Primitive Recursive Realizability and Basic Propositional Logic
نویسنده
چکیده
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 strictly primitive recursively realizable. Another variant of primitive recursive realizability was introduced by S. Salehi in 2000. This kind of realizability is defined for the formulas of Basic Arithmetic introduced by W. Ruitenburg in 1998. We prove that these two notions of primitive recursive realizability are essentially different. Namely there exists arithmetic sentence being also a sentence of Basic Arithmetic which is strictly primitive recursively realizable but is not realizable by Salehi. The negation of such a sentence is realizable by Salehi but is not strictly primitive recursively realizable. The relation between Basic Propositional Logic and strictly primitive recursive realizability is studied. We consider a sequent variant of Basic Propositional Calculus. Notions of strictly primitive recursive realizability for arithmetic and propositional sequents are defined. We prove that every sequent deducible in Basic Propositional Calculus is strictly primitive recursively realizable. An example of a sequent which is deducible in Intuitionistic Propositional Calculus but is not strictly primitive recursively realizable is proposed.
منابع مشابه
Polynomially Bounded Recursive Realizability
A polynomially bounded recursive realizability, in which the recursive functions used in Kleene’s realizability are restricted to polynomially bounded functions, is introduced. It is used to show that provably total functions of Ruitenburg’s Basic Arithmetic are polynomially bounded (primitive) recursive functions. This sharpens our earlier result where those functions were proved to be primiti...
متن کاملProvably total functions of Basic Arithmetic
It is shown that all the provably total functions of Basic Arithmetic , a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which is sound, is introduced. This realizability is similar to Kleene’s recursive realizability, except that recursive functions are restricted to primitive recurs...
متن کاملEquality propositional logic and its extensions
We introduce a new formal logic, called equality propositional logic. It has two basic connectives, $boldsymbol{wedge}$ (conjunction) and $equiv$ (equivalence). Moreover, the $Rightarrow$ (implication) connective can be derived as $ARightarrow B:=(Aboldsymbol{wedge}B)equiv A$. We formulate the equality propositional logic and demonstrate that the resulting logic has reasonable properties such a...
متن کاملSafe Positive Induction in the Programming Logic TK
We describe an alternative schema of induction for the programming logic TK based on safe positive induction. This replaces the original schema based on the well founded part of a relation. We show how the new schema can be included into the realizability definition and how the soundness of realizability can be extended to allow for the derivation of recursive programs from proofs of specificat...
متن کاملPavel Naumov On Modal Logics of Partial Recursive Functions
The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and non-deterministic functions are considered and for both...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007