A "Game Semantical" Intuitionistic Realizability Validating Markov's Principle
نویسندگان
چکیده
We propose a very simple modification of Kreisel’s modified realizability in order to computationally realize Markov’s Principle in the context of Heyting Arithmetic. Intuitively, realizers correspond to arbitrary strategies in Hintikka-Tarski games, while in Kreisel’s realizability they can only represent winning strategies. Our definition, however, does not employ directly game semantical concepts and remains in the style of functional interpretations. As term calculus, we employ a purely functional language, which is Gödel’s System T enriched with some syntactic sugar. 1998 ACM Subject Classification F.4.1 Mathematical Logic
منابع مشابه
Lifschitz realizability for intuitionistic Zermelo-Fraenkel set theory
A variant of realizability for Heyting arithmetic which validates Church’s thesis with uniqueness condition, but not the general form of Church’s thesis, was introduced by V. Lifschitz in [15]. A Lifschitz counterpart to Kleene’s realizability for functions (in Baire space) was developed by van Oosten [19]. In that paper he also extended Lifschitz’ realizability to second order arithmetic. The ...
متن کاملModified Realizability Toposes and Strong Normalization Proofs
This paper is motivated by the discovery that an appropriate quotient SN 3 of the strongly normalising untyped 3-terms (where 3 is just a formal constant) forms a partial applicative structure with the inherent application operation. The quotient structure satises all but one of the axioms of a partial combinatory algebra (pca). We call such partial applicative structures conditionally partial ...
متن کاملClassical logic as limit completion
We define a constructive model for ∆2-maps, that is, maps recursively definable from a map deciding the halting problem. Our model refines existing constructive interpretation for classical reasoning over one-quantifier formulas: it is compositional (Modus Ponens is interpreted as an application) and semantical (rather than translating classical proofs into intuitionistic ones, we define a math...
متن کاملMarkov's Principle for Propositional Type Theory
In this paper we show how to extend a constructive type theory with a principle that captures the spirit of Markov’s principle from constructive recursive mathematics. Markov’s principle is especially useful for proving termination of specific computations. Allowing a limited form of classical reasoning we get more powerful resulting system which remains constructive and valid in the standard c...
متن کاملPrimitive Recursive Realizability and Basic Propositional Logic
Two notions of primitive recursive realizability for arithmetic sentences are considered. The first one is strictly primitive recursive realizability introduced by Z. Damnjanovic in 1994. We prove that intuitionistic predicate logic is not sound with this kind of realizability. Namely there exists an arithmetic sentence which is deducible in the intuitionistic predicate calculus but is not stri...
متن کامل