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Register a new userRecursion-Theoretic Ranking and Compression
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Added by
Lane A. Hemaspaandra
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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For which sets A does there exist a mapping, computed by a total or partial recursive function, such that the mapping, when its domain is restricted to A, is a 1-to-1, onto mapping to $Sigma^*$? And for which sets A does there exist such a mapping that respects the lexicographical ordering within A? Both cases are types of perfect, minimal hash functions. The complexity-theoret
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The locality of a graph problem is the smallest distance $T$ such that each node can choose its own part of the solution based on its radius-$T$ neighborhood. In many settings, a graph problem can be solved efficiently with a distributed or parallel algorithm if and only if it has a small locality. In this work we seek to automate the study of solvability and locality: given the description of a graph problem $Pi$, we would like to determine if $Pi$ is solvable and what is the asymptotic locality of $Pi$ as a function of the size of the graph. Put otherwise, we seek to automatically synthesize efficient distributed and parallel algorithms for solving $Pi$. We focus on locally checkable graph problems; these are problems in which a solution is globally feasible if it looks feasible in all constant-radius neighborhoods. Prior work on such problems has brought primarily bad news: questions related to locality are undecidable in general, and even if we focus on the case of labeled paths and cycles, determining locality is $mathsf{PSPACE}$-hard (Balliu et al., PODC 2019). We complement prior negative results with efficient algorithms for the cases of unlabeled paths and cycles and, as an extension, for rooted trees. We introduce a new automata-theoretic perspective for studying locally checkable graph problems. We represent a locally checkable problem $Pi$ as a nondeterministic finite automaton $mathcal{M}$ over a unary alphabet. We identify polynomial-time-computable properties of the automaton $mathcal{M}$ that near-completely capture the solvability and locality of $Pi$ in cycles and paths, with the exception of one specific case that is $mbox{co-$mathsf{NP}$}$-complete.
We show that there are $Sigma_3^0$-complete languages of infinite words accepted by non-deterministic Petri nets with Buchi acceptance condition, or equivalently by Buchi blind counter automata. This shows that omega-languages accepted by non-deterministic Petri nets are topologically more complex than those accepted by deterministic Petri nets.
We introduce a new game-theoretic semantics (GTS) for the modal mu-calculus. Our so-called bounded GTS replaces parity games with alternative evaluation games where only finite paths arise; infinite paths are not needed even when the considered transition system is infinite. The novel games offer alternative approaches to various constructions in the framework of the mu-calculus. For example, they have already been successfully used as a basis for an approach leading to a natural formula size game for the logic. While our main focus is introducing the new GTS, we also consider some applications to demonstrate its uses. For example, we consider a natural model transformation procedure that reduces model checking games to checking a single, fixed formula in the constructed models, and we also use the GTS to identify new alternative variants of the mu-calculus with PTime model checking.
In this paper I distinguish two (pre)congruence requirements for semantic equivalences and preorders on processes given as closed terms in a system description language with a recursion construct. A lean congruence preserves equivalence when replacing closed subexpressions of a process by equivalent alternatives. A full congruence moreover allows replacement within a recursive specification of subexpressions that may contain recursion variables bound outside of these subexpressions. I establish that bisimilarity is a lean (pre)congruence for recursion for all languages with a structural operational semantics in the ntyft/ntyxt format. Additionally, it is a full congruence for the tyft/tyxt format.
A data tree is an unranked ordered tree whose every node is labelled by a letter from a finite alphabet and an element (datum) from an infinite set, where the latter can only be compared for equality. The article considers alternating automata on data trees that can move downward and rightward, and have one register for storing data. The main results are that nonemptiness over finite data trees is decidable but not primitive recursive, and that nonemptiness of safety automata is decidable but not elementary. The proofs use nondeterministic tree automata with faulty counters. Allowing upward moves, leftward moves, or two registers, each causes undecidability. As corollaries, decidability is obtained for two data-sensitive fragments of the XPath query language.
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