Circuits and Expressions withNon-Associative GatesJoshua BermanArthur DriskoFrancois LemieuxCristopher MooreDenis Therien

SFI WORKING PAPER: 1997-01-007

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SANTA FE INSTITUTE

Circuits and Expressions with

Non�Associative Gates

Joshua Berman�� Arthur Drisko�� Francois Lemieux�� Cristopher Moore�� andDenis Th�erien�

� State University of New York at Binghamton� National Security Agency� USA

� McGill University� Montr�eal� Qu�ebec� The Santa Fe Institute� Santa Fe� New Mexico �corresponding author�

Abstract� We consider circuits and expressions whose gates carry outmultiplication in a non�associative algebra such as a quasigroup or loop�We de�ne a class we call the polyabelian algebras� formed by iteratedquasidirect products of Abelian groups� We show that a quasigroup canexpress arbitrary Boolean functions if and only if it is not polyabelian�in which case its Expression Evaluation and Circuit Value prob�lems are NC��complete and P�complete respectively� This is not true foralgebras in general� and we give a counter�example�

We show that Expression Evaluation is also NC��complete if thealgebra has a non�solvable multiplication group or semigroup� but is inTC� if the algebra is both polyabelian and has a solvable multiplicationsemigroup� e�g� for a nilpotent loop or group�

Thus� in the non�associative case� earlier results about the role of solv�ability in circuit complexity generalize in several dierent ways�

� Introduction� algebraic circuits and expressions

Boolean expressions and circuits are well�known constructs in logic and computerscience� A Boolean expression � is either a variable xi� or is formed from shorterexpressions as ��� � ���� ��� � ��� or ���� If all the xi have truth values trueor false� then ��s truth value is determined by interpreting �� � and � as and�or and not respectively�

A Boolean circuit is an acyclic directed graph with source nodes �inputs� xiand a single sink node �output�� and three kinds of intermediate nodes andand or gates with two inputs� and not gates with one input� �Expressions aresimply circuits whose graph is a tree�� Then the truth value of the output isdened 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 respectively� These problems are deeply related totwo important complexity classes� NC� and P�

NC� is the set of problems solvable by parallel circuits of polynomial size andlogarithmic depth as a function of length of the input� P is the set of problems

solvable in polynomial time by a deterministic serial computer such as a Turingmachine�

More generally� circuits of polynomial size and polylogarithmic depth �O�logk n�for inputs of size n� recognize the following complexity classes ��� ��� where thefan�in is the number of inputs to each gate

� NCk if the gates are and�s and or�s with fan�in �� ACk if the gates are and�s and or�s with unbounded fan�in� TCk if the gates are threshold gates with unbounded fan�in� �t � true if tor more of their inputs are true

� ACCk�p� if the gates are and�s� or�s� and �sum mod p� with unboundedfan�in

� ACCk � �pACCk�p��

The union of any of these classes over all k is NC� the class of problems solvableby parallel circuits of polylogarithmic depth �which is also polylogarithmic timeon an idealized parallel computer��

Since combinations of threshold gates can compute any function that dependsonly on the sum of the inputs� including sum mod k� we have

NCk � ACk � ACCk � TCk � NCk��

for all k� The classes we�ll be 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 areinherently sequential problems in P that cannot be e�ciently parallelized� Somesmall progress has been made towards proving this � � �� ��� ��� parity is inACC���� but not AC�� ACC��p� and ACC��q� are incomparable if p and q aredistinct primes� and majority is in TC� but not ACC����� Thus the rst twoinclusions in this series are proper� but ACC���� and P �or even NP� could beidentical for all anyone has been able to prove�

A reduction from a problem A to a problem B is a mapping of instances of Ato instances of B� If the mapping is computationally easy compared to B� thenany fast algorithm for B becomes a fast algorithm for A� thus B is at least ashard as A� A problem B is complete for a complexity class if every other problemin that class can be reduced to it�

For P it is common to use LOGSPACE �O�log n� memory in a Turingmachine� or NC� reductions� for NC we will use AC� or NC� reductions� thelatter being essentially local replacement rules�

We also have

NC� � DET � NC�

where DET is the class of problems NC��reducible to calculating the determi�nant of an integer matrix� DET is not known to be comparable with AC� orACC��

Then we have the classical results that� for Boolean gates� ExpressionEvaluation and Circuit Value are NC��complete and P�complete respec�tively� under AC� and NC� reductions ��� ���

We will consider circuits and expressions where the sole operation is multi�plication in some nite algebra �A� ��� rather than the usual Boolean operations�Thus our expressions are polynomials like �x� � x�� � �x� � x��� and our circuitshave one kind of node whose output is the product a � b of its two inputs�

Then depending on the algebraic properties of �A� ��� such as associativity�commutativity� solvability and so on� Expression Evaluation and Circuit

Value can have varying complexities� Previous results for the associative case�groups and semigroups� include the following ��� �� ��

Expression Evaluation Circuit Value

non�solvable NC��complete P�complete

solvable ACC� ACC� DET

In this paper� we will extend these results to non�associative algebras suchas quasigroups and loops� and to some extent to algebras in general� We willshow that the idea of solvability generalizes in two important ways in the non�associative case polyabelianness� the property of being an iterated quasidirectproduct of Abelian groups� and M�solvability� the property of having a solvablemultiplication group or semigroup�

We hope that these results are interesting in their own right� and that theymight help illuminate the internal structure of P and NC and the relationshipbetween di�erent circuit models�

In addition� the problem of predicting a cellular automaton for a polynomialnumber of steps corresponds to a special case of Circuit Value where thecircuit has a periodic structure� Thus these results will also help us tell whenthere are fast algorithms for predicting cellular automata whose rules correspondto certain algebras� as in � �� ���

� Algebraic preliminaries

We will use the following terms� For the theory of quasigroups and loops� werecommend � �� as an introduction�

An algebra �also called a groupoid or magma� �G� �� is a binary operationf G G � G� written f�a� b� � a � b or simply ab� The order of an algebra isthe number of elements in G� written jGj� Throughout the paper� we will assumethat our algebras are nite�

A quasigroup is an algebra whose multiplication table is a Latin square� inwhich each symbol occurs once in each row and each column� Equivalently� forevery a� b there are unique elements a�b and anb such that �a�b� � b � a anda � �anb� � b� thus the left �right� cancellation 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 isa quasigroup with an identity� In an Abelian group we will call the identity �instead of �

In a loop� 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 themboth as a���

An algebra is associative if a � �b � c� � �a � b� � c for all a� b� c� A semigroup isan associative algebra� and a monoid is a semigroup with identity� A group is anassociative quasigroup� groups have inverses and an identity�

Two elements a� b commute if a � b � b � a� An algebra is commutative if allpairs of elements commute� Commutative groups are also called Abelian� We willuse � instead of � for products in an Abelian group�

In a group� the order of an element a is the smallest p � � such that ap � �or pn � � in an Abelian group��

A homomorphism is a function � from one algebra �A� �� to another �B� ��such that ��a � b� � ��a� � ��b�� An isomorphism is a one�to�one and onto homo�morphism� we will write A �� B if A and B are isomorphic� Homomorphisms andisomorphisms from an algebra into itself are called endomorphisms and automor�phisms respectively� the automorphisms of an algebra A form a group Aut�A��Endomorphisms of Abelian groups can be represented by matrices�

A subalgebra �subquasigroup� subloop� etc�� of G is a subset H � G such thatb� � b� H for all b�� b� H � The subalgebra generated by a set S� consisting ofall possible products of elements in S� is written hSi�

The left �right� cosets of a subloop H � G are the sets aH � fah jh Hgand Ha � fha jh Hg for each a G� A subloop H is normal if the followinghold for all a� b G

aH � Ha� a�bH� � �ab�H� and �aH�b � a�Hb�

Then the set of cosets of H is the quotient loop or factor G�H � it has identity H � H � and is the image of G under the homomorphism ��a� � aH �

A subloop of G is proper if it is neither f g nor all of G� A minimal normalsubloop of G is one which does not properly contain any proper normal subloopsof G� A simple loop is one with no proper normal subloops�

The commutator of two elements in a loop is �a� b� � ab � ba� and the as�sociator of three elements is �a� b� c� � �ab�c � a�bc�� The subloop generated byall possible commutators and associators in a loop G is called the commutator�associator subloop or derived subloop G�� It is normal� and it is the smallestsubloop such that the quotient G�G� is an Abelian group�

A loop G is solvable if its derived series G � G� � G� � � � �� where Gi�� � G�ifor all i� ends in Gk � f g after a nite number of steps� Any non�solvable loopcontains a simple loop H for which H � � H � More generally� we call an algebrasolvable if it contains no non�solvable loops�

The center of a loop is the set of elements that associate and commute witheverything� Z�G� � fc j cx � xc� c�xy� � �xc�y � x�yc� for all x� y Gg� It is anormal subloop of G�

The upper central series of a loop is f g � Z� � Z� � � � � where Zi���Zi

is the center of G�Zi� A loop is nilpotent �of class k� if Zk � G for some k�Inductively� G is nilpotent if it has a nontrivial center Z�G�� and G�Z�G� has anon�trivial center� and so on until we get an Abelian group for which Z�G� � G�The nilpotent loops are a proper subclass of the solvable ones�

A variety is a class of algebras V such that subgroups� factors� and directproducts of algebras in V are also in V � A pseudovariety is closed under nitedirect products� but not necessarily innite ones� Solvable and nilpotent loopsboth form pseudovarieties�

In a quasigroup Q� we can dene left and right multiplication as functionsLa�b� � a � b and Ra�b� � b � a� These are permutations on Q �the rows andcolumns of the multiplication table�� and the multiplication group M�Q� is thegroup of permutations they generate� More generally� any algebra has a mul�tiplication semigroup generated by the La and Ra� which are not necessarilyone�to�one functions on Q� If we have more than one operation we will refer toL�a � M�Q���� and so on�

Finally� we refer to the identity function ��x� � x� the cyclic group Zp �f�� � � � � � p� g with addition mod p� and the groups Sn and An of permutationsand even permutations respectively on n elements�

� Solvability and Boolean�completeness in groups and

loops

Let us dene the set of functions that can be expressed as circuits whose gatescarry out multiplication in an algebra A� and whose inputs can be variablesor constant elements of A� This is equivalent to the set of polynomial ex�pressions denable in A with constants and variables� such as ��x�� x�� x�� ��x��ax����x�b�

�

De�nition� The polynomial closure of an algebra A is the smallest set P�A�of functions � on an arbitrary number of variables x�� � � � � xk containing thefollowing

� �constants� a for all a A�

� �projections� xi for all i�

� �products� �� � �� for all ��� �� P�A��

Since P�A� is closed under composition and substitution of one functionfor a variable of another� it is a clone ����� We will sometimes refer to the set offunctions on k variables as Pk�A� for instance� P��A� contains the multiplicationsemigroup M�A�� as well as functions like ��x� � x��

We are interested in whether an algebra can express arbitrary Boolean func�tions� For instance� in the quasigroup

� � � � � � �� � � � � � � �� � � �

if false � and true � �� then

a � b � �a � b�� and �a � � � � � a�

are expressions of these Boolean functions as polynomials in �� We can combinethese to make any other Boolean function� Formally

De�nition� An algebra �A� �� is strongly Boolean�complete if there exist ele�ments true and false in A� and functions ��� �� in P�A�� such that ���a� b� �a � b and ���a� � �a whenever a� b ftrue� falseg�

In general� we will allow true and false to be sets� rather than singleelements

De�nition� An algebra �A� �� is Boolean�complete if there exist disjoint subsetsT� F � A of �true� and �false� elements respectively� and functions ��� �� inP�A� such that ���a� b� � a � b and ���a� � �a whenever a� b T � F � wherex � y if x and y are both true or both false�

Then a strongly Boolean�complete algebra is a Boolean�complete one whereT and F are the singletons ftrueg and ffalseg�

Lemma�� If an algebra is Boolean�complete� then its Expression Evalua�

tion and Circuit Value problems are NC��complete and P�complete �underAC� and NC� reductions� respectively�

Proof� Boolean Expression Evaluation and Circuit Value problems arereducible to their algebraic counterparts� since a local rule can replace and� orand not gates with complexes of algebraic gates� ut

Then we brie�y re�state a theorem of Barrington ���� with a slightly di�erentproof� The construction hinges on the fact that the commutator has the characterof an and gate if false � � then �a� b� is false if either a or b is� since theidentity commutes with everything�

First we show two useful lemmas

Lemma�� If Q is a �nite quasigroup� the divisions a�b and anb are in P��Q� asfunctions of a and b� Therefore� functions that yield the commutator� associator�and left and right inverses are in P as well�

Proof� Recall the denition of La and Ra above� If Q is a quasigroup of order n�La and Ra are permutations on its elements� and Ln�

a and Rn�a are the identity

�� Then aLn���a �b� � Ln�

a �b� � b� so

anb � Ln���a �b� � a�a�� � � �a� �z �

n��� times

� b���

is in P�Q�� Similarly for a�b� then by composition we can dene �a� b� � ab � ba��a� b� c� � �ab�c � a�bc�� a� � �a and a� � an � ut

Lemma�� If G is a simple loop or group� for any x� y �� there is a function�x�y in P��G� that sends x to y and keeps the identity �xed�

Proof� Let U be the set of functions � in P��G� such that �� � � � For anelement x G� let U�x� � f��x� j� Ug be the set of y�s that we can send xto� while xing �

Then U�x� is a subloop� since ��� � � if we dene ���a� � ���a� � ���a��Moreover� U�x� is a normal subloop� since we can fulll each requirement fornormality with a suitable denition of ��

aU�x� � U�x� a ���x� � a ��x� � a

a�b U�x�� � �ab�U�x� ���x� � �ab� n a�b ��x��

a�U�x� b� � �aU�x��b ���x� � a n�a���x� b� � b

�In each case �� is in P�G� by lemma �� and in U since ��� � � if and only if�� � � �

If G is simple� then there are no proper normal subloops and U�x� � G forany x �� � Thus for any y� there exists � U such that ��x� � y� this is our�x�y� ut

Theorem� Barrington� Non�solvable groups are strongly Boolean�complete�

Proof� Assume without loss of generality that G is simple and non�Abelian�so that G � G�� Then choose any non�commuting pair of elements x� y with�x� y� �� � Let false � and true be any non�identity element t �� � anddene

a � b � ��x�y�t

���t�x�a�� �t�y�b��

�This expression evaluates to t if a � b � t� and if either a or b is � in otherwords� it is an and gate� Finally� we can express negation as �a � t � a��� ut

By using an associator instead of a commutator� this generalizes to loops� ��� Like the commutator� the associator �x� y� z� is if any of its arguments is � since the identity associates with everything�

Theorem�� Non�solvable loops are strongly Boolean�complete�

Proof� Assume without loss of generality that G is simple and non�associative�since theorem � treats the associative case� Choose a triplet of non�associatingelements x� y� z G with �x� y� z� �� � Let false � and true � t �� asbefore� Then

a � b � ��x�y�z�t

���t�x�a�� �t�y�b�� z�

�and �a � t�a or t � a�� ut

In fact� simple non�Abelian groups � ��� simple loops � ��� and non�a�nesimple quasigroups � �� have a stronger property� that their polynomial closurecontains all possible n�ary functions on their elements� This is called functionalcompleteness� and is of interest in the eld of multi�valued logic �� � ���� However�Boolean�completeness is su�cient for our purposes�

Since the Circuit Value problem for solvable groups is in NC ��� ��� non�solvability is both necessary and su�cient for Boolean�completeness in the caseof groups �or semigroups� in fact�� This also means that in the associative case�Boolean�completeness and strong Boolean�completeness are equivalent�

However� a loop can be solvable and still be �strongly� Boolean�complete�Let �G� �� be

� � � � � � � � � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � � � � � � � � � �� � � � � � � �

Here G� is the normal subloop f � �� �� �g �� Z�� But the lower right�hand blockis the Boolean�complete quasigroup � given above� and � can be expressed inP�G� as a � b � �� � a� � �� � b�� Then if false � and true � � as before� wecan write a � b � �� � ���a���b���� and G is Boolean�complete� We also note thatthis loop has a solvable multiplication group� which we will discuss below�

A loop has the inverse property � �� if a���ab� � �ba�a�� � b� where a�� �a� � a�� Solvable loops with the inverse property can also be Boolean�completewe believe the smallest example is

� � � � � � � � � � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �

Here G� � f � �� �� �� �g �� Z� so G is solvable� However� if false � andtrue � �� we can dene a � b � �a� b� �� since ��� �� �� � � �and �a � ��a��

In both these examples� the lower right�hand block can be any quasigroupwhatsoever clearly the standard denition of �solvable� for loops is not a verymeaningful constraint for our purposes� In the next section we will show thatfor loops and quasigroups� a property slightly more subtle than solvability is thedividing line between Boolean�completeness and �incompleteness�

� Polyabelian algebras

��� De�nition and properties

The direct product of two algebras AB is the set of pairs �a� b� with pairwisemultiplication� �a�� b�� � �a�� b�� � �a�a�� b�b��� Consider the following general�ization

De�nition� A quasidirect product ��� of two algebras A and B is the set of pairs�a� b�� under an operation of the form

�a�� b�� � �a�� b�� � �a�a�� b� �a��a� b��

where each a�� a� denes a local operation �a��a� on B� We will denote such aproduct A�B�

If the local operations are of the form

b� �a��a� b� � fa��a��b�� � ga��a��b��

we will call them separable� Furthermore� if B is an Abelian group and the ��sare of the form

b� �a��a� b� � fa��a��b�� � ga��a��b�� � ha��a�

where f and g are endomorphisms on B and h is an element of B� dependingarbitrarily on a� and a�� then we will call them a�ne� We will call a quasidirectproduct A�B separable or a�ne on B if all its local operations are�

Lemma�� If A�B is a quasigroup� then A and B are quasigroups and all the�s are quasigroup operations on B� If A � B is a quasigroup and is a�ne onB� then all the f s and gs are automorphisms on B� If A � B is a loop a�neon �B���� then A is a loop� and for all a A we have fa�� � g��a � � andha�� � h��a � � �here we are using for the identities of both A and B�� Thusb� ���� b� � b� � b� for b�� b� B� and B is a subloop of A�B on which � and� coincide�

Proof� Easy� left to the reader� ut

The quasidirect product is a rather general way of extending to a loop froma normal subloop

Lemma � If a loop G has a normal subloop N � then G is isomorphic to aquasidirect product �G�N��N � Furthermore� all the local operations� and �if Nis Abelian and G is a�ne on N� the f s� gs and hs� are expressible in P�G��

Proof� Choose a set T with one element in each coset of N � and dene an oper�ation � on T where t� � t� is the element of T in the same coset as t� � t�� Thenclearly T �� G�N �

Every element can be uniquely written g � tn where t T and n N � Then

�t�n�� � �t�n�� � �t� � t�� � �n� �t��t� n��

wheren� �t��t� n� � �t� � t�� n ��t�n�� � �t�n���

which is in N since N is normal� Thus G is a quasidirect product T �N � andall the local operations are in P�G�� If N is an Abelian group and G is a�ne onN � then

fa��a��n� � �n�a��a� ��� ���a��a� ��

ga��a��n� � ���a��a� n�� ���a��a� ��

ha��a� � ���a��a� ��

where we abuse notation by writing � and �� instead of � and � for products inN � Finally� �N��� is in P�G� since a���� b � a� b by lemma �� ut

Then dene the following class of algebras

De�nition� An algebra is polyabelian if it is an iterated quasidirect product ofAbelian groups Ai

��A� � A���A��� � � � �Ak

where all the products are a�ne�

It is easy to show that subalgebras� factors� and nite direct products ofpolyabelian algebras are polyabelian� so this class forms a pseudovariety� Thenext few lemmas show inclusions between the polyabelian loops and some com�mon classes of groups and loops�

Lemma�� Polyabelian loops are solvable�

Proof� Let Hi � �Ai � Ai��� � � � � � Ak with H� � G� Then the reader canshow that all the Hi are normal subloops of G� and Hi�Hi�� � Ai is Abelian�Therefore� H �

i � Hi�� and the derived series ends after at most k steps� ut

The converse is not true for loops in general �for instance� the solvableBoolean�complete loops above� since the local operations in their lower right�hand blocks are not a�ne� but it is true for groups

Lemma�� Solvable groups are polyabelian�

Proof� Any solvable group G has a normal subgroupN which is Abelian� namelythe last non�trivial group in its derived series such that N � � f g� Then the localoperation in G�N �N �from lemma �� can be written

n� �t��t� n� � �t� � t����t�n�t�n�

� ��t� � t����t�t�� � �t��

� n�t�� � n�

where we use � for products within N � Thus G is a�ne on N where

ft��t��n� � t��� nt�

gt��t��n� � n

ht��t� � �t� � t����t�t�

Then G�N has an Abelian normal subgroup� and so on� by induction G is polya�belian�

�If t� � t� � t�t� so that h � �� then T is a subgroup of G isomorphic to G�N �the quasidirect product reduces to the semidirect product on groups� and G is asplit extension of N by T ����� In � �� we dened polyabelianness with semidirectproducts only� in which case any solvable group is a subgroup of a polyabeliangroup by a theorem regarding wreath products�� ut

Lemma��� Nilpotent loops are polyabelian�

Proof� Let G be a nilpotent loop with center Z�G�� Then the local operation inG�Z�G�� Z�G� is

n� �t��t� n� � ��t� � t��nt�t�� � n� � n�

since n� and n� associate and commute with everything� So G is a�ne on Z�G�with f � g � � and h � �t� � t��nt�t�� Then G�Z�G� has a non�trivial center�and so on� by induction G is polyabelian� ut

Thus polyabelianness coincides with solvability for groups� and lies properlybetween nilpotence and solvability for loops�

We wish to show that� for purposes of Boolean�completeness� polyabeliannessis the correct generalization of solvability in the non�associative case that is� analgebra is Boolean�complete if and only if it is not polyabelian� This will turnout to be true for loops and quasigroups� but not for algebras in general�

��� Polyabelian algebras aren�t Boolean�complete

In one direction� we can prove this for all algebras� In � �� we show that CircuitValue for polyabelian algebras is in ACC�� and a simple modication of theproof for solvable semigroups in ��� shows that it is also in DET� We now showdirectly that polyabelian algebras cannot express the and function� First� twolemmas from ����

De�nition� A function � on variables x�� � � � � xn is a�ne if there exists anAbelian groupA� endomorphisms f�� � � � � fn� and an element h such that ��x�� � � � � xn� �P

i fi�xi� � h�

Lemma��� If A is an Abelian group� then any function in P�A� is a�ne� andthe a�ne functions are closed under composition�

Proof� Obvious ��a� b� � a � b is a�ne� and if �� and �� are both a�ne� thenso are �� � �� and �� � ��� ut

Lemma��� If ��a� b� is an a�ne function� then ��a�� b�� � ��a�� b�� if andonly if ��a�� b�� � ��a�� b�� for any four elements a�� a�� b�� b��

Proof� We can write ��a� b� � f�a��g�b��h where f and g are endomorphisms�Then ��a�� b�� � ��a�� b�� implies that g�b�� � g�b��� which in turn implies that��a�� b�� � ��a�� b�� for any a�� ut

Theorem��� Polyabelian algebras cannot express the and function� and so arenot Boolean�complete�

Proof� If G is Boolean�complete� then it can express an n�ary and function forany n� i�e� ��a�� a�� � � � � an� T if and only if ai T for all i �assuming thatai T �F for all i�� We will show that this is impossible for n su�ciently large�

If G � �A��A���� � ��Ak� then any g G has a unique vector of components�g�� g�� � � � � gk� where gi Ai for all i� Call gi the Ai�component of g� We willproceed through the Ai by induction� showing that there are elements of T andF matching on all their components� and therefore equal� then T and F are notdisjoint� a contradiction�

We will use the following fact if �a�� � � � � an� �P

i fi�ai� � h is an n�ary a�ne function on an Abelian group A of order p� and if A has k distinctendomorphisms �for instance� k � pm

�

for A � Zmp since its endomorphisms are

m m matrices with entries in Zp�� and if n is greater than �p � �k� then atleast p of the variables have the same fi � f � Then if these p variables are allequal� they contribute nothing to since pf��� In particular� if the n� p othervariables are true� has the same value whether these p variables are true orfalse� As shorthand for this� we write �fptn�p� � �tn�� Thus cannot be anand function�

So assume that there is an n�ary and function � in P�G�� To start theinduction� since A� is a factor of G� then ��s A��component �� is a function ofthe A��components of the ai� expressible in P�A�� and therefore a�ne by lemma � Choose t T and f F � then for n su�ciently large ���f

ptn�p� � ���tn��

Let f� � ��fptn�p� and t� � ��tn�� then t� T and f� F by hypothesis� andthey have the same A��component�

Now suppose that tm T and fm F agree on their Aj�components forall j � m� Think of � as a tree where each node outputs the product of twosubexpressions according to some local product� Now the Aj �components at eachnode depend only on the Aj� �components of its two subexpressions for j� � j

�since A� � � � � � Aj is a factor of G for all j� and tm and fm have the sameAj �component for all j � m� so inductively � and each of its subexpressionshave constant Aj�components for j � m when restricted to inputs in ftm� fmg�

Furthermore� each node applies an a�ne local operation on Am��� and whichone it applies depends only on its subexpressions� Aj�components for j � m�Since these are constant in this restriction� each node always applies the samelocal operation� the composition of all of these make �m�� an a�ne function onthe Am���components of its inputs�

Then if we let fm�� � ��fpmtn�pm � F and tm�� � ��tnm� T � we see that

fm�� and tm�� agree on their Aj�components for all j � m� � After k steps ofthis induction� tk and fk agree on all their components� and so are equal� so Tand F are not disjoint� ut

Theorem � and lemma � establish the converse� namely that non�polyabelianalgebras are Boolean�complete� in the case of groups� we will now show this forloops and then for quasigroups� using slightly di�erent techniques�

��� Non�polyabelian loops and quasigroups are Boolean�complete

First we need the following lemma

Lemma��� If a factor of G is Boolean�complete� then so is G�

Proof� If a homomorphic image ��G� is Boolean�complete with T and F � simplylet T � and F � in G be the inverse images ����T � and ����F �� ut

Corollary� The set of non�Boolean�complete algebras forms a pseudovariety�

Proof� It is easy to show that subalgebras and direct products of non�Boolean�complete algebras are non�Boolean�complete� lemma � completes the proof�since it shows that �contrapositively� factors of non�Boolean�complete algebrasare non�Boolean�complete� ut

Then

Theorem��� Non�polyabelian loops are Boolean�complete�

Proof� Let H be the smallest non�polyabelian factor of G� We will show that His strongly Boolean�complete�

Assume without loss of generality that H is solvable� since we have alreadytreated the non�solvable case with theorem �� Then H has a normal subloop Kwhich is an Abelian group� namely the last non�trivial subloop in its derivedseries with K � � f g� Let N be a minimal normal subloop of H contained inK� then N is also Abelian� We know that H is not a�ne on N � otherwise H�Nwould be a smaller non�polyabelian factor of G�

Recall the denition of U�x� from lemma �� Since N is minimal� U�n� � Nfor any n N � otherwise U�n�N would be a smaller normal subloop since the

intersection of normal subloops is normal� So for any n�� n� N � there exists afunction �n��n� P�H� that sends n� to n� and preserves the identity�

Since H is not a�ne on N � some local operation � is either not separable ornot a�ne� Dene the separator

K��n�� n�� � �n� � n��� �n� � ��� ��� n�� � ��� ��

where we use � and� for products inN � If� is not separable� thenK��n�� n�� �k �� � for some n�� n�� however� K���� n� � K��n� �� � � for any n� But thisgives us our and gate let false � � and choose true � t N � and let

a � b � �k�t

�K���t�n��a�� �t�n� �b��

�If all the local operations are separable� then one must not be a�ne that is�

some f or g is not a endomorphism of N � Let f�n� � �n � �� � �� � �� as inlemma �� and dene the a�nator

Lf �n�� n�� � f�n� � n��� f�n��� f�n��

If f is not a endomorphism� then Lf �n�� n�� � k �� � for some n�� n�� butLf �n� �� � Lf ��� n� � � for all n� so

a � b � �k�t

�Lf ��t�n� �a�� �t�n��b��

�is an and gate� Similarly if some g is not a endomorphism�

So any nonlinearity in the local operations can be used to construct an and

gate� and we can express negation �a � t�a as before� Thus H is stronglyBoolean�complete� and G is Boolean�complete by lemma �� ut

For the quasigroup case� we need some additional denitions and lemmas�The idea of a normal subloop generalizes in the following way

De�nition� A normal congruence on a quasigroup Q is an equivalence � suchthat if a� � a� and b� � b�� then a�a� � b�b�� In other words� the equivalenceclass of the product depends only on the equivalence classes of a and b� so themap from Q to the set of equivalence classes Q�� is a homomorphism� �Thisdenition works for nite quasigroups� for innite ones� additional requirementsare needed to make sure Q�� is also a quasigroup � ���� For a loop with a normalsubloop� for instance� the equivalence classes are simply its cosets�

A proper congruence is one other than the identity �a � b only if a � b� or allof Q �a � b for all a� b�� A quasigroup is simple if it has no proper congruences�

A normal congruence � is minimal if there is no proper normal congruencewhose equivalence classes are properly contained in those of ��

Then lemma � generalizes in the following way

Lemma��� If Q is a simple quasigroup� then for any x� y� z� w with x �� z� thereexists a function �x�y�z�w in P��Q� that sends x to y and z to w�

Proof� Fix z� and let Uz�w be the set of functions � in P��Q� such that ��z� � w�Then Uz�w�x� is the set of y�s that we can send x to� while sending z to w�

Suppose y� Uz�w��x� and y� Uz�w�

�x�� that is� suppose there are��� �� P�Q� such that ���x� � y�� ���z� � w�� ���x� � y� and ���z� � w��Then if ���a� � ���a� � ���a�� we have ���x� � y� � y� and ���z� � w� � w��Therefore� �� Uz�w��w�

and y� � y� Uz�w��w��x��

So for each x and z� the sets Uz�w�x� for di�erent w are equivalence classesof a normal congruence� Moreover� each one has more than one element forinstance� w and �w�z�x are both in Uz�w�x� by the functions ��a� � w and�w�z�a� and are distinct in a quasigroup if x �� z�

Since Q is simple� Uz�w�x� � Q� so for every y� there exists a function� Uz�w�x� that sends x to y� which is our �x�y�z�w� ut

Two more lemmas will prove useful�

Lemma� � If a quasigroup �Q� �� is separable on a loop operation �� i�e� ifa � b � f�a� � g�b� for some f� g� then a � b is expressible in P��Q� ���

Proof� Let the identity of � be � Then f�a� � a � g��� � and g�a� � f��� � � a�Since Q is a quasigroup� f and g are one�to�one� and f�� � fn��� and g�� �gn��� if Q has order n� Then a �b � f���a��g���b�� �Note that it is not necessaryfor f and g to be automorphisms�� ut

Lemma��� Simple non�a�ne quasigroups are strongly Boolean�complete�

Proof� This is just a special case of the functional completeness result mentionedabove � ��� We rely on the following fact� the proof of which can be found is���� if Q is not a�ne� then there is a function � P�Q� that violates thestatement of lemma �� i�e� ��a�� b�� � ��a�� b�� but ��a�� b�� �� ��a�� b�� forsome a�� a�� b�� b��

Then choose an element x� and dene

�a� b� � �x � ��a� b�����a� b��

The reader can check that �a� b� � x for three out of four combinations of a��a�� b� and b�� except �a�� b�� which is some y �� x� This has the shape of anand gate� and using lemma � we can dene

a � b � �x�false�y�true

���false�a��true�a��a�� �false�b��true�b��b��

�and

�a � �true�false�false�true�a�

of which one example derived from a Malcev operation ���� is

�true�true�n��true�a� � false�

since �t�t�nt � �t�t�n��t�t�t� � t� ut

Finally� we note that lemma � holds for quasigroups as well if Q has a normalcongruence�� then Q �� �Q����N whereN is any one of the equivalence classes�and the local operations on N are in P�Q�� �Unlike the loop case� N might notbe a subquasigroup of Q�� Lemma � also holds if Q�� is Boolean�complete�then so is Q� Then we can prove

Theorem��� Non�polyabelian quasigroups are Boolean�complete�

Proof� As before� replace Q with its smallest non�polyabelian factor R� which wewill show is strongly Boolean�complete� Since lemma � takes care of the casewhen R is simple� we assume without loss of generality that R has a minimalnormal congruence � such that R is not a�ne on its equivalence classes�

If there is a particular local operation which is not a�ne� then it is a simplequasigroup operation expressible in P�R�� and lemma � applies� However� itcould be the case that every local operation is a�ne but not all on an identicalAbelian group� For instance� in the Boolean�complete loop of order � above�every local operation is isomorphic to Z� but in a way which is not a�ne oneach other the lower right�hand block is the upper left�hand block with � and �switched� but there is no automorphism of Z that only switches two elements�

In this case� we use lemma � if any of the operations is a�ne on an Abeliangroup A� then �A��� is expressible in P�R�� Then if any other local operationis not also a�ne on A� we use its separator or a�nator with respect to � toconstruct an and gate as in theorem ��

Then R is strongly Boolean�complete and Q is Boolean�complete by lemma �� ut

Corollary� The non�Boolean�complete quasigroups and the polyabelian quasi�groups form the same pseudovariety�

��� Algebras in general

It would be very nice if non�polyabelianness were the criterion for Boolean�completeness for algebras in general �presumably with �Abelian group� replacedby �Abelian semigroup� in the denition�� However� we can give a counterexam�ple� Let �A� �� be

� � � �� � � � � � �� � � � �� � � �

Theorem��� �A� �� is non�polyabelian� but cannot express the and function�

Proof� The equivalence classes f�� g and f�� �g form the only proper normalcongruence � of A� There is no Abelian semigroup on which the lower�right andupper�left local operations are both a�ne� so A is not polyabelian�

We could not have true and false in di�erent equivalence classes� since�A��� �� Z�� So we will assume rst that true� false f�� g�

The lower right�hand block looks like an and gate with true � andfalse � �� in which case we could use the upper left�hand block to say �a � �a�But there is no way to map� say� � to � and to �� since a�b � b�a � a whenevera f�� �g and b f�� g� that is� � and � dominate � and �

Formally� consider a polynomial expression � in P�A�� The following rulespreserve the value of � on inputs restricted to f�� g

� Replace a � � and � � a with a if a f�� �g and � contains only elements off�� g� and

� Replace a � b with the appropriate constant in f�� g if a� b f�� �g�

Applying these rules repeatedly will leave us either with a constant in f�� �g orwith variables and constants in f�� g� on which � is a subalgebra isomorphic toZ�� So any function in P�A� is constant or a�ne when restricted to variables inf�� g� and cannot express an and�

Finally� assume that true� false f�� �g� Any node whose output is � or� is equal to either its left or its right input �whichever one is � or ��� whichin turn is equal to one of its inputs� and so on back up to a single constant orvariable� Since � is normal� for inputs in f�� �g the same subexpressions alwayshave values in f�� �g� and this path always leads back to the same input orconstant� So any function in P�A� with an output in f�� �g is either constant orequal to one of its inputs when restricted to variables in f�� �g� and cannot bean and� ut

Note that we have not shown that this algebra�s Expression Evaluation

or Circuit Value problem is in NC� simply that it is not Boolean�complete�Perhaps the reader can nd some more subtle analog of polyabelianness thatworks for all algebras�

� M�solvability and expression evaluation

We now consider the relationship between expressions in an algebra and wordsin that algebra�s multiplication group �or semigroup��

De�nition� An algebra isM�solvable if its multiplication semigroup is solvable�

For groups� solvability and M�solvability coincide since one can show thatM�G� is isomorphic to �G G��Z�G� and is solvable if G is� For loops� M�solvability implies solvability by a recent result of Vesanen ����� and nilpotentloops are M�solvable by a theorem of Bruck ���� Thus� like polyabelianness� M�solvability is a generalization of solvability which lies properly between nilpotenceand solvability for loops�

However� theM�solvable loops and polyabelian loops are incomparable� Firstwe show that M�solvability is related to the solvability of the automorphismsgenerated by the f �s and g�s in a quasidirect product

De�nition� For an a�ne quasidirect product A � B� dene hfg�B�i as thesubgroup of Aut�B� generated by all the f �s and g�s in the local operations� anddene hfgh�B�i as the group of a�ne operations on B generated by all the rowsand columns of the local operations�

We need the following denition from � ��

De�nition� The wreath product of B by A� written A o B� is a particular kindof semidirect product A �BA where BA is the set of functions from A to B�with multiplication dened as

�a�� �� � �a�� �� � �a�a�� �� where ��a� � ��a�a� � ��a�

In other words� each element of A o B consists of an element of A and a vector of jAj elements of B� the �s are multiplied componentwise� but with thecomponents of � permuted by a�� Thus the A�component is a�ected by a A�while the B�component within a block fag B is a�ected by �a� BA�

Then we have

Lemma��� If G � A�B is a quasigroup� then M�B��� for every local oper�ation � is contained in a factor of a subgroup of M�G�� Furthermore� if G isa�ne on B� the following are equivalent�

� G is M�solvable� hfg�B�i is solvable� hfgh�B�i is solvable�

Proof� Let �G� �� � A�B� Let H be the subgroup of M�G� consisting of thosemultiplications that preserve the blocks fagB �i�e� that leave the A�componentunchanged�� and let N be the subgroup of H that also xes the B�componentsof elements in a particular block fa�g B� It is easy to see that N is normalin H � and that H�N is isomorphic to the set of permutations of B that can becarried out with multiplications in G�

Although� unlike the loop case� fa�gB might not be a subalgebra� we canidentify b with �a�� b� and dene the local operations as b��b� � a n �a�b���a�b��where a is chosen so that the A�component of b�� b� is a�� Then multiplicationsin � are compositions of multiplications in G for instance� R�b � L��

a R�a�b�La� �So M�B��� � H�N �

Now suppose G is a�ne on B� Since the rows and columns of all the ��sgenerate exactly hfgh�B�i� we have hfgh�B�i � H�N � The subgroup of hfgh�B�ithat preserves the identity of B is simply hfg�B�i� Since subgroups and factorsof solvable groups are solvable� both these groups are solvable if M�G� is�

Conversely� since elements of M�G� permute the blocks with each other byelements of A while permuting them internally by the rows and columns of the��s� M�G� is a subgroup of the wreath product M�A� o hfgh�B�i� and thewreath product of two solvable groups is solvable�

Finally� since two a�ne functions ���b� � f��b� � h� and ���b� � f��b� � h�compose as

��� � ����b� � �f� � f���b� � h� � ���h��

we see that hfgh�B�i is a semidirect product hfg�B�i�B� The semidirect prod�uct of two solvable groups is solvable� and B is Abelian� so hfgh�B�i is solvableif hfg�B�i is� ut

Theorem��� The classes of polyabelian and M�solvable loops are incompara�ble�

Proof� First� we construct a polyabelian loop which is not M�solvable� Let Bbe an Abelian group with a non�solvable automorphism group for instance�Aut�Z�

�� is the simple group of order ��� and is generated by two elements ����

f �

��� �� � �

�A and g �

��� � � �� �

�A

If we let G � Z��Z�� where b� ���� b� � f�b�� � g�b��� then hfg�B�i � Aut�B�

is non�solvable and G is not M�solvable by lemma � � �This produces a loop oforder �� since Abelian groups smaller than Z�

� all have solvable automorphismgroups� we believe this is the smallest possible��

Conversely� the Boolean�complete loop of order � given in section � aboveis M�solvable M�G� consists of those permutations of eight elements whicheither preserve or switch the blocks f � �� �� �g and f�� �� �� �g� This is the wreathproduct Z�oS�� and its derived subgroupM�G�� is the subset of S�S� consistingof pairs of permutations whose total parity is even� The rest of the derived seriesis

M�G�� � A� A� � Z�� � f g

Thus non�polyabelian� and Boolean�complete� loops can be M�solvable� ut

We now give a simple result relating expressions in M�G� to expressions inG�

Lemma��� For any algebra G� the Expression Evaluation problem ofM�G�is NC��reducible to that of G�

Proof� Suppose we have a word in M�G� such as LaRbLc� If G has n elements�the product is determined by its action on each element � � � � � n� But this justmeans evaluating n expressions in G� namely a��c �b�� a��c��b�� � � � � a��cn�b��This is easily seen to be an NC� reduction� ut

Then we immediately have

Theorem��� The Expression Evaluation problem for non�M�solvable al�gebras is NC��complete under AC� reductions�

Proof� By lemma �� and the fact that Expression Evaluation for non�solvablesemigroups is NC��complete under AC� reductions ���� ut

No analogue of lemma �� seems to exist in the case of circuits since CircuitValue is P�complete for non�solvable algebras but in ACC� for non�M�solvableones that are polyabelian� circuits over M�G� are not easy to simulate withcircuits over G� or vice versa� unless P � ACC��

As mentioned above� Expression Evaluation is in ACC� for solvablegroups ���� In this case� the correct generalization of solvability in the non�associative case consists of being both polyabelian andM�solvable� but the needto parse the expression makes the problem somewhat harder

Theorem��� For an algebra which is both polyabelian and M�solvable� Ex�pression Evaluation is in TC�� or in ACC� if the expressions parse tree isalready known�

Proof� Consider a polyabelian algebra A � �A� �A��� � � � �Ak � An expressionsuch as � � �x� ��x� �x����x� on variables x�� � � � � xn can be inductively calculatedin the following way �similar to the algorithm in � �� for Circuit Value� theA��component is just a sum in an Abelian group� and the Am�� component isan expression with a�ne local operations �i�

�x� �� �x� �� x����� x�

where the �i are determined by the Am�components of the subexpressions totheir left and right�

If a�i b � fi�a� � gi�b� � hi for all i� this is a sum

nXi �

Fi�xi� �

n��Xj �

Gj�hj�

where the Fi and Gj are endomorphisms of Am�� composed of less than n ofthe f �s and g�s� In this case� the reader can verify that

F� � f�f� G� � f�F� � f�g�f� G� � f�g�F� � f�g�g� G� � �F� � g�

Each one of these is a word in hfg�Am���i� by lemma � this is solvable if Ais M�solvable� Since Expression Evaluation is in ACC� for solvable groups���� these words and sums can be evaluated in ACC�� We can calculate all of��s components with k induction steps� and we�re done�

However� how the f �s and g�s contribute to the F �s and G�s depends on theexpression�s parse tree� For each subexpression ���k�

�� that xi is contained in� Figains an fk or gk depending on whether xi is to the left or right of �k� Similarly�Gi gains an fk or gk for each subexpression ���k �

�� that �i�s subexpression iscontained in�

The set of well�formed expressions is a structured context�free language � ��and such expressions can be parsed in TC� as shown in ���� In particular� thresh�old gates can count the ascent of a string in the parse tree� dened as the numberof ��s minus the number of ��s� then x is in the subexpression to the left of � ifthe ascent of the string between them is at least as great as the ascent of any ofits initial substrings�

Thus we can compute � by rst parsing it with a TC� circuit� and thencalculating the above sum with k levels of ACC� circuits�

�Since the expression only has to be parsed once� the majority�depth of thecircuit� dened as the maximum number of threshold gates that any path tra�

verses� is constant for all algebras� Thus this problem is in dTC�

k for some smallk as dened in � ���� ut

Since nilpotent loops are both polyabelian �by lemma �� and M�solvable���� we have the corollary

Corollary� Expression Evaluation is in TC�� or ACC� if the expressionsparse tree is already known� for nilpotent loops�

Cases where the parse tree is already known could include uniform families ofexpressions� one of each length� For instance� the problem of predicting a cellularautomaton amounts to evaluating a uniform family of circuits� one of each size�This uniformity can signicantly simplify the Circuit Value problem as in � ���where cellular automata based on nilpotent groups are shown to be predictablein ACC��

� Conclusion and directions for further work

We have shown that the relationship between solvability and circuit complexitygeneralizes in non�trivial ways in the non�associative case solvability becomespolyabelianness for Boolean�completeness and Circuit Value� and a combina�tion of polyabelianness and M�solvability for Expression Evaluation� Thetable given in the Introduction becomes the following in the case of quasigroupsor loops

Expression Evaluation Circuit Value

non�polyabelian NC��complete P�completepolyabelian butnot M�solvable NC��complete ACC� DETpolyabelian and

M�solvable TC� ACC� DET�including nilpotent�

For algebras in general� non�polyabelianness needs to be replaced with somefurther generalization as the necessary and su�cient condition for Boolean�completeness� However� the second and third rows of this table hold for allalgebras�

These results also have a language�theoretic interpretation� A regular lan�guage can be characterized by its syntactic monoid� the semigroup of allowedtransitions of its nite�state machine� Thus the regular languages that areNC��complete are exactly those whose syntactic monoid is non�solvable� Similarly�the set of expressions in a non�associative algebra that evaluate to a particu�lar element is a structured context�free language� generated by the productionsa � �bc� for all b� c such that b � c � a� Thus it seems that we may be close toshowing exactly which �structured� context�free languages are NC��complete�

Overall� the fact that algebras with di�ering properties have ExpressionEvaluation and Circuit Value problems with �probably� di�ering circuitcomplexities may help us learn more about the internal structure of NC� andhopefully make some progress towards proving that ACCk� TCk� and NCk

form rich� distinct hierarchies within P� rather than all being equal to it�Acknowledgements� J�B� assisted with this work as an intern at the Santa

Fe Institute� funded by the Research Experience for Undergraduates programof the National Science Foundation� A�D� is grateful to the Santa Fe Institutefor an enjoyable visit� C�M� is grateful to McGill University for their hospitality�and to Claire Riley for spicing the visit up a bit�

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