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Phase semantic cut-elimination and normalization proofs of first- and higher-order linear logic

Mitsuhiro Okada
1999 Theoretical Computer Science  
Then we extend this idea to a phase semantic cut-elimination proof for higher order linear logic.  ...  We give a natural extension of Girard's phase semantic completeness proof of the (ÿrst order) linear logic Girard (Theoret. Comput. Sci., 1987) to a phase semantic cut-elimination proof.  ...  Max Kanovitch and Dr. Andre Scedrov for their careful reading of an earlier version and for their invaluable comments, stimulating discussions during the preparation of this paper.  ... 
doi:10.1016/s0304-3975(99)00058-4 fatcat:raudiqxk6rfutchsap3rcvh2lq

A uniform semantic proof for cut-elimination and completeness of various first and higher order logics

Mitsuhiro Okada
2002 Theoretical Computer Science  
Then we show that this semantic framework allows us to derive a uniform semantic proof of the (ÿrst order and) higher order cut-elimination theorem (as well as a (ÿrst order and) higher order phase-semantic  ...  We present a natural generalization of Girard's (ÿrst order) phase semantics of linear logic (Theoret. Comput. Sci. 50 (1987)) to intuitionistic and higher-order phase semantics.  ...  This paper is the full and revised version of the ÿrst part of Okada [20] (the extended abstract "Phase Semantics for Higher Order Completeness, Cut-Elimination and Normalization Proofs" appeared in  ... 
doi:10.1016/s0304-3975(02)00024-5 fatcat:dwlmhsceknfytalgc5pud5cuum

Page 3585 of Mathematical Reviews Vol. , Issue 2001F [page]

2001 Mathematical Reviews  
Andreja Prijatelj (SV-LJUB; Ljubljana) 2001f:03120 03F52 03F0s Okada, Mitsuhiro (J-KEIO-Q; Minato) Phase semantic cut-elimination and normalization proofs of first- and higher-order linear logic.  ...  This paper is about using phase semantics to prove cut elimination for higher-order linear logic, and the normalizability of some languages of proof structures.  ... 

Contents and abstracts of the electronic notes in theoretical computer science vol. 3

1998 Theoretical Computer Science  
Preface The field of linear logic has developed very rapidly during the last ten years. Linear logic is now one of the most active research areas in Logic and in Theoretical Computer Science.  ...  The Linear Logic 96 Tokyo Meeting provided a forum for an exchange of research results and a discussion of the future directions in the field.  ...  The model takes into account the irreversibility of changes in state, and makes explicit the difference between copying and sharing of entities.  ... 
doi:10.1016/s0304-3975(97)00246-6 fatcat:2skydvbtyvdq3mmxtqwo2hncgi

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  ...  These more expressive logic programming languages allow for capturing stateful computations and rich forms of abstractions, including higher-order programming, modularity, and abstract data types.  ...  Acknowledgments: I thank Miroslaw Truszczynski and several anonymous reviewers for their comments on an earlier version of this paper.  ... 
arXiv:2109.01483v2 fatcat:fp3c746m5zaavd56ltn2d43kp4

Semantic Cut Elimination in the Intuitionistic Sequent Calculus [chapter]

Olivier Hermant
2005 Lecture Notes in Computer Science  
Cut elimination is a central result of the proof theory.  ...  We also give an example of rewrite system for which cut elimination holds but that doesn't enjoys proof normalization.  ...  for intuitionistic Linear Logic (first and higher-order).  ... 
doi:10.1007/11417170_17 fatcat:iszbedmxpjctnnlxslrnyu2txu

Least and Greatest Fixed Points in Linear Logic

David Baelde
2012 ACM Transactions on Computational Logic  
The first-order theory of MALL (multiplicative, additive linear logic) over only equalities is an interesting but weak logic since it cannot capture unbounded (infinite) behavior.  ...  In particular, µMALL = satisfies cut-elimination, which implies consistency, and has a complete focused proof system.  ...  Acknowledgments We thank Alexis Saurin for helpful discussions and the anonymous reviewers of a previous draft of this paper for their comments, which helped us to reorganize this paper.  ... 
doi:10.1145/2071368.2071370 fatcat:zicllwzchvbjpopgi2uf3swaae

Linear logic

Jean-Yves Girard
1987 Theoretical Computer Science  
The familiar connective of negation is broken into two operations: linear negation which is the purely negative part of negation and the modality "of course" which has the meaning of a reaffirmation.  ...  Following this basic discovery, a completely new approach to the whole area between constructive logics and programmation is initiated.  ...  Acknowledgment The author is indebted to Jean-Louis Krivine, whose simultaneous work on lambda-calculus and its execution had some crucial influence on earlier versions of linear logic.  ... 
doi:10.1016/0304-3975(87)90045-4 fatcat:2pz66b65i5fzjaggrqhwyeimki

On structuring proof search for first order linear logic

Paola Bruscoli, Alessio Guglielmi
2006 Theoretical Computer Science  
Full first-order linear logic can be presented as an abstract logic programming language in Miller's system Forum, which yields a sensible operational interpretation in the 'proof search as computation  ...  We further improve on Forum by restricting the class of formulae allowed, in a system we call G-Forum, which is still equivalent to full first-order linear logic.  ...  The two phases of elimination of classical cuts and contractions are best interpreted as bookkeeping phases around the central phase of elimination of linear cuts.  ... 
doi:10.1016/j.tcs.2005.11.047 fatcat:4k52tzyz4vekfnu3iskr7oweha

On Structuring Proof Search for First Order Linear Logic [chapter]

Paola Bruscoli, Alessio Guglielmi
2003 Lecture Notes in Computer Science  
Full first-order linear logic can be presented as an abstract logic programming language in Miller's system Forum, which yields a sensible operational interpretation in the 'proof search as computation  ...  We further improve on Forum by restricting the class of formulae allowed, in a system we call G-Forum, which is still equivalent to full first-order linear logic.  ...  The two phases of elimination of classical cuts and contractions are best interpreted as bookkeeping phases around the central phase of elimination of linear cuts.  ... 
doi:10.1007/978-3-540-39813-4_28 fatcat:xenaqnjqw5emlf2jflyzy7cy4m

Computational Complexity of Deciding Provability in Linear Logic and its Fragments [article]

Florian Chudigiewitsch
2021 arXiv   pre-print
Linear logic has, for example, found applications in proof theory, quantum logic, and the theory of programming languages.  ...  To present these questions and give new perspectives, this thesis consists of three main parts which build on each other: We present the syntax, proof theory, and various approaches to a semantics for  ...  write linear logic, we always mean propositional linear logic. When adding first-order (or higher order) predicates, we speak of first-order (or higher-order) logic explicitly.  ... 
arXiv:2110.00562v1 fatcat:a6sidawnbjh5jpf2cvrermfrcy

Phase semantics for light linear logic

Max I Kanovich, Mitsuhiro Okada, Andre Scedrov
2003 Theoretical Computer Science  
Strong completeness is established, with a purely semantic proof of cut elimination as a consequence.  ...  A number of mathematical examples of ÿbred phase spaces are presented that illustrate subtleties of light linear logic.  ...  It would also be interesting to see if such semantic methods can also establish the stronger version of cut elimination that proof normalization reductions terminate.  ... 
doi:10.1016/s0304-3975(01)00177-3 fatcat:6zzngao3xjg2himig3mnw3lbku

Phase semantics and decidability of elementary affine logic

U DALLAGO
2004 Theoretical Computer Science  
Light, elementary and soft linear logics are formal systems derived from Linear Logic, enjoying remarkable normalization properties.  ...  Phase models for Light A ne Logic and Soft Linear Logic are also deÿned and shown complete.  ...  Acknowledgements Yves Lafont's [14] has been a continuous source of inspiration; we are also happy to thank Yves for the many e-mail exchanges on phase semantics.  ... 
doi:10.1016/s0304-3975(04)00136-7 fatcat:n6msrc4shbfsfgb2nj6kdncgae

Phase semantics and decidability of elementary affine logic

Ugo Dal Lago, Simone Martini
2004 Theoretical Computer Science  
Light, elementary and soft linear logics are formal systems derived from Linear Logic, enjoying remarkable normalization properties.  ...  Phase models for Light A ne Logic and Soft Linear Logic are also deÿned and shown complete.  ...  Acknowledgements Yves Lafont's [14] has been a continuous source of inspiration; we are also happy to thank Yves for the many e-mail exchanges on phase semantics.  ... 
doi:10.1016/j.tcs.2004.02.037 fatcat:lv2qb64ii5hlthe75y5eygovmq

Proof-Theoretic and Higher-Order Extensions of Logic Programming [chapter]

Alberto Momigliano, Mario Ornaghi
2010 Lecture Notes in Computer Science  
We review the Italian contribution to proof-theoretic and higher-order extensions of logic programming; this originated from the realization that Horn clauses lacked standard abstraction mechanisms such  ...  as higher-order programming, scoping constructs and forms of information hiding.  ...  This survey owes to many of Miller's papers, especially "An Overview of Linear Logic Programming" [50] .  ... 
doi:10.1007/978-3-642-14309-0_12 fatcat:qj4tb5daj5a75npxcpagh6mltm
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