227 research outputs found

    О Π·Π°Π΄Π°Ρ‡Π΅ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°Ρ†ΠΈΠΈ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½Ρ‹Ρ… ΠΏΡ€ΠΎΠ³Ρ€Π°ΠΌΠΌ

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    First-order program schemata is one of the simplest models of sequential imperative programs intended for solving verification and optimization problems. We consider the decidable relation of logical-thermal equivalence of these schemata and the problem of their size minimization while preserving logical-thermal equivalence. We prove that this problem is decidable. Further we show that the first-order program schemata supplied with logical-thermal equivalence and finite state deterministic transducers operating over substitutions are mutually translated into each other. This relationship implies that the equivalence checking problem and the minimization problem for these transducers are also decidable. In addition, on the basis of the discovered relationship, we have found a subclass of firstorder program schemata such that their minimization can be performed in polynomial time by means of known techniques for minimization of finite state transducers operating over semigroups. Finally, we demonstrate that in general case the minimization problem for finite state transducers over semigroups may have several non-isomorphic solutions.Π‘Ρ‚Π°Π½Π΄Π°Ρ€Ρ‚Π½Ρ‹Π΅ схСмы ΠΏΡ€ΠΎΠ³Ρ€Π°ΠΌΠΌ β€” это ΠΎΠ΄Π½Π° ΠΈΠ· Π½Π°ΠΈΠ±ΠΎΠ»Π΅Π΅ простых ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½Ρ‹Ρ… ΠΈΠΌΠΏΠ΅Ρ€Π°Ρ‚ΠΈΠ²Π½Ρ‹Ρ… ΠΏΡ€ΠΎΠ³Ρ€Π°ΠΌΠΌ, прСдназначСнная для Ρ€Π΅ΡˆΠ΅Π½ΠΈΡ Π·Π°Π΄Π°Ρ‡ ΠΎΠΏΡ‚ΠΈΠΌΠΈΠ·Π°Ρ†ΠΈΠΈ ΠΈ Π²Π΅Ρ€ΠΈΡ„ΠΈΠΊΠ°Ρ†ΠΈΠΈ ΠΏΡ€ΠΎΠ³Ρ€Π°ΠΌΠΌ. ΠœΡ‹ рассматриваСм Ρ€Π°Π·Ρ€Π΅ΡˆΠΈΠΌΠΎΠ΅ ΠΎΡ‚Π½ΠΎΡˆΠ΅Π½ΠΈΠ΅ Π»ΠΎΠ³ΠΈΠΊΠΎ-Ρ‚Π΅Ρ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠΉ эквивалСнтности стандартных схСм ΠΏΡ€ΠΎΠ³Ρ€Π°ΠΌΠΌ ΠΈ Π·Π°Π΄Π°Ρ‡Ρƒ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°Ρ†ΠΈΠΈ ΠΈΡ… Ρ€Π°Π·ΠΌΠ΅Ρ€Π° ΠΏΡ€ΠΈ условии сохранСния ΠΎΡ‚Π½ΠΎΡˆΠ΅Π½ΠΈΡ Π»ΠΎΠ³ΠΈΠΊΠΎ-Ρ‚Π΅Ρ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠΉ эквивалСнтности. Нами Π΄ΠΎΠΊΠ°Π·Π°Π½ΠΎ, Ρ‡Ρ‚ΠΎ эта Π·Π°Π΄Π°Ρ‡Π° являСтся алгоритмичСски Ρ€Π°Π·Ρ€Π΅ΡˆΠΈΠΌΠΎΠΉ. Π”Π°Π»Π΅Π΅ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, Ρ‡Ρ‚ΠΎ стандартныС схСмы ΠΏΡ€ΠΎΠ³Ρ€Π°ΠΌΠΌ с ΠΎΡ‚Π½ΠΎΡˆΠ΅Π½ΠΈΠ΅ΠΌ Π»ΠΎΠ³ΠΈΠΊΠΎ-Ρ‚Π΅Ρ€ΠΌΠ°Π»ΡŒΠ½ΠΎΠΉ эквивалСнтности ΠΈ ΠΊΠΎΠ½Π΅Ρ‡Π½Ρ‹Π΅ Π΄Π΅Ρ‚Π΅Ρ€ΠΌΠΈΠ½ΠΈΡ€ΠΎΠ²Π°Π½Π½Ρ‹Π΅ Π°Π²Ρ‚ΠΎΠΌΠ°Ρ‚Ρ‹-ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚Π΅Π»ΠΈ, Ρ€Π°Π±ΠΎΡ‚Π°ΡŽΡ‰ΠΈΠ΅ Π½Π°Π΄ ΠΏΠΎΠ»ΡƒΠ³Ρ€ΡƒΠΏΠΏΠ°ΠΌΠΈ подстановок, Π²Π·Π°ΠΈΠΌΠ½ΠΎ Ρ‚Ρ€Π°Π½ΡΠ»ΠΈΡ€ΡƒΡŽΡ‚ΡΡ Π΄Ρ€ΡƒΠ³ Π² Π΄Ρ€ΡƒΠ³Π°. ΠžΡ‚ΡΡŽΠ΄Π° слСдуСт, Ρ‡Ρ‚ΠΎ Ρ‚Π°ΠΊΠΆΠ΅ Ρ€Π°Π·Ρ€Π΅ΡˆΠΈΠΌΡ‹ Π·Π°Π΄Π°Ρ‡ΠΈ ΠΏΡ€ΠΎΠ²Π΅Ρ€ΠΊΠΈ эквивалСнтности ΠΈ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°Ρ†ΠΈΠΈ для ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚Π΅Π»Π΅ΠΉ ΡƒΠΊΠ°Π·Π°Π½Π½ΠΎΠ³ΠΎ Π²ΠΈΠ΄Π°. ΠšΡ€ΠΎΠΌΠ΅ Ρ‚ΠΎΠ³ΠΎ, Π½Π° основС ΠΎΠ±Π½Π°Ρ€ΡƒΠΆΠ΅Π½Π½ΠΎΠΉ взаимосвязи Π²Ρ‹Π΄Π΅Π»Π΅Π½ подкласс стандартных схСм ΠΏΡ€ΠΎΠ³Ρ€Π°ΠΌΠΌ, минимизация ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… осущСствима Π·Π° полиномиальноС врСмя ΠΏΡ€ΠΈ ΠΏΠΎΠΌΠΎΡ‰ΠΈ Ρ€Π°Π½Π΅Π΅ извСстных ΠΌΠ΅Ρ‚ΠΎΠ΄ΠΎΠ² ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°Ρ†ΠΈΠΈ Π°Π²Ρ‚ΠΎΠΌΠ°Ρ‚ΠΎΠ²-ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚Π΅Π»Π΅ΠΉ, Ρ€Π°Π±ΠΎΡ‚Π°ΡŽΡ‰ΠΈΡ… Π½Π°Π΄ ΠΏΠΎΠ»ΡƒΠ³Ρ€ΡƒΠΏΠΏΠ°ΠΌΠΈ. Π’ Π·Π°ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΠΈ ΠΏΡ€ΠΈΠ²Π΅Π΄Π΅Π½ ΠΏΡ€ΠΈΠΌΠ΅Ρ€, ΡΠ²ΠΈΠ΄Π΅Ρ‚Π΅Π»ΡŒΡΡ‚Π²ΡƒΡŽΡ‰ΠΈΠΉ ΠΎ Ρ‚ΠΎΠΌ, Ρ‡Ρ‚ΠΎ Π² ΠΎΠ±Ρ‰Π΅ΠΌ случаС Π·Π°Π΄Π°Ρ‡Π° ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°Ρ†ΠΈΠΈ Π°Π²Ρ‚ΠΎΠΌΠ°Ρ‚ΠΎΠ²- ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚Π΅Π»Π΅ΠΉ Π½Π°Π΄ ΠΏΠΎΠ»ΡƒΠ³Ρ€ΡƒΠΏΠΏΠΎΠΉ подстановок ΠΌΠΎΠΆΠ΅Ρ‚ ΠΈΠΌΠ΅Ρ‚ΡŒ нСсколько Π½Π΅ΠΈΠ·ΠΎΠΌΠΎΡ€Ρ„Π½Ρ‹Ρ… Ρ€Π΅ΡˆΠ΅Π½ΠΈΠΉ.

    Algebraic Recognition of Regular Functions

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    We consider regular string-to-string functions, i.e. functions that are recognized by copyless streaming string transducers, or any of their equivalent models, such as deterministic two-way automata. We give yet another characterization, which is very succinct: finiteness-preserving functors from the category of semigroups to itself, together with a certain output function that is a natural transformation

    О ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°Ρ†ΠΈΠΈ ΠΊΠΎΠ½Π΅Ρ‡Π½Ρ‹Ρ… Π°Π²Ρ‚ΠΎΠΌΠ°Ρ‚ΠΎΠ²-ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚Π΅Π»Π΅ΠΉ Π½Π°Π΄ ΠΏΠΎΠ»ΡƒΠ³Ρ€ΡƒΠΏΠΏΠ°ΠΌΠΈ

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    Finite state transducers over semigroups are regarded as a formal model of sequentialΒ reactive programs that operate in the interaction with the environment. At receiving a piece of data aΒ program performs a sequence of actions and displays the current result. Such programs usually arise atΒ implementation of computer drivers, on-line algorithms, control procedures. In many cases verificationΒ of such programs can be reduced to minimization and equivalence checking problems for finite stateΒ transducers. Minimization of a transducer over a semigroup is performed in three stages. At firstΒ the greatest common left-divisors are computed for all states of the transducer, next the transducer isΒ brought to a reduced form by pulling all such divisors ”upstream”, and finally a minimization algorithmΒ for finite state automata is applied to the reduced transducer.Автоматы-ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚Π΅Π»ΠΈ Π½Π°Π΄ ΠΏΠΎΠ»ΡƒΠ³Ρ€ΡƒΠΏΠΏΠ°ΠΌΠΈ ΠΌΠΎΠΆΠ½ΠΎ ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΠΎΠ²Π°Ρ‚ΡŒ Π² качСствС модСли ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½Ρ‹Ρ… Ρ€Π΅Π°Π³ΠΈΡ€ΡƒΡŽΡ‰ΠΈΡ… ΠΏΡ€ΠΎΠ³Ρ€Π°ΠΌΠΌ, Ρ€Π°Π±ΠΎΡ‚Π°ΡŽΡ‰ΠΈΡ… Π² постоянном взаимодСйствии со своим ΠΎΠΊΡ€ΡƒΠΆΠ΅Π½ΠΈΠ΅ΠΌ. ΠŸΠΎΠ»ΡƒΡ‡ΠΈΠ² ΠΎΡ‡Π΅Ρ€Π΅Π΄Π½ΡƒΡŽ ΠΏΠΎΡ€Ρ†ΠΈΡŽ Π΄Π°Π½Π½Ρ‹Ρ…, Ρ€Π΅Π°Π³ΠΈΡ€ΡƒΡŽΡ‰Π°Ρ ΠΏΡ€ΠΎΠ³Ρ€Π°ΠΌΠΌΠ° выполняСт Π½Π΅ΠΊΠΎΡ‚ΠΎΡ€ΡƒΡŽ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒΠ½ΠΎΡΡ‚ΡŒ дСйствий ΠΈ ΠΏΡ€Π΅Π΄ΡŠΡΠ²Π»ΡΠ΅Ρ‚ Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚. Π’Π°ΠΊΠΈΠ΅ ΠΏΡ€ΠΎΠ³Ρ€Π°ΠΌΠΌΡ‹ Π²ΠΎΠ·Π½ΠΈΠΊΠ°ΡŽΡ‚ ΠΏΡ€ΠΈ ΠΏΡ€ΠΎΠ΅ΠΊΡ‚ΠΈΡ€ΠΎΠ²Π°Π½ΠΈΠΈ ΠΊΠΎΠΌΠΏΡŒΡŽΡ‚Π΅Ρ€Π½Ρ‹Ρ… Π΄Ρ€Π°ΠΉΠ²Π΅Ρ€ΠΎΠ², Π°Π»Π³ΠΎΡ€ΠΈΡ‚ΠΌΠΎΠ², Ρ€Π°Π±ΠΎΡ‚Π°ΡŽΡ‰ΠΈΡ… Π² ΠΎΠΏΠ΅Ρ€Π°Ρ‚ΠΈΠ²Π½ΠΎΠΌ Ρ€Π΅ΠΆΠΈΠΌΠ΅, сСтСвых ΠΊΠΎΠΌΠΌΡƒΡ‚Π°Ρ‚ΠΎΡ€ΠΎΠ². Π’ΠΎ ΠΌΠ½ΠΎΠ³ΠΈΡ… случаях ΠΏΡ€ΠΎΠ±Π»Π΅ΠΌΠ° Π²Π΅Ρ€ΠΈΡ„ΠΈΠΊΠ°Ρ†ΠΈΠΈ ΠΏΡ€ΠΎΠ³Ρ€Π°ΠΌΠΌ Ρ‚Π°ΠΊΠΎΠ³ΠΎ Ρ€ΠΎΠ΄Π°Β ΠΌΠΎΠΆΠ΅Ρ‚ Π±Ρ‹Ρ‚ΡŒ свСдСна ΠΊ Π·Π°Π΄Π°Ρ‡Π°ΠΌ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°Ρ†ΠΈΠΈ ΠΈ ΠΏΡ€ΠΎΠ²Π΅Ρ€ΠΊΠΈ эквивалСнтности ΠΊΠΎΠ½Π΅Ρ‡Π½Ρ‹Ρ… Π°Π²Ρ‚ΠΎΠΌΠ°Ρ‚ΠΎΠ²ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚Π΅Π»Π΅ΠΉ. ΠœΠΈΠ½ΠΈΠΌΠΈΠ·Π°Ρ†ΠΈΡ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚Π΅Π»Π΅ΠΉ Π½Π°Π΄ ΠΏΠΎΠ»ΡƒΠ³Ρ€ΡƒΠΏΠΏΠ°ΠΌΠΈ проводится Π² Ρ‚Ρ€ΠΈ этапа.Β Π’Π½Π°Ρ‡Π°Π»Π΅ для всСх состояний прСобразоватСля Π²Ρ‹Ρ‡ΠΈΡΠ»ΡΡŽΡ‚ΡΡ наибольшиС ΠΎΠ±Ρ‰ΠΈΠ΅ Π»Π΅Π²Ρ‹Π΅ Π΄Π΅Π»ΠΈΡ‚Π΅Π»ΠΈ.Β Π—Π°Ρ‚Π΅ΠΌ всС вычислСнныС Π΄Π΅Π»ΠΈΡ‚Π΅Π»ΠΈ β€ΠΏΠΎΠ΄Π½ΠΈΠΌΠ°ΡŽΡ‚ΡΡ ввСрх” ΠΏΠΎ ΠΏΠ΅Ρ€Π΅Ρ…ΠΎΠ΄Π°ΠΌ прСобразоватСля, ΠΈ Π² Ρ€Π΅Π·ΡƒΠ»ΡŒΡ‚Π°Ρ‚Π΅ образуСтся ΠΏΡ€ΠΈΠ²Π΅Π΄Π΅Π½Π½Ρ‹ΠΉ ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚Π΅Π»ΡŒ. НаконСц, для ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°Ρ†ΠΈΠΈ ΠΏΡ€ΠΈΠ²Π΅Π΄Π΅Π½Π½Ρ‹Ρ… ΠΏΡ€Π΅ΠΎΠ±Ρ€Π°Π·ΠΎΠ²Π°Ρ‚Π΅Π»Π΅ΠΉ ΠΏΡ€ΠΈΠΌΠ΅Π½ΡΡŽΡ‚ΡΡ ΠΌΠ΅Ρ‚ΠΎΠ΄Ρ‹ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°Ρ†ΠΈΠΈ классичСских ΠΊΠΎΠ½Π΅Ρ‡Π½Ρ‹Ρ… Π°Π²Ρ‚ΠΎΠΌΠ°Ρ‚ΠΎΠ²-распознаватСлСй

    On the Finiteness Problem for Automaton (Semi)groups

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    This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskii, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure
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