Journal of Symbolic Logic 71 (2):399 - 424 (2006)
We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory △ possesses a polynomial space computable model which is exponential space decidable and thus △ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
References found in this work BETA
Polynomial-Time Versus Recursive Models.Douglas Cenzer & Jeffrey Remmel - 1991 - Annals of Pure and Applied Logic 54 (1):17-58.
Polynomial-Time Abelian Groups.Douglas Cenzer & Jeffrey Remmel - 1992 - Annals of Pure and Applied Logic 56 (1-3):313-363.
Feasible Graphs and Colorings.Douglas Cenzer & Jeffrey Remmel - 1995 - Mathematical Logic Quarterly 41 (3):327-352.
Complexity-Theoretic Algebra II: Boolean Algebras.A. Nerode & J. B. Remmel - 1989 - Annals of Pure and Applied Logic 44 (1-2):71-99.
Citations of this work BETA
No citations found.
Similar books and articles
Complexity Results for Modal Dependence Logic.Peter Lohmann & Heribert Vollmer - 2013 - Studia Logica 101 (2):343-366.
A Proof of Completeness for Continuous First-Order Logic.Arthur Paul Pedersen & Itai Ben Yaacov - unknown
On Polynomial Time Computation Over Unordered Structures.Andreas Blass, Yuri Gurevich & Saharon Shelah - 2002 - Journal of Symbolic Logic 67 (3):1093-1125.
Polynomial Time Operations in Explicit Mathematics.Thomas Strahm - 1997 - Journal of Symbolic Logic 62 (2):575-594.
Tailoring Recursion for Complexity.Erich Grädel & Yuri Gurevich - 1995 - Journal of Symbolic Logic 60 (3):952-969.
Every Polynomial-Time 1-Degree Collapses If and Only If P = Pspace.Stephen A. Fenner, Stuart A. Kurtz & James S. Royer - 2004 - Journal of Symbolic Logic 69 (3):713-741.
Initial Segments of the Lattice of Π01 Classes.Douglas Cenzer & Andre Nies - 2001 - Journal of Symbolic Logic 66 (4):1749 - 1765.
Degree Spectra of Prime Models.Barbara F. Csima - 2004 - Journal of Symbolic Logic 69 (2):430 - 442.
Infinite Time Decidable Equivalence Relation Theory.Samuel Coskey & Joel David Hamkins - 2010 - Notre Dame Journal of Formal Logic 52 (2):203-228.
Bounding Prime Models.Barbara F. Csima, Denis R. Hirschfeldt, Julia F. Knight & Robert I. Soare - 2004 - Journal of Symbolic Logic 69 (4):1117 - 1142.
Added to index2010-08-24
Total downloads35 ( #143,129 of 2,152,003 )
Recent downloads (6 months)5 ( #162,637 of 2,152,003 )
How can I increase my downloads?
There are no threads in this forum
Nothing in this forum yet.