Logic of Non-monotonic Interactive Proofs

Kramer, Simon

doi:10.1007/978-3-642-36039-8_16

Cited by 3 publications

(18 citation statements)

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“…LIiP, the result of the construction, however is independent of LiP. Further note that like in [Kra12a], [Kra12c,Kra13c], and [Kra12b,Kra13b], we still understand interactive proofs as sufficient evidence for intended resource-unbounded proof-checking agents (who are though unable to guess), and leave probabilistic and polynomial-time resource bounded agents for future work. Finally note that we choose our meta-logic to be classical (singleton meta-universe or meta-world unicity, cf.…”

Section: Introductionmentioning

confidence: 92%

“…• Like in [Kra12c,Kra13c] and [Kra12b,Kra13b], we endow the proof modality with a standard Kripke-semantics [BvB07], but first define its accessibility relation M R CM constructively in terms of elementary set-theoretic constructions, 3 namely as M R CM (cf. Section 2.2.1), and then match it to an abstract semantic interface in standard form (which abstractly stipulates the characteristic properties of the accessibility relation [Fit07]).…”

Section: Contributionmentioning

confidence: 99%

“…The reader of [Kra12a,Kra13c] will recognise similar but simpler definitions here; the one of M R CM could be even simpler (cf. Fact 1.2), but is as now in order to allow for a simpler, pattern-matching comparison with the corresponding one in [Kra12a,Kra13c]. (We wanted to show how to produce LIiP from LiP.)…”

Section: Semanticallymentioning

confidence: 99%

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

Kramer¹

2015

ACM Trans. Comput. Logic

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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).

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Section: Introductionmentioning

confidence: 92%

Section: Contributionmentioning

confidence: 99%

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

Kramer¹

2015

ACM Trans. Comput. Logic

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“…Our immediate motivation for LDiiP is first the theoretical concept and second the practical application of a negation-complete variant of our interactive proofs [Kra13,Kra12c,Kra12a]. The overarching motivation for LDiiP is to serve in an intuitionistic foundation of interactive computation.…”

Section: Motivationmentioning

confidence: 99%

“…Our goal here is to produce LDiiP axiomatically as well as semantically from LiiP. Note that like in [Kra13,Kra12c,Kra12a], we still understand interactive proofs as sufficient evidence for intended resource-unbounded proofchecking agents (who are though unable to guess), and leave probabilistic and polynomial-time resource bounds for future work.…”

Section: Introductionmentioning

confidence: 99%

Logic of Negation-Complete Interactive Proofs (Formal Theory of Epistemic Deciders)

Kramer

2014

Electronic Notes in Theoretical Computer Science

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We produce a decidable classical normal modal logic of internalised negation-complete and thus disjunctive non-monotonic interactive proofs (LDiiP) from an existing logical counterpart of non-monotonic or instant interactive proofs (LiiP). LDiiP internalises agent-centric proof theories that are negation-complete (maximal) and consistent (and hence strictly weaker than, for example, Peano Arithmetic) and enjoy the disjunction property (like Intuitionistic Logic). In other words, internalised proof theories are ultrafilters and all internalised proof goals are definite in the sense of being either provable or disprovable to an agent by means of disjunctive internalised proofs (thus also called epistemic deciders). Still, LDiiP itself is classical (monotonic, non-constructive), negation-incomplete, and does not have the disjunction property. The price to pay for the negation completeness of our interactive proofs is their non-monotonicity and non-communality (for singleton agent communities only). As a normal modal logic, LDiiP enjoys a standard Kripke-semantics, which we justify by invoking the Axiom of Choice on LiiP's and then construct in terms of a concrete oracle-computable function. LDiiP's agent-centric internalised notion of proof can also be viewed as a negation-complete disjunctive explicit refinement of standard KD45-belief, and yields a disjunctive but negation-incomplete explicit refinement of S4-provability.

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

Krämer¹

2012

Preprint

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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.

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Logic of Non-monotonic Interactive Proofs

Cited by 3 publications

References 6 publications

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

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

Logic of Negation-Complete Interactive Proofs (Formal Theory of Epistemic Deciders)

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

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