Answers to two old questions are given in this paper. 1. Modal logic S4, which was informally specified by Gödel in 1933 as a logic for provability, meets its exact provability interpretation. 2. Brouwer-Heyting-Kolmogorov realizing operations (1931-32) for intuitionistic logic Int also get exact interpretation as corresponding propositional operations on proofs; both S4 and Int turn out to be complete with respect to this proof realization. These results are based on operational reading of S4, where a modality is split into three operations. The logic of proofs with these operations is shown to be arithmetically complete with respect to the intended provability semantics and sufficient to realize every operation on proofs admitting propositional specification in arithmetic.

The true belief components of Plato's tripartite definition of knowledge as justified true belief are represented in formal epistemology by modal logic and its possible worlds semantics. At the same time, the justification component of Plato's definition did not have a formal representation. This

Classical logic has so far been the logic of choice in formal hardware verification. This paper proposes the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes. The model-theoretic properties are exploited to handle the second-order nature of bounded delays in a purely propositional setting without need to introduce explicit time and temporal operators. The proof theoretic properties are exploited to extract quantitative timing information and to reintroduce explicit time in a convenient and systematic way. We present a natural Kripke-style semantics for intuitionistic propositional logic, as a special case of a Kripke constraint model for Propositional Lax Logic [15], in which validity is validity up to stabilisation, and implication oe comes out as "boundedly gives rise to." We show that this semantics is equivalently characterised by a notion of realisability with stabilisation bounds as realisers...

Research supported by NSF grants DMS-0600823 and DMS-0652637.

Explicit modal logic was first sketched by Gödel 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.

In 1933 Godel introduced a modal logic of provability (S4) and left open the problem of a formal provability semantics for this logic. Since then numerous attempts have been made to give an adequate provability semantics to Godel's provability logic with only partial success. In this paper we give the complete solution to this problem in the Logic of Proofs (LP). LP implements Godel's suggestion (1938) of replacing formulas "F is provable" by the propositions for explicit proofs "t is a proof of F" (t : F ). LP admits the reflection of explicit proofs t : F ! F thus circumventing restrictions imposed on the provability operator by Godel's second incompleteness theorem. LP formalizes the Kolmogorov calculus of problems and proves the Kolmogorov conjecture that intuitionistic logic coincides with the classical calculus of problems.

Note to the Reader. I see six constituencies in CSE: computer, mathematical, and physical scientists; engineers; and technical and non-technical managers. I have adopted a conversational tone to make this article as widely accessible as possible. Ihave also provided a Bibliography.

. The goal of foundational thinking in computer science is to understand the methods and practices of working programmers; we might even be able to improve upon those practices. The investigation outlined here applies the methods of constructive mathematics 'a l`a A. N. Kolmogoroff, L. E. J. Brouwer and Errett Bishop to contemporary computer science. The major approach is to use Kolmogoroff's interpretation of the predicate calculus. This investigation includes an attempt to merge contemporary thoughts on computability and computing semantics with the language of mental constructions proposed by Brouwer. This necessarily forces us to ask about the psychology of language. I present a definition of algorithms that links language, constructive mathematics, and logic. Using the concept of an abstract family of algorithms (Hennie) and principles of constructivity, a definition of problem solving. The constructive requirements for an algorithm are developed and presented. Given this framewor...

The language of the basic logic of proofs extends the usual propositional language by forming sentences of the sort x is a proof of F for any sentence F. In this paper a complete axiomatization for the basic logic of proofs in Heyting Arithmetic HA was found. 1 Introduction. The classical logic of proofs LP inspired by the works by Kolmogorov [24] and Gödel [16, 17] was found in [3, 4] (see also surveys [6, 8, 12]). LP is a natural extension of the classical propositional logic in a language of proof-carrying formulas. LP axiomatizes all valid logical principles concerning propositions and proofs with a fixed sufficiently

An issue of an epistemic logic with justification has been discussed since the early 1990s. Such a logic, along with the usual knowledge operator ✷F (F is known), should contain assertions t:F (t is a justification for F), which gives a more nuanced and realistic model of knowledge. In this paper, we build two systems of epistemic logic with justification: the minimal one—S4LP—which is an extension of the basic epistemic logic S4 by an appropriate calculus of justification corresponding to the logic of proofs LP, and S4LPN—which is S4LP augmented by the explicit negative introspection principle ¬(t:F) → ✷¬(t:F). Epistemic semantics for both systems are suggested. Completeness and specific properties of S4LP and S4LPN, reflecting the explicit character of those systems, are established. 1