Operations on proofs that can be specified by means of modal logic 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 Collections: Computer Technologies and Information Sciences