85 research outputs found
Marking Shortest Paths On Pushdown Graphs Does Not Preserve MSO Decidability
In this paper we consider pushdown graphs, i.e. infinite graphs that can be
described as transition graphs of deterministic real-time pushdown automata. We
consider the case where some vertices are designated as being final and we
built, in a breadth-first manner, a marking of edges that lead to such vertices
(i.e., for every vertex that can reach a final one, we mark all out-going edges
laying on some shortest path to a final vertex).
Our main result is that the edge-marked version of a pushdown graph may
itself no longer be a pushdown graph, as we prove that this enrich graph may
have an undecidable MSO theory.
In this paper we consider pushdown graphs, i.e. infinite graphs that can be
described as transition graphs of deterministic real-time pushdown automata. We
consider the case where some vertices are designated as being final and we
build, in a breadth-first manner, a marking of edges that lead to such vertices
(i.e., for every vertex that can reach a final one, we mark all out-going edges
laying on some shortest path to a final vertex).
Our main result is that the edge-marked version of a pushdown graph may
itself no longer be a pushdown graph, as we prove that the MSO theory of this
enriched graph may be undecidable.Comment: 11 pages, 2 figure
The FC-rank of a context-free language
We prove that the finite condensation rank (FC-rank) of the lexicographic
ordering of a context-free language is strictly less than
An analysis of the equational properties of the well-founded fixed point
Well-founded fixed points have been used in several areas of knowledge
representation and reasoning and to give semantics to logic programs involving
negation. They are an important ingredient of approximation fixed point theory.
We study the logical properties of the (parametric) well-founded fixed point
operation. We show that the operation satisfies several, but not all of the
equational properties of fixed point operations described by the axioms of
iteration theories
On Long Words Avoiding Zimin Patterns
A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern p is unavoidable if, over every finite alphabet, every sufficiently long word encounters p. A theorem by Zimin and independently by Bean, Ehrenfeucht and McNulty states that a pattern over n distinct variables is unavoidable if, and only if, p itself is encountered in the n-th Zimin pattern. Given an alphabet size k, we study the minimal length f(n,k) such that every word of length f(n,k) encounters the n-th Zimin pattern. It is known that f is upper-bounded by a tower of exponentials. Our main result states that f(n,k) is lower-bounded by a tower of n-3 exponentials, even for k=2. To the best of our knowledge, this improves upon a previously best-known doubly-exponential lower bound. As a further result, we prove a doubly-exponential upper bound for encountering Zimin patterns in the abelian sense
The Caucal hierarchy of infinite graphs in terms of logic and higher-order pushdown automata
In this paper we give two equivalent characterizations of the Caucal hierarchy, a hierarchy of infinite graphs with a decidable monadic second-order (MSO) theory. It is obtained by iterating the graph transformations of unfolding and inverse rational mapping. The first characterization sticks to this hierarchical approach, replacing the language-theoretic operation of a rational mapping by an MSO-transduction and the unfolding by the treegraph operation. The second characterization is non-iterative. We show that the family of graphs of the Caucal hierarchy coincides with the family of graphs obtained as the ε-closure of configuration graphs of higher-order pushdown automata. While the different characterizations of the graph family show their robustness and thus also their importance, the characterization in terms of higher-order pushdown automata additionally yields that the graph hierarchy is indeed strict
How Good Is a Strategy in a Game with Nature?
International audienceWe consider games with two antagonistic players — Éloïse (modelling a program) and Abélard (modelling a byzantine environment) — and a third, unpredictable and uncontrollable player, that we call Nature. Motivated by the fact that the usual probabilistic semantics very quickly leads to undecidability when considering either infinite game graphs or imperfect information, we propose two alternative semantics that leads to decidability where the probabilistic one fails: one based on counting and one based on topology
Optimal Strategies in Pushdown Reachability Games
An algorithm for computing optimal strategies in pushdown reachability games was given by Cachat. We show that the information tracked by this algorithm is too coarse and the strategies constructed are not necessarily optimal. We then show that the algorithm can be refined to recover optimality. Through a further non-trivial argument the refined algorithm can be run in 2EXPTIME by bounding the play-lengths tracked to those that are at most doubly exponential. This is optimal in the sense that there exists a game for which the optimal strategy requires a doubly exponential number of moves to reach a target configuration
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