Abstract. We study infinite games where one of the players always has a positional (memory-less) winning strategy, while the other player may use a history-dependent strategy. We investigate winning conditions which guarantee such a property for all arenas, or all finite arenas. We establish some closure properties of such conditions, and discover some common reasons behind several known and new positional determinacy results. We exhibit several new classes of winning conditions having this property: the class of concave conditions (for finite arenas) and the classes of monotonic conditions and geometrical conditions (for all arenas).

Parikh's Theorem states that semilinear sets are effectively equivalent with the Parikh images of regular languages and those of context-free languages. In this paper, we study the complexity of Parikh's Theorem over any fixed alphabet size d. We prove various normal form theorems in the case of NFAs and CFGs. In particular, the normal form theorems ensure that a union of linear sets with d generators suffice to express such Parikh images, which in the case of NFAs can further be computed in polynomial time. We then apply apply our results to derive: (1) optimal complexity for decision problems concerning Parikh images (e.g. membership, universality, equivalence, and disjointness), (2) a new polynomial fragment of integer programming, (3) an answer to an open question about PAC-learnability of semilinear sets, and (4) an optimal algorithm for verifying LTL over discrete-timed reversal-bounded counter systems.

Abstract. We consider commutative regular and context-free grammars, or, in other words, Parikh images of regular and context-free languages. By using linear algebra and a branching analog of the classic Euler theorem, we show that, under an assumption that the terminal alphabet is fixed, the membership problem for regular grammars (given v in binary and a regular commutative grammar G, does G generate v?) is P, and that the equivalence problem for context free grammars (do G1 and G2 generate the same language?) is in Π P 2 . IntroductionLet Σ be a finite alphabet. By Σ * we denote the set of words over Σ, or finite sequences of elements of Σ. For a word w ∈ Σ * , by Ψ(w) (the Parikh image of w) we denote the function from Σ to non-negative integers N, such that eachContext free and regular languages are one of the most important classes of languages in computer science [HU79]. By a famous result of Parikh [Par66], a subset of N Σ is a Parikh image of a context free language if and only if it is a semilinear set, or a union of finitely many linear sets.In this paper, we explore the complexity of various problems related to Parikh images of context free languages, such as the following:• Membership: Given a context-free grammar G and v ∈ N Σ (given in binary). Is v a member of the Ψ(G), the Parikh image of the language generated by G? • Universality: Given two context-free grammars G, is Ψ(G) equal to N Σ ?• Inclusion: Given two context-free grammars G 1 and G 2 , does Ψ(G 1 ) ⊆ Ψ(G 2 )?• Equality: Given two context-free grammars G 1 and G 2 , does Ψ(G 1 ) = Ψ(G 2 )?• Disjointness: Given two context-free grammars G 1 and G 2 , is Ψ(G 1 ) ∩ Ψ(G 2 ) nonempty?Since in this paper we are never interested in the order of terminals or non-terminals, we treat everything in a commutative way. This allows us to identify the commutative languages (subsets of Σ * ) with their Parikh images (subsets of N Σ ).

The fixed-parameter tractability of graph isomorphism is an open problem with respect to a number of natural parameters, such as tree-width, genus and maximum degree. We show that graph isomorphism is fixed-parameter tractable when parameterized by the tree-depth of the graph. We also extend this result to a parameter generalizing both tree-depth and max-leaf-number by deploying new variants of cops-androbbers games.

Gossip protocols deal with a group of communicating agents, each holding a private information, and aim at arriving at a situation in which all the agents know each other secrets. Distributed epistemic gossip protocols are particularly simple distributed programs that use formulas from an epistemic logic. Recently, the implementability of these distributed protocols was established (which means that the evaluation of these formulas is decidable), and the problems of their partial correctness and termination were shown to be decidable, but their exact computational complexity was left open. We show that for any monotonic type of calls the implementability of a distributed epistemic gossip protocol is a P NPcomplete problem, while the problems of its partial correctness and termination are in coNP NP .