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Computation over algebraic structures and a classification of undecidable problems

  • CHRISTINE GAßNER (a1)
Abstract

We consider a uniform model of computation over algebraic structures resulting from a generalization of the Turing machine and the BSS model of computation. This model allows us to gain more insight into the reasons for unsolvability of algorithmic decision problems from different perspectives. For example, classes of undecidable problems can be introduced in several ways by analogy with the classical arithmetical hierarchy and, for many structures, the different definitions lead to different hierarchies of undecidable problems. Here, we will investigate some classes of a hierarchy that is defined semantically by our deterministic oracle machines and that can be syntactically characterized by formulas whose quantifiers range only over an enumerable set. Starting from machines over algebraic structures endowed with some relations and containing an infinite recursively enumerable sequence of individuals, we will also consider this hierarchy for BSS RAM's over the reals and some undecidable problems defined by algebraic properties of the real numbers.

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Asveld P.R.J. and Tucker J.V. (1982). Complexity theory and the operational structure of algebraic programming systems. Acta Informatica 17 451476. http://dx.doi.org/10.1007/BF00264163.
Blum L., Cucker F., Shub M. and Smale S. (1998). Complexity and Real Computation, Springer.
Blum L., Shub M. and Smale S. (1989). On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society 21 146. http://projecteuclid.org/euclid.bams/1183555121.
Börger E. (1992). Berechenbarkeit, Komplexität, Logik, Vieweg.
Bournez O., Cucker F., de Naurois P.J. and Marion J.-Y. (2003). Computability over an arbitrary structure: Sequential and parallel polynomial time. In: Proceeding FOSSACS'03/ETAPS'03 185–199. http://dx.doi.org/10.1007/3-540-36576-1_12.
Bournez O., Cucker F., de Naurois P.J. and Marion J.-Y. (2006). Implicit complexity over an arbitrary structure: Quantifier alternations. Information and Computation 202 (2) 210230. http://dx.doi.org/10.1016/j.ic.2005.07.005
Calvert W., Kramer K. and Miller R. (2011). Noncomputable functions in the Blum-Shub-Smale model. Special Issue for the “Seventh International Conference on Computability and Complexity in Analysis (CCA 2010)”; Logical Methods in Computer Science 7 (2:15) (2011), 120. http://dx.doi.org/10.2168/LMCS-7(2:15)2011.
Cucker F. (1992). The arithmetical hierarchy over the reals. Journal of Logic and Computation 2 (3) 375395. http://dx.doi.org/10.1093/logcom/2.3.375.
Cucker F. and Koiran P. (1995). Computing over the reals with addition and order: Higher complexity classes. Journal of Complexity 11 358376. http://dx.doi.org/10.1006/jcom.1995.1018.
Dobkin D. and Lipton R.J. (1978). A lower bound of inline-graphic $\frac12n^2$ on linear search programs for the knapsack problem. Journal of Computer and System Sciences 16 (3) 413417. http://dx.doi.org/10.1016/0022-0000(78)90026-0.
Friedman H. and Mansfield R. (1992). Algorithmic procedure. Transactions of the AMS 332 297312.
Gaßner C. (1997). On NP-completeness for linear machines. Journal of Complexity 13 259271. http://dx.doi.org/10.1006/jcom.1997.0444.
Gaßner C. (2009). Oracles and relativizations of the P =? NP question for several structures. Journal of Universal Computer Science 15 (6) 11861205. http://dx.doi.org/10.3217/jucs-015-06-1186.
Gaßner C. (2013). Strong Turing degrees for additive BSS RAM's. Logical Methods in Computer Science 9 (4:25) 118. http://dx.doi.org/10.2168/LMCS-9(4:25)2013.
Grädel E. (2007). Finite model theory and descriptive complexity. In: Grädel E., Kolaitis P.G., Libkin L., Marx M., Spencer J., Vardi M.Y., Venema Y. and Weinstein S. (eds.) Finite Model Theory and Its Applications, Springer, 125230.
Hemmerling A. (1998). Computability of string functions over algebraic structures. Mathematical Logic Quarterly 44 144. http://dx.doi.org/10.1002/malq.19980440102.
Kleene S.C. (1952). Introduction to Metamathematics, North-Holland Publ. Co.
Koiran P. (1995). Computing over the reals with addition and order. Theoretical Computer Science 133 3547. http://dx.doi.org/10.1016/0304-3975(93)00063-B.
Meer K. and Ziegler M. (2008). An explicit solution to post's problem over the reals. Journal of Complexity 24 315. http://dx.doi.org/10.1016/j.jco.2006.09.004.
Moschovakis Y.N. (1969). Abstract first order computability. I. Transactions of the American Mathematical Society 138 427464. http://dx.doi.org/10.2307/1994926.
Moschovakis Y.N. (1980). Descriptive set theory. Studies in Logic and the Foundations of Mathematics 100, North-Holland.
Poizat B. (1995). Les Petits Cailloux. Aléas.
Preparata F.P. and Shamos M.I. (1985). Computational Geometry: An Introduction, Springer.
Scott D. (1967). Some definitional suggestions for automata theory. Journal of Computer and System Sciences 1 (2) 187212. http://dx.doi.org/10.1016/S0022-0000(67)80014-X.
Soare R.I. (1987). Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Springer.
Spreen D. (1995). On some decision problems in programming. Information and Computation 122 120139. http://dx.doi.org/10.1006/inco.1995.1143. Corrigendum ibid. 148 (1999) 241–244. http://dx.doi.org/10.1006/inco.1998.2758
Spreen D. (2010). Every Δ0 2-set is natural, up to Turing equivalence. In: Ferreira F. et al., (eds.) CiE2010, Lecture Notes in Computer Science 6158, Springer (2010), 386393. http://dx.doi.org/10.1007/978-3-642-13962-8_43
Tavana N. and Weihrauch K. (2011). Turing machines on represented sets, a model of computation for analysis. Logical Methods in Computer Science 7 (2) 121. http://dx.doi.org/10.2168/LMCS-7(2:19)2011.
Tucker J.V. and Zucker J.I. (2001). Computable functions and semicomputable sets on many-sorted algebras. Handbook of Logic in Computer Science 5 397525.
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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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