5,612 research outputs found
Mean-payoff Automaton Expressions
Quantitative languages are an extension of boolean languages that assign to
each word a real number. Mean-payoff automata are finite automata with
numerical weights on transitions that assign to each infinite path the long-run
average of the transition weights. When the mode of branching of the automaton
is deterministic, nondeterministic, or alternating, the corresponding class of
quantitative languages is not robust as it is not closed under the pointwise
operations of max, min, sum, and numerical complement. Nondeterministic and
alternating mean-payoff automata are not decidable either, as the quantitative
generalization of the problems of universality and language inclusion is
undecidable.
We introduce a new class of quantitative languages, defined by mean-payoff
automaton expressions, which is robust and decidable: it is closed under the
four pointwise operations, and we show that all decision problems are decidable
for this class. Mean-payoff automaton expressions subsume deterministic
mean-payoff automata, and we show that they have expressive power incomparable
to nondeterministic and alternating mean-payoff automata. We also present for
the first time an algorithm to compute distance between two quantitative
languages, and in our case the quantitative languages are given as mean-payoff
automaton expressions
Hyperbolic tilings and formal language theory
In this paper, we try to give the appropriate class of languages to which
belong various objects associated with tessellations in the hyperbolic plane.Comment: In Proceedings MCU 2013, arXiv:1309.104
Representing a P-complete problem by small trellis automata
A restricted case of the Circuit Value Problem known as the Sequential NOR
Circuit Value Problem was recently used to obtain very succinct examples of
conjunctive grammars, Boolean grammars and language equations representing
P-complete languages (Okhotin, http://dx.doi.org/10.1007/978-3-540-74593-8_23
"A simple P-complete problem and its representations by language equations",
MCU 2007). In this paper, a new encoding of the same problem is proposed, and a
trellis automaton (one-way real-time cellular automaton) with 11 states solving
this problem is constructed
Comparator automata in quantitative verification
The notion of comparison between system runs is fundamental in formal
verification. This concept is implicitly present in the verification of
qualitative systems, and is more pronounced in the verification of quantitative
systems. In this work, we identify a novel mode of comparison in quantitative
systems: the online comparison of the aggregate values of two sequences of
quantitative weights. This notion is embodied by {\em comparator automata}
({\em comparators}, in short), a new class of automata that read two infinite
sequences of weights synchronously and relate their aggregate values.
We show that {aggregate functions} that can be represented with B\"uchi
automaton result in comparators that are finite-state and accept by the B\"uchi
condition as well. Such {\em -regular comparators} further lead to
generic algorithms for a number of well-studied problems, including the
quantitative inclusion and winning strategies in quantitative graph games with
incomplete information, as well as related non-decision problems, such as
obtaining a finite representation of all counterexamples in the quantitative
inclusion problem.
We study comparators for two aggregate functions: discounted-sum and
limit-average. We prove that the discounted-sum comparator is -regular
iff the discount-factor is an integer. Not every aggregate function, however,
has an -regular comparator. Specifically, we show that the language of
sequence-pairs for which limit-average aggregates exist is neither
-regular nor -context-free. Given this result, we introduce the
notion of {\em prefix-average} as a relaxation of limit-average aggregation,
and show that it admits -context-free comparators
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