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

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

Synthesis of recursive state machines from libraries of game modules

2013 - 2014This thesis is focused on synthesis. In formal veri cation synthesis can be referred to the controller synthesis and the system synthesis. This work combines both this area of research. First we focus on synthesizing modular controllers considering game on recursive game graph with the requirement that the strategy for the protagonist must be modular. A recursive game graph is composed of a set of modules, whose vertices can be standard vertices or can correspond to invocations of other modules and the standard and the set of vertices is split into two sets each controlled by one of the players. A strategy is modular if it is local to a module and is oblivious to previous module invocations, and thus does not depend on the context of invocation. We study for the rst time modular strategies with respect to winning conditions that can be expressed languages of pushdown automata. We show that pushdown modular games are undecidable in general, and become decidable for visibly pushdown automata speci cations. We carefully characterize the computational complexity of the considered decision problem. In particular, we show that modular games with a universal B uchi or co-B uchi visibly pushdown winning condition are Exptime-complete, and when the winning condition is given as 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 di erent synthesis algorithm that runs faster than known solutions for large speci cations and many exits. In the second part of this thesis, we introduce and solve a new componentbased synthesis problem that subsumes the synthesis from libraries of recursive components introduced by Lustig and Vardi with the modular synthesis introduced by Alur et al. for recursive game graphs. We model the components of our libraries as game modules of a recursive game graph with unmapped boxes, and consider as correctness speci cation a target set of vertices. To solve this problem, we give an exponential-time xed-point algorithm that computes annotations for the vertices of the library components by exploring them backwards. We show a matching lower-bound via a direct reduction from linear-space alternating Turing machines, thus proving Exptime-completeness. We also give a second algorithm that solves this problem by annotating in a table the result of many local reachability game queries on each game component. This algorithm is exponential only in the number of the exits of the game components, and thus shows that the problem is xed-parameter tractable. Finally, we study a more general synthesis problem for component-based pushdown systems, the modular synthesis from a library of components (Lms). We model each component as a game graph with boxes as placeholders for calls to components, as in the previous model, but now the library is equipped also with a box-to-component map that is a partial function from boxes to components. An instance of a component C is essentially a copy of C along with a local strategy that resolves the nondeterminism of pl 0. An RSM S synthesized from a library is a set of instances along with a total function that maps each box in S to an instance of S and is consistent with the box-to-component map of the library. We give a solution to the Lms problem with winning conditions given as internal reachability objectives, or as external deterministic nite automata (FA) and deterministic visibly pushdown automata (VPA) (6). We show that the Lms problem is Exptime-complete for any of the considered speci cations. [edited by Author]XIII n.s

Visibly Linear Dynamic Logic

We introduce Visibly Linear Dynamic Logic (VLDL), which extends Linear Temporal Logic (LTL) by temporal operators that are guarded by visibly pushdown languages over finite words. In VLDL one can, e.g., express that a function resets a variable to its original value after its execution, even in the presence of an unbounded number of intermediate recursive calls. We prove that VLDL describes exactly the $\omega$-visibly pushdown languages. Thus it is strictly more expressive than LTL and able to express recursive properties of programs with unbounded call stacks. The main technical contribution of this work is a translation of VLDL into $\omega$-visibly pushdown automata of exponential size via one-way alternating jumping automata. This translation yields exponential-time algorithms for satisfiability, validity, and model checking. We also show that visibly pushdown games with VLDL winning conditions are solvable in triply-exponential time. We prove all these problems to be complete for their respective complexity classes.Comment: 25 Page

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

Automata Tutor v3

Computer science class enrollments have rapidly risen in the past decade. With current class sizes, standard approaches to grading and providing personalized feedback are no longer possible and new techniques become both feasible and necessary. In this paper, we present the third version of Automata Tutor, a tool for helping teachers and students in large courses on automata and formal languages. The second version of Automata Tutor supported automatic grading and feedback for finite-automata constructions and has already been used by thousands of users in dozens of countries. This new version of Automata Tutor supports automated grading and feedback generation for a greatly extended variety of new problems, including problems that ask students to create regular expressions, context-free grammars, pushdown automata and Turing machines corresponding to a given description, and problems about converting between equivalent models - e.g., from regular expressions to nondeterministic finite automata. Moreover, for several problems, this new version also enables teachers and students to automatically generate new problem instances. We also present the results of a survey run on a class of 950 students, which shows very positive results about the usability and usefulness of the tool

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