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Topic: Sequent calculus

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	Sequent calculus - Wikipedia, the free encyclopedia
		In proof theory and mathematical logic, the sequent calculus is a widely known deduction system for first-order logic (and propositional logic as a special case of it).
		The system LK A (formal) proof in this calculus is a sequence of sequents, where each of the elements is derivable from a number of sequents, that appear earlier in the sequence, by using one of the below rules.
		The Sequent calculus LK has been introduced by Gerhard Gentzen as a tool for studying natural deduction (which has been around before, although not quite as formal).
	en.wikipedia.org /wiki/Sequent_calculus (1562 words)

	Sequent - Wikipedia, the free encyclopedia
		In proof theory, a sequent is a formalized statement of provability that is frequently used when specifying calculi for deduction.
		In a sequent, Γ is called the antecedent and Σ is said to be the succedent of the sequent.
		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.
	en.wikipedia.org /wiki/Sequent (665 words)

	M. E. Szabo: The Collected Works of Gerhard Gentzen (Site not responding. Last check: 2007-11-05)
		After developing and examining the 'natural calculus' Gentzen conjectured that proofs should have a "normal form" in which "all concepts required for the proof would in some sense appear in the conclusion of the proof." This is the subformula property.
		In #8 [1938], Gentzen gives a new proof of consistency of elementary number theory based on his sequent calculus LK (where the one in #4 was based on natural deduction).
		The rank of the derivation is the sum of the left rank and right rank, where the left rank is the longest path which ends with the left-hand upper sequent of the mix and all the sequents on the path contain the mix formula in the succedent.
	www.andrew.cmu.edu /user/cebrown/notes/szabo.html (3057 words)

	Sequent Calculus Primer
		Sequents are expressions of the form Γ - Δ, where Γ and Δ are (possibly empty) sequences of logical formulas.
		This compactness of proofs in sequent calculus is in a sharp contrast with Hilbert-style proofs.
		Sequent calculus formulation G4 introduced by Kleene is perhaps the most known formulation after the Gentzen's.
	sakharov.net /sequent.html (2754 words)

	The Calculus of Structures - FAQ
		No, the calculus of structures uses deep inference, abolishes the distinction between formula and sequent and exploits a symmetry between premise and conclusion in inference rules: these are not features of the sequent calculus and they require new techniques.
		Sequent calculus rules only see the root of formulae, what we call `shallow inference´; rules in the calculus of structures, instead, can rewrite at any position in the formula tree.
		In this case one can trivially translate a system in the sequent calculus into an isomorphic one in the calculus of structures which has an isomorphic proof theory: it would just be a change of notation.
	www.ki.inf.tu-dresden.de /~guglielm/Research/faq.html (1486 words)

	Sequent Calculus
		Sequent calculus is usually viewed as being applied in a backward manner to construct proofs for sequents
		of sequents, by the forward application of the rules of the sequent calculus.
		In that context, the cut rule becomes the main working horse, and its combination with the substitution rule, which is an inverse form of certain quantifier rules, is called resolution.
	www.mpi-sb.mpg.de /SATURATE/doc/Saturate/node15.html (349 words)

	AMCA: A Model Based Cut Elimination Proof by Olivier Hermant
		The sequent calculus modulo, introduced in [1] is a deduction system based on the fact that some axioms can be successfully replaced by rewrite rules on terms an on propositions.
		Sequent calculus is a proof method for propositions that is equivalent to natural deduction.
		The proof of a sequent is made with the help of deduction rules and the cut rule is the only rule where we have a proposition P in premises that doesn't appears in the conclusion.
	at.yorku.ca /c/a/j/y/09.htm (1124 words)

	definitions in sequent calculus (Site not responding. Last check: 2007-11-05)
		A notion of definition has been introduced in sequent calculus in recent years by, for example, Eriksson, Girard, Hallnas, Schroeder-Heister, and Staerk.
		This paper proves that sequent calculus for intuitionistic logic extended with definitions and natural number induction satisfies cut-elimination.
		Thus it is possible, for example, to provide formal, sequent calculus proofs of meta-level statements such as subject-reduction and determinacy of evaluation.
	www.cis.upenn.edu /~bcpierce/types/archives/1997-98/msg00070.html (294 words)

	Sequent calculi
		Gentzen therefore introduced a related system, called the sequent calculus, in which there are rules for introduction of the logical constants on the right of the turnstile, and (instead of elimination rules) rules for introduction of the constants on the left.
		To understand how MacLogic works, it is vital to recognise that the sequent calculus rules can be regarded, upside down, as tactics for decomposing problems: and that a sequent calculus proof can be regarded (at least in the intuitionistic case) as a recipe for constructing a natural deduction proof.
		This view of sequent calculus rules as tactics for use of natural deduction rules works well for the intuitionistic case: we leave it to the interested reader to work out the correspondence for the other logical constants as an exercise.
	www.dcs.st-and.ac.uk /~rd/logic/mac/docs/sec4.3.html (1083 words)

	paper announcement (Site not responding. Last check: 2007-11-05)
		Well-known features of LC are: a new sequent calculus, with a mixed (classical and linear) consequence relation, a translation into linear logic, a denotational semantics and a refinement of Goedel's double negation interpretation into intuitionistic logic.
		Given the sequent calculus LC and its interpretation in the sequent calculus LL for linear logic, it is not too difficult to transfer the formalism of proof-nets with additive and exponential boxes from LL to LC.
		These are the situations where the flow of information changes; in the corresponding inferences of the sequent calculus sequents without privileged occurrence are required.
	www.seas.upenn.edu /~sweirich/types/archive/1994/msg00091.html (544 words)

	Logic and Proof
		A sequent is of the form "Gamma - phi" where Gamma is a set of formulas and phi is a single formula.
		It is fairly straightforward to convert the rules of natural deduction from the book to those of the sequent calculus.
		Express the other ten rules of the book's system of natural deduction in the sequent calculus and use the resulting rules to prove the theorem p->q - ~p v q in the sequent calculus.
	www.mathcs.duq.edu /simon/Fall04/cpma511hw2.html (896 words)

	A Sequent Calculus for a (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		The reasons for this lack of interest in exploring sequent calculus presentations for nonmonotonic logics may certainly be found in the context sensitiveness that is required for nonmonotonic inference, which seems to be in direct opposition to the local character of sequent...
		6 A Sequent Calculus of the Logic of Epistemic Inconsistency - Martins - 1994
		A Sequent Calculus for the Logic of Epistemic Inconsistency - Martins, Pequeno (1994)
	citeseer.ist.psu.edu /605379.html (387 words)

	1. Orientation, Inference, and Implication
		This informal account of a classical sequent calculus naturally is at odds with the so-called paradoxes of logical implication that LK contains.
		A sequent calculus will not count as a genuine calculus of logical inference unless all thinning has to result from an explicit introduction of formulae, i.e.
		applications of thinning, have introduced irrelevant propositions such that by inserting a logical constant, a sequent lacking these formulae is derivable, for instance by introducing conjunction in the antecedent; or a rule of LKA permits all by itself the introduction of a logical constant such that the resulting sequent cannot stand for an inference rule.
	www.hf.uio.no /filosofi/njpl/vol4no1/connexive/node2.html (997 words)

	[No title]
		The sequents are built out of finite sequences (empty included) of formulas, i.e.
		the nodes are sequents such that each sequent on the tree follows from the ones immediately preceding it by one of the rules.
		We picture, and write our proof-trees with the node on the top, and leafs on the very bottom, instead of more common way, where the leafs are on the top and root is on the bottom of the tree.
	www.cs.sunysb.edu /~cse537/Chapter2/RS_System_For_Propositional_Logic2.html (744 words)

	A Sequent Calculus for Circumscription (ResearchIndex)
		In this paper, we introduce a sequent calculus CIRC for propositional Circumscription.
		This work is part of a larger project, aiming at a uniform proof-theoretic reconstruction of the major families of non-monotonic logics.
		Among the novelties of the calculus, we mention that CIRC is analytic and comprises an axiomatic rejection method, which allows for a fully detailed formalization of the nonmonotonic aspects of inference.
	citeseer.ist.psu.edu /411191.html (603 words)

	Logic and Semantics Seminar - 4th June, 2004: Stéphane Lengrand
		As opposed to Gentzen-style sequent calculi, Herbelin's formalism is a permutation-free sequent calculus, which avoids redundency in proof-search, so that cut-free proofs are in one-to-one correspondence with normal forms of natural deduction.
		The formalism is actually closer than natural deduction to the representation of proofs in software like Coq, and the proof-search tactics currently implemented in the latter more closely correspond to the typing rules of sequent calculus than those of natural deduction.
		As sequent calculus is also a neat way of presenting classical logic, we would also like to define a classical version of the above systems, and I will mention some issues about doing so.
	www.cl.cam.ac.uk /Research/LS/Talks/2003_04/2004-06-04.Lengrand.html (259 words)

	Conservative extensions of the lambda-calculus for the computational interpretation of sequent calculus
		This thesis offers a study of the Curry-Howard correspondence for a certain fragment (the canonical fragment) of sequent calculus based on an investigation of the relationship between cut elimination in that fragment and normalisation.
		The output of this study may be summarised in a new assignment Theta, to proofs in the canonical fragment, of terms from certain conservative extensions of the lambda-calculus.
		Next, a comprehensive study of the relationship between normalisation and these calculi of cut-elimination is done, producing several new insight of independent interest, particularly concerning a generalisation of Prawitz's mapping of normal natural deduction proofs into sequent calculus.
	www.lfcs.inf.ed.ac.uk /reports/02/ECS-LFCS-02-430 (372 words)

	Herbrand methods in Linear Logic (revised)
		While I only treat intuitionistic sequent calculus, I believe that the technique is easily adapted for other cut-free sequent calculi that have conventional quantifier rules by identifying the impermutable pairs of inference rules.
		The definition is based on the view that the proof-theoretic role of Herbrand functions (to replace universal quantifiers), and of unification (to find instances corresponding to existential quantifiers), is to ensure that the eigenvariable conditions on a sequent proof are respected.
		The propositional impermutabilities that arise in the intuitionistic but not the classical sequent calculus motivate a generalization of the classical notion of Herbrand functions.
	www.seas.upenn.edu /~sweirich/types/archive/1992/msg00032.html (423 words)

	The Curry/Howard correspondence and some of its extensions to sequent calculus (Abstract) (Site not responding. Last check: 2007-11-05)
		We review natural deduction and sequent calculus for intuitionistic logic, with emphasis on two issues: (i) permutability of inferences in sequent calculus and the permutability theorem establishing that two sequent calculus derivations may be interpreted as the same natural deduction proof iff they are inter-permutable; (ii) the relationship between normalisation and cut-elimination.
		We survey some of the attempts to extend the Curry/Howard correspondence to sequent calculus, with special focus on Herbelin's work.
		We show how to view this system as a sequent calculus with term annotations, where the rule for typing the new form of application, when viewed as an inference rule, encompasses both a form of left introduction and a class of cuts.
	www.math.uminho.pt /~dil2004/jeslp-abstract-html (300 words)

	Citebase - A classical sequent calculus free of structural rules
		A classical sequent calculus free of structural rules
		This paper presents a classical sequent calculus which is also free of contraction and weakening, but more symmetrically: both contraction and weakening are absorbed into conjunction, leaving the axiom rule untouched.
		We prove a minimality theorem for hybrid logic: any sequent calculus (within a standard class of right-sided calculi) is complete iff it contains hybrid logic.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0506463 (335 words)

	Citebase - A Sequent Calculus and a Theorem Prover for Standard Conditional Logics
		A Sequent Calculus and a Theorem Prover for Standard Conditional Logics
		In this paper we present a cut-free sequent calculus, called SeqS, for some standard conditional logics, namely CK, CK+ID, CK+MP and CK+MP+ID. The calculus uses labels and transition formulas and can be used to prove decidability and space complexity bounds for the respective logics.
		A Sequent Calculus and a Theorem Prover for Standard Conditional Logics 45 Pozzato, G. Deduzione automatica per logiche condizionali:Analisi e sviluppo di un theorem prover.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:cs/0407064 (867 words)

	CPSC 629: Deductive Systems (Site not responding. Last check: 2007-11-05)
		In this lecture we present the Curry-Howard algorithm as a relation between the simply typed lambda calculus and the natural deduction calculus.
		First, we show that propositions are represented as types, proofs as terms, local reductions as beta, cut elimination as normalization etc. Second, we define a term language for natural deduction: lambda calculus plus first-class continuations (for negation), i.e.
		And forth, we give a translation from the former to the latter and its correspondence to Thm 3.10 (completeness of sequent calculus w/ cut) of Pfenning's notes on sequent calculus.
	www.cs.yale.edu /homes/carsten/classes/f00/16.html (111 words)

	Sequent Calculus and the Specification of Computation
		The sequent calculus has been used to for many purposes in recent years within theoretical computer science.
		No advanced knowledge of the sequent calculus or of linear logic will be assumed, although some familiarity with their elementary syntactic properties will be useful.
		After providing an overview of sequent calculus principles, we shall develop the notion of goal directed search for a variety of logics, starting with the intuitionistic logic theory of Horn clauses and hereditary Harrop formulas.
	www.disi.unige.it /person/PalamidessiC/dale.html (747 words)

	The Calculus of Structures
		The calculus of structures is a new proof theoretical formalism, introduced by myself in 1999 and mostly developed by members of our group in Dresden since 2000.
		We can present deductive systems in the calculus of structures and analyse their properties, as we do in the sequent calculus, natural deduction and proof nets.
		The systems so obtained cannot be presented in the sequent calculus, but they enjoy the usual properties of locality, decomposition and cut elimination available in the calculus of structures.
	www.ki.inf.tu-dresden.de /~guglielm/Research/research.html (2085 words)

	Journal of Logic and Computation, Volume 11, Issue 5, pp. pp. 671-689.: Abstract. (Site not responding. Last check: 2007-11-05)
		A sequent calculus for hybrid logics is developed from a calculus for classical predicate logic by a series of transformations.
		We formalize the semantic theory of hybrid logic using a sequent calculus for predicate logic plus axioms.
		The unattractive features are removed one-by-one, until the final vestiges of the metalanguage can be set aside to reveal a fully internalized calculus.
	www.mi.sanu.ac.yu /~uros.m/logcom/hdb/Volume_11/Issue_05/110671.sgm.abs.html (141 words)

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