10 research outputs found
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Approximate comparison of distance automata
Distance automata are automata weighted over the semiring (NâȘ {â}, min,+) (the tropical semiring). Such automata compute functions from words to N
âȘ{â} such as the number of occurrences of a given letter. It is known that testing f 0 and two functions f,g computed by distance automata, answers "yes" if f <= (1-Δ ) g, "no" if f \not\leq g, and may answer "yes" or "no" in all other cases. This result highly refines previously known decidability results of the same type. The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation to the closure under products of sets of matrices over the tropical semiring. We also provide another theorem, of affine domination, which shows that previously known decision procedures for cost-automata have an improved precision when used over distance automata
Safety and Liveness of Quantitative Automata
The safety-liveness dichotomy is a fundamental concept in formal languages which plays a key role in verification. Recently, this dichotomy has been lifted to quantitative properties, which are arbitrary functions from infinite words to partially-ordered domains. We look into harnessing the dichotomy for the specific classes of quantitative properties expressed by quantitative automata. These automata contain finitely many states and rational-valued transition weights, and their common value functions Inf, Sup, LimInf, LimSup, LimInfAvg, LimSupAvg, and DSum map infinite words into the totally-ordered domain of real numbers. In this automata-theoretic setting, we establish a connection between quantitative safety and topological continuity and provide an alternative characterization of quantitative safety and liveness in terms of their boolean counterparts. For all common value functions, we show how the safety closure of a quantitative automaton can be constructed in PTime, and we provide PSpace-complete checks of whether a given quantitative automaton is safe or live, with the exception of LimInfAvg and LimSupAvg automata, for which the safety check is in ExpSpace. Moreover, for deterministic Sup, LimInf, and LimSup automata, we give PTime decompositions into safe and live automata. These decompositions enable the separation of techniques for safety and liveness verification for quantitative specifications
Regular Cost Functions, Part I: Logic and Algebra over Words
The theory of regular cost functions is a quantitative extension to the
classical notion of regularity. A cost function associates to each input a
non-negative integer value (or infinity), as opposed to languages which only
associate to each input the two values "inside" and "outside". This theory is a
continuation of the works on distance automata and similar models. These models
of automata have been successfully used for solving the star-height problem,
the finite power property, the finite substitution problem, the relative
inclusion star-height problem and the boundedness problem for monadic-second
order logic over words. Our notion of regularity can be -- as in the classical
theory of regular languages -- equivalently defined in terms of automata,
expressions, algebraic recognisability, and by a variant of the monadic
second-order logic. These equivalences are strict extensions of the
corresponding classical results. The present paper introduces the cost monadic
logic, the quantitative extension to the notion of monadic second-order logic
we use, and show that some problems of existence of bounds are decidable for
this logic. This is achieved by introducing the corresponding algebraic
formalism: stabilisation monoids.Comment: 47 page
Safety and Liveness of Quantitative Automata
The safety-liveness dichotomy is a fundamental concept in formal languages
which plays a key role in verification. Recently, this dichotomy has been
lifted to quantitative properties, which are arbitrary functions from infinite
words to partially-ordered domains. We look into harnessing the dichotomy for
the specific classes of quantitative properties expressed by quantitative
automata. These automata contain finitely many states and rational-valued
transition weights, and their common value functions Inf, Sup, LimInf, LimSup,
LimInfAvg, LimSupAvg, and DSum map infinite words into the totally-ordered
domain of real numbers. In this automata-theoretic setting, we establish a
connection between quantitative safety and topological continuity and provide
an alternative characterization of quantitative safety and liveness in terms of
their boolean counterparts. For all common value functions, we show how the
safety closure of a quantitative automaton can be constructed in PTime, and we
provide PSpace-complete checks of whether a given quantitative automaton is
safe or live, with the exception of LimInfAvg and LimSupAvg automata, for which
the safety check is in ExpSpace. Moreover, for deterministic Sup, LimInf, and
LimSup automata, we give PTime decompositions into safe and live automata.
These decompositions enable the separation of techniques for safety and
liveness verification for quantitative specifications.Comment: Full version of the paper to appear in CONCUR 202
Distance Desert Automata and Star Height Substitutions
We introduce the notion of nested distance desert automata as a joint generalization and further development of distance automata and desert automata. We show that limitedness of nested distance desert automata is PSPACE-complete. As an application, we show that it is decidable in 22O(n2) space whether the language accepted by an n-state non-deterministic automaton is of a star height less than a given integer h (concerning rational expressions with union, concatenation and iteration). We also show some decidability results for some substitution problems for recognizable languages
A Burnside Approach to the Termination of Mohriâs Algorithm for Polynomially Ambiguous Min-Plus-Automata
We show that the termination of Mohri's algorithm is decidable for polynomially ambiguous weighted finite automata over the tropical semiring which gives a partial answer to a question by Mohri [29]. The proof relies on an improvement of the notion of the twins property and a Burnside type characterization for the finiteness of the set of states produced by Mohri's algorithm