Visibly Pushdown Modular Games

Games on recursive game graphs can be used to reason about the control flow of sequential programs with recursion. In games over recursive game graphs, the most natural notion of strategy is the modular strategy, i.e., a strategy that is local to a module and is oblivious to previous module invocations, and thus does not depend on the context of invocation. In this work, we study for the first time modular strategies with respect to winning conditions that can be expressed by a pushdown automaton. We show that such games are undecidable in general, and become decidable for visibly pushdown automata specifications. Our solution relies on a reduction to modular games with finite-state automata winning conditions, which are known in the literature. We carefully characterize the computational complexity of the considered decision problem. In particular, we show that modular games with a universal Buchi or co Buchi visibly pushdown winning condition are EXPTIME-complete, and when the winning condition is given by a CARET or NWTL temporal logic formula the problem is 2EXPTIME-complete, and it remains 2EXPTIME-hard even for simple fragments of these logics. As a further contribution, we present a different solution for modular games with finite-state automata winning condition that runs faster than known solutions for large specifications and many exits.Comment: In Proceedings GandALF 2014, arXiv:1408.556

LIPIcs

We study two-player zero-sum games over infinite-state graphs equipped with ωB and finitary conditions. Our first contribution is about the strategy complexity, i.e the memory required for winning strategies: we prove that over general infinite-state graphs, memoryless strategies are sufficient for finitary Büchi, and finite-memory suffices for finitary parity games. We then study pushdown games with boundedness conditions, with two contributions. First we prove a collapse result for pushdown games with ωB-conditions, implying the decidability of solving these games. Second we consider pushdown games with finitary parity along with stack boundedness conditions, and show that solving these games is EXPTIME-complete

Hyperplane Separation Technique for Multidimensional Mean-Payoff Games

We consider both finite-state game graphs and recursive game graphs (or pushdown game graphs), that can model the control flow of sequential programs with recursion, with multi-dimensional mean-payoff objectives. In pushdown games two types of strategies are relevant: global strategies, that depend on the entire global history; and modular strategies, that have only local memory and thus do not depend on the context of invocation. We present solutions to several fundamental algorithmic questions and our main contributions are as follows: (1) We show that finite-state multi-dimensional mean-payoff games can be solved in polynomial time if the number of dimensions and the maximal absolute value of the weight is fixed; whereas if the number of dimensions is arbitrary, then problem is already known to be coNP-complete. (2) We show that pushdown graphs with multi-dimensional mean-payoff objectives can be solved in polynomial time. (3) For pushdown games under global strategies both single and multi-dimensional mean-payoff objectives problems are known to be undecidable, and we show that under modular strategies the multi-dimensional problem is also undecidable (whereas under modular strategies the single dimensional problem is NP-complete). We show that if the number of modules, the number of exits, and the maximal absolute value of the weight is fixed, then pushdown games under modular strategies with single dimensional mean-payoff objectives can be solved in polynomial time, and if either of the number of exits or the number of modules is not bounded, then the problem is NP-hard. (4) Finally we show that a fixed parameter tractable algorithm for finite-state multi-dimensional mean-payoff games or pushdown games under modular strategies with single-dimensional mean-payoff objectives would imply the solution of the long-standing open problem of fixed parameter tractability of parity games.Comment: arXiv admin note: text overlap with arXiv:1201.282

Symbolic construction of GR(1) contracts for systems with full information

This work proposes a symbolic algorithm for the construction of assume-guarantee specifications that allow multiple agents to cooperate. Each agent is assigned goals expressed in a fragment of linear temporal logic known as generalized Streett with one pair, GR(1). These goals may be unrealizable, unless each agent makes additional assumptions, about the behavior of other agents. The algorithm constructs a contract among the agents, in that only the infinite behavior of the given goals is constrained, known as liveness, not the finite one, known as safety. This defers synthesis to a later stage of refinement, modularizing the design process. We prove that there exist GR(1) games that do not admit any refining GR(1) contract. For this reason, we formulate contracts with nested GR(1) properties and auxiliary communication variables, and prove that they always exist. The algorithm’s fixpoint structure is similar to GR(1) synthesis, enjoying time complexity polynomial in the number of states, and linear in number of recurrence goals

IST Austria Technical Report

Two-player games on graphs are central in many problems in formal verification and program analysis such as synthesis and verification of open systems. In this work we consider solving recursive game graphs (or pushdown game graphs) that can model the control flow of sequential programs with recursion. While pushdown games have been studied before with qualitative objectives, such as reachability and ω-regular objectives, in this work we study for the first time such games with the most well-studied quantitative objective, namely, mean-payoff objectives. In pushdown games two types of strategies are relevant: (1) global strategies, that depend on the entire global history; and (2) modular strategies, that have only local memory and thus do not depend on the context of invocation, but only on the history of the current invocation of the module. Our main results are as follows: (1) One-player pushdown games with mean-payoff objectives under global strategies are decidable in polynomial time. (2) Two- player pushdown games with mean-payoff objectives under global strategies are undecidable. (3) One-player pushdown games with mean-payoff objectives under modular strategies are NP- hard. (4) Two-player pushdown games with mean-payoff objectives under modular strategies can be solved in NP (i.e., both one-player and two-player pushdown games with mean-payoff objectives under modular strategies are NP-complete). We also establish the optimal strategy complexity showing that global strategies for mean-payoff objectives require infinite memory even in one-player pushdown games; and memoryless modular strategies are sufficient in two- player pushdown games. Finally we also show that all the problems have the same complexity if the stack boundedness condition is added, where along with the mean-payoff objective the player must also ensure that the stack height is bounded

Faster Algorithms for Algebraic Path Properties in Recursive State Machines with Constant Treewidth

Interprocedural analysis is at the heart of numerous applications in programming languages, such as alias analysis, constant propagation, etc. Recursive state machines (RSMs) are standard models for interprocedural analysis. We consider a general framework with RSMs where the transitions are labeled from a semiring, and path properties are algebraic with semiring operations. RSMs with algebraic path properties can model interprocedural dataflow analysis problems, the shortest path problem, the most probable path problem, etc. The traditional algorithms for interprocedural analysis focus on path properties where the starting point is fixed as the entry point of a specific method. In this work, we consider possible multiple queries as required in many applications such as in alias analysis. The study of multiple queries allows us to bring in a very important algorithmic distinction between the resource usage of the one-time preprocessing vs for each individual query. The second aspect that we consider is that the control flow graphs for most programs have constant treewidth. Our main contributions are simple and implementable algorithms that support multiple queries for algebraic path properties for RSMs that have constant treewidth. Our theoretical results show that our algorithms have small additional one-time preprocessing, but can answer subsequent queries significantly faster as compared to the current best-known solutions for several important problems, such as interprocedural reachability and shortest path. We provide a prototype implementation for interprocedural reachability and intraprocedural shortest path that gives a significant speed-up on several benchmarks

Faster Algorithms for Dynamic Algebraic Queries in Basic RSMs with Constant Treewidth

Interprocedural analysis is at the heart of numerous applications in programming languages, such as alias analysis, constant propagation, and so on. Recursive state machines (RSMs) are standard models for interprocedural analysis. We consider a general framework with RSMs where the transitions are labeled from a semiring and path properties are algebraic with semiring operations. RSMs with algebraic path properties can model interprocedural dataflow analysis problems, the shortest path problem, the most probable path problem, and so on. The traditional algorithms for interprocedural analysis focus on path properties where the starting point is fixed as the entry point of a specific method. In this work, we consider possible multiple queries as required in many applications such as in alias analysis. The study of multiple queries allows us to bring in an important algorithmic distinction between the resource usage of the one-time preprocessing vs for each individual query. The second aspect we consider is that the control flow graphs for most programs have constant treewidth. Our main contributions are simple and implementable algorithms that support multiple queries for algebraic path properties for RSMs that have constant treewidth. Our theoretical results show that our algorithms have small additional one-time preprocessing but can answer subsequent queries significantly faster as compared to the current algorithmic solutions for interprocedural dataflow analysis. We have also implemented our algorithms and evaluated their performance for performing on-demand interprocedural dataflow analysis on various domains, such as for live variable analysis and reaching definitions, on a standard benchmark set. Our experimental results align with our theoretical statements and show that after a lightweight preprocessing, on-demand queries are answered much faster than the standard existing algorithmic approaches

Nash Equilibria in Symmetric Graph Games with Partial Observation

International audienceWe investigate a model for representing large multiplayer games, which satisfy strong symmetry properties. This model is made of multiple copies of an arena; each player plays in his own arena, and can partially observe what the other players do. Therefore, this game has partial information and symmetry constraints, which make the computation of Nash equilibria difficult. We show several undecidability results, and for bounded-memory strategies, we precisely characterize the complexity of computing pure Nash equilibria for qualitative objectives in this game model