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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/01/15 17:22:23」(JST)

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For other uses, see Sequent (disambiguation).

In proof theory, a sequent is a formalized statement of provability that is frequently used when specifying calculi for deduction. In the sequent calculus, the name sequent is used for the construct which can be regarded as a specific kind of judgment, characteristic to this deduction system.

Explanation[edit]

A sequent has the form

where both Γ and Σ are sequences of logical formulae (i.e., both the number and the order of the occurring formulae matter). The symbol is usually referred to as turnstile or tee and is often read, suggestively, as "yields" or "proves". It is not a symbol in the language, rather it is a symbol in the metalanguage used to discuss proofs. In a sequent, Γ is called the antecedent and Σ is said to be the succedent of the sequent.

Intuitive meaning[edit]

The intuitive meaning of the sequent is that under the assumption of Γ the conclusion of Σ is provable. Classically, the formulae on the left of the turnstile can be interpreted conjunctively while the formulae on the right can be considered as a disjunction. This means that, when all formulae in Γ hold, then at least one formula in Σ also has to be true. If the succedent is empty, this is interpreted as falsity, i.e. means that Γ proves falsity and is thus inconsistent. On the other hand an empty antecedent is assumed to be true, i.e., means that Σ follows without any assumptions, i.e., it is always true (as a disjunction). A sequent of this form, with Γ empty, is known as a logical assertion.

Of course, other intuitive explanations are possible, which are classically equivalent. For example, can be read as asserting that it cannot be the case that every formula in Γ is true and every formula in Σ is false (this is related to the double-negation interpretations of classical intuitionistic logic, such as Glivenko's theorem).

In any case, these intuitive readings are only pedagogical. Since formal proofs in proof theory are purely syntactic, the meaning of (the derivation of) a sequent is only given by the properties of the calculus that provides the actual rules of inference.

Barring any contradictions in the technically precise definition above we can describe sequents in their introductory logical form. represents a set of assumptions that we begin our logical process with, for example "Socrates is a man" and "All men are mortal". The represents a logical conclusion that follows under these premises. For example "Socrates is mortal" follows from a reasonable formalization of the above points and we could expect to see it on the side of the turnstile. In this sense, means the process of reasoning, or "therefore" in English.

Example[edit]

A typical sequent might be:

This claims that either or can be derived from and .

Property[edit]

Since every formula in the antecedent (the left side) must be true to conclude the truth of at least one formula in the succedent (the right side), adding formulas to either side results in a weaker sequent, while removing them from either side gives a stronger one.

Rules[edit]

Most proof systems provide ways to deduce one sequent from another. These inference rules are written with a list of sequents above and below a line. This rule indicates that if everything above the line is true, so is everything under the line.

A typical rule is:

This indicates that if we can deduce that yields , and that yields , then we can also deduce that yields .

Variations[edit]

The general notion of sequent introduced here can be specialized in various ways. A sequent is said to be an intuitionistic sequent if there is at most one formula in the succedent (although multi-succedent calculi for intuitionistic logic is also possible). Similarly, one can obtain calculi for dual-intuitionistic logic (a type of paraconsistent logic) by requiring that sequents be singular in the antecedent.

In many cases, sequents are also assumed to consist of multisets or sets instead of sequences. Thus one disregards the order or even the number of occurrences of the formulae. For classical propositional logic this does not yield a problem, since the conclusions that one can draw from a collection of premises does not depend on these data. In substructural logic, however, this may become quite important.

History[edit]

Historically, sequents have been introduced by Gerhard Gentzen in order to specify his famous sequent calculus. In his German publication he used the word "Sequenz". However, in English, the word "sequence" is already used as a translation to the German "Folge" and appears quite frequently in mathematics. The term "sequent" then has been created in search for an alternative translation of the German expression.

This article incorporates material from Sequent on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

External links[edit]

Hazewinkel, Michiel, ed. (2001), "Sequent (in logic)", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

English Journal

CuO-ZnO Micro/Nanoporous Array-Film-Based Chemosensors: New Sensing Properties to H2 S.

Xu Z1, Duan G, Li Y, Liu G, Zhang H, Dai Z, Cai W.Author information 1Key Lab of Materials Physics Anhui Key Lab of Nanomaterials and Nanotechnology, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031 (P.R. China), Fax: (+86) 551-65591434.AbstractCuO-ZnO micro/nanoporous array-films are synthesized by transferring a solution-dipped self-organized colloidal template onto a device substrate and sequent heat treatment. Their morphologies and structures are characterized by X-ray diffraction, field-emission scanning electron microscopy, transmission electron microscopy, and X-ray photoelectron spectrum analysis. Based on the sensing measurement, it is found that the CuO-ZnO films prepared with the composition of [Cu2+ ]/[Zn2+ ]=0.005, 0.01, and 0.05 all show a nice sensitivity to 10 ppm H2 S. Interestingly, three different zones exist in the patterns of gas responses versus H2 S concentrations: a platform zone, a rapidly increasing zone, and a slowly increasing zone. Further experiments show that the hybrid CuO-ZnO porous film sensor exhibits shorter recovery time and better selectivity to H2 S gas against other interfering gases at a concentration of 10 ppm. These new sensing properties may be due to a depletion layer induced by p-n junction between p-type CuO and n-type ZnO and high chemical activity of CuO to H2 S. This work will provide a new construction route of ZnO-based sensing materials, which can be used as H2 S sensors with high performances.
Chemistry (Weinheim an der Bergstrasse, Germany).Chemistry.2014 Apr 7. doi: 10.1002/chem.201304722. [Epub ahead of print]
CuO-ZnO micro/nanoporous array-films are synthesized by transferring a solution-dipped self-organized colloidal template onto a device substrate and sequent heat treatment. Their morphologies and structures are characterized by X-ray diffraction, field-emission scanning electron microscopy, transmis
PMID 24711055

A randomized pilot study of stochastic vibration therapy in spinocerebellar ataxia.

Kaut O1, Jacobi H, Coch C, Prochnicki A, Minnerop M, Klockgether T, Wüllner U.Author information 1Department of Neurology, University of Bonn, Sigmund-Freud-Strasse 25, 53105, Bonn, Germany.AbstractWhole body vibration (WBV) is a biomechanical treatment used widely in professional sports and rehabilitation. We examined the effect of stochastic WBV on ataxia in spinocerebellar ataxia types 1, 2, 3, and 6 (SCA 1, 2, 3 and 6) in a single-center double-blind sham-controlled study. Stochastic WBV was applied on four sequent days, each treatment consisting of five stimulus trains of 60-s duration at a frequency of 6.5 Hz and 60-s resting time between stimuli (n = 17). Patients allocated to the sham group received the same treatment with 1 Hz (n = 15). All patients were rated at baseline and after the last treatment using clinical scores (SARA, SCAFI, and INAS). After treatment, we found significant improvements of gait, posture, and speed of speech in the verum group while limb kinetics and ataxia of speech did not respond. Stochastic WBV might act on proprioceptive mechanisms and could also stimulate non-cerebellar/compensatory mechanisms. But at present, the involved cellular mechanism and the presumed neuronal loops cannot be deciphered. Thus, future work is needed to understand the mechanisms of whole body vibration. Finally, the use of stochastic WBV could provide a supplementation to treat ataxia in SCA and can be combined with physiotherapeutical motor training.
Cerebellum (London, England).Cerebellum.2014 Apr;13(2):237-42. doi: 10.1007/s12311-013-0532-5.
Whole body vibration (WBV) is a biomechanical treatment used widely in professional sports and rehabilitation. We examined the effect of stochastic WBV on ataxia in spinocerebellar ataxia types 1, 2, 3, and 6 (SCA 1, 2, 3 and 6) in a single-center double-blind sham-controlled study. Stochastic WBV w
PMID 24197754

Prospective 'real world' registry for the use of the 'PCB only' strategy in small vessel de novo lesions.

Zeymer U1, Waliszewski M, Spiecker M, Gastmann O, Faurie B, Ferrari M, Alidoosti M, Palmieri C, Heang TN, Ong PJ, Dietz U.Author information 1Klinikum Ludwigshafen, , Ludwigshafen, Germany.AbstractBACKGROUND: This prospective registry assessed the safety and efficacy of paclitaxel coated balloon (PCB) angioplasty for small vessel coronary artery disease in Europe and Asia with the intention to treat lesions without additional stenting. The use of PCBs in small vessels seems to be associated with favourable outcomes; however, prospective data for the use of PCBs without stenting are limited.
Heart (British Cardiac Society).Heart.2014 Feb;100(4):311-6. doi: 10.1136/heartjnl-2013-304881. Epub 2013 Nov 26.
BACKGROUND: This prospective registry assessed the safety and efficacy of paclitaxel coated balloon (PCB) angioplasty for small vessel coronary artery disease in Europe and Asia with the intention to treat lesions without additional stenting. The use of PCBs in small vessels seems to be associated w
PMID 24281754

Japanese Journal

背理法を用いない効率的な証明のシーケントによる計画法

中西泰雄
東京都立産業技術高等専門学校研究紀要 (5), 88-98, 2011-03
NAID 40018890507

Novel Multiple-Valued Logic Design Using BiCMOS-Based Negative Differential Resistance Circuit Biased by Two Current Sources

GAN Kwang-Jow,LIANG Dong-Shong,CHEN Yan-Wun
IEICE transactions on information and systems 93(8), 2068-2072, 2010-08-01
… In measuring, we can obtain eight logic states at the output according to a sequent control of two current sources on and off in order. …
NAID 10027364509

Cut elimination and strong separation for substructural logics: An algebraic approach

Galatos Nikolaos,Ono Hiroakira
Annals of Pure and Applied Logic 161(9), 1097-1133, 2010-04-17
… We present a Gentzen-style sequent system GL that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of FL. …
NAID 120002511606

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