A Logic of Interactive Proofs (Formal Theory of Knowledge Transfer)

We propose a logic of interactive proofs as a framework for an intuitionistic foundation for interactive computation, which we construct via an interactive analog of the Goedel-McKinsey-Tarski-Artemov definition of Intuitionistic Logic as embedded into a classical modal logic of proofs, and of the Curry-Howard isomorphism between intuitionistic proofs and typed programs. Our interactive proofs effectuate a persistent epistemic impact in their intended communities of peer reviewers that consists in the induction of the (propositional) knowledge of their proof goal by means of the (individual) knowledge of the proof with the interpreting reviewer. That is, interactive proofs effectuate a transfer of propositional knowledge (knowable facts) via the transmission of certain individual knowledge (knowable proofs) in multi-agent distributed systems. In other words, we as a community can have the formal common knowledge that a proof is that which if known to one of our peer members would induce the knowledge of its proof goal with that member. Last but not least, we prove non-trivial interactive computation as definable within our simply typed interactive Combinatory Logic to be nonetheless equipotent to non-interactive computation as defined by simply typed Combinatory Logic.

💡 Research Summary

The paper introduces a formal system called the Logic of Interactive Proofs (LiP) that serves as a foundation for interactive computation in multi‑agent distributed environments. The authors begin by motivating the need for a theory that captures not merely the transmission of data, as in Shannon’s information theory, but the transmission of knowledge: the induction of propositional knowledge (facts) in a recipient agent through the delivery of individual knowledge (proofs) as messages. They distinguish between individual knowledge (an agent knows a particular message M, denoted a k M) and propositional knowledge (an agent knows that a formula φ is true, denoted Kₐ(φ)).

The central idea is that an interactive proof is a message that, when received and interpreted by an intended reviewer, forces the reviewer to know the proof’s goal. Formally, the proof‑as‑message construct ⟨M⟩ₐ φ expresses “agent a can convince the interpreter that φ holds by sending message M.” This leads to a modal operator □ₐ φ meaning “agent a has a proof of φ.”

LiP is built on top of Artemov’s classical Logic of Proofs (LP) and the Gödel‑McKinsey‑Tarski embedding of intuitionistic logic into S4. The construction proceeds in five layers:

1. LiP – a classical modal logic enriched with proof‑transmission terms, agent‑indexed modalities, and epistemic axioms that capture knowledge transfer.
2. iS4 – an epistemically guarded first‑order extension (egFOLiP) that embeds LiP into a version of S4, ensuring that provability behaves like necessity.
3. iIL – an interactive intuitionistic logic obtained by embedding iS4 back into intuitionistic logic, mirroring the traditional Gödel‑McKinsey‑Tarski correspondence but in an interactive setting.
4. iCL – an untyped interactive combinatory logic that generalises classic combinatory logic (CL) to a multi‑agent context, adding operations for pairing, signing, and encryption of messages.
5. TiCL – a simply‑typed version of iCL obtained via a Curry‑Howard‑style isomorphism from iIL, thus aligning proof terms with program types.

Semantically, the authors provide both a concrete Kripke‑style model and an abstract categorical model. In the concrete model, each world records each agent’s individual knowledge set and propositional knowledge set; the accessibility relation Rₐ captures the effect of receiving and verifying a message. The abstract model treats proof terms as λ‑terms and types as propositions, establishing a Curry‑Howard correspondence that validates the typing rules of TiCL.

Key technical results include:

• Structural Laws (Theorem 1) – proof‑transmission operators satisfy associativity, commutativity, and identity, guaranteeing that message composition behaves predictably in concurrent settings.
• Logical Laws (Theorem 2) – the interaction between □ₐ and ⟨M⟩ₐ φ obeys S4‑style axioms (K, T, 4) and intuitionistic negation principles, ensuring soundness of knowledge induction.
• Completeness (Section 2.5, Appendix A) – every semantically valid LiP formula is derivable in the proof system, establishing that the axioms are sufficient for the intended epistemic semantics.
• Oracle‑Computation Interpretation (Section 2.4) – the framework accommodates external oracles (e.g., cryptographic keys, trusted third parties) as special agents that supply otherwise unavailable individual knowledge, thereby modelling non‑deterministic or secret‑dependent computations.
• Equi‑potency with Classical Computation (Section 3.3) – TiCL is shown to be computationally equivalent to simply‑typed combinatory logic; any program expressible in classic CL can be translated into an interactive proof term and vice‑versa.

The paper also discusses practical examples such as digital signatures, contract verification, and secure communication protocols, illustrating how LiP can formalise the epistemic impact of cryptographic artifacts. Related work is surveyed, contrasting LiP with traditional proof systems, modal logics of knowledge, and prior attempts at interactive Turing machines.

In conclusion, the authors argue that LiP provides a rigorous, epistemic‑centric foundation for interactive computation: proofs become first‑class communicative objects that transfer knowledge, and the logic captures both the syntactic manipulation of proofs and the semantic effect on agents’ knowledge states. Future directions include extending the framework to dynamic agent creation, richer trust models, and implementation of concrete verification tools based on TiCL.