Sergei Vorobyov

A Discrete Subexponential Algorithm for Parity Games

We suggest a new randomized algorithm for solving parity games with worst case time complexity roughly min(O(n3 ? (n/k+ 1)k), 2O(?n log n), where n is the number of vertices and k the number of colors of the game. This is comparable with the previously known algorithms when the number of colors is small. However, the subexponential bound is an advantage when the number of colors is large, k = ?(n1/2+?).

Memoryless determinacy of parity and mean payoff games: a simple proof

We give a simple, direct, and constructive proof of memoryless determinacy for parity and mean payoff games. First, we prove by induction that the finite duration versions of these games, played until some vertex is repeated, are determined and both players have memoryless winning strategies. In contrast to the proof of Ehrenfeucht and Mycielski, Internat. J. Game Theory, 8 (1979) 109-113, our proof does not refer to the infinite-duration versions. Second, we show that memoryless determinacy straightforwardly generalizes to infinite duration versions of parity and mean payoff games.

Complexity of nonrecursive logic programs with complex values

values should be treated. There are two major approaches. We investigate complexity of the SUCCESS problem for logic query languages with complex values: check whether a query defines a nonempty set. The SUCCESS problem for recursive query languages with complex values is undecidable, so WC study the complexity of nonrecursive queries. By complcx values we understand values such as trees, finite sets, and multlscts. Due to the well-known correspondence between relational query languages and datalog, our results can be considered as results about relational query languages with complex values. The paper gives a complete complexity classification of the SUCCESS problem for nonrecursive logic programs over trees depending on the underlying signature, presence of negation, and range restrictedness. We also prove several results about finite sets and multisets. 1. In constmint logic programming [41,42] and constraa’nt databases [33] any value is identified by the set of constraints true on this value. The addition of a new type of values requires the addition of new constraint predicates. A similar approach to relational query languages was also considered in [lo]. 2. Another approach to adding complex values, which can be called structuml, requires that values be represented by means of their structure. For example, to represent sets one may enrich the language with constant 0 to denote the empty set and the set constructor {sit} denoting the addition of an element s to the set t. Then the set (tl,. . . , {tll . . . {tnlO}. . .}. t,,} will be denoted by the term The only changes to the semantics of logic programming are the changes in the treatment of equality, since new predicate symbols are not free constructors. Such an approach is considered in a number of papers, for example [25,37,9,50,22,21,19].

Combinatorial structure and randomized subexponential algorithms for infinite games

The complexity of solving infinite games, including parity, mean payoff, and simple stochastic, is an important open problem in verification, automata, and complexity theory. In this paper, we develop an abstract setting for studying and solving such games, based on function optimization over certain discrete structures. We introduce new classes of recursively local-global (RLG) and partial recursively local-global (PRLG) functions, and show that strategy evaluation functions for simple stochastic, mean payoff, and parity games belong to these classes.In this setting, we suggest randomized subexponential algorithms appropriate for RLG-and PRLG-function optimization. We show that the subexponential algorithms for combinatorial linear programming, due to Kalai and Matousek, Sharir, Welzl, can be adapted for optimizing the RLG-and PRLG-functions.

Cyclic games and linear programming

New efficient algorithms for solving infinite-duration two-person adversary games with the decision problem inNP@?coNP, based on linear programming (LP), LP-representations, combinatorial LP, linear complementarity problem (LCP), controlled LP are surveyed.

Complexity of Model Checking by Iterative Improvement: The Pseudo-Boolean Framework

We present several new algorithms as well as new lower and upper bounds for optimizing functions underlying infinite games pertinent to computer-aided verification.

An Improved Lower Bound for the Elementary Theories of Trees

The first-order theories of finite and rational, constructor and feature trees possess complete axiomatizations and are decidable by quantifier elimination [15, 13, 14, 5, 10, 3, 20, 4, 2]. By using the uniform inseparability lower bounds techniques due to Compton and Henson [6], based on representing large binary relations by means of short formulas manipulating with high trees, we prove that all the above theories, as well as all their subtheories, are non-elementary in the sense of Kalmar, i.e., cannot be decided within time bounded by a k-story exponential function exp k (n) for any fixed k. Moreover, for some constant d>0 these decision problems require nondeterministic time exceeding exp∞ (⌊dn⌋) infinitely often.

Linear complementarity and p-matrices for stochastic games

We define the first nontrivial polynomially recognizable subclass of P-matrixGeneralized Linear Complementarity Problems (GLCPs) with a subexponential pivot rule. No such classes/rules were previously known. We show that a subclass of Shapley turn-based stochastic games, subsuming Condons simple stochastic games, is reducible to the new class of GLCPs. Based on this we suggest the new strongly subexponential combinatorial algorithms for these games.

The "Hardest" natural decidable theory

We prove that any decision procedure for a modest fragment of L. Henkins theory of pure propositional types requires time exceeding a tower of 2s of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linearly high towers of 2s and since mid-seventies it was an open problem whether natural decidable theories requiring more than that exist. We give the affirmative answer. As an application of this todays strongest lower bound we improve known and settle new lower bounds for several problems in the simply typed lambda calculus.

On the arithmetic inexpressiveness of term rewriting systems

Unquantified Presburger arithmetic is proved to be nonaxiomatizable by a canonical (i.e. Noetherian and confluent) term-rewriting system, if Boolean connectives are not allowed in the left-hand sides of the rewrite rules. It is conjectured that the same is true if the number of Boolean connectives in left-hand sides of the rules is uniformly bounded by an arbitrary natural number.<<ETX>>