D. Barrington and D. Therien, \Finite monoids and the ne structure of NC ." Journal of the ACM 35 (1988) 941-952.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

.... exactly the complexity of recognizing a regular set [13] In the algebraic theory of circuits the relationship between NC 1 and ACC is striking: NC 1 can be characterized by computations over non solvable algebras [13] while ACC can be characterized by computations over solvable algebras [18]. As di erent as that makes them appear, it is not known if ACC and NC 1 are distinct. Indeed, ACC lies right at the frontier of what is known in circuit complexity. To be explicit, note that it is easy to see that AC 0 ACC[2] ACC NC 1 NC In fact, as indicated, the rst two ....

D. A. Barrington and D. Therien, \Finite monoids and the ne structure of NC

....groups is NP complete [9] On the other hand, if a groupoid lacks this expressive power, all these problems may be signi cantly easier. Languages recognized by solvable groups have simple combinatorial descriptions [18, 21] and circuits over them can be evaluated quickly in parallel [2, 3]. Similarly, cellular automata de ned with polyabelian operations can be predicted much more quickly than by explicit simulation [14] Thus the algebraic properties of a groupoid are intimately linked to its computational complexity. This paper is organized as follows. Section 2 gives an ....

D. Barrington and D. Therien, \Finite monoids and the ne structure of NC ." Journal of the ACM 35 (1988) 941-952.

....of computational complexity, especially low level parallel complexity classes. For instance, expressions and circuits over solvable groups can be evaluated in the classes ACC 0 and ACC 1 , while over non solvable groups these problems are NC 1 complete and P complete respectively (see [3, 4, 15] for de nitions of these classes and proofs of these results) Similarly, equations over nilpotent groups can be solved in polynomial time, while for non solvable groups this problem is NP complete [11] and for solvable groups quasipolynomial time is believed to suce. Finally, languages de ned ....

D.A. Mix Barrington and D. Therien, \Finite monoids and the ne structure of NC 1 ." Journal of the ACM 35 (1988) 941-952.

....problem is NP complete. Furthermore interesting connections to the well studied Constraint Satisfiability Problem are uncovered and exploited. 1 Introduction Using ideas and tools from algebraic automata theory, a number of algebraic characterizations of complexity classes have been uncovered ([1, 7] among many others) This has increased the importance of the study of problems whose computational complexity is parametrized by the properties of an underlying algebraic structure [6, 2, 11] In [6] Goldmann and Russell studied the computational complexity of solving single equations and ....

D. A. M. Barrington and D. Therien. Finite monoids and the ne structure of NC 1 . Journal of the ACM, 35(4):941-952, Oct. 1988.

....non solvable groups is NP complete [9] On the other hand, if a groupoid lacks this expressive power, all these problems may be signi cantly easier. Languages recognized by solvable groups have simple combinatorial descriptions [18, 21] and circuits over them can be evaluated quickly in parallel [2, 3]. Similarly, cellular automata de ned with polyabelian operations can be predicted much more quickly than by explicit simulation [14] Thus the algebraic properties of a groupoid are intimately linked to its computational complexity. This paper is organized as follows. Section 2 gives an ....

D. Barrington and D. Therien, \Finite monoids and the ne structure of NC 1 ." Journal of the ACM 35 (1988) 941-952.

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