Computation over algebraic structures and a classification of undecidable problems

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.

[1]  Egon Börger,et al.  Berechenbarkeit Komplexität Logik , 1985 .

[2]  B. Poizat.  Les petits cailloux : une approche modèle-théorique de l'algorithmie , 1995 .

[3]  Robert I. Soare,et al.  Recursively enumerable sets and degrees - a study of computable functions and computability generated sets , 1987, Perspectives in mathematical logic.

[4]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[5]  Christine Gaßner.  On NP-Completeness for Linear Machines , 1997, J. Complex..

[6]  Pascal Koiran.  Computing over the Reals with Addition and Order , 1994, Theor. Comput. Sci..

[7]  Maarten Marx,et al.  Finite Model Theory and Its Applications , 2007, Texts in Theoretical Computer Science. An EATCS Series.

[8]  Wesley Calvert,et al.  Noncomputable functions in the Blum-Shub-Smale model , 2011, Log. Methods Comput. Sci..

[9]  Alexandra Silva,et al.  Generalizing determinization from automata to coalgebras , 2013, Log. Methods Comput. Sci..

[10]  Dana S. Scott.  Some Definitional Suggestions for Automata Theory , 1967, J. Comput. Syst. Sci..

[11]  Yiannis N. Moschovakis,et al.  Abstract first order computability. II , 1969 .

[12]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[13]  Dieter Spreen.  On Some Decision Problems in Programming , 1995, Inf. Comput..

[14]  DAVID DOBKIN,et al.  A Lower Bound of the ½n² on Linear Search Programs for the Knapsack Problem , 1978, J. Comput. Syst. Sci..

[15]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[16]  J. V. Tucker,et al.  Computable functions and semicomputable sets on many-sorted algebras , 2001, Logic in Computer Science.

[17]  Christine Gaßner.  Strong Turing Degrees for Additive BSS RAM's , 2013, Log. Methods Comput. Sci..

[18]  Christine Gaßner.  Oracles and Relativizations of the P =? NP Question for Several Structures , 2009, J. Univers. Comput. Sci..

[19]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[20]  Felipe Cucker.  The Arithmetical Hierarchy over the Reals , 1992, J. Log. Comput..

[21]  Armin Hemmerling.  Computability of String Functions Over Algebraic Structures , 1998, Math. Log. Q..

[22]  Felipe Cucker,et al.  Computability over an Arbitrary Structure. Sequential and Parallel Polynomial Time , 2003, FoSSaCS.

[23]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[24]  Felipe Cucker,et al.  Implicit complexity over an arbitrary structure: Quantifier alternations , 2006, Inf. Comput..

[25]  Felipe Cucker,et al.  Computing over the Reals with Addition and Order: Higher Complexity Classes , 1995, J. Complex..

[26]  J. V. Tucker,et al.  Complexity theory and the operational structure of algebraic programming systems , 1982, Acta Informatica.

[27]  Erich Grädel,et al.  Finite Model Theory and Descriptive Complexity , 2007 .

[28]  Klaus Weihrauch,et al.  Turing machines on represented sets, a model of computation for Analysis , 2011, Log. Methods Comput. Sci..

[29]  Y. Moschovakis.  Descriptive Set Theory , 1980 .

[30]  Klaus Meer,et al.  An explicit solution to Post's Problem over the reals , 2005, J. Complex..

[31]  Pawel Sobocinski,et al.  Being Van Kampen is a universal property , 2011, Log. Methods Comput. Sci..