Abstract
The computational complexity of the circuit and expression evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or a multiplicative identity. The following dichotomy is shown: If a finite semiring is such that (i) the multiplicative semigroup is solvable and (ii) it does not contain a subsemiring with an additive identity 0 and a multiplicative identity 1 ≠ 0, then the circuit evaluation problem is in DET ⊆ NC2, and the expression evaluation problem for the semiring is in TC0. For all other finite semirings, the circuit evaluation problem is P-complete and the expression evaluation problem is NC1-complete. As an application, we determine the complexity of intersection non-emptiness problems for given context-free grammars (regular expressions) with a fixed regular language.
- Manindra Agrawal and Somenath Biswas. 2003. Primality and identity testing via chinese remaindering. J. Assoc. Comput. Mach. 50, 4 (2003), 429--443. Google ScholarDigital Library
- Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, and Peter Bro Miltersen. 2009. On the complexity of numerical analysis. SIAM J. Comput. 38, 5 (2009), 1987--2006. Google ScholarDigital Library
- Eric Allender, Jia Jiao, Meena Mahajan, and V. Vinay. 1998. Non-commutative arithmetic circuits: Depth reduction and size lower bounds. Theor. Comput. Sci. 209, 1--2 (1998), 47--86. Google ScholarDigital Library
- Jorge Almeida, Stuart Margolis, Benjamin Steinberg, and Mikhail Volkov. 2009. Representation theory of finite semigroups, semigroup radicals and formal language theory. Trans. Amer. Math. Soc. 361, 3 (2009), 1429--1461.Google ScholarCross Ref
- Sanjeev Arora and Boaz Barak. 2009. Computational Complexity - A Modern Approach. Cambridge University Press. Google ScholarDigital Library
- Karl Auinger and Benjamin Steinberg. 2005. Constructing divisions into power groups. Theor. Comput. Sci. 341, 1--3 (2005), 1--21. Google ScholarDigital Library
- D. A. M. Barrington. 1989. Bounded-width polynomial-size branching programs recognize exactly those languages in NC<sup>1</sup>. J. Comput. System Sci. 38 (1989), 150--164. Google ScholarDigital Library
- D. A. M. Barrington, N. Immerman, and H. Straubing. 1990. On uniformity within NC<sup>1</sup>. J. Comput. Syst. Sci. 41 (1990), 274--306. Google ScholarDigital Library
- David A. Mix Barrington and Denis Thérien. 1988. Finite monoids and the fine structure of NC<sup>1</sup>. J. ACM 35, 4 (1988), 941--952. Google ScholarDigital Library
- Martin Beaudry and Markus Holzer. 2007. The complexity of tensor circuit evaluation. Comput. Complex. 16, 1 (2007), 60--111. Google ScholarDigital Library
- Martin Beaudry and Pierre McKenzie. 1995. Circuits, matrices, and nonassociative computation. J. Comput. System Sci. 50, 3 (1995), 441--455. Google ScholarDigital Library
- Martin Beaudry, Pierre McKenzie, Pierre Péladeau, and Denis Thérien. 1997. Finite monoids: From word to circuit evaluation. SIAM J. Comput. 26, 1 (1997), 138--152. Google ScholarDigital Library
- Samuel R. Buss. 1987. The boolean formula value problem is in ALOGTIME. In Proceedings of the 19th Annual Symposium on Theory of Computing (STOC’87). ACM Press, New York, NY, 123--131. Google ScholarDigital Library
- Ashok K. Chandra, Steven Fortune, and Richard J. Lipton. 1985. Unbounded fan-in circuits and associative functions. J. Comput. Syst. Sci. 30, 2 (1985), 222--234.Google ScholarCross Ref
- Ashok K. Chandra, Larry J. Stockmeyer, and Uzi Vishkin. 1984. Constant depth reducibility. SIAM J. Comput. 13, 2 (1984), 423--439.Google ScholarCross Ref
- Stephen A. Cook. 1985. A taxonomy of problems with fast parallel algorithms. Inf. Control 64, 1--3 (1985), 2--21. Google ScholarDigital Library
- Stephen A. Cook and Lila Fontes. 2012. Formal theories for linear algebra. Logic. Methods Computer Science 8, 1 (2012).Google Scholar
- M. Elberfeld, A. Jakoby, and T. Tantau. 2012. Algorithmic meta theorems for circuit classes of constant and logarithmic depth. In Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS’12), Vol. 14. Schloss Dagstuhl--Leibniz-Zentrum für Informatik, 66--77.Google Scholar
- Kousha Etessami. 1997. Counting quantifiers, successor relations, and logarithmic space. J. Comput. System Sci. 54, 3 (1997), 400--411. Google ScholarDigital Library
- Moses Ganardi and Markus Lohrey. 2017. A universal tree balancing theorem. CoRR abs/1704.08705 (2017). http://arxiv.org/abs/1704.08705Google Scholar
- Jonathan S. Golan. 1999. Semirings and Their Applications. Springer.Google Scholar
- Raymond Greenlaw, H. James Hoover, and Walter L. Ruzzo. 1995. Limits to Parallel Computation: P-Completeness Theory. Oxford University Press. Google ScholarDigital Library
- Oscar H. Ibarra and Shlomo Moran. 1983. Probabilistic algorithms for deciding equivalence of straight-line programs. J. ACM 30, 1 (1983), 217--228. Google ScholarDigital Library
- Neil D. Jones and William T. Laaser. 1976. Complete problems for deterministic polynomial time. Theor. Comput. Sci. 3, 1 (1976), 105--117.Google ScholarCross Ref
- Daniel König and Markus Lohrey. 2017. Evaluation of circuits over nilpotent and polycyclic groups. Algorithmica (2017).Google Scholar
- S. Rao Kosaraju. 1990. On parallel evaluation of classes of circuits. In Proceedings of the 10th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS’90). Lecture Notes in Computer Science, Vol. 472. Springer, 232--237. Google ScholarDigital Library
- Richard E. Ladner. 1975. The circuit value problem is log space complete for P. SIGACT News 7, 1 (1975), 18--20. Google ScholarDigital Library
- Markus Lohrey. 2001. On the parallel complexity of tree automata. In Proceedings of the 12th International Conference on Rewrite Techniques and Applications (RTA’01), Aart Middeldorp (Ed.), Lecture Notes in Computer Science, Vol. 2051. Springer, 201--215. Google ScholarDigital Library
- Pierre McKenzie and Klaus W. Wagner. 2007. The complexity of membership problems for circuits over sets of natural numbers. Comput. Complex. 16, 3 (2007), 211--244. Google ScholarDigital Library
- Gary L. Miller, Vijaya Ramachandran, and Erich Kaltofen. 1988. Efficient parallel evaluation of straight-line code and arithmetic circuits. SIAM J. Comput. 17, 4 (1988), 687--695. Google ScholarDigital Library
- Gary L. Miller and Shang-Hua Teng. 1999. The dynamic parallel complexity of computational circuits. SIAM J. Comput. 28, 5 (1999), 1664--1688. Google ScholarDigital Library
- Cristopher Moore, Denis Thérien, François Lemieux, Joshua Berman, and Arthur Drisko. 2000. Circuits and expressions with nonassociative gates. J. Comput. Syst. Sci. 60, 2 (2000), 368--394. Google ScholarCross Ref
- John Rhodes and Benjamin Steinberg. 2008. The q-theory of Finite Semigroups. Springer. Google ScholarDigital Library
- Alexander A. Rubtsov and Mikhail N. Vyalyi. 2015. Regular realizability problems and context-free languages. In Proceedings of the 17th International Workshop on Descriptional Complexity of Formal Systems (DCFS’15), Lecture Notes in Computer Science, Vol. 9118. Springer, 256--267.Google Scholar
- Amir Shpilka and Amir Yehudayoff. 2010. Arithmetic circuits: A survey of recent results and open questions. Found. Trends Theor. Comput. Sci. 5, 3–4 (2010), 207--388. Google ScholarDigital Library
- Imre Simon. 1975. Piecewise testable events. In Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages, 1975, Lecture Notes in Computer Science, Vol. 33. Springer, 214--222. Google ScholarDigital Library
- Stephen D. Travers. 2006. The complexity of membership problems for circuits over sets of integers. Theor. Comput. Sci. 369, 1–3 (2006), 211--229. Google ScholarDigital Library
- Leslie G. Valiant, Sven Skyum, S. Berkowitz, and Charles Rackoff. 1983. Fast parallel computation of polynomials using few processors. SIAM J. Comput. 12, 4 (1983), 641--644.Google ScholarDigital Library
- Heribert Vollmer. 1990. The gap-language-technique revisited. In Proceedings of the 4th Workshop on Computer Science Logic (CSL’90), Lecture Notes in Computer Science, Vol. 533. Springer, 389--399. Google ScholarDigital Library
- Heribert Vollmer. 1999. Introduction to Circuit Complexity. Springer. Google ScholarDigital Library
Index Terms
- Circuits and Expressions over Finite Semirings
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