H.Caussinus and F.Lemieux, The Complexity of Computing over Quasigroups, Proc. FST& TCS, 1994, pp. 36-47.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of groupoids and classes of languages, and research has been done in this direction. In a rst approach, ad hoc examples of groupoids whose recognition power coincides with several signi cant subclasses of CFL s were presented in [3, 14] Another approach was initiated and partially explored in [8, 2]; it consists in classifying groupoids in terms of the properties of their multiplication monoid (denoted M(G) for de nition see Subsection 2.3) and to look at the languages recognized by the groupoids in a given class. An obvious case is the quasigroups, which are those groupoids such that ....

....monoid (denoted M(G) for de nition see Subsection 2.3) and to look at the languages recognized by the groupoids in a given class. An obvious case is the quasigroups, which are those groupoids such that M(G) is a group: it was shown recently that they recognize exactly the open regular languages [8, 4]. The study of the aperiodic groupoids, namely those for which M(G) is a group free monoid, began when it was observed that there exist aperiodic universal groupoids [6] it was then shown in [2] that groupoids such that M(G) is J trivial with threshold 2 are powerful enough to recognize the ....

H.Caussinus and F.Lemieux, The Complexity of Computing over Quasigroups, Proc. FST& TCS, 1994, pp. 36-47.

....L a and R a are the identity 1. Then aL a (b) L a (b) b, so anb = L a (b) a(a( a z n 1 times b) is in P(Q) Similarly for a=b; then by composition we can de ne [a; b] ab = ba, a; b; c] ab)c = a(bc) a = 1=a and a = an1. Then we have the following [6, 11]: Theorem 3.5 If G is a nite simple loop that is not an Abelian group then G is functionally complete. Proof. Let g 1 ; g 2 2 G and g 1 6= 1. By Lemma 3.3, there exists a function g1 g 2 (x) that xes the identity and maps g 1 to g 2 and that is expressible in G. Because it is simple and ....

H. Caussinus and F. Lemieux, \The complexity of computing over quasigroups. " In Proc. 14th annual FST&TCS (1994) 36-47.

....separations between them. Most of this work has dealt with associative structures, namely groups, semigroups and monoids, largely because the idea of a syntactic monoid is familiar from the theory of nite state automata. However, some progress has been made in the non associative case as well [5, 6, 13, 10, 17]. Here, concepts such as solvability generalize in several competing ways, and nding the appropriate one for a given problem can be dicult. For instance, the complexity of circuit evaluation and expression evaluation over loops is determined by two di erent generalizations of solvability, which ....

H. Caussinus and F. Lemieux, \The Complexity of Computing over Quasigroups." Proc. 14th annual FST&TCS (1994) 36-47.

.... any w 2 A we have that w 2 L if and only if G( w) F 6= When G is associative, this de nition corresponds to the standard recognition by monoid (e.g. see [9] Many recent papers have investigated the applications of nite groupoids in complexity theory and in formal language theory (e.g. [3, 5, 1, 7, 2]) A fundamental result (implicit in [11] is that a language is context free if and only if it is recognized by a nite groupoid (see also [3] For all a in a groupoid G, we de ne two functions LG (a) G G and RG (a) G G such that xLG (a) ax and xRG (a) xa. Whenever there is no ....

H. Caussinus and F. Lemieux, The Complexity of Computing over Quasigroups, In the Proceedings of the 14th annual FST&TCS Conference, pp.3647, 1994.

....that 1 x y k, let S(x; y) Q x Gamma1 i=1 T i Delta Q y i=x T i Delta Q k i=y 1 T i Delta R, ignoring the first term (the next to last term) whenever x = 1 (y = k) Observe that S(x; y) is always yield equivalent to T . The following result is due to Herv e Caussinus. Lemma 4.2. 1 ([19]) There exist two integers a; b such that 1 a b k and such that v(T ) v(S(a; b) Proof. Define S(1; 0) T . Since jQj k, there exist, by the pigeon hole principle, two integers a; b such that 1 a b k and such that v(S(1; a Gamma 1) v 0 a Gamma1 Y i=1 T i Delta b Y i=a ....

....(In general this is not true since no word over a weakly linear groupoid can be evaluated to a nonzero element unless a linear evaluation tree be used. This is formalized in the following theorem. The following proof is a simplification of a result from Herv e Caussinus. Theorem 4.5. 1 ([19]) Let Q be a quasigroup of order q. For any n 0, any w 2 Q n and any T with yield w, there exists a yield equivalent tree S of depth smaller than 3q log 2 n such that v(T ) v(S) Proof. Let n 0 be the root of T and suppose that T has a path of length d 3q log 2 n. It is possible to ....

H. Caussinus and F. Lemieux, The Complexity of Computing over Quasigroups, In the Proceedings of the 14th annual FST&TCS Conference, pp.36-47, 1994.

....a rich theory with many deep results and applications, and it remains an active field that continues to challenge researchers. This makes more striking the observation that no such theory exists for context free languages. Nevertheless, this topic has been the subjet of recent investigations (e.g. [18, 21, 10, 13, 19, 7, 8, 20]) that we briefly describe here. A groupoid G is a set with a binary operation that can be non associative. All groupoids considered in this paper are finite. Groupoids can be used as language recognizers as follows. For any w 2 G , denote with G(w) the set of all elements g 2 G such that w can ....

....has a right and a left inverse (not necessarily identical) We observe that the multiplication table of a finite loop is such that every row and column is a permutation. Hence, a group is an associative loop but not all loops are associative (the smallest example is the loop B 5 of Section 4) In [13], it has been shown that any language recognized by a finite loop is regular. Despite this lack of power of finite loops, their investigation is essential in order to better understand the non associativity of general groupoids and the languages they recognize. This result has been refined in [9] ....

H. Caussinus and F. Lemieux, The Complexity of Computing over Quasigroups, In the Proceedings of the 14th annual FST&TCS Conference, LNCS 1256, SpringerVerlag 1994, pp.36-47.

....be a group such as SL(2; 5) whose simple divisor PSL(2; 5) A 5 is not a subgroup [33] recall that a divisor is not generally a subgroup) In this case, PSL(2; 5) is strongly Boolean complete, while SL(2; 5) is not. By using an associator instead of a commutator, this generalizes to loops [18, 11]. Like the commutator, the associator [x; y; z] is 1 if any of its arguments is 1, since the identity associates with everything. Theorem 6. Simple non Abelian loops are strongly Boolean complete, and nonsolvable loops are Boolean complete. Proof. Assume without loss of generality that G is ....

....these are not the same for all a unless the loop has the right inverse property) So simple non Abelian loops are strongly Boolean complete; and since non solvable loops have simple nonAbelian divisors, they are Boolean complete by Lemma 1. ut In fact, simple non Abelian groups [20] simple loops [18, 11], and non ane simple quasigroups [21] have a stronger property, that their closure contains all possible n ary functions on their elements. This is called functional completeness, and is of interest in the eld of multi valued logic [27, 31] However, Booleancompleteness is sucient for our ....

H. Caussinus and F. Lemieux, \The Complexity of Computing over Quasigroups." In Proc. 14th annual FST&TCS (1994) 36-47.

....a are the identity 1. Then aL n 1 a (b) L n a (b) b, so anb = L n 1 a (b) a(a( a z n 1 times b) is in P(Q) Similarly for a=b; then by composition we can de ne [a; b] ab = ba, a; b; c] ab)c = a(bc) a = 1=a and a = an1. Then we have the following [6, 11]: Theorem 3.5 If G is a nite simple loop that is not an Abelian group then G is functionally complete. Proof. Let g 1 ; g 2 2 G and g 1 6= 1. By Lemma 3.3, there exists a function g1 g 2 (x) that xes the identity and maps g 1 to g 2 and that is expressible in G. Because it is simple and ....

H. Caussinus and F. Lemieux, \The complexity of computing over quasigroups. " In Proc. 14th annual FST&TCS (1994) 36-47.

....author: School of Computer Science, McGill University, 3480 rue University, Montr eal (Qc) Canada, H3A 2A7, denis cs.mcgill.ca [7, 8, 1, 6] consists of loops, i.e. groupoids with an identity and for which every row and every column of the multiplication table contains every element. In [9] it was proved that any language recognized by a finite loop must be regular. The main result of our paper gives an exact characterization of which languages can be recognized by loops. The answer is surprizing and elegant: a language L A can be recognized by a finite loop iff L is regular ....

....Q j except that (a; b) c; d) 0 whenever a 2 F i and d 2 F j . Then, Q recognizes the language L k = A a 1 A Delta Delta Delta A a k A with 0 as the accepting element. By Lemma 3.3, L is also recognized by a finite loop. 4 Finite loops recognize only open regular languages In [9], it is shown that finite loops only recognize regular languages. In this section we refine this result by showing that only open regular languages can be recognized by such algebras. The following can be observed. Lemma 4.1 Any language L A of the form L 0 Delta Delta Delta L k , where L ....

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H. Caussinus and F. Lemieux, The complexity of computing over quasigroups, Proc. 14th annual FST&TCS, 1994, pp.36-47.

....Evaluation and Circuit Value problems are NC 1 complete and P complete respectively. Proof. Boolean Expression Evaluation and Circuit Value are reducible to their algebraic counterparts, since a local transformation can replace and, or and not gates with complexes of algebraic gates. ut In [11] (see also [17] it is shown that any non solvable groupoid is Boolean complete. Since the circuit value problem for solvable semigroups is in NC [6, 20] non solvability is both necessary and sufficient for Boolean completeness in the case of semigroups (assuming NC ( P) However, a loop can be ....

....Nf s g ; then clearly the subsets T 0 = fnt i ; n 2 N; 1 i rg and F 0 = fnf i ; n 2 N; 1 i sg of G satisfy the requirements for Boolean completeness. ut We can now show: Theorem 12. Non polyabelian loops are Boolean complete. Proof. For non solvable loops, the result has been proved in [11]. Assume now that G is solvable, and let H be the smallest non polyabelian factor of G. Then H has a normal subloop K which is an Abelian group, namely the last non trivial subloop in its derived series with K 0 = f1g. Let N be a minimal normal subloop of H contained in K; then N is also ....

H. Caussinus and F. Lemieux, "The Complexity of Computing over Quasigroups", In the Proceedings of the 14th annual FST&TCS Conference, pp.36-47, 1994.

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H. Caussinus and F. Lemieux, The complexity of computing over quasigroups, Proc. 14th annual FST&TCS, 1994, pp.36-47.

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