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### Page 4021 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews
The phase semantics developed by Girard for linear logic is the basic tool for a unified approach to cut-elimination theorems for various logic systems.  ...  Intuitionistic negation fares much better. Not only do the intuitionistic rules have a natural semantics, that semantics amounts to familiar intuitionistic truth conditions.  ...

### Reasoning about Resource-Sensitive Multi-Agents [chapter]

Norihiro Kamide
2011 Multi-Agent Systems - Modeling, Control, Programming, Simulations and Applications
The difference between such a semantics and the original semantics for the intuitionistic linear logic is only the definition of the valuations: whereas the original semantics for the intuitionistic linear  ...  Thus, to prove the cut-elimination theorem effectively, we need the phase semantic cut-elimination method proposed by Okada. Phase semantics We now define a phase semantics for MILL.  ...  A multi-agent system (MAS) is a system composed of multiple interacting intelligent agents.  ...

### Page 370 of Mathematical Reviews Vol. 53, Issue 2 [page]

1977 Mathematical Reviews
The approach to almost all systems is three-pronged.  ...  The first four sections deal with the positive and the full propo- sitional logic, both intuitionist and classical.  ...

### Kripke semantics and proof systems for combining intuitionistic logic and classical logic

Chuck Liang, Dale Miller
2013 Annals of Pure and Applied Logic
We combine intuitionistic logic and classical logic into a new, first-order logic called Polarized Intuitionistic Logic.  ...  The second proof system is based on a semantic tableau and extends Dragalin's multiple-conclusion version of intuitionistic sequent calculus.  ...  In this submodel, these classical connectives are interpreted almost as if they were intuitionistic ones (thinking of classical implication as A ⊥ ∨ e B).  ...

### On denotational completeness extended abstract

Jean-Yves Girard
1996 Electronical Notes in Theoretical Computer Science
This idea is at work in the original denotational semantics of linear logic, coherent spaces, but also in the phase semantics of linear logic, where the A bilinear form B which induces the duality is nothing  ...  The rather crude phase semantics has the advantage of being complete, and against all predictions, this kind of semantics had some applications.  ...  This result holds of course for classical logic it can be extended to other logical systems : for instance intuitionistic logic is sound and complete w.r.t.  ...

### CONSTRUCTIVE CLASSICAL LOGIC AS CPS-CALCULUS

ICHIRO OGATA
2000 International Journal of Foundations of Computer Science
Constructive classical logic we refer to are LKT and LKQ introduced by Danos et al.(1993).  ...  We establish the Curry-Howard isomorphism between constructive classical logic and CPS-calculus. CPS-calculus exactly means the target language of Continuation Passing Style(CPS) transforms.  ...  Denition B.1 (explicit substitution) Explicit substitution is of the form t hx A := si. In this framework, -contraction divided into two phases.  ...

### Concurrent constraint programming and non-commutative logic [chapter]

Paul Ruet, François Fages
1998 Lecture Notes in Computer Science
This paper presents a connection between the intuitionistic fragment of a non-commutative version of linear logic introduced by the rst author (NLI) and concurrent constraint programming (CC).  ...  characterizations of operational aspects of CC, by providing a logical interpretation of ner observable properties of CC programs, namely stores, successes and suspensions. possible next rule in the proof is  ...  This can be proved as a consequence of the completeness of the phase semantics 28] : the cut rule is sound and the cut-free calculus is complete.  ...

### A Linear Logic Based Approach to Timed Petri Nets [chapter]

Norihiro Kamide
2008 Petri Net, Theory and Applications
phase semantics.  ...  Although in (Kanovich & Ito, 1998) , the phase semantic methods for both classical and intuitionistic cases were intensively investigated, other semantic methods and their applications to concurrency  ...  The main attraction of Petri nets is the way in which the basic aspects of concurrent systems are captured both conceptually and mathematically.  ...

### A Semantic Framework for Proof Evidence

Zakaria Chihani, Dale Miller, Fabien Renaud
2016 Journal of automated reasoning
The FPC framework is described for both classical and intuitionistic logics and for proof structures as diverse as resolution refutations, natural deduction, Frege proofs, and equality proofs.  ...  We propose the foundational proof certificates (FPC) framework for defining the semantics of a broad range of proof evidence.  ...  Acknowledgments This paper is an extension of the conference paper  by the authors. This work has been funded by the ERC Advanced Grant ProofCert.  ...

### A Survey of the Proof-Theoretic Foundations of Logic Programming [article]

Dale Miller
2021 arXiv   pre-print
Researchers have been using this foundation for the past 35 years to elevate logic programming from its roots in first-order classical logic into higher-order versions of intuitionistic and linear logic  ...  Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming.  ...  In a sense, Horn clauses form a setting that is so weak that it cannot distinguish between classical and intuitionistic provability.  ...

### Substructural logics on display

R Gore
1998 Logic Journal of the IGPL
We give algebraic semantics for the Bi-Lambek logics and prove that our calculi are sound and complete with respect to these semantics.  ...  Each of these logics also has a classical counterpart, and some even have a "cyclic" counterpart. These (bi-intuitionistic and bi-classical) extensions of Bi-Lambek logic are not so well understood.  ...  That is, (our formulation of) Grishin's rules convert every intuitionistic substructural logic into its classical counterpart.  ...

### From IF to BI

Samson Abramsky, Jouko Väänänen
2008 Synthese
We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O'Hearn, which combines intuitionistic and multiplicative connectives.  ...  As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that  ...  As a commutative quantale, L(M ) is a model of intuitionistic linear logic (phase semantics (Yetter, 1990; Rosenthal, 1990; Girard, 1987) ). 8 In particular, we have A ⊗ B = ↓{m + n | m ∈ A ∧ n ∈ B}  ...

### From IF to BI: a tale of dependence and separation [article]

Samson Abramsky, Jouko Vaananen
2011 arXiv   pre-print
We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O'Hearn, which combines intuitionistic and multiplicative connectives.  ...  As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that  ...  As a commutative quantale, L(M ) is a model of intuitionistic linear logic (phase semantics (Yetter, 1990; Rosenthal, 1990; Girard, 1987) ). 8 In particular, we have A ⊗ B = ↓{m + n | m ∈ A ∧ n ∈ B}  ...

### Page 800 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews
step resolution phase can be performed by OTTER after a simple translation into classical propositional logic.  ...  The paper presents a new denotational model for untyped lambda- calculus based on game semantics.  ...

### Page 4420 of Mathematical Reviews Vol. , Issue 84k [page]

1984 Mathematical Reviews
A new 34-page chapter on intuitionistic logic has been added to the second edition, covering intuitionistic propositional and predicate logic, Gédel’s translation of classical logic into intuitionistic  ...  logic, the soundness and completeness of Kripke’s semantics for intuitionistic logic, the disjunction and existence properties of intuitionistic logic, and the theory of apartness. 84k:03001 84k:03002  ...
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