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2008
After examining the {\bf P} versus {\bf NP} problem against the Kleene-Rosser paradox of the lambda\lambdalambda-calculus [94], it was found that it represents a counter-example to NP-completeness. We prove that it contradicts the proof of Cook's theorem. A logical formalization of the liar's paradox leads to the same result. This formalization of the liar's paradox into a computable form is a 2-valued instance of a fuzzy logic programming paradox discovered in the system of [90]. Three proofs that show that {\bf SAT} is (NOT) NP-complete are presented. The counter-example classes to NP-completeness are also counter-examples to Fagin's theorem [36] and the Immermann-Vardi theorem [89,110], the fundamental results of descriptive complexity. All these results show that {\bf ZF$\not$C} is inconsistent.
A Paradox Implies SAT is (NOT) NP-complete and ZFC is Inconsistent
Computing Research Repository, 2008
After examining the {\bf P} versus {\bf NP} problem against the Kleene-Rosser paradox of the lambda\lambdalambda-calculus [94], it was found that it represents a counter-example to NP-completeness. We prove that it contradicts the proof of Cook's theorem. A logical formalization of the liar's paradox leads to the same result. This formalization of the liar's paradox into a computable form is a 2-valued instance of a fuzzy logic programming paradox discovered in the system of [90]. Three proofs that show that {\bf SAT} is (NOT) NP-complete are presented. The counter-example classes to NP-completeness are also counter-examples to Fagin's theorem [36] and the Immermann-Vardi theorem [89,110], the fundamental results of descriptive complexity. All these results show that {\bf ZF$\not$C} is inconsistent.
Decidability and Undecidability Results for Nelson-Oppen and Rewrite-Based Decision Procedures
2006
In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in finite and infinite models, respectively. We exhibit a theory T 1 such that satisfiability is decidable, but satisfiability in infinite models is undecidable. It follows that satisfiability in T 1 ∪ T 2 is undecidable, whenever T2 has only infinite models, even if signatures are disjoint and satisfiability in T2 is decidable.
On The Complexity of Bounded Time Reachability for Piecewise Affine Systems
Lecture Notes in Computer Science, 2014
Reachability for piecewise affine systems is known to be undecidable, starting from dimension 2. In this paper we investigate the exact complexity of several decidable variants of reachability and control questions for piecewise affine systems. We show in particular that the region to region bounded time versions leads to NP -complete or co-NP-complete problems, starting from dimension 2.
What Do We Learn from Experimental Algorithmics?
2000
Experimental Algorithmics is concerned with the design, implementation, tuning, debugging and performance analysis of computer programs for solving algorithmic problems. It provides methodologies and tools for designing, developing and experimentally analyzing efficient algorithmic codes and aims at integrating and reinforcing traditional theoretical approaches for the design and analysis of algorithms and data structures.