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	Proof theory - Wikipedia, the free encyclopedia
		Proof theory can also be considered a branch of philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.
		Structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof.
		Structural proof theory is connected to type theory by means of the Curry-Howard correspondence, which observes a structural analogy between the process of normalisation in the natural deduction calculus and beta reduction in the typed lambda calculus.
	en.wikipedia.org /wiki/Proof_theory (1014 words)

	More on Proof Theory
		Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques.
		The subject of proof theory has a significant if somewhat opaque prehistory as metamathematics, the proposed theory under development since the start of the twentieth century, which was, for a generation, under the influence of David Hilbert.
		proof of theories for the extinction of dinosaurs
	www.artilifes.com /proof-theory.htm (1247 words)

	Structural proof theory - Wikipedia, the free encyclopedia
		In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof.
		The notion of analytic proof was introduced by Gerhard Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free.
		His natural deduction calculus also supports a notion of analytic proof, as was shown by Dag Prawitz; the definition is slightly more complex — we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting.
	en.wikipedia.org /wiki/Structural_proof_theory (330 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		This book is, as stated on the cover, "an introduction to the basic ideas of structural proof theory,..., with cut elimination and normalization as central tools".
		Structural proof theory, as conceived here, is the study of sequent calculi (i.e.
		Chapter 9 is an introduction to the proof theory of modal and linear logics, a welcome supplement (in the first case) to the extensive material elsewhere on their model theory.
	www.dcs.st-and.ac.uk /~rd/publications/BPT-review.txt (775 words)

	Purity through Unravelling -- Abtract (Site not responding. Last check: 2007-10-14)
		We divide attempts to give the structural proof theory of modal logics into two kinds, those pure formulations whose inference rules characterise modality completely by means of manipulations of boxes and diamonds, and those labelled formulations that leverage the use of labels in giving inference rules.
		The widespread adoption of labelled formulations is driven by their ability to model features of the model theory of modal logic in its proof theory.
		We describe here an approach to the structural proof theory of modal logic that aims to bring under one roof the benefits of both the pure and the labelled formulations.
	www.wv.inf.tu-dresden.de /~hein/abstract-sd05.html (152 words)

	Proof Theory and Programming
		A consequence of this approach is that we consider different formal systems from type theory and intuitionistic, classical or linear logics, that allow to represent and to analyze concepts, at a logical level, such as communication, concurrency, sequentiality, control or verification of systems properties.
		A main point, that is the interest and the difficulty of the approach, is the strong interaction between the works on the semantics and structure of proofs, on the algorithmic content of proofs and on the algorithms for efficiency proof search.
		The results will be applied and specialized in domains as the design of process calculi based on proof theory of linear logic, the automated analysis of sentences in natural language, the diagnosis of actions and the synchronization of activities in a network.
	www.ercim.org /publication/Ercim_News/enw23/galmiche.html (740 words)

	Proof Theory on the eve of Year 2000 (Site not responding. Last check: 2007-10-14)
		Add to this that it is closely connected with the proof theory of feasible arithmetic, and it seems clear to me that it is a classic problem of proof theory, though one that was quite unconsidered by the early pioneers.
		By the former, I mean proof theory as the formal study of mathematical practice, with an eye to illuminating aspects of that practice; by the latter, I mean the more general mathematical study of deductive systems and their properties.
		Proof theorists, having failed in analysing proofs in mathematics, went on to apply their skills (somewhat opportunistically in my mind) in logical systems different from the two canonical ones, intuitionistic and classical.
	www-logic.stanford.edu /proofsurvey.html (19489 words)

	Proof theory (via CobWeb/3.1 planetlab1.isi.jhu.edu) (Site not responding. Last check: 2007-10-14)
		Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Peano, Russell and Dedekind, conventionally the story of modern Proof theory is seen as being established by David Hilbert, who initiated what is called Hilbert's program in the Foundations of mathematics.
		Main article: Consistency proof As we have discussed, the spur for the mathematical investigation of proofs in formal theories was Hilbert's program.
		Main article: Structural Proof theory Structural Proof theory is the subdiscipline of Proof theory that studies Proof calculi that support a notion of analytic proof.
	proof-theory.iqnaut.net.cob-web.org:8888 (1095 words)

	Summer School and Workshop on Proof Theory, Computation and Complexity
		Friedman's proof is one of the first applications of the idea of "logical relation", later extended and developed by Plotkin and Statman.
		Proof theory has been developed for formalisms that adopt shallow inference, like the sequent calculus, natural deduction and proof nets.
		Structural proof analysis in natural deduction, in contrast to sequent calculus, soon leads to very complicated considerations.
	www.ki.inf.tu-dresden.de /~guglielm/WPT2 (1485 words)

	Lutz Strassburger (Site not responding. Last check: 2007-10-14)
		proof theory, which is a very vivid and active field.
		The calculus of structures is a proof formalism that pursues this idea further, but in an orthogonal way: it leaves the connectives alone and chops the classical proof rules into pieces.
		This might be the place where structural proof theory (which is mostly concerned with propositional logics) meets the other areas of proof theory: constructivism, logical complexity, and ordinal analysis.
	ps.uni-sb.de /~lutz/diploma-topics.html (1068 words)

	Structural Proof Theory - Cambridge University Press
		Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs.
		This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists.
		Back to natural deduction; Conclusion: diversity and unity in structural proof theory; Appendix A. Simple type theory and categorical grammar; Appendix B. Proof theory and constructive type theory; Appendix C. A proof editor for sequent calculus.
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521793076 (231 words)

	Proof Theory Forum (Site not responding. Last check: 2007-10-14)
		Links: A course on structural proof theory given by Roy Dyckhoff at the University of Dresden.
		Papers on proof theory by Sara Negri and Jan von Plato.
		Text of the tutorial (in pdf format) "Five Lectures on Proof Analysis" given by Sara Negri at the Summer School on Proof Theory, Computation and Complexity, Dresden 2003.
	www.helsinki.fi /~negri/ptforum.html (214 words)

	DI & CoS - Modal Logic
		This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a Hilbert-Lewis style and conditions on the accessibility relation on frames.
		By contrast, the usual structural proof theory of modal logic, as given in Gentzen systems, is ad-hoc.
		Alex Simpson, in his 1993 PhD thesis, introduced a labelled proof theory for modal logic that that allows cut-elimination for a class of modal logics, which is characterised by so called geometric theories.
	alessio.guglielmi.name /res/cos/ML (1073 words)

	A systematic proof theory for several modal logics (Site not responding. Last check: 2007-10-14)
		The family of normal propositional modal logic systems are given a highly systematic organisation by their model theory.
		This model theory is generally given using Kripkean frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a Hilbert style and conditions on the accessibility relation on frames.
		By contrast, the usual structural proof theory of modal logic, as given in Gentzen calculi, is ad-hoc.
	www.linearity.org /cas/papers/sysptf.html (259 words)

	Amazon.com: Basic Proof Theory (Cambridge Tracts in Theoretical Computer Science): Books: A. S. Troelstra,H. ... (Site not responding. Last check: 2007-10-14)
		Proof theory may be roughly divided into two parts: structural proof theory and interpretational proof theory.
		Considerations about lengths of proofs are undeniably important when the proofs in question are infinitely long; yet students of the subject should be allowed to see that the considerations that apply here are just generalizations of the same considerations as they apply to finitely long proofs.
		People doing research in proof theory might also welcome the fact that the authors discuss quite a wide variety of logical systems, thus giving the reader a chance to weigh up the merits and disadvantages of each.
	www.amazon.com /Cambridge-Tracts-Theoretical-Computer-Science/dp/0521779111 (1640 words)

	Deep Inference and the Calculus of Structures
		Proof nets are not deductive systems, because they cannot be checked by local inspection, but they play a crucial role in understanding the semantics of proofs, which is one of the most active research areas in proof theory, in close connection with theoretical computer science.
		The calculus of structures is a milestone in the development of deep inference, because of its simplicity and its resemblance to traditional formalisms.
		Non-commutativity and MELL in the Calculus of Structures
	alessio.guglielmi.name /res/cos/index.html (5328 words)

	Just What is it that Makes Martin-Lof's Type Theory so Different, so Appealing? \| Lambda the Ultimate
		Constructive proofs are important for several reasons: they are usually clearer than pulling a rabbit out of a hat, they state a "cause" for truth, they form programs, and they are required in some instances (as for example Montague semantics).
		A notion of proof equivalence is important because it tells you, if someone claims to have found a "new" proof, whether it is genuinely new or just a different perspective on an old proof.
		Since you seem knowledgeable about ML type theory, maybe you can clarify a puzzle for me. Whenever I read about W-types, I see mysterious comments to the effect that there are problems with their induction principles due to too much "junk" being around.
	lambda-the-ultimate.org /node/view/1078 (3049 words)

	ProofTheory.ORG (Site not responding. Last check: 2007-10-14)
		There is a mailing list for discussions about proof theory: click here to subscribe or here to unsubscribe; alternatively, send a message to Majordomo@Janeway.Inf.TU-Dresden.DE whose body contains `subscribe pt' or `unsubscribe pt' without quotes.
		There is also a smaller and more active mailing list, called Frogs, specifically devoted to structural proof theory: click here to subscribe or here to unsubscribe.
		Proof Theory on the Eve of Year 2000
	prooftheory.org /index.html (128 words)

	Proof, Computation, Complexity
		The aim of PCC is to stimulate research in proof theory, computation, and complexity, focusing on issues which combine logical and computational aspects.
		Specific areas of interest are (non-exhaustively listed) foundations for specification and programming languages, logical methods in specification and program development, new developments in structural proof theory, and implicit computational complexity.
		Unity in structural proof theory and structural extensions of the lambda-calculus
	cmaf.lmc.fc.ul.pt /~isarocha/pcc05.html (362 words)

	Proof Theory Landscape (Site not responding. Last check: 2007-10-14)
		Depending on the particular rules and connectives chosen, any of the following styles of rule systems may be used to define minimal, intuitionistic, or classical versions of propositional or predicate logic.
		The strength of the logic refers to the set of formulas that are provable in the system rather than the method of proof.
		This historically preceded thinking of proofs as trees and would have been the original method of doing Hilbert-style proofs; it is trivally applicable to any rule system since there is no substantial difference from trees: do a topological sort of the nodes of the proof tree and use pointers to identify parent nodes
	www.cis.upenn.edu /~plclub/cis700-009-s05/proof-theory.html (289 words)

	Structural Proof Theory by Sara Negri, Jan von Plato, Aarne Ranta, New, Used Books, Cheap Prices, ISBN 0521793076
		Structural Proof Theory (By Sara Negri,Jan von Plato)
		Proof Theory and Logical Complexity (Studies in Pr...
		Handbook of Proof Theory (By Samuel R. Buss)
	www.bookfinder4u.com /detail/0521793076.html (368 words)

	Philosophical Dictionary: Price-Pythagoras
		British philosopher who defended a comprehensive theory of the relation between sense-data and material objects in
		The primary qualities are intrinsic features of the thing itself (its size, shape, internal structure, mass, and momentum, for example), while the secondary qualities are merely its powers to produce sensations in us (its color, odor, sound, and taste, for example).
		This distinction was carefully drawn by Galileo, Descartes, Boyle, and Locke, whose statement of the distinction set the tone for future scientific inquiry.
	www.philosophypages.com /dy/p9.htm (1196 words)

	Amazon.ca: Basic Proof Theory: Books: A. S. Troelstra,H. Schwichtenberg (Site not responding. Last check: 2007-10-14)
		It is written by two of the experts in the field and comes up to their usual standards of precision and care.' Ray Turner, Computer Journal
		This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of first-order logic formalization.
		Examples are given of several areas of application, namely: the metamathematics of pure first-order logic, logic programming theory, category theory, modal logic, linear logic, first-order arithmetic and second-order logic.
	www.amazon.ca /Basic-Proof-Theory-S-Troelstra/dp/0521779111 (510 words)

	PhD Opportunities in Computational Logic at St Andrews
		One of the most fruitful ideas in logic has been the correspondence between proofs and programs, observed by Curry, Howard and others, with a similar correspondence between logical formulae and type systems for programming languages.
		Dependent type theory (of Martin-Löf) is an example of a strong type system in this line of development.
		Ideas on proof search in dependent type theory from [2] would probably be exploited.
	www.dcs.st-and.ac.uk /~rd/PhD.html (812 words)

	Minicourse on Structural Graph Theory (Site not responding. Last check: 2007-10-14)
		Structural Graph Theory at the Euler Institute for Discrete Mathematics in
		The emergence of planarity in graph structure theory, the two-paths problem.
		Edmonds' matching theorem, the linear hull of perfect matchings, the matching structure: decomposition into bricks and braces, the matching lattice, Pfaffian orientations and their use, Pfaffian orientations of bipartite graphs, Polya's permanent problem, the even directed cycle problem, sign-nonsingular matrices, applications in economics.
	www.math.gatech.edu /~thomas/SLIDE/EIDMA (375 words)

	Amazon.com: Structural Proof Theory: Books: Sara Negri,Jan von Plato,Aarne Ranta (Site not responding. Last check: 2007-10-14)
		Proof Theory: An Introduction (Lecture Notes in Mathematics) by Wolfram Pohlers
		Basic Proof Theory (Cambridge Tracts in Theoretical Computer Science) by A. Troelstra on page 24, and Back Matter
		Proof Theory and Logical Complexity : Volume I (Studies in Proof Theory) by Jean-Yves Girard in Back Matter
	www.amazon.com /Structural-Proof-Theory-Sara-Negri/dp/0521793076 (730 words)

	PESCA (Site not responding. Last check: 2007-10-14)
		Companion to the book Structural Proof Theory by Sara Negri and Jan von Plato, Cambridge University Press, 2001.
		PESCA* implements a mechanical proof search in elementary number theory, using cuts, which are not present in plain PESCA.
		, which shows the current proof tree together with the subnode numbering.
	www.cs.chalmers.se /~aarne/pesca (132 words)

	Publications (Site not responding. Last check: 2007-10-14)
		Confluence and strong normalization of the generalised multiary lambda-calculus, with Luís Pinto, in Stefano Berardi, Mario Coppo, Ferruccio Damiani (eds.), Revised selected papers from the International Workshop TYPES 2003, Torino, Italy, April 30 – May 4 2003, LNCS vol.
		Structural proof theory as rewriting, with Maria João Frade and Luís Pinto, in Proceedings of 17
		Unity in structural proof theory and structural extensions of the λ-calculus, July 2005, ps file
	www.math.uminho.pt /~jes/Publications.htm (221 words)

	Amazon.fr : Basic Proof Theory: Livres en anglais: Anne S. Troelstra,A. S. Troelstra,H. Schwichtenberg (Site not responding. Last check: 2007-10-14)
		Amazon.fr : Basic Proof Theory: Livres en anglais: Anne S. Troelstra,A. Troelstra,H. Schwichtenberg
		In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence.
		Introduces the basic ideas of structural proof theory.
	www.amazon.fr /Basic-Proof-Theory-Anne-Troelstra/dp/0521779111 (369 words)

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