227 research outputs found
Π Π·Π°Π΄Π°ΡΠ΅ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΡΡ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ
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‘ΡΠ°Π½Π΄Π°ΡΡΠ½ΡΠ΅ ΡΡ
Π΅ΠΌΡ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ β ΡΡΠΎ ΠΎΠ΄Π½Π° ΠΈΠ· Π½Π°ΠΈΠ±ΠΎΠ»Π΅Π΅ ΠΏΡΠΎΡΡΡΡ
ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΡΡ
ΠΈΠΌΠΏΠ΅ΡΠ°ΡΠΈΠ²Π½ΡΡ
ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ, ΠΏΡΠ΅Π΄Π½Π°Π·Π½Π°ΡΠ΅Π½Π½Π°Ρ Π΄Π»Ρ ΡΠ΅ΡΠ΅Π½ΠΈΡ Π·Π°Π΄Π°Ρ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΠΈ Π²Π΅ΡΠΈΡΠΈΠΊΠ°ΡΠΈΠΈ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ. ΠΡ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΠΌ ΡΠ°Π·ΡΠ΅ΡΠΈΠΌΠΎΠ΅ ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΠ΅ Π»ΠΎΠ³ΠΈΠΊΠΎ-ΡΠ΅ΡΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½ΠΎΡΡΠΈ ΡΡΠ°Π½Π΄Π°ΡΡΠ½ΡΡ
ΡΡ
Π΅ΠΌ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ ΠΈ Π·Π°Π΄Π°ΡΡ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΠΈΡ
ΡΠ°Π·ΠΌΠ΅ΡΠ° ΠΏΡΠΈ ΡΡΠ»ΠΎΠ²ΠΈΠΈ ΡΠΎΡ
ΡΠ°Π½Π΅Π½ΠΈΡ ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΡ Π»ΠΎΠ³ΠΈΠΊΠΎ-ΡΠ΅ΡΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½ΠΎΡΡΠΈ. ΠΠ°ΠΌΠΈ Π΄ΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΡΠΎ ΡΡΠ° Π·Π°Π΄Π°ΡΠ° ΡΠ²Π»ΡΠ΅ΡΡΡ Π°Π»Π³ΠΎΡΠΈΡΠΌΠΈΡΠ΅ΡΠΊΠΈ ΡΠ°Π·ΡΠ΅ΡΠΈΠΌΠΎΠΉ. ΠΠ°Π»Π΅Π΅ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΡΠΎ ΡΡΠ°Π½Π΄Π°ΡΡΠ½ΡΠ΅ ΡΡ
Π΅ΠΌΡ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ Ρ ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΠ΅ΠΌ Π»ΠΎΠ³ΠΈΠΊΠΎ-ΡΠ΅ΡΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½ΠΎΡΡΠΈ ΠΈ ΠΊΠΎΠ½Π΅ΡΠ½ΡΠ΅ Π΄Π΅ΡΠ΅ΡΠΌΠΈΠ½ΠΈΡΠΎΠ²Π°Π½Π½ΡΠ΅ Π°Π²ΡΠΎΠΌΠ°ΡΡ-ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»ΠΈ, ΡΠ°Π±ΠΎΡΠ°ΡΡΠΈΠ΅ Π½Π°Π΄ ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏΠ°ΠΌΠΈ ΠΏΠΎΠ΄ΡΡΠ°Π½ΠΎΠ²ΠΎΠΊ, Π²Π·Π°ΠΈΠΌΠ½ΠΎ ΡΡΠ°Π½ΡΠ»ΠΈΡΡΡΡΡΡ Π΄ΡΡΠ³ Π² Π΄ΡΡΠ³Π°. ΠΡΡΡΠ΄Π° ΡΠ»Π΅Π΄ΡΠ΅Ρ, ΡΡΠΎ ΡΠ°ΠΊΠΆΠ΅ ΡΠ°Π·ΡΠ΅ΡΠΈΠΌΡ Π·Π°Π΄Π°ΡΠΈ ΠΏΡΠΎΠ²Π΅ΡΠΊΠΈ ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½ΠΎΡΡΠΈ ΠΈ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ Π΄Π»Ρ ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»Π΅ΠΉ ΡΠΊΠ°Π·Π°Π½Π½ΠΎΠ³ΠΎ Π²ΠΈΠ΄Π°. ΠΡΠΎΠΌΠ΅ ΡΠΎΠ³ΠΎ, Π½Π° ΠΎΡΠ½ΠΎΠ²Π΅ ΠΎΠ±Π½Π°ΡΡΠΆΠ΅Π½Π½ΠΎΠΉ Π²Π·Π°ΠΈΠΌΠΎΡΠ²ΡΠ·ΠΈ Π²ΡΠ΄Π΅Π»Π΅Π½ ΠΏΠΎΠ΄ΠΊΠ»Π°ΡΡ ΡΡΠ°Π½Π΄Π°ΡΡΠ½ΡΡ
ΡΡ
Π΅ΠΌ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ, ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΡ ΠΊΠΎΡΠΎΡΡΡ
ΠΎΡΡΡΠ΅ΡΡΠ²ΠΈΠΌΠ° Π·Π° ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠΈΠ°Π»ΡΠ½ΠΎΠ΅ Π²ΡΠ΅ΠΌΡ ΠΏΡΠΈ ΠΏΠΎΠΌΠΎΡΠΈ ΡΠ°Π½Π΅Π΅ ΠΈΠ·Π²Π΅ΡΡΠ½ΡΡ
ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ Π°Π²ΡΠΎΠΌΠ°ΡΠΎΠ²-ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»Π΅ΠΉ, ΡΠ°Π±ΠΎΡΠ°ΡΡΠΈΡ
Π½Π°Π΄ ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏΠ°ΠΌΠΈ. Π Π·Π°ΠΊΠ»ΡΡΠ΅Π½ΠΈΠΈ ΠΏΡΠΈΠ²Π΅Π΄Π΅Π½ ΠΏΡΠΈΠΌΠ΅Ρ, ΡΠ²ΠΈΠ΄Π΅ΡΠ΅Π»ΡΡΡΠ²ΡΡΡΠΈΠΉ ΠΎ ΡΠΎΠΌ, ΡΡΠΎ Π² ΠΎΠ±ΡΠ΅ΠΌ ΡΠ»ΡΡΠ°Π΅ Π·Π°Π΄Π°ΡΠ° ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ Π°Π²ΡΠΎΠΌΠ°ΡΠΎΠ²- ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»Π΅ΠΉ Π½Π°Π΄ ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏΠΎΠΉ ΠΏΠΎΠ΄ΡΡΠ°Π½ΠΎΠ²ΠΎΠΊ ΠΌΠΎΠΆΠ΅Ρ ΠΈΠΌΠ΅ΡΡ Π½Π΅ΡΠΊΠΎΠ»ΡΠΊΠΎ Π½Π΅ΠΈΠ·ΠΎΠΌΠΎΡΡΠ½ΡΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ.
Algebraic Recognition of Regular Functions
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
Π ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΠΊΠΎΠ½Π΅ΡΠ½ΡΡ Π°Π²ΡΠΎΠΌΠ°ΡΠΎΠ²-ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»Π΅ΠΉ Π½Π°Π΄ ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏΠ°ΠΌΠΈ
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 veriο¬cationΒ of such programs can be reduced to minimization and equivalence checking problems for ο¬nite stateΒ transducers. Minimization of a transducer over a semigroup is performed in three stages. At ο¬rstΒ 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 βupstreamβ, and ο¬nally a minimization algorithmΒ for ο¬nite state automata is applied to the reduced transducer.ΠΠ²ΡΠΎΠΌΠ°ΡΡ-ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»ΠΈ Π½Π°Π΄ ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏΠ°ΠΌΠΈ ΠΌΠΎΠΆΠ½ΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°ΡΡ Π² ΠΊΠ°ΡΠ΅ΡΡΠ²Π΅Β ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΡΡ
ΡΠ΅Π°Π³ΠΈΡΡΡΡΠΈΡ
ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ, ΡΠ°Π±ΠΎΡΠ°ΡΡΠΈΡ
Π² ΠΏΠΎΡΡΠΎΡΠ½Π½ΠΎΠΌ Π²Π·Π°ΠΈΠΌΠΎΠ΄Π΅ΠΉΡΡΠ²ΠΈΠΈΒ ΡΠΎ ΡΠ²ΠΎΠΈΠΌ ΠΎΠΊΡΡΠΆΠ΅Π½ΠΈΠ΅ΠΌ. ΠΠΎΠ»ΡΡΠΈΠ² ΠΎΡΠ΅ΡΠ΅Π΄Π½ΡΡ ΠΏΠΎΡΡΠΈΡ Π΄Π°Π½Π½ΡΡ
, ΡΠ΅Π°Π³ΠΈΡΡΡΡΠ°Ρ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠ° Π²ΡΠΏΠΎΠ»Π½ΡΠ΅Ρ Π½Π΅ΠΊΠΎΡΠΎΡΡΡ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΡ Π΄Π΅ΠΉΡΡΠ²ΠΈΠΉ ΠΈ ΠΏΡΠ΅Π΄ΡΡΠ²Π»ΡΠ΅Ρ ΡΠ΅Π·ΡΠ»ΡΡΠ°Ρ. Π’Π°ΠΊΠΈΠ΅ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΡ Π²ΠΎΠ·Π½ΠΈΠΊΠ°ΡΡ ΠΏΡΠΈ ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ
Π΄ΡΠ°ΠΉΠ²Π΅ΡΠΎΠ², Π°Π»Π³ΠΎΡΠΈΡΠΌΠΎΠ², ΡΠ°Π±ΠΎΡΠ°ΡΡΠΈΡ
Π² ΠΎΠΏΠ΅ΡΠ°ΡΠΈΠ²Π½ΠΎΠΌ ΡΠ΅ΠΆΠΈΠΌΠ΅, ΡΠ΅ΡΠ΅Π²ΡΡ
ΠΊΠΎΠΌΠΌΡΡΠ°ΡΠΎΡΠΎΠ². ΠΠΎ ΠΌΠ½ΠΎΠ³ΠΈΡ
ΡΠ»ΡΡΠ°ΡΡ
ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ° Π²Π΅ΡΠΈΡΠΈΠΊΠ°ΡΠΈΠΈ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌ ΡΠ°ΠΊΠΎΠ³ΠΎ ΡΠΎΠ΄Π°Β ΠΌΠΎΠΆΠ΅Ρ Π±ΡΡΡ ΡΠ²Π΅Π΄Π΅Π½Π° ΠΊ Π·Π°Π΄Π°ΡΠ°ΠΌ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΠΈ ΠΏΡΠΎΠ²Π΅ΡΠΊΠΈ ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½ΠΎΡΡΠΈ ΠΊΠΎΠ½Π΅ΡΠ½ΡΡ
Π°Π²ΡΠΎΠΌΠ°ΡΠΎΠ²ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»Π΅ΠΉ. ΠΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΡ ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»Π΅ΠΉ Π½Π°Π΄ ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏΠ°ΠΌΠΈ ΠΏΡΠΎΠ²ΠΎΠ΄ΠΈΡΡΡ Π² ΡΡΠΈ ΡΡΠ°ΠΏΠ°.Β ΠΠ½Π°ΡΠ°Π»Π΅ Π΄Π»Ρ Π²ΡΠ΅Ρ
ΡΠΎΡΡΠΎΡΠ½ΠΈΠΉ ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»Ρ Π²ΡΡΠΈΡΠ»ΡΡΡΡΡ Π½Π°ΠΈΠ±ΠΎΠ»ΡΡΠΈΠ΅ ΠΎΠ±ΡΠΈΠ΅ Π»Π΅Π²ΡΠ΅ Π΄Π΅Π»ΠΈΡΠ΅Π»ΠΈ.Β ΠΠ°ΡΠ΅ΠΌ Π²ΡΠ΅ Π²ΡΡΠΈΡΠ»Π΅Π½Π½ΡΠ΅ Π΄Π΅Π»ΠΈΡΠ΅Π»ΠΈ βΠΏΠΎΠ΄Π½ΠΈΠΌΠ°ΡΡΡΡ Π²Π²Π΅ΡΡ
β ΠΏΠΎ ΠΏΠ΅ΡΠ΅Ρ
ΠΎΠ΄Π°ΠΌ ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»Ρ, ΠΈ Π² ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠ΅ ΠΎΠ±ΡΠ°Π·ΡΠ΅ΡΡΡ ΠΏΡΠΈΠ²Π΅Π΄Π΅Π½Π½ΡΠΉ ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»Ρ. ΠΠ°ΠΊΠΎΠ½Π΅Ρ, Π΄Π»Ρ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΠΏΡΠΈΠ²Π΅Π΄Π΅Π½Π½ΡΡ
ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°ΡΠ΅Π»Π΅ΠΉ ΠΏΡΠΈΠΌΠ΅Π½ΡΡΡΡΡ ΠΌΠ΅ΡΠΎΠ΄Ρ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΠΊΠ»Π°ΡΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΊΠΎΠ½Π΅ΡΠ½ΡΡ
Π°Π²ΡΠΎΠΌΠ°ΡΠΎΠ²-ΡΠ°ΡΠΏΠΎΠ·Π½Π°Π²Π°ΡΠ΅Π»Π΅ΠΉ
On the Finiteness Problem for Automaton (Semi)groups
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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