On the Borel Inseparability of Game Tree Languages

The game tree languages can be viewed as an automata-theoretic counterpart of parity games on graphs. They witness the strictness of the index hierarchy of alternating tree automata, as well as the fixed-point hierarchy over binary trees. We consider a game tree language of the first non-trivial level, where Eve can force that 0 repeats from some moment on, and its dual, where Adam can force that 1 repeats from some moment on. Both these sets (which amount to one up to an obvious renaming) are complete in the class of co-analytic sets. We show that they cannot be separated by any Borel set, hence {\em a fortiori} by any weakly definable set of trees. This settles a case left open by L.Santocanale and A.Arnold, who have thoroughly investigated the separation property within the $\mu$-calculus and the automata index hierarchies. They showed that separability fails in general for non-deterministic automata of type $\Sigma^{\mu}_{n}$, starting from level $n=3$, while our result settles the missing case $n=2$

On the separation question for tree languages

We show that the separation property fails for the classes Sigma_n of the Rabin-Mostowski index hierarchy of alternating automata on infinite trees. This extends our previous result (obtained with Szczepan Hummel) on the failure of the separation property for the class Sigma_2 (i.e., for co-Buchi sets). It remains open whether the separation property does hold for the classes Pi_n of the index hierarchy. To prove our result, we first consider the Rabin-Mostowski index hierarchy of deterministic automata on infinite words, for which we give a complete answer (generalizing previous results of Selivanov): the separation property holds for Pi_n and fails for Sigma_n-classes. The construction invented for words turns out to be useful for trees via a suitable game

The Ackermann Award 2012

Report on the Ackermann Award 2012

The longest path problem is polynomial on interval graphs.

The longest path problem is the problem of finding a path of maximum length in a graph. Polynomial solutions for this problem are known only for small classes of graphs, while it is NP-hard on general graphs, as it is a generalization of the Hamiltonian path problem. Motivated by the work of Uehara and Uno in [20], where they left the longest path problem open for the class of interval graphs, in this paper we show that the problem can be solved in polynomial time on interval graphs. The proposed algorithm runs in O(n 4) time, where n is the number of vertices of the input graph, and bases on a dynamic programming approach

Fixed Point Characterization of Infinite Behavior of Finite State Systems

Infinite behavior of nondeterministic finite state automata running over infinite trees or more generally over elements of an arbitrary algebraic structure is characterized by a calculus of fixed point terms interpreted in powerset algebras. These terms involve the least and greatest fixed point operators and disjunction as the only logical operation. A tight correspondence is established between a hierarchy of Rabin indices of automata and a hierarchy induced by alternation of the least and greatest fixed point operators. It is shown that, in the powerset algebra of trees constructed from a set of functional symbols, the fixed point hierarchy is infinite unless all the symbols are unary (i.e. trees are words). It is also shown that an interpretation of a closed fixed point term in any powerset algebra can be factorized through the interpretation of this term in the powerset algebra of trees, from which it is deduced that the question whether a term denotes always ; can be answered in ..

Continuous separation of game languages

We show that a family of tree languages W(';^), previously used by J. Brado/eld, and by the o/rst author to show the strictness of the Mostowski index hierarchy of alternating tree automata, forms a hierarchy w.r.t. the Wadge reducibility. That is, W(';^) ^W W('0;^0) if and only if the index ('0; ^0) is above ('; ^). This is one of the few separation results known so far, concerning the topological complexity of nondeterministically recognizable tree languages, and one of the few results about finite-state recognizable non-Borel sets of trees. The interest of the result is reinforced by the fact that a related family M(';^), witnessing the strictness of the index hierarchy of non-deterministic automata, does not have a similar property

First-Order Queries over Temporal Databases Inexpressible in Temporal Logic

. Queries over temporal databases involve references to time. We study differences between two approaches of including such references into a first-order query language (e.g., relational calculus): explicit (using typed variables and quantifiers) vs. implicit (using a finite set of modal connectives). We also show that though the latter approach---a firstorder query language with implicit references to time---is appealing by its simplicity and ease of use, it cannot express all queries expressible using the first one in general. This result also settles a longstanding open problem about the expressive power of first-order temporal logic. A consequence of this result is that there is no first-order complete query language subquery-closed with respect to a uniform database schema, and thus we cannot use temporal relational algebra over uniform relations to evaluate all first-order definable queries. 1 Introduction In the last several years, various languages for querying temporal databa..

On the Feasibility of Checking Temporal Integrity Constraints

We analyze the computational feasibility of checking temporal integrity constraints formulated in some sublanguages of first-order temporal logic. Our results illustrate the impact of the quantifier pattern on the complexity of this problem. The presence of a single quantifier in the scope of a temporal operator makes the problem undecidable. On the other hand, if no quantifiers are in the scope of a temporal operator and all the quantifiers are universal, temporal integrity checking can be done in exponential time. 1 Introduction As temporal databases become more widely used in practice [27, 28], the need arises to address database integrity issues that are specific to such databases. In particular, it is necessary to generalize the standard notion of static integrity (involving single database states) to temporal integrity (involving sequences of database states). This work is the first attempt to date to analyze the computational feasibility of checking temporal integrity constrain..