Paper detail

Computation Environments (2) Persistently Evolutionary Semantics

In the manuscript titled "Computation environment (1)", we introduced a notion called computation environment as an interactive model for computation and complexity theory. In this model, Turing machines are not autonomous entities and find their meanings through the interaction between a computist and a universal processor, and thus due to evolution of the universal processor, the meanings of Turing machines could change. In this manuscript, we discuss persistently evolutionary intensions. We introduce a new semantics, called persistently evolutionary semantics, for predicate logic that the meaning of function and predicate symbols are not already predetermined, and predicate and function symbols find their meaning through the interaction of the subject with the language. In (classic) model theory, the mathematician who studies a structure is assumed as a god who lives out of the structure, and the study of the mathematician does not effect the structure. The meaning of predicate and function symbols are assumed to be independent of the mathematician who does math. The persistently evolutionary semantics could be regarded as a start of "Interactive Model Theory" as a new paradigm in model theory (similar to the paradigm of interactive computation). In interactive model theory, we suppose that a mathematical structure should consist of two parts: 1) an intelligent agent (a subject), and 2) an environment (language), and every things should find its meaning through the interaction of these two parts. We introduce persistently evolutionary Kripke structure for propositional and predicate logic. Also, we propose a persistently evolutionary Kripke semantics for the notion of computation, where the intension of a code of a Turing machine persistently evolve. We show that in this Kripke model the subject can never know P = NP.