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Circuit equivalence in 2-nilpotent algebras
[article]

2019
*
arXiv
*
pre-print

We start a systematic study of this open case, proving that the

arXiv:1909.12256v1
fatcat:5t5ttzgblvazjlsn3m3ntte62u
*circuit**equivalence*problem is*in*P for*2*-*nilpotent*such*algebras*. ... This problem not just generalises the*equivalence*problem for Boolean*circuits*, but is also of high interest*in*universal*algebra*, as it models the problems of checking identities*in*A. ...*Circuit**equivalence**In*two previous sections we have investigated the structure of*2*-*nilpotent**algebras*. We know that every such*algebra*A is of the form L ⊗ F U. ...##
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CC-circuits and the expressive power of nilpotent algebras
[article]

2022
*
arXiv
*
pre-print

Furthermore, we investigate the complexity of deciding identities and solving equations

arXiv:1911.01479v9
fatcat:tw3lcx6syvgwhipifs4cyf6q5u
*in*a fixed*nilpotent**algebra*. ... We show that CC-*circuits*of bounded depth have the same expressive power as*circuits*over finite*nilpotent**algebras*from congruence modular varieties. ... Acknowledgments I would like to thank the anonymous reviewers for their many helpful remarks,*in*particular for pointing out several inaccurate references. ...##
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Expressive Power, Satisfiability and Equivalence of Circuits over Nilpotent Algebras

2018
*
International Symposium on Mathematical Foundations of Computer Science
*

Unfortunately that paper leaves a gap for

doi:10.4230/lipics.mfcs.2018.17
dblp:conf/mfcs/IdziakKK18
fatcat:tlfybhasbzberbhmby5kdiw7we
*nilpotent*but not supernilpotent*algebras*.*In*this paper we discuss possible attacks on filling this gap. ... A project of characterizing finite*algebras*A with polynomial time algorithms deciding satisfiability of*circuits*over A has been undertaken*in*[12] . ... , c U ). (7) and*Equivalence*of*Circuits*over*Nilpotent**Algebras*...##
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Intermediate problems in modular circuits satisfiability
[article]

2020
*
arXiv
*
pre-print

*In*arXiv:1710.08163 a generalization of Boolean

*circuits*to arbitrary finite

*algebras*had been introduced and applied to sketch P versus NP-complete borderline for

*circuits*satisfiability over

*algebras*... h measures how much a

*nilpotent*

*algebra*fails to be supernilpotent. ...

*algebra*A, • Ceqv(A) -

*circuits*

*equivalence*over the

*algebra*A. ...

##
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The equation solvability problem over nilpotent Mal'cev algebras
[article]

2018
*
arXiv
*
pre-print

By a result of Horv\'ath the equation solvability problem over finite

arXiv:1710.03083v3
fatcat:3wvd6t6whngstczgkpxible3yy
*nilpotent*groups and rings is*in*P. ... We generalize his result, showing that the equation solvability over every finite supernilpotent Mal'cev*algebra*is*in*P. ...*In*other words every*nilpotent**algebra*A = (A, F ) with Mal'cev term is polynomially*equivalent*to a*nilpotent*loop (A, ·, \, /) expanded by additional operations F . ...##
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The Power of Depth 2 Circuits over Algebras

2009
*
Foundations of Software Technology and Theoretical Computer Science
*

The

doi:10.4230/lipics.fsttcs.2009.2333
dblp:conf/fsttcs/SahaSS09
fatcat:gn3ossdnyfch5lb7not57ooxmq
*equivalence*further implies that PIT of ΣΠΣ*circuits*reduces to PIT of width-*2*commutative*Algebraic*Branching Programs(ABP). ... We show that identity testing of depth 3 (ΣΠΣ) arithmetic*circuits*over a field F is polynomial time*equivalent*to identity testing of depth*2*(ΠΣ) arithmetic*circuits*over U*2*(F), the*algebra*of upper-triangular ... If all non-units*in*R are*nilpotents*then R is a local ring and the set of*nilpotents*forms the unique maximal ideal. Suppose, there is a non-*nilpotent*non-unit z*in*R. ...##
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Satisfiability in multi-valued circuits
[article]

2017
*
arXiv
*
pre-print

Satisfiability of Boolean

arXiv:1710.08163v1
fatcat:yxmb7d24tvbrtmyjybnb4a3q4m
*circuits*is among the most known and important problems*in*theoretical computer science. ... We are also looking for polynomial time algorithms that decide if two*circuits*over a finite*algebra*compute the same function. ...*Equivalence*of*Circuits*This section considers the*equivalence*of*circuits*as defined*in*the problem Ceqv. Our results*in*this direction are covered by Theorem 2.11. Theorem 2.11. ...##
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The Power of Depth 2 Circuits over Algebras
[article]

2009
*
arXiv
*
pre-print

The

arXiv:0904.2058v1
fatcat:up42y4zzb5fsrfevtuef2kk7iq
*equivalence*further shows that PIT of depth 3 arithmetic*circuits*reduces to PIT of width-*2*planar commutative*Algebraic*Branching Programs (ABP). ... Thus, identity testing for commutative ABPs is interesting even*in*the case of width-*2*. ...*In*section*2*we prove the*equivalence*of identity testing between depth 3*circuits*and depth*2**circuits*over U*2*(F) (Theorem 1.4), and show how it connects to width-*2*ABPs (Corollary 1.5). ...##
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Even faster algorithms for CSAT over supernilpotent algebras
[article]

2020
*
arXiv
*
pre-print

*In*this paper two algorithms solving

*circuit*satisfiability problem over supernilpotent

*algebras*are presented. ...

*circuit*satisfiability problem for group G is either tractable

*in*probabilistic linear time if G is

*nilpotent*or is NP-complete if G fails to be

*nilpotent*. ... For a finite

*algebra*A from a congruence modular variety the following conditions are

*equivalent*: (1) A is k-supernilpotent, (

*2*) A is k-

*nilpotent*, decomposes into a direct product of

*algebras*of prime ...

##
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Nilpotent Quantum Mechanics: Analogs and Applications

2017
*
Frontiers in Physics
*

The most significant characteristic of

doi:10.3389/fphy.2017.00028
fatcat:2vxi2dwpyvadljd7pcc5fjgc5q
*nilpotent*quantum mechanics is that the quantum system (fermion state) and its environment (vacuum) are,*in*mathematical terms, mirror images of each other. ... The*nilpotent*structure has also been identified as being constructed from two commutative vector spaces. ... of the*circuit*. ...##
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Page 685 of Mathematical Reviews Vol. , Issue 87b
[page]

1987
*
Mathematical Reviews
*

*In*this note we prove that if a ring R contains a nonzero T-

*nilpotent*left ideal, then R contains a nonzero T-

*nilpotent*ideal. ... A,B € U(M) implies A = B or there exists a

*circuit*, containing both A and B), then the 1-flats and

*2*-flats of this space form the points and lines of a Desarguesian projective geometry PG(M). ...

##
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Even Faster Algorithms for CSAT Over supernilpotent Algebras

2020
*
International Symposium on Mathematical Foundations of Computer Science
*

This, together with the algorithm for

doi:10.4230/lipics.mfcs.2020.55
dblp:conf/mfcs/KawalekK20
fatcat:lylyi2xwzvdmxkzop2yqgxo6yq
*nilpotent*but not supernilpotent*algebras*presented*in*[Paweł M. ... Recently, a few papers considering the polynomial equation satisfiability problem and the*circuit*satisfiability problem over finite supernilpotent*algebras*from so called congruence modular varieties ... Kearnes [22] , and has been observed*in*[*2*] . Theorem 2.1. For a finite*algebra*A from a congruence modular variety the following conditions are*equivalent*: 1. A is supernilpotent,*2*. ...##
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Page 2809 of Mathematical Reviews Vol. , Issue 91E
[page]

1991
*
Mathematical Reviews
*

It is shown that solvability and

*nilpotency*can be decided*in*polynomial time for finite*algebras*. Also, for*algebras*over Q,*nilpotency*can be decided efficiently. ... Programming 13 (1990), no.*2*-3, 133-180. ...##
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The April meeting in Stanford University

1949
*
Bulletin of the American Mathematical Society
*

Hamilton: Transformations topologically

doi:10.1090/s0002-9904-1949-09256-x
fatcat:tzyqelql5ba7tawohp36nlj3lq
*equivalent*to isometric transformations*in*Hilbert space. ... It is shown that if T is a pointwise periodic transformation of a point set*in*an w-dimensional Euclidean space, then T is topologically*equivalent*to a transformation of a point set*in*a Hilbert space ... Albert, Non-associative*algebras*I, Ann. of Math. vol. 43 (1942) pp. 685-707) is applied to Ghent's classification of*nilpotent**algebras*of order ^4 (A note on*nilpotent**algebras**in*four units, Bull ...##
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Solving systems of equations in supernilpotent algebras
[article]

2019
*
arXiv
*
pre-print

Kompatscher proved that for each finite supernilpotent

arXiv:1901.07862v1
fatcat:vslz6yznobbtnlq3nbloygaxv4
*algebra*A*in*a congruence modular variety, there is a polynomial time algorithm to solve polynomial equations over this*algebra*. ... Szabó had used to solve equations over finite*nilpotent*rings, we show that for A, there is c ∈N such that a solution of every system of s equations*in*n variables can be found by testing at most c n^sd ... Kompatscher for dicussions on solving equations over*nilpotent**algebras*. These discussions took place during a workshop organized by P. Aglianò at the University of Siena*in*June 2018. ...
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