Papers by Hervé Caussinus
The Complexity of Computing over Quasigroups
In [7] the notions of recognition by semigroups and by programs over semigroups were extended to ... more In [7] the notions of recognition by semigroups and by programs over semigroups were extended to groupoids. This led to a new characterization of the context-free languages and the class SAC¹. In this paper, we investigate the classes of languages obtained when the groupoids are restricted to be quasigroups (i.e. the multiplication table forms a latin square). We prove that languages recognized by quasigroups are regular and that programs over quasigroups characterize NC¹. We introduce the notions of linear recognition by groupoids and by programs over groupoids, and characterize the linear context-free languages and NL. Here again, when quasigroups are used, only regular languages and languages in NC¹ can be obtained. We also consider the problem of evaluating a well-parenthesized expression over a finite loop (a quasigroup with an identity). This problem is in NC¹ for any finite loop, and we give algebraic conditions for its completeness. In particular, we prove that it is suffici...
We define the counting classes #NC¹, GapNC¹, PNC¹ and C=NC¹. We prove that boolean circuits, alge... more We define the counting classes #NC¹, GapNC¹, PNC¹ and C=NC¹. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC¹ ` #L, that PNC¹ ` L, and that C=NC¹ ` L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separations of ACC⁰ from MOD-PH and that of TC⁰ from the counting hierarchy. Moreover we obtain that if dlogtime-uniformity and logspace-uniformity for AC⁰ coincide then the polynomial time hierarchy equals PSPACE.
A note on a theorem of Barrington, Straubing and Thérien
Information Processing Letters, Apr 8, 1996
NondeterministicNC1Computation
We define the counting classes #NC1, GapNC1, PNC1 and C=NC1. We prove that boolean circuits, alge... more We define the counting classes #NC1, GapNC1, PNC1 and C=NC1. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC1 subseteq #L and that C=NC1 subseteq L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separation of ACC0 from ModPH as well as that of TC0 from the counting hierarchy. Moreover we obtain that dlogtime-uniformity and logspace-uniformity for AC0 coincide if and only if the polynomial time hierarchy equals PSPACE.
The Complexity of Computing over Quasigroups
Fsttcs, 1994
In [7] the notions of recognition by semigroups and by programs over semigroups were extended to ... more In [7] the notions of recognition by semigroups and by programs over semigroups were extended to groupoids. As a consequence of this transformation, the induced classes of languages became CFL instead of REG, in the first case, and SAC 1 instead of NC 1 in the ...
Uploads
Papers by Hervé Caussinus