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Circuit equivalence in 2-nilpotent algebras [article]

Piotr Kawałek, Michael Kompatscher, Jacek Krzaczkowski
2019 arXiv   pre-print
We start a systematic study of this open case, proving that the 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.  ... 
arXiv:1909.12256v1 fatcat:5t5ttzgblvazjlsn3m3ntte62u

CC-circuits and the expressive power of nilpotent algebras [article]

Michael Kompatscher
2022 arXiv   pre-print
Furthermore, we investigate the complexity of deciding identities and solving equations 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.  ... 
arXiv:1911.01479v9 fatcat:tw3lcx6syvgwhipifs4cyf6q5u

Expressive Power, Satisfiability and Equivalence of Circuits over Nilpotent Algebras

Pawel M. Idziak, Piotr Kawalek, Jacek Krzaczkowski, Michael Wagner
2018 International Symposium on Mathematical Foundations of Computer Science  
Unfortunately that paper leaves a gap for 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  ... 
doi:10.4230/lipics.mfcs.2018.17 dblp:conf/mfcs/IdziakKK18 fatcat:tlfybhasbzberbhmby5kdiw7we

Intermediate problems in modular circuits satisfiability [article]

Paweł M. Idziak, Piotr Kawałek, Jacek Krzaczkowski
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.  ... 
arXiv:2002.08626v2 fatcat:aq6cg4ks2jahjaxfhfywm52oju

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

Michael Kompatscher
2018 arXiv   pre-print
By a result of Horv\'ath the equation solvability problem over finite 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 .  ... 
arXiv:1710.03083v3 fatcat:3wvd6t6whngstczgkpxible3yy

The Power of Depth 2 Circuits over Algebras

Chandan Saha, Ramprasad Saptharishi, Nitin Saxena, Marc Herbstritt
2009 Foundations of Software Technology and Theoretical Computer Science  
The 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.  ... 
doi:10.4230/lipics.fsttcs.2009.2333 dblp:conf/fsttcs/SahaSS09 fatcat:gn3ossdnyfch5lb7not57ooxmq

Satisfiability in multi-valued circuits [article]

Paweł M. Idziak, Jacek Krzaczkowski
2017 arXiv   pre-print
Satisfiability of Boolean 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.  ... 
arXiv:1710.08163v1 fatcat:yxmb7d24tvbrtmyjybnb4a3q4m

The Power of Depth 2 Circuits over Algebras [article]

Chandan Saha, Ramprasad Saptharishi, Nitin Saxena
2009 arXiv   pre-print
The 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).  ... 
arXiv:0904.2058v1 fatcat:up42y4zzb5fsrfevtuef2kk7iq

Even faster algorithms for CSAT over supernilpotent algebras [article]

Piotr Kawałek, Jacek Krzaczkowski
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  ... 
arXiv:2002.08634v1 fatcat:pytb23sorneozh53yv34lipbqy

Nilpotent Quantum Mechanics: Analogs and Applications

Peter Marcer, Peter Rowlands
2017 Frontiers in Physics  
The most significant characteristic of 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.  ... 
doi:10.3389/fphy.2017.00028 fatcat:2vxi2dwpyvadljd7pcc5fjgc5q

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).  ... 

Even Faster Algorithms for CSAT Over supernilpotent Algebras

Piotr Kawałek, Jacek Krzaczkowski, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science  
This, together with the algorithm for 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.  ... 
doi:10.4230/lipics.mfcs.2020.55 dblp:conf/mfcs/KawalekK20 fatcat:lylyi2xwzvdmxkzop2yqgxo6yq

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.  ... 

The April meeting in Stanford University

J. W. Green
1949 Bulletin of the American Mathematical Society  
Hamilton: Transformations topologically 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  ... 
doi:10.1090/s0002-9904-1949-09256-x fatcat:tzyqelql5ba7tawohp36nlj3lq

Solving systems of equations in supernilpotent algebras [article]

Erhard Aichinger
2019 arXiv   pre-print
Kompatscher proved that for each finite supernilpotent 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.  ... 
arXiv:1901.07862v1 fatcat:vslz6yznobbtnlq3nbloygaxv4
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