Logic of Intuitionistic Interactive Proofs (Formal Theory of Perfect Knowledge Transfer)

Abstract

We produce a decidable super-intuitionistic normal modal logic of internalised intuitionistic (and thus disjunctive and monotonic) interactive proofs (LIiP) from an existing classical counterpart of classical monotonic non-disjunctive interactive proofs (LiP). Intuitionistic interactive proofs effect a durable epistemic impact in the possibly adversarial communication medium CM (which is imagined as a distinguished agent) and only in that, that consists in the permanent induction of the perfect and thus disjunctive knowledge of their proof goal by means of CM's knowledge of the proof: If CM knew my proof then CM would persistently and also disjunctively know that my proof goal is true. So intuitionistic interactive proofs effect a lasting transfer of disjunctive propositional knowledge (disjunctively knowable facts) in the communication medium of multi-agent distributed systems via the transmission of certain individual knowledge (knowable intuitionistic proofs). Our (necessarily) CM-centred notion of proof is also a disjunctive explicit refinement of KD45-belief, and yields also such a refinement of standard S5-knowledge. Monotonicity but not communality is a commonality of LiP, LIiP, and their internalised notions of proof. As a side-effect, we offer a short internalised proof of the Disjunction Property of Intuitionistic Logic (originally proved by Gödel).

Logic of Non-monotonic Interactive Proofs

“…For LiP, we provide a substantially simplified semantic interface and a slightly simplified axiomatisation, which is a nice side-effect of obtaining LiiP + . The Kripke-semantics for LiiP (like for LiP [Kra12]) is knowledge-constructive in the sense that (cf. Fact 1) our interactive proofs induce the knowledge of their proof goal (say φ) in their intended interpreting agents (say a) such that the induced knowledge (K a (φ)) is knowledge in the sense of the standard modal logic of knowledge S5 [FHMV95,MV07,HR10].…”

“…In contrast [Kra12], the epistemic impact of persistent interactive proofs is durable in the sense of being the case necessarily at the instant of learning the proof and henceforth, where time can be present implicitly (such as here) or explicitly (in future work). In other words, when a persistent proof can prove a certain statement, the proof will always be able to robustly do so, independently of whether or not more messages (data) than just the proof are learnt.…”

“…The subject matter of this paper is modal logic of interactive proofs, i.e., a novel logic of non-monotonic or instant interactive proofs (LiiP) as well as an existing logic of monotonic or persistent interactive proofs (LiP) [Kra12]. (We abbreviate interactivity-related adjectives with lower-case letters.)…”

“…See Appendix B for formal application examples. Like in LiP [Kra12], we understand interactive proofs as sufficient evidence to intended resource-unbounded (though unable to guess) proof-and signature-checking agents (designated verifiers).…”

“…The overarching motivation for L(i)iP is to serve in an intuitionistic foundation of interactive computation. See [Kra12] for a programmatic and methodological motivation.…”