Computation Environments, An Interactive Semantics for Turing Machines (which P is not equal to NP considering it)

Abstract

To scrutinize notions of computation and time complexity, we introduce and formally define an interactive model for computation that we call it the computation environment. A computation environment consists of two main parts: i) a universal processor and ii) a computist who uses the computability power of the universal processor to perform effective procedures. The notion of computation finds it meaning, for the computist, through his interaction with the universal processor. We are interested in those computation environments which can be considered as alternative for the real computation environment that the human being is its computist. These computation environments must have two properties: 1-being physically plausible, and 2-being enough powerful. Based on Copeland' criteria for effective procedures, we define what a physically plausible computation environment is. By being enough powerful, we mean that the universal processor of the computation environment, must be in a way that the computist can carry out (via the universal processor) all logical deductions that the human being can carry out. We construct two physically plausible and enough powerful computation environments: 1-the Turing computation environment, denoted by E T , and 2-a persistently evolutionary computation environment, denoted by E e , which persistently evolve in the course of executing the computations. We prove that the equality of complexity classes P and NP in the computation environment E e conflicts with the free will of the computist. We provide an axiomatic system T for Turing computability and prove that ignoring just one of the axiom of T , it would not be possible to derive P = NP from the rest of axioms. We prove that the computist who lives inside the environment E T , can never be confident that whether he lives in a static environment or a persistently evolutionary one.

Computation Environments (2) Persistently Evolutionary Semantics

“…It is easy to check that the meaning function π j behaves similar to the persistently evolutionary Turing machine P T 1 introduced in example 4.6 in [7]. The next theorem is a formal version of the theorem 4.9 in [7]. Let K ′ be the persistently evolutionary Kripke model which the set of its meaning function Π is {π j }.…”

“…One of our purpose of proposing persistently evolutionary semantics is to provide a framework to formalize the notion of free will. We discussed the notion of free will in section 4.3 of [7]. In this part, we repeat the same discussion using persistently evolutionary Kripke structures.…”

“…In section 4, and section 5, we introduce persistently evolutionary semantics for propositional and predicate logic. In section 6, using persistently evolutionary semantics for predicate logic, we formalize the argument of section 7 of the manuscript [7].…”

“…Then the two followings are straightforward. 1-( a, i) ∼ ag ( a, j), and 2-(K ce , ( a, j)) |= (P = NP) One should verify that at the state ( a, j)), the formula P = NP conflicts with the free will of the agent (see the proof of theorem 5.8 of [7]), and thus we have (K ce , ( a, j)) |= (P = NP). Therefore, (K ce , ( , i)) |= ¬K ag (P = NP).…”

“…Let IN ST s and CON F s be two set introduced in the Turing computation environment (see example 3.4 of [7]). The set of actions is defined to be…”