Recursive Realizability and Iterated Limiting PCAs

• Recursive realizability is a framework that associates logical formulas with recursively computable objects via partial combinatory algebras and their limiting extensions.
• It employs the limiting PCA construction to systematically extend the representable functions, aligning computational strength with increasing logical complexity in the arithmetical hierarchy.
• The method bridges constructive arithmetic and classical systems, using techniques like parametrized bar recursion to interpret choice principles and extract computational content from proofs.

Recursive realizability encapsulates a suite of techniques and models for interpreting classical and constructive arithmetic by associating logical formulas with recursively computable objects or operations. Central among these is the use of partial combinatory algebras (PCAs) and their systematic extensions to capture successively higher fragments of arithmetic, culminating in sound computational models for Peano Arithmetic (PA) through iterated limiting constructions. The development of recursive realizability has yielded a stratified correspondence between logical systems—including Heyting Arithmetic (HA), fragments with specific classical principles, and PA—via hierarchies of realizability models that reflect the structure of the arithmetical hierarchy.

1. Partial Combinatory Algebras and Representability

A partial combinatory algebra (PCA) is defined as a set $A$ equipped with a partially defined binary operation $\cdot : A \times A \rightharpoonup A$ called application, along with distinguished elements $k,s \in A$ satisfying the fundamental combinatory equations:

• $k a b = a$
• $(s a b) c \simeq a c (b c)$, with $(s a b) c\downarrow$

Church-numerals $\overline{n} \in A$ for $n \in \mathbb{N}$ provide a canonical representation of natural numbers inside the algebra. A partial function $f \colon \mathbb{N}^k \rightharpoonup \mathbb{N}$ is $A$-representable if there exists $\cdot : A \times A \rightharpoonup A$0 such that $\cdot : A \times A \rightharpoonup A$1 whenever $\cdot : A \times A \rightharpoonup A$2, with divergence otherwise. The class $\cdot : A \times A \rightharpoonup A$3 collects all such representable functions. This notion forms the foundation for constructing realizability interpretations that reflect computability-theoretic constraints at the base of arithmetic.

2. The Limiting PCA Construction

Given a PCA $\cdot : A \times A \rightharpoonup A$4, the limiting PCA, denoted $\cdot : A \times A \rightharpoonup A$5, enlarges the class of representable functions to include stable limits of $\cdot : A \times A \rightharpoonup A$6-computations up to finite error. Employing the cofinite filter $\cdot : A \times A \rightharpoonup A$7 on $\cdot : A \times A \rightharpoonup A$8, equivalence classes $\cdot : A \times A \rightharpoonup A$9 are identified such that $k,s \in A$0 and $k,s \in A$1 agree for cofinitely many $k,s \in A$2. Application in $k,s \in A$3 is defined pointwise using combinator sequences.

A function $k,s \in A$4 yields a limiting partial function $k,s \in A$5 if, for each input $k,s \in A$6, there exists $k,s \in A$7 such that for all $k,s \in A$8, $k,s \in A$9. The representable functions in $k a b = a$0 are precisely these limiting partial functions where the approximants themselves are $k a b = a$1-representable: $k a b = a$2 This construction enables a controlled escalation of computational power, necessary for interpreting fragments of arithmetic containing more classical principles.

3. Iterated Limiting PCAs and the Arithmetical Hierarchy

Iterating the limiting operation yields a hierarchy, with $k a b = a$3-times iterated limiting PCAs defined as: $k a b = a$4 Inductive analysis shows that $k a b = a$5 represents exactly those $k a b = a$6-fold iterated limits of $k a b = a$7-representable partial functions. Notably, for the standard PCA of partial recursive functions,

$k a b = a$8

The inductive limit $k a b = a$9 encompasses all arithmetical partial functions, matching the full expressive power of arithmetical PA.

PCA Variant	Represented Function Class	Logical Principles Realized
$(s a b) c \simeq a c (b c)$0 (Base PCA)	$(s a b) c \simeq a c (b c)$1-representable (partial recursive)	Heyting Arithmetic (HA)
$(s a b) c \simeq a c (b c)$2	$(s a b) c \simeq a c (b c)$3-iterated limiting partial recursive	$(s a b) c \simeq a c (b c)$4-DNE over HA
$(s a b) c \simeq a c (b c)$5	Arithmetical partial functions	Full Peano Arithmetic (PA)

4. Realizability Interpretations via Iterated Limiting PCAs

For any PCA $(s a b) c \simeq a c (b c)$6, the realizability relation $(s a b) c \simeq a c (b c)$7 is defined inductively on the syntax of $(s a b) c \simeq a c (b c)$8. Notably, the $(s a b) c \simeq a c (b c)$9-iterated limiting PCA $(s a b) c\downarrow$0 realizes exactly those logical principles provable from HA together with double-negation elimination for $(s a b) c\downarrow$1 formulas: $(s a b) c\downarrow$2 Thus, HA + $(s a b) c\downarrow$3-DNE is sound with respect to realizability in $(s a b) c\downarrow$4. At the limit, all axioms and rules of PA are realized in $(s a b) c\downarrow$5, recovering Kleene-style classical realizability for PA when $(s a b) c\downarrow$6 is the PCA of partial recursive functions (Akama, 2013).

5. Intermediate Logical Systems and Hierarchical Structure

Intermediate systems between HA and PA emerge via fragments: $(s a b) c\downarrow$7 Each $(s a b) c\downarrow$8 realizes the system with double-negation elimination up to complexity $(s a b) c\downarrow$9; the logical power strictly increases with $\overline{n} \in A$0. The prenex normal form theorem holds in these fragments: any first-order formula with at most $\overline{n} \in A$1 quantifiers is equivalent (in $\overline{n} \in A$2) to a prenex sentence with exactly $\overline{n} \in A$3 quantifier alternations.

Notably, independence-of-premise schemes $\overline{n} \in A$4-IP are not derivable from $\overline{n} \in A$5, nor is $\overline{n} \in A$6 derivable from the independence-of-premise scheme, as demonstrated by realizability counterexamples in the respective $\overline{n} \in A$7. The inductive limit $\overline{n} \in A$8 is essential: no finite fragment suffices to recover all of PA.

6. Connections to Realizability in Classical Analysis

Parametrised bar recursion extends the reach of recursive realizability frameworks to interpret principles of classical analysis, especially variants of dependent choice. By design, bar recursion realizes the negative translation of countable and dependent choice in extended systems of primitive recursive functionals. For instance, the functional schema $\overline{n} \in A$9 parametrizes earlier bar-recursive realizers (BBC-functional, modified bar recursion, products of selection functions) by suitable choices of higher-order parameters and well-foundedness conditions (Powell, 2014).

The general schema: $n \in \mathbb{N}$0 accommodates a family of recursive realizers, each assigned to variants of choice principles and tuned by altering the parameter selections and well-founded relations. Soundness for negative translations of choice principles is obtained by backward induction, reflecting a uniform method for interpreting these principles computationally.

7. Significance and Ongoing Developments

Recursive realizability, through iterated limiting PCAs and uniform bar recursion, provides powerful tools for structuring the computational interpretation of classical and semi-classical arithmetic. The explicit alignment with the hierarchy of arithmetic and logical fragments offers sharp characterizations of what computational content is extractable from proofs in various systems. The parametrized approach to bar recursion signifies a unifying trend, enabling a general analysis of realizers for strong choice principles and fostering the systematic discovery of new program extraction techniques (Powell, 2014).

Ongoing research addresses further extensions, granular refinements of the arithmetical hierarchy, and broadened frameworks to accommodate yet more general principles, with recursive realizability continuing as a core construct in proof theory and the foundations of constructive mathematics.