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Summary: Logic of knowledge with justi cations
S. Artemov E. Kazakov & D. Shapiro
Department of Computer Science Department of Mathematics
Cornell University Moscow University
Ithaca, NY 14853, USA Moscow, 119899, Russia
1 Introduction
How could one describe in logic of knowledge (cf. [4]) the following 1 : an agent receives a product of
two very large prime integers. In what sense does the agent know those primes? Another well-known
problem: in what sense does an agent know all the derivable formulas regardless to the complexity
of their justi cations? These and similar problems could be approached in a framework of reasoning
about explicit knowledge justi cations with atoms p : F (p is a justi cation for F) vs. the traditional
2F (F is known). An adequate set of justi cation terms and operations on them would enable us,
in partucular, to evaluate costs of knowledge extraction from a data given. The problem of nding
systems for knowledge representation with justi cations was discussed by van Benthem in [3]. We wish
to think that this paper makes a meaningful step toward to developing such a system.
In this paper we introduce the justi cation calculus LPS5 corresponding to the modal logic S5.
Justi cation terms in LPS5 are called extended proof polynomials. They may be regarded as gen-
eralization of combinatory terms (therefore typed -terms) and naturally subsume proof polynomials
for S4 from [1], [2]. The nonboolean atoms in LPS5 have form p : F were p is an extended proof
polynomial and F is an arbitrary formula, possibly containing other atoms of this kind. The intended
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