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Π Π³Π΅Π½Π΅ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡΠΈ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΠΈΠ·ΠΎΠΌΠΎΡΡΠΈΠ·ΠΌΠ° ΠΊΠΎΠ½Π΅ΡΠ½ΡΡ ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏ
ΠΠ·ΡΡΠ°Π΅ΡΡΡ Π³Π΅Π½Π΅ΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡΡ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΠΈΠ·ΠΎΠΌΠΎΡΡΠΈΠ·ΠΌΠ° ΠΊΠΎΠ½Π΅ΡΠ½ΡΡ
ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏ: ΠΏΠΎ Π»ΡΠ±ΡΠΌ Π΄Π²ΡΠΌ ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏΠ°ΠΌ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²ΠΎΠ³ΠΎ ΠΏΠΎΡΡΠ΄ΠΊΠ°, Π·Π°Π΄Π°Π½Π½ΡΠΌ ΡΠ°Π±Π»ΠΈΡΠ°ΠΌΠΈ ΡΠΌΠ½ΠΎΠΆΠ΅Π½ΠΈΡ, ΡΡΠ΅Π±ΡΠ΅ΡΡΡ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΠΈΡΡ, ΡΠ²Π»ΡΡΡΡΡ Π»ΠΈ ΠΎΠ½ΠΈ ΠΈΠ·ΠΎΠΌΠΎΡΡΠ½ΡΠΌΠΈ. Π ΡΡΠΎΠΉ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ΅ ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠΈΠ°Π»ΡΠ½ΠΎ ΡΠ²ΠΎΠ΄ΠΈΡΡΡ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ° ΠΈΠ·ΠΎΠΌΠΎΡΡΠΈΠ·ΠΌΠ° ΠΊΠΎΠ½Π΅ΡΠ½ΡΡ
Π³ΡΠ°ΡΠΎΠ². Π’Π°ΠΊΠΈΠΌ ΠΎΠ±ΡΠ°Π·ΠΎΠΌ, ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ° ΠΈΠ·ΠΎΠΌΠΎΡΡΠΈΠ·ΠΌΠ° ΠΊΠΎΠ½Π΅ΡΠ½ΡΡ
ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏ Ρ Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΠΎΠΉ ΡΠΎΡΠΊΠΈ Π·ΡΠ΅Π½ΠΈΡ Π½Π΅ ΠΏΡΠΎΡΠ΅ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΠΈΠ·ΠΎΠΌΠΎΡΡΠΈΠ·ΠΌΠ° Π³ΡΠ°ΡΠΎΠ². ΠΡΠ΅Π΄Π»Π°Π³Π°Π΅ΡΡΡ Π³Π΅Π½Π΅ΡΠΈΡΠ΅ΡΠΊΠΈΠΉ ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠΈΠ°Π»ΡΠ½ΡΠΉ Π°Π»Π³ΠΎΡΠΈΡΠΌ Π΄Π»Ρ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΠΈΠ·ΠΎΠΌΠΎΡΡΠΈΠ·ΠΌΠ° ΠΊΠΎΠ½Π΅ΡΠ½ΡΡ
ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏ. Π Π΅Π³ΠΎ ΠΎΡΠ½ΠΎΠ²Π΅ Π»Π΅ΠΆΠΈΡ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΠ·Π°ΡΠΈΡ ΠΏΠΎΡΡΠΈ Π²ΡΠ΅Ρ
ΠΊΠΎΠ½Π΅ΡΠ½ΡΡ
ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏ ΠΊΠ°ΠΊ 3-Π½ΠΈΠ»ΡΠΏΠΎΡΠ΅Π½ΡΠ½ΡΡ
ΠΏΠΎΠ»ΡΠ³ΡΡΠΏΠΏ ΡΠΏΠ΅ΡΠΈΠ°Π»ΡΠ½ΠΎΠ³ΠΎ Π²ΠΈΠ΄Π°, Π° ΡΠ°ΠΊΠΆΠ΅ ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠΈΠ°Π»ΡΠ½ΡΠΉ Π°Π»Π³ΠΎΡΠΈΡΠΌ ΠΠΎΠ»Π»ΠΎΠ±Π°ΡΠ°, ΡΠ΅ΡΠ°ΡΡΠΈΠΉ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΠΈΠ·ΠΎΠΌΠΎΡΡΠΈΠ·ΠΌΠ° Π΄Π»Ρ ΠΏΠΎΡΡΠΈ Π²ΡΠ΅Ρ
ΡΠΈΠ»ΡΠ½ΠΎ ΡΠ°Π·ΡΠ΅ΠΆΠ΅Π½Π½ΡΡ
Π³ΡΠ°ΡΠΎΠ². Generic-case approach to algorithmic problems was suggested by A. Miasnikov, V. Kapovich, P. Schupp, and V. Shpilrain in 2003. This approach studies behavior of an algorithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we study the generic complexity of the isomorphism problem for finite semigroups. In this problem, for any two semigroups of the same order, given by their multiplication tables, it is required to determine whether they are isomorphic. V. Zemlyachenko, N. Korneenko, and R. Tyshkevich in 1982 proved that the graph isomorphism problem polynomially reduces to this problem. The graph isomorphism problem is a well-known algorithmic problem that has been actively studied since the 1970s, and for which polynomial algorithms are still unknown. So from a computational point of view the studied problem is no simpler than the graph isomorphism problem. We present a generic polynomial algorithm for the isomorphism problem of finite semigroups. It is based on the characterization of almost all finite semigroups as 3-nilpotent semigroups of a special form, established by D. Kleitman, B. Rothschild, and J. Spencer, as well as the Bollobas polynomial algorithm, which solves the isomorphism problem for almost all strongly sparse graphs
The conjugacy problem for automorphism groups of countable homogeneous structures
We consider the conjugacy problem for the automorphism groups of a number of
countable homogeneous structures. In each case we find the precise complexity
of the conjugacy relation in the sense of Borel reducibility
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