Interactive Realizers and Monads

We propose a realizability interpretation of a system for quantifier free arithmetic which is equivalent to the fragment of classical arithmetic without “nested” quantifiers, called here EM1-arithmetic. We interpret classical proofs as interactive learning strategies, namely as processes going through several stages of knowledge and learning by interacting with the “environment” and with each other. We give a categorical presentation of the interpretation through the construction of two suitable monads.

💡 Research Summary

The paper introduces a novel realizability interpretation for a fragment of classical arithmetic called EM1‑arithmetic, which consists of quantifier‑free formulas together with a single layer of quantifiers but forbids nesting of quantifiers. This fragment is expressive enough to capture many elementary number‑theoretic statements while remaining amenable to a constructive analysis. Traditional realizability approaches—such as Kreisel’s modified realizability, the Dialectica interpretation, or Kleene’s recursive realizability—are designed for intuitionistic systems and break down when applied directly to EM1‑arithmetic because classical principles like the law of excluded middle (LEM) and choice introduce non‑constructive information that cannot be represented by a single static witness.

To overcome this obstacle, the authors propose interactive realizers, a dynamic notion of a realizer that evolves through a sequence of knowledge states. An interactive realizer is not a single function mapping inputs to outputs; instead it is a process that, at each stage, receives a question from the environment, produces a response, and updates its internal state accordingly. The interaction proceeds until the target formula is “realized” in the sense that the process has accumulated enough evidence to certify the formula’s truth. This view aligns with learning‑in‑the‑limit and dialogue‑based proof interpretations, where the proof object learns from counterexamples supplied by the environment and refines its strategy.

From a categorical perspective the paper formalizes this dynamics using two monads. The first monad M encapsulates the evolution of the knowledge state. Its underlying functor maps a type X to a type of stateful computations State → (X × State). The unit η_M injects a pure value with an unchanged state, while the multiplication μ_M composes two stateful steps into a single step, satisfying the usual monad laws (associativity and identity). The second monad N captures the interaction effect with the environment. Its functor sends X to a type of question‑answer interactions Question → (X × Feedback). The bind operation of N threads the feedback from one interaction into the next, ensuring that later stages can depend on earlier answers.

The composition N ∘ M yields a combined monad that models an interactive realizer in its entirety: a computation that carries a mutable state while simultaneously exchanging information with an external oracle. The authors prove that the combined monad satisfies the monad laws, guaranteeing that sequential compositions of interactive realizers are well‑behaved and that different parenthesizations of a proof strategy are equivalent. This categorical structure provides a clean algebraic account of the learning process underlying classical proofs in EM1‑arithmetic.

The paper also relates interactive realizability to classical realizability. In the traditional setting a realizer for a formula φ is a static term r such that, for every realizer of the premises, r computes a witness for φ. In the interactive setting a realizer is a term r̂ of type Input → (Output × Feedback). The extra Feedback component records the environment’s responses (e.g., counterexamples to a tentative witness). When the environment supplies a counterexample, the realizer can revise its guess and continue; when no counterexample appears, the realizer can safely conclude that the guessed witness satisfies φ. This mechanism effectively internalizes the law of excluded middle: to realize A ∨ ¬A, the realizer first attempts to produce a witness for A; if the environment disproves it, the realizer switches to a proof of ¬A, and the interaction terminates once one side succeeds.

Beyond the theoretical contribution, the authors sketch concrete applications. By interpreting the monads in a functional language such as Haskell, one can implement interactive realizers as free monads or effect handlers. This enables the construction of proof‑search programs that interact with external data sources, user input, or even machine‑learning components. Moreover, the monadic framework suggests a systematic way to transform static realizers (obtained from intuitionistic fragments) into interactive ones that can handle classical reasoning, opening a pathway for integrating classical proof assistants with learning‑oriented AI systems.

In summary, the paper delivers a robust categorical model—via two carefully designed monads—for interpreting classical proofs in EM1‑arithmetic as interactive learning processes. It bridges the gap between non‑constructive logical principles and constructive computational content, offering both deep theoretical insight and practical avenues for implementation in modern functional programming and automated reasoning environments.