Models for the Logic of Proofs
LFCS '97 Proceedings of the 4th International Symposium on Logical Foundations of Computer Science
On the lengths of proofs in the propositional calculus (Preliminary Version)
STOC '74 Proceedings of the sixth annual ACM symposium on Theory of computing
On the complexity of the reflected logic of proofs
Theoretical Computer Science - Clifford lectures and the mathematical foundations of programming semantics
JELIA '08 Proceedings of the 11th European conference on Logics in Artificial Intelligence
Pillars of computer science
Self-referentiality of justified knowledge
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
Fields of logic and computation
Prehistoric phenomena and self-referentiality
CSR'10 Proceedings of the 5th international conference on Computer Science: theory and Applications
The Ontology of Justifications in the Logical Setting
Studia Logica
Knowledge, Time, and the Problem of Logical Omniscience
Fundamenta Informaticae - Logic, Language, Information and Computation
Prehistoric Graph in Modal Derivations and Self-Referentiality
Theory of Computing Systems
Hi-index | 0.00 |
Artemov's logic of proofs LP is a complete calculus of propositions and proofs, which is now becoming a foundation for the evidence-based approach to reasoning about knowledge. Additional atoms in LP have form t: F, read as "t is a proof of F" (or, more generally, as "t is an evidence for F") for an appropriate system of terms t called proof polynomials. In this paper, we answer two well-known questions in this area. One of the main features of LP is its ability to realize modalities in any S4-derivation by proof polynomials thus revealing a statement about explicit evidences encoded in that derivation. We show that the original Artemov's algorithm of building such realizations can produce proof polynomials of exponential length in the size of the initial S4-derivation. We modify the realization algorithm to produce proof polynomials of at most quadratic length. We also found a modal formula, any realization of which necessarily requires self-referential constants of type c: A(c). This demonstrates that the evidence-based reasoning encoded by the modal logic S4 is inherently self-referential.