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97-01-007.Pdf Circuits and Expressions with Non-Associative Gates Joshua Berman Arthur Drisko Francois Lemieux Cristopher Moore Denis Therien SFI WORKING PAPER: 1997-01-007 SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. ©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. www.santafe.edu SANTA FE INSTITUTE Circuits and Expressions with NonAsso ciative Gates Joshua Berman Arthur Drisko Francois Lemieux Cristopher Mo ore and Denis Therien State UniversityofNewYork at Binghamton National Security AgencyUSA McGill UniversityMontreal Queb ec The Santa Fe Institute Santa Fe New Mexico corresp onding author Abstract We consider circuits and expressions whose gates carry out multiplication in a nonasso ciative algebra such as a quasigroup or lo op We dene a class we call the polyabelian algebras formed by iterated quasidirect pro ducts of Ab elian groups Weshow that a quasigroup can express arbitrary Bo olean functions if and only if it is not p olyab elian in which case its Expression Evaluation and Circuit Value prob lems are NC complete and Pcomplete resp ectively This is not true for algebras in general and wegiveacoun terexample We show that Expression Evaluation is also NC complete if the algebra has a nonsolvable multiplication group or semigroup but is in TC if the algebra is b oth p olyab elian and has a solvable multiplication semigroup eg for a nilp otent lo op or group Thus in the nonasso ciative case earlier results ab out the role of solv ability in circuit complexity generalize in several dierentways Intro duction algebraic circuits and expressions Bo olean expressions and circuits are wellknown constructs in logic and computer science A Bo olean expression is either a variable x or is formed from shorter i expressions as or If all the x have truth values true i or falsethens truth value is determined byinterpreting and as and ely or and not resp ectiv A Bo olean circuit is an acyclic directed graph with source no des inputs x i and a single sink no de output and three kinds of intermediate no des and and or gates with two inputs and not gates with one input Expressions are simply circuits whose graph is a tree Then the truth value of the output is dened in the obvious way Given an expression or circuit and the truth values of its variables or inputs determining the truth value of its output is called the Expression Evaluation or Circuit Value problem resp ectively These problems are deeply related to two imp ortant complexity classes NC and P NC is the set of problems solvable by parallel circuits of p olynomial size and logarithmic depth as a function of length of the input P is the set of problems solvable in p olynomial time by a deterministic serial computer suchasaTuring machine k More generally circuits of p olynomial size and p olylogarithmic depth O log n for inputs of size n recognize the following complexity classes where the fanin is the numberofinputstoeach gate k NC if the gates are ands and ors with fanin k AC if the gates are ands and ors with unb ounded fanin k TC if the gates are threshold gates with unb ounded fanin true if t t or more of their inputs are true k ACC p if the gates are ands ors and sum mo d pwith unb ounded fanin k k ACC ACC p p ver all k is NC the class of problems solvable The union of any of these classes o by parallel circuits of p olylogarithmic depth which is also p olylogarithmic time on an idealized parallel computer Since combinations of threshold gates can compute any function that dep ends only on the sum of the inputs including sum mo d k wehave k k k k k NC AC ACC TC NC for all k The classes well b e interested in are AC ACC ACC TC NC ACC NC P It is believed but not known that P NC in other words that there are inherently sequential problems in P that cannot b e eciently parallelized Some small progress has b een made towards proving this parity is in ACC but not AC ACC pandACC q are incomparable if p and q are distinct primes and ma jorityis in TC but not ACC Thus the rst two inclusions in this series are prop er but ACC and P or even NPcouldbe identical for all anyone has b een able to prove A reduction from a problem A to a problem B is a mapping of instances of A to instances of B If the mapping is computationally easy compared to B then any fast algorithm for B b ecomes a fast algorithm for Athus B is at least as hard as A A problem B is complete for a complexity class if every other problem in that class can b e reduced to it For P it is common to use LOGSPACE O log n memory in a Turing machine or NC reductions for NC wewilluseAC or NC reductions the latter b eing essentially lo cal replacement rules We also have NC DET NC where DET is the class of problems NC reducible to calculating the determi nantofaninteger matrix DET is not known to b e comparable with AC or ACC Then we have the classical results that for Bo olean gates Expression Evaluation and Circuit Value are NC complete and Pcomplete resp ec tively under AC and NC reductions We will consider circuits and expressions where the sole op eration is multi plication in some nite algebra A rather than the usual Bo olean op erations Thus our expressions are p olynomials like x x x x and our circuits have one kind of no de whose output is the pro duct a b of its two inputs Then dep ending on the algebraic prop erties of A such as asso ciativity commutativity solvability and so on Expression Evaluation and Circuit Value can havevarying complexities Previous results for the asso ciativecase groups and semigroups include the following Expression Evaluation Circuit Value nonsolvable NC complete Pcomplete solvable ACC ACC DET In this pap er wewill extend these results to nonasso ciative algebras such as quasigroups and lo ops and to some extent to algebras in general We will show that the idea of solvability generalizes in two imp ortantways in the non asso ciative case polyabelianness the prop erty of being an iterated quasidirect pro duct of Ab elian groups and Msolvability the prop ertyofhaving a solvable multiplication group or semigroup We hop e that these results are interesting in their own right and that they might help illuminate the internal structure of P and NC and the relationship between dierent circuit mo dels In addition the problem of predicting a cellular automaton for a p olynomial number of steps corresp onds to a sp ecial case of Circuit Value where the circuit has a p erio dic structure Thus these results will also help us tell when there are fast algorithms for predicting cellular automata whose rules corresp ond to certain algebras as in Algebraic preliminaries We will use the following terms For the theory of quasigroups and lo ops we recommend asanintro duction An algebra also called a groupoid or magma G is a binary op eration f G G G written f a ba b or simply ab The order of an algebra is the numb er of elements in G written jGj Throughout the pap er we will assume that our algebras are nite A quasigroup is an algebra whose multiplication table is a Latin square in hrow and each column Equivalentlyfor which each symb ol o ccurs once in eac every a b there are unique elements ab and anb such that ab b a and a anbbthus the left right cancel lation property holds that bc bd resp cd bd implies c d An identity is an element such that a a a for all a A loop is a quasigroup with an identity In an Ab elian group we will call the identity instead of In a lo op the left right inverse of an element a is a a resp a an so that a a resp a a If these are the same we will refer to them both as a An algebra is associative if a b ca b c for all a b cA semigroup is an asso ciative algebra and a monoid is a semigroup with identityA group is an asso ciative quasigroup groups haveinverses and an identity Two elements a b commute if a b b a An algebra is commutative if all pairs of elements commute Commutative groups are also called AbelianWe will use instead of for pro ducts in an Ab elian group p In a group the order of an element a is the smallest psuchthat a or pn in an Ab elian group A homomorphism is a function from one algebra A to another B such that a ba b An isomorphism is a onetoone and onto homo B if A and B are isomorphic Homomorphisms and morphism we will write A isomorphisms from an algebra into itself are called endomorphisms and automor phisms resp ectively the automorphisms of an algebra A form a group Aut A Endomorphisms of Ab elian groups can b e represented by matrices A subalgebra subquasigroup subloop etc of G is a subset H G suchthat b b H for all b b H The subalgebra generated by a set S consisting of
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