A Deep Inference System for the Modal Logic S5

Stouppa, Phiniki

doi:10.1007/s11225-007-9028-y

Cited by 33 publications

(28 citation statements)

References 16 publications

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“…Several logics which were lacking analytic Gentzen proof systems have been shown to enjoy very simple analytic proof systems in deep inference. This is especially true of modal logics [SS05,Sto07,Brü06], but also of Yetter's non-commutative logic [DG04], and there is work in progress for several intermediate logics.…”

Section: The Calculus Of Structures and Deep Inferencementioning

confidence: 99%

A system of interaction and structure V: the exponentials and splitting

Guglielmi¹,

Straβburger²

2011

Math. Struct. Comp. Sci.

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System NEL is the mixed commutative/non-commutative linear logic BV augmented with linear logic's exponentials, or, equivalently, it is MELL augmented with the non-commutative self-dual connective seq. System NEL is Turingcomplete, it is able to directly express process algebra sequential composition and it faithfully models causal quantum evolution. In this paper, we show cut elimination for NEL, based on a property that we call splitting. NEL is presented in the calculus of structures, which is a deep-inference formalism, because no Gentzen formalism can express it analytically. The splitting theorem shows how and to what extent we can recover a sequent-like structure in NEL proofs. Together with the decomposition theorem, proved in the previous paper of the series, this immediately leads to a cut-elimination theorem for NEL.

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Section: The Calculus Of Structures and Deep Inferencementioning

confidence: 99%

A system of interaction and structure V: the exponentials and splitting

Guglielmi¹,

Straβburger²

2011

Math. Struct. Comp. Sci.

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“…The most prominent example of a formalism employing deep inference is the calculus of structures. It has successfully been employed to give new presentations for many logics, including classical logic [BT01,Brü03b], minimal logic [Brü03c], intuitionistic logic [Tiu06], modal logics [SS05,Sto07,Brü06a,BS09], linear logic [Str03,Str02], and various non-commutative logics [DG04,Gug07,GS02].…”

Section: Deep Inference For Classical Logicmentioning

confidence: 99%

From Deep Inference to Proof Nets via Cut Elimination

Straßburger

2009

Journal of Logic and Computation

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This paper shows how derivations in the deep inference system SKS for classical propositional logic can be translated into proof nets. Since an SKS derivation contains more information about a proof than the corresponding proof net, we observe a loss of information which can be understood as "eliminating bureaucracy". Technically this is achieved by cut reduction on proof nets. As an intermediate step between the two extremes, SKS derivations and proof nets, we will see proof graphs representing derivations in "Formalism A".

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“…It is a methodology according to which several formalisms can be defined with excellent structural properties. The calculus of structures [Gug07] is one of them and is now well developed for classical [Brü03,Brü06a,Brü06d,BT01,Brü06b], intuitionistic [Tiu06a], linear [Str02,Str03b], modal [Brü06c,GT07,Sto07] and commutative/non-commutative logics [Gug07, Tiu06b, Str03a, Bru02, DG04, GS01, GS02, GS07, Kah06, Kah07b]. The basic proof complexity properties of the calculus of structures are known [BG08].…”

Section: Background On Deep Inferencementioning

confidence: 99%

Normalisation Control in Deep Inference via Atomic Flows

Guglielmi¹,

Gundersen²

2008

Logical Methods in Computer Science

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Abstract. We introduce 'atomic flows': they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a new and very general normalisation theorem, which contains cut elimination as a special case. We operate in deep inference, which is more general than other syntactic paradigms, and where normalisation is more difficult to control. We argue that atomic flows are a significant technical advance for normalisation theory, because 1) the technique they support is largely independent of syntax; 2) indeed, it is largely independent of logical inference rules; 3) they constitute a powerful geometric formalism, which is more intuitive than syntax.

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A Deep Inference System for the Modal Logic S5

Cited by 33 publications

References 16 publications

A system of interaction and structure V: the exponentials and splitting

A system of interaction and structure V: the exponentials and splitting

From Deep Inference to Proof Nets via Cut Elimination

Normalisation Control in Deep Inference via Atomic Flows

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