Denis Kuperberg

Quasi-Weak Cost Automata: A New Variant of Weakness

Cost automata have a finite set of counters which can be manipulated on each transition but do not aect control flow. Based on the evolution of the counter values, these automata define functions from a domain like words or trees to N[{1}, modulo an equivalence relation which ignores exact values but preserves boundedness properties. These automata have been studied by Colcombet et al. as part of a “theory of regular cost functions”, an extension of the theory of regular languages which retains robust equivalences, closure properties, and decidability like the classical theory. We extend this theory by introducing quasi-weak cost automata. Unlike traditional weak automata which have a hard-coded bound on the number of alternations between accepting and rejecting states, quasi-weak automata bound the alternations using the counter values (which can vary across runs). We show that these automata are strictly more expressive than weak cost automata over infinite trees. The main result is a Rabin-style characterization theorem: a function is quasi-weak definable if and only if it is definable using two dual forms of nondeterministic Buchi cost automata. This yields a new decidability result for cost functions over infinite trees. 1998 ACM Subject Classification F.1.1 Models of Computation

Two-way cost automata and cost logics over infinite trees

Regular cost functions provide a quantitative extension of regular languages that retains most of their important properties, such as expressive power and decidability, at least over finite and infinite words and over finite trees. Much less is known over infinite trees. We consider cost functions over infinite trees defined by an extension of weak monadic second-order logic with a new fixed-point-like operator. We show this logic to be decidable, improving previously known decidability results for cost logics over infinite trees. The proof relies on an equivalence with a form of automata with counters called quasi-weak cost automata, as well as results about converting two-way alternating cost automata to one-way alternating cost automata.

Nondeterminism in the presence of a diverse or unknown future

Choices made by nondeterministic word automata depend on both the past (the prefix of the word read so far) and the future (the suffix yet to be read). In several applications, most notably synthesis, the future is diverse or unknown, leading to algorithms that are based on deterministic automata. Hoping to retain some of the advantages of nondeterministic automata, researchers have studied restricted classes of nondeterministic automata. Three such classes are nondeterministic automata that are good for trees (GFT; i.e., ones that can be expanded to tree automata accepting the derived tree languages, thus whose choices should satisfy diverse futures), good for games (GFG; i.e., ones whose choices depend only on the past), and determinizable by pruning (DBP; i.e., ones that embody equivalent deterministic automata). The theoretical properties and relative merits of the different classes are still open, having vagueness on whether they really differ from deterministic automata. In particular, while DBP ⊆ GFG ⊆ GFT, it is not known whether every GFT automaton is GFG and whether every GFG automaton is DBP. Also open is the possible succinctness of GFG and GFT automata compared to deterministic automata. We study these problems for ω-regular automata with all common acceptance conditions. We show that GFT=GFG⊃DBP, and describe a determinization construction for GFG automata.

On Determinisation of Good-for-Games Automata

In this work we study Good-For-Games GFG automata over

Varieties of Cost Functions.

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Linear Temporal Logic for Regular Cost Functions

-words: non-deterministic automata where the non-determinism can be resolved by a strategy depending only on the prefix of the

Trading Bounds for Memory in Games with Counters

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