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

Abstract

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 Gödel-McKinsey-Tarski-Artëmov 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 noninteractive computation as defined by simply typed Combinatory Logic.Keywords agents as knowledge processors, proof-and signature-checkers, and reduction relations; communication channels as variable bindings; constructive Kripke-semantics; cryptographic and interpreted communication; interactive and oracle computation; proof equality; proofs as certificates, messages, and sufficient evidence; propositions as types.

Denial Logic

“…Recently, Artemov provided Justification Logic with a natural semantics in which justifications are interpreted as sets of formulas [7]. In addition to its applications in epistemic logic, modal logic and proof theory, LP has been generalized to interactive multi-agent computation [25]. Studies of justification logics have tended to focus on justified true belief, provability and logical omniscience [19,3,2].…”