[Paper Review] Learning, Realizability and Games in Classical Arithmetic

Federico Aschieri|arXiv (Cornell University)|Dec 22, 2010

Computability, Logic, AI Algorithms7 references3 citations

TL;DR

This paper introduces Interactive Learning-Based Realizability, a novel computational semantics for classical arithmetic that models proofs as self-correcting programs learning from errors. It establishes a full soundness and completeness correspondence between learning-based realizability and 1-Backtracking Coquand game semantics, and provides a constructive analysis of learning processes in classical arithmetic, including bounds on learning length and a transfinite extension of update procedures for second-order arithmetic.

ABSTRACT

In this dissertation we provide mathematical evidence that the concept of learning can be used to give a new and intuitive computational semantics of classical proofs in various fragments of Predicative Arithmetic. First, we extend Kreisel modified realizability to a classical fragment of first order Arithmetic, Heyting Arithmetic plus EM1 (Excluded middle axiom restricted to Sigma^0_1 formulas). We introduce a new realizability semantics we call "Interactive Learning-Based Realizability". Our realizers are self-correcting programs, which learn from their errors and evolve through time. Secondly, we extend the class of learning based realizers to a classical version PCFclass of PCF and, then, compare the resulting notion of realizability with Coquand game semantics and prove a full soundness and completeness result. In particular, we show there is a one-to-one correspondence between realizers and recursive winning strategies in the 1-Backtracking version of Tarski games. Third, we provide a complete and fully detailed constructive analysis of learning as it arises in learning based realizability for HA+EM1, Avigad's update procedures and epsilon substitution method for Peano Arithmetic PA. We present new constructive techniques to bound the length of learning processes and we apply them to reprove - by means of our theory - the classic result of Godel that provably total functions of PA can be represented in Godel's system T. Last, we give an axiomatization of the kind of learning that is needed to computationally interpret Predicative classical second order Arithmetic. Our work is an extension of Avigad's and generalizes the concept of update procedure to the transfinite case. Transfinite update procedures have to learn values of transfinite sequences of non computable functions in order to extract witnesses from classical proofs.

Motivation & Objective

• To provide a constructive, learning-based computational interpretation of classical proofs in fragments of arithmetic, particularly Heyting Arithmetic plus EM₁.
• To extend Kreisel's modified realizability to classical logic by modeling realizers as self-correcting programs that evolve through error-driven learning.
• To establish a complete correspondence between learning-based realizability and 1-Backtracking game semantics, proving soundness and completeness.
• To give a constructive analysis of learning processes in classical arithmetic, including bounds on the length of learning in terms of Gödel's system T.
• To extend Avigad’s update procedures to the transfinite case for interpreting predicative classical second-order arithmetic.

Proposed method

• Introduce a new realizability semantics—Interactive Learning-Based Realizability—where atomic realizers are modeled as learning agents that correct themselves based on failures.
• Define realizers as self-correcting programs that perpetually test, question, and extend knowledge, with classical principles enabling this learning behavior over intuitionistic logic.
• Construct a correspondence between learning-based realizers and recursive winning strategies in 1-Backtracking Tarski games, proving soundness and completeness.
• Develop a non-standard model of Gödel’s system T using pairs of non-standard natural numbers and moduli of convergence to bound learning process lengths.
• Extend Avigad’s finite update procedures to transfinite ordinals (e.g., ω + k) to interpret classical second-order arithmetic, using epsilon substitution and critical formula analysis.
• Use bar recursion and type-0 recursion to analyze termination and convergence of learning processes, ensuring constructive bounds in system T.

Experimental results

Research questions

• RQ1 How can classical proofs in Heyting Arithmetic plus EM₁ be given a constructive, learning-based computational interpretation?
• RQ2 What is the precise correspondence between learning-based realizability and 1-Backtracking game semantics?
• RQ3 Can the length of learning processes in classical arithmetic be constructively bounded, and if so, how?
• RQ4 How can Avigad’s finite update procedures be generalized to transfinite ordinals for second-order arithmetic?
• RQ5 What is the role of classical principles in enabling self-correcting, learning-based computation in intuitionistic frameworks?

Key findings

• Interactive Learning-Based Realizability provides a sound and complete interpretation of classical proofs in HA + EM₁, with realizers corresponding exactly to recursive winning strategies in 1-Backtracking Tarski games.
• The paper constructs a non-standard model of Gödel’s system T where learning processes are bounded by moduli of convergence, and these bounds are themselves representable in system T.
• The length of learning processes in the epsilon substitution method and update procedures is constructively bounded, and this bound is provable in system T.
• The transfinite extension of update procedures to ordinals ω + k provides a constructive interpretation of predicative classical second-order arithmetic.
• The theory re-proves Gödel’s result that provably total functions of PA are representable in system T, using a new constructive method based on learning and convergence moduli.
• The adequacy of the update procedure of ordinal ω + k is proven, showing that if all such procedures have finite zeros, then Elementary Analysis is 1-consistent.