Topological Complexity of Context-Free omega-Languages: A Survey

Highly Undecidable Problems For Infinite Computations

We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $Π_2^1$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $Π_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, c…

An omega-Power of a Finitary Language Which is a Borel Set of Infinite Rank

Omega-powers of finitary languages are omega languages in the form V^omega, where V is a finitary language over a finite alphabet X. Since the set of infinite words over X can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers naturally arises and has been raised by Niwinski, by Simonnet, and by Staiger. It has been recently proved that for each integer n > 0, there exist some omega-powers of context free languages which are Pi^0_n-complete Borel sets, and that there exists a context free language L such that L^omega is analytic but not Borel…

On Omega Context Free Languages which are Borel Sets of Infinite Rank

This paper is a continuation of the study of topological properties of omega context free languages (omega-CFL). We proved before that the class of omega-CFL exhausts the hierarchy of Borel sets of finite rank, and that there exist some omega-CFL which are analytic but non Borel sets. We prove here that there exist some omega context free languages which are Borel sets of infinite (but not finite) rank, giving additional answer to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621].

Topology and Ambiguity in Omega Context Free Languages

We study the links between the topological complexity of an omega context free language and its degree of ambiguity. In particular, using known facts from classical descriptive set theory, we prove that non Borel omega context free languages which are recognized by Büchi pushdown automata have a maximum degree of ambiguity. This result implies that degrees of ambiguity are really not preserved by the operation of taking the omega power of a finitary context free language. We prove also that taking the adherence or the delta-limit of a finitary language preserves neither unambiguity nor inheren…

Borel Ranks and Wadge Degrees of Context Free Omega Languages

We show that, from a topological point of view, considering the Borel and the Wadge hierarchies, 1-counter Büchi automata have the same accepting power than Turing machines equipped with a Büchi acceptance condition. In particular, for every non null recursive ordinal alpha, there exist some Sigma^0_alpha-complete and some Pi^0_alpha-complete omega context free languages accepted by 1-counter Büchi automata, and the supremum of the set of Borel ranks of context free omega languages is the ordinal gamma^1_2 which is strictly greater than the first non recursive ordinal. This very surprising res…

Extending Büchi Automata with Constraints on Data Values

Recently data trees and data words have received considerable amount of attention in connection with XML reasoning and system verification. These are trees or words that, in addition to labels from a finite alphabet, carry data values from an infinite alphabet (data). In general it is rather hard to obtain logics for data words and trees that are sufficiently expressive, but still have reasonable complexity for the satisfiability problem. In this paper we extend and study the notion of Büchi automata for omega-words with data. We prove that the emptiness problem for such extension is decidable…

Approximate Automata for Omega-Regular Languages

Automata over infinite words, also known as omega-automata, play a key role in the verification and synthesis of reactive systems. The spectrum of omega-automata is defined by two characteristics: the acceptance condition (e.g. Büchi or parity) and the determinism (e.g., deterministic or nondeterministic) of an automaton. These characteristics play a crucial role in applications of automata theory. For example, certain acceptance conditions can be handled more efficiently than others by dedicated tools and algorithms. Furthermore, some applications, such as synthesis and probabilistic model ch…

并发博弈中纯策略相关均衡的复杂性

The Complexity of Pure Strategy Relevant Equilibria in Concurrent Games

本文研究并发博弈中满足社会福利或帕累托最优条件的纯策略纳什均衡的存在性复杂性，证明社会福利条件与受约束纳什均衡问题计算效率相同，而帕累托条件通常上界更高，但在 Büchi 和 Muller 博弈中分别属于 P 类和 PSPACE-complete。

Liveness Verification and Synthesis: New Algorithms for Recursive Programs

We consider the problems of liveness verification and liveness synthesis for recursive programs. The liveness verification problem (LVP) is to decide whether a given omega-context-free language is contained in a given omega-regular language. The liveness synthesis problem (LSP) is to compute a strategy so that a given omega-context-free game, when played along the strategy, is guaranteed to derive a word in a given omega-regular language. The problems are known to be EXPTIME-complete and EXPTIME-complete, respectively. Our contributions are new algorithms with optimal time complexity. For LVP,…