 
Summary: Operations on proofs that can be specified
by means of modal logic
Sergei N. Artemov \Lambda
Abstract
Explicit modal logic was first sketched by GĻodel in [16] as the logic with the atoms
``t is a proof of F''. The complete axiomatization of the Logic of Proofs LP was found in
[4] (see also [6],[7],[18]). In this paper we establish a sort of a functional completeness
property of proof polynomials which constitute the system of proof terms in LP. Proof
polynomials are built from variables and constants by three operations on proofs: ``\Delta''
(application), ``!'' (proof checker), and ``+'' (choice). Here constants stand for canonical
proofs of ``simple facts'', namely instances of propositional axioms and axioms of LP in a
given proof system. We show that every operation on proofs that (i) can be specified in
a propositional modal language and (ii) is invariant with respect to the choice of a proof
system is realized by a proof polynomial.
Introduction
The intended meaning of the intuitionistic logic was informally explained first in terms of
operations on proofs due to Brouwer, Heyting and Kolmogorov (cf. [43],[44],[12]). This
interpretation is widely known as the BHK semantics of intuitionistic logic. However, despite
some similarities in the informal description of the functions assigned to the intuitionistic
connective, the Heyting semantics and the Kolmogorov semantics have fundamentally different
