses them at lines 16,19, and 20. However, since the user may have forgotten to pass the required arguments, the programmer must explicitly check whether the memory accesses at lines 16,19,20 are safe in order to prevent potentially disastrous buffer overflow or underflow errors.…”

“…As the first abduction-based approach to synthesis, our work is algorithmically very different from prior methods in this area. A concrete benefit is that, unlike prior constraint-based approaches to synthesis [10,11,18,19], our method does not require a template for the expressions being synthesized. A second benefit is that we can show the synthesized expressions to be optimal relative to loop invariants.…”