Kleene's Realizability Research Articles

Two-level realization of logical formulas for deductive program synthesis

This paper presents a novel approach to interpreting logical formulas for synthesizing algorithms and programs. The proposed method combines features of Kleene realizability and Gödel's “dialectica” interpretation but does not rely on them directly. A simple version of positive predicate logic without functions is considered, including conjunction, disjunction, implication, and universal and existential quantifiers. A new realizability semantics for formulas and sequents is described, which considers not just a realization of a formula, but a realization with additional support. The realization roughly corresponds to Kleene realizability. The support provides additional data in favor of the correctness of the realization. The support must confirm that the realization works correctly for the formula under any valid conditions of application. A proof language is presented for which a correctness theorem is proved showing that any derivable sequent has a realization and support confirming that this realization works correctly for this formula under any valid conditions with a suitable interpreter for the programs used.

Realizability interpretation of PA by iterated limiting PCA

For any partial combinatory algebra (PCA for short) $\mathcal{A}$, the class of $\mathcal{A}$-representable partial functions from $\mathbb{N}$ to $\mathcal{A}$ quotiented by the filter of cofinite sets of $\mathbb{N}$ is a PCA such that the representable partial functions are exactly the limiting partial functions of $\mathcal{A}$-representable partial functions (Akama 2004). The n-times iteration of this construction results in a PCA that represents any n-iterated limiting partial recursive function, and the inductive limit of the PCAs over all n is a PCA that represents any arithmetical partial function. Kleene's realizability interpretation over the former PCA interprets the logical principles of double negation elimination for Σ0n-formulae, and over the latter PCA, it interprets Peano's arithmetic (PA for short). A hierarchy of logical systems between Heyting's arithmetic (HA for short) and PA is used to discuss the prenex normal form theorem, relativised independence-of-premise schemes, and the statement ‘PA is an unbounded extension of HA’.

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 Lifschitz (Proc Am Math Soc 73:101---106, 1979). A Lifschitz counterpart to Kleene's realizability for functions (in Baire space) was developed by van Oosten (J Symb Log 55:805---821, 1990). In that paper he also extended Lifschitz' realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo---Fraenkel set theory, IZF. The machinery would also work for extensions of IZF with large set axioms. In addition to separating Church's thesis with uniqueness condition from its general form in intuitionistic set theory, we also obtain several interesting corollaries. The interpretation repudiates a weak form of countable choice, AC ?,? , asserting that a countable family of inhabited sets of natural numbers has a choice function. AC ?,? is validated by ordinary Kleene realizability and is of course provable in ZF. On the other hand, a pivotal consequence of AC ?,? , namely that the sets of Cauchy reals and Dedekind reals are isomorphic, remains valid in this interpretation. Another interesting aspect of this realizability is that it validates the lesser limited principle of omniscience.

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 primitive recursive. Also a polynomially bounded schema of Church's Thesis is shown to be polynomially bounded realizable. So the schema is consistent with Basic Arithmetic, whereas it is inconsistent with Heyting Arithmetic.

A partial analysis of modified realizability

A formalized version of Kleene realizability for intuitionistic first-order arithmetic HA was axiomatically characterized by Troelstra (see [2, 3.2]) as follows: for an arbitrary HA-sentence φ, HA ⊢ ∃x(x realizes φ) if and only if HA + ECT0 ⊢ φ.Many notions of realizability have been characterized in this fashion: see [2] or [3] for details. For some notions, for example extensional realizability, it is necessary to pass to an extension of HA: realizability in HA is characterized by deducibility from certain axioms in an extension of HA.The present note is concerned with modified realizability, seen as interpretation for HA. From semantical considerations (see [4]) it follows that this interpretation can be constructed as a combination of three ingredients:i) Kleene realizability;ii) Kripke forcing over a 2-element linear order P;iii) The Friedman translation [1].This will be shown in section 2. The Friedman translation (in the way we use it) introduces a new propositional constant V; hence we move to an extension HA(V) of HA. We must then define Kleene realizability and forcing for the extended language. Now let (φ)v be the result of the Friedman translation applied to φ. We obtain, in HA, that the sentence saying that φ is modified-realizable, is equivalent to the sentence which says that the statement “(φ)v is Kleene-realizable” is forced in P (see section 2).

Logical operations and Kolmogorov complexity

Conditional Kolmogorov complexity K(x|y) can be understood as the complexity of the problem “ Y → X ”, where X is the problem “construct x” and Y is the problem “construct y”. Other logical operations ( ∧,∨,↔) can be interpreted in a similar way, extending Kolmogorov interpretation of intuitionistic logic and Kleene realizability. This leads to interesting problems in algorithmic information theory. Some of these questions are discussed.

Extensional realizability

Two straightforward “extensionalisations” of Kleene's realizability are considered; denoted r e and e. It is shown that these realizabilities are not equivalent. While the r e -notion is (as a relation between numbers and sentences) a subset of Kleene's realizability, the e-notion is not. The problem of an axiomatization of e-realizability is attacked and one arrives at an axiomatization over a conservative extension of arithmetic, in a language with variables for finite sets. A derived rule for arithmetic is obtained by the use of a q-variant of e-realizability; this rule subsumes the well-known Extended Church's Rule. The second part of the paper focuses on toposes for these realizabilities. By a relaxation of the notion of partial combinatory algebra, a new class of realizability toposes emerges. Relationships between the various realizability toposes are given, and results analogous to Robinson and Rosolini's characterization of the effective topos, are obtained for a topos generalizing e-realizability.

Realizing Brouwer's sequences

When Kleene extended his recursive realizability interpretation from intuitionistic arithmetic to analysis, he was forced to use more than recursive functions to interpret sequences and conditional constructions. In fact, he used what classically appears to be the full continuum. We describe here a generalization to higher type of Kleene's realizability, one case of which, ( U, R)-realizability, uses general recursive functions throughout, both to realize theorems and to interpret choice sequences. ( U, R)-realizability validates a version of the bar theorem and the usual continuity principles, while also providing naturally, as Kleene's 1965 realizability does not, for versions of lawless sequence axioms, as well as of Church's Thesis.

Axiomatizing higher-order Kleene realizability

Kleene's realizability interpretation (1945) for first-order arithmetic was shown by Hyland (1982) to fit into the internal logic of an elementary topos, the “Effective topos” Eff . In this paper it is shown, that there is an internal realizability definition in Eff , i.e. a syntactical translation of the internal language of Eff into itself of form “ n realizes ϕ” (where n is a variable of type the natural numbers object of Eff ), which extends Kleene's definition, and such that for sentences ϕ, the equivalence ϕ[harr]∃ n( n realizes ϕ) is true in Eff . The internal realizability definition depends on finding separated covers for non-separated objects of Eff . However, for the objects arising in higher-order arithmetic, canonical covers are available, which are definable in higher-order arithmetic. These canonical covers yield “covering axioms”. It is shown that these covering axioms, together with uniformity principles and Extended Church's Thesis, axiomatize a formalized extension of Kleene realizability to a constructive system of higher-order arithmetic. The details are worked out for second and third-order arithmetic, but the method can be extended to any order. As an application, it is shown in the final section that a certain completeness property of the (internal) category of “modest sets” can be derived in third-order arithmetic from the realizability axioms.

Goodman’s theorem and beyond

modified realizability: specific realizabilities in HRO, HEO as X: Kleene's recursive realizability. Clearly X is an abstract interpretation of HA in a suitable theory of partial functionals; then Kleene's realizability will result by an interpretation analogous to HRO. And, the very realizability which we need will result by an interpretation analogous to HEO. The details of this program are so similar to what is known already that we give only a sketch. First we describe a suitable theory HA of partial functionals. These are to be partial functionals of partial arguments, not only partial functionals of total arguments. We have the same type structure as for HA. We have an relation App(/, x, y) at each sensible triple of types, which is meant to stand for f(x) ^ y. So we need the axiom App(/, x, y) and App(/, x, z) —> x = z. Exactly as in Feferman's theories, we introduce application or for short, terms, which are not officially part of the language. These terms are built up from variables and constants; we include the same constants as in