Strategy synthesis for linear arithmetic games

We present a causality-based algorithm for solving two-player reachability games represented by logical constraints. These games are a useful formalism to model a wide array of problems arising, e.g., in program synthesis. Our technique for solving these games is based on the notion of subgoals, which are slices of the game that the reachability player necessarily needs to pass through in order to reach the goal. We use Craig interpolation to identify these necessary sets of moves and recursively slice the game along these subgoals. Our approach allows us to infer winning strategies that are structured along the subgoals. If the game is won by the reachability player, this is a strategy that progresses through the subgoals towards the final goal; if the game is won by the safety player, it is a permissive strategy that completely avoids a single subgoal. We evaluate our prototype implementation on a range of different games. On multiple benchmark families, our prototype scales dramatically better than previously available tools.

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Causality-Based Game Solving

Baier

Coenen

Finkbeiner

et al. 2021

“…Another setting with a decidable synthesis result over unbounded domains is work on strategy synthesis for linear arithmetic satisfiability games [17]. There it is shown that for a satisfiability game, in which two players (SAT and UNSAT) play to prove a formula is satisfiable (where the formula is interpreted over the theory of linear rational arithmetic), if the SAT player has a winning strategy then a strategy can be synthesized.…”

Section: Related Workmentioning

confidence: 99%

Decidable Synthesis of Programs with Uninterpreted Functions

Krogmeier

Mathur

Murali

et al. 2020

We identify a decidable synthesis problem for a class of programs of unbounded size with conditionals and iteration that work over infinite data domains. The programs in our class use uninterpreted functions and relations, and abide by a restriction called coherence that was recently identified to yield decidable verification. We formulate a powerful grammar-restricted (syntax-guided) synthesis problem for coherent uninterpreted programs, and we show the problem to be decidable, identify its precise complexity, and also study several variants of the problem.

“…To our knowledge, this paper is the first one to automatically learn abstract domains and transformers that are useful for program synthesis. We also believe it is the first to apply interpolation to program synthesis, although interpolation has been used to synthesize other artifacts such as circuits [23] and strategies for infinite games [24]. In what follows, we briefly survey existing work related to program synthesis, abstraction learning, and abstract transformer computations.…”

Section: Related Workmentioning

confidence: 99%

Learning Abstractions for Program Synthesis

Wang

Anderson

Dillig

et al. 2018

Many example-guided program synthesis techniques use abstractions to prune the search space. While abstraction-based synthesis has proven to be very powerful, a domain expert needs to provide a suitable abstract domain, together with the abstract transformers of each DSL construct. However, coming up with useful abstractions can be non-trivial, as it requires both domain expertise and knowledge about the synthesizer. In this paper, we propose a new technique for learning abstractions that are useful for instantiating a general synthesis framework in a new domain. Given a DSL and a small set of training problems, our method uses tree interpolation to infer reusable predicate templates that speed up synthesis in a given domain. Our method also learns suitable abstract transformers by solving a certain kind of second-order constraint solving problem in a data-driven way. We have implemented the proposed method in a tool called Atlas and evaluate it in the context of the Blaze meta-synthesizer. Our evaluation shows that (a) Atlas can learn useful abstract domains and transformers from few training problems, and (b) the abstractions learned by Atlas allow Blaze to achieve significantly better results compared to manually-crafted abstractions. Abstraction learner Synthesizer ATLAS approach Current abstractionSynthesized programs DSL + training problemsAbstraction ( predicates + transformers )