 
Summary: Logic of knowledge with justications
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 wellknown
problem: in what sense does an agent know all the derivable formulas regardless to the complexity
of their justications? These and similar problems could be approached in a framework of reasoning
about explicit knowledge justications with atoms p : F (p is a justication for F) vs. the traditional
2F (F is known). An adequate set of justication 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 justications 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 justication calculus LPS5 corresponding to the modal logic S5.
Justication 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
