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29
Operational Modal Logic
, 1995
"... 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. BrouwerHeytingKolmogorov realizing operations (193132) for intuitionistic logic Int also get exact in ..."
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Cited by 79 (28 self)
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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. BrouwerHeytingKolmogorov realizing operations (193132) 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.
FirstOrder Logic of Proofs
, 2011
"... The propositional logic of proofs LP revealed an explicit provability reading of modal logic S4 which provided an indented provability semantics for the propositional intuitionistic logic IPC and led to a new area, Justification Logic. In this paper, we find the firstorder logic of proofs FOLP capa ..."
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Cited by 27 (11 self)
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The propositional logic of proofs LP revealed an explicit provability reading of modal logic S4 which provided an indented provability semantics for the propositional intuitionistic logic IPC and led to a new area, Justification Logic. In this paper, we find the firstorder logic of proofs FOLP capable of realizing firstorder modal logic S4 and, therefore, the firstorder intuitionistic logic HPC. FOLP enjoys a natural provability interpretation; this provides a semantics of explicit proofs for firstorder S4 and HPC compliant with BrouwerHeytingKolmogorov requirements. FOLP opens the door to a general theory of firstorder justification.
On Epistemic Logic with Justification
 NATIONAL UNIVERSITY OF SINGAPORE
, 2005
"... 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 representat ..."
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Cited by 26 (9 self)
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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
Timing Analysis of Combinational Circuits in Intuitionistic Propositional Logic
 Formal Methods in System Design
, 1999
"... 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 modeltheoretic properties are exploited to handle the s ..."
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Cited by 7 (1 self)
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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 modeltheoretic properties are exploited to handle the secondorder 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 Kripkestyle 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...
Mass problems and measuretheoretic regularity
, 2009
"... Research supported by NSF grants DMS0600823 and DMS0652637. ..."
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Cited by 6 (3 self)
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Research supported by NSF grants DMS0600823 and DMS0652637.
Mass problems and intuitionism
, 2007
"... Let Pw be the lattice of Muchnik degrees of nonempty Π 0 1 subsets of 2 ω. The lattice Pw has been studied extensively in previous publications. In this note we prove that the lattice Pw is not Brouwerian. 1 ..."
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Cited by 4 (3 self)
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Let Pw be the lattice of Muchnik degrees of nonempty Π 0 1 subsets of 2 ω. The lattice Pw has been studied extensively in previous publications. In this note we prove that the lattice Pw is not Brouwerian. 1
The Basic Intuitionistic Logic of Proofs
, 2005
"... 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 p ..."
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Cited by 2 (0 self)
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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 proofcarrying formulas. LP axiomatizes all valid logical principles concerning propositions and proofs with a fixed sufficiently
Operations on Proofs That Can Be Specified By Means of Modal Logic
"... 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 polynom ..."
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Cited by 2 (2 self)
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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.
A critical look at design, verification, and validation of large scale simulations
"... Note to the Reader. I see six constituencies in CSE: computer, mathematical, and physical scientists; engineers; and technical and nontechnical managers. I have adopted a conversational tone to make this article as widely accessible as possible. Ihave also provided a Bibliography. ..."
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Cited by 1 (0 self)
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Note to the Reader. I see six constituencies in CSE: computer, mathematical, and physical scientists; engineers; and technical and nontechnical managers. I have adopted a conversational tone to make this article as widely accessible as possible. Ihave also provided a Bibliography.
Proof Realization of Intuitionistic and Modal Logics
, 1996
"... Logic of Proofs (LP) has been introduced in [2] as a collection of all valid formulas in the propositional language with labeled logical connectives [[t]](\Delta) where t is a proof term with the intended reading of [[t]]F as "t is a proof of F". LP is supplied with a natural axiom system, ..."
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Cited by 1 (1 self)
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Logic of Proofs (LP) has been introduced in [2] as a collection of all valid formulas in the propositional language with labeled logical connectives [[t]](\Delta) where t is a proof term with the intended reading of [[t]]F as "t is a proof of F". LP is supplied with a natural axiom system, completeness and decidability theorems. LP may express some constructions of logic which have been formulated or/and interpreted in an informal metalanguage involving the notion of proof, e.g. the intuitionistic logic and its BrauwerHeytingKolmogorov semantics, classical modal logic S4, etc (cf. [2]). In the current paper we demonstrate how the intuitionistic propositional logic Int can be directily realized into the Logic of Proofs. It is shown, that the proof realizability gives a fair semantics for Int.