en>Cuzkatzimhut: /* top */ removed 15-month old merger template, as per consensus to ''not'' merge.

2014-12-05T20:11:41Z

top: removed 15-month old merger template, as per consensus to ''not'' merge.

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	~~Hello. Allow me introduce the writer. Her title~~ is Refugia ~~Shryock~~. ~~North~~ Dakota ~~is our beginning location~~. ~~Managing individuals~~ is ~~what I do and the salary has been really fulfilling~~. ~~Body building~~ is 1 of the things I adore most.<br><br>~~Also visit my webpage over the counter std test -~~ [http://~~youplaisir~~.com//~~blog/57249 see here now~~] -		Nice to satisfy you, my name is Refugia. For a whilst I've been in South Dakota and my mothers and fathers live nearby. My day occupation is a meter reader. To collect coins is one of the things I adore most.<br><br>My web page: healthy food delivery ([http://www.Myprgenie.com/view-publication/weight-loss-getting-you-down-these-tips-can-help Continuing])

en>Vieque: Disambiguated: state vector → Quantum state

2014-02-14T21:27:59Z

Disambiguated: state vector → Quantum state

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	In [[logic]], a '''logical framework''' provides a means to define (or present) a logic as a signature in a higher-order [[type theory]] in such a way that [[provability]] of a formula in the original logic reduces to a [[type inhabitation]] problem in the framework type theory.<ref name="Jacobs2001">{{cite book\|author=Bart Jacobs\|title=Categorical Logic and Type Theory\|year=2001\|publisher=Elsevier\|isbn=978-0-444-50853-9\|page=598}}</ref><ref name="Gabbay1994">{{cite book\|editor=Dov M. Gabbay\|title=What is a logical system?\|url=http://books.google.com/books?id=XqCu4XjHrIQC&pg=PA382\|year=1994\|publisher=Clarendon Press\|isbn=978-0-19-853859-2\|page=382}}</ref> This approach has been used successfully for (interactive) [[automated theorem proving]]. The first logical framework was [[Automath]], however the name of the idea comes from the more widely known Edinburgh Logical Framework, '''LF'''. Several more recent proof tools like [[Isabelle (theorem prover)\|Isabelle]] are based on this idea.~~<ref name="Jacobs2001"/> Unlike a direct embedding, the logical framework approach allows many logics to be embedded in~~ the ~~same type system~~.~~<ref name="BoveBarbosa2009">{{cite book\|author1=Ana Bove\|author2=Luis Soares Barbosa\|author3=Alberto Pardo\|~~title=Language Engineering and Rigourous (sic) Software Development: International LerNet ALFA Summer School 2008, Piriapolis, Uruguay, February 24 - March 1, 2008, Revised, Selected Papers\|url=http://books.google.com/books?id=YOqHiA5MYpEC&pg=PA48\|year=2009\|publisher=Springer\|isbn=978-3-642-03152-6\|pages=48}}</ref>		Hello. Allow me introduce the writer. Her title is Refugia Shryock. North Dakota is our beginning location. Managing individuals is what I do and the salary has been really fulfilling. Body building is 1 of the things I adore most.<br><br>Also visit my webpage over the counter std test - [http://youplaisir.com//blog/57249 see here now] -

	A logical framework is based on a general treatment of syntax, rules and proofs by means of a [[dependent type theory\|dependently typed lambda calculus]]. Syntax is treated in a style similar to, but more general than [[Per Martin-Löf]]'s system of arities.

	~~To describe a logical framework, one must provide the following:~~

	~~# A characterization of the class of object-logics to be represented;~~
	~~# An appropriate meta-language;~~
	~~# A characterization of the mechanism by which object-logics are represented.~~

	~~This is summarized by:~~

	~~''‘Framework = Language + Representation’''.~~

	~~== LF ==~~

	~~In the case of the '''LF logical framework''', the meta-language is the [[λΠ-calculus\|<math>\lambda\Pi</math>-calculus]]. This~~ is ~~a system of first-order dependent function types which are related by the [[propositions as types principle]] to first-order minimal logic~~. ~~The key features of the <math>\lambda\Pi</math>-calculus are that it consists of entities of three levels: objects, types and families of types. It~~ is [[Impredicativity \| predicative]], all well-typed terms are [[strongly normalizing]] and [[Church-Rosser]] and the property of being well-typed is [[Decidability (logic)\|decidable]]. However, [[type inference]] is [[Decidability (logic)\|undecidable]].

	~~A logic is represented in the '''LF logical framework''' by the judgements-as-types representation mechanism~~. ~~This~~ is inspired by [[Per Martin-Löf]]'s development of [[Immanuel Kant\|Kant]]'s notion of [[judgement (mathematical logic)\|judgement]], in the 1983 Siena Lectures. The two higher-order judgements, the hypothetical <math>J\vdash K</math> and the ~~general, <math>\Lambda x\in J~~. K(x)</math>, correspond to the ordinary and dependent function space, respectively. The methodology of judgements-as-types is that judgements are represented as the types of their proofs. A [[logical system]] <math>{\mathcal L}</math> is represented by its signature which assigns kinds and types to a finite set of constants that represents its syntax, its judgements and its rule schemes. An object-logic's rules and proofs are seen as primitive proofs of hypothetico-general judgements <math>\Lambda x\in C. J(x)\vdash K</math>.

	~~An implementation of the LF logical framework~~ is ~~provided by the [[Twelf]] system at [[Carnegie Mellon University]]. Twelf includes~~
	~~:* a logic programming engine~~
	~~:* meta-theoretic reasoning about logic programs (termination, coverage, etc.)~~
	~~:* an inductive [[meta-logic]]al theorem prover~~

	~~==References==~~
	~~{{reflist}}~~

	~~==See also==~~

	* [[Grammatical Framework]]

	~~==Further reading==~~
	* {{cite book\|editor=Helmut Schwichtenberg, Ralf Steinbrüggen\|title=Proof and system-reliability\|chapter=Logical frameworks—a brief introduction\|year=2002\|publisher=Springer\|isbn=978-1-4020-0608-1~~\|author=[[Frank Pfenning]]\|url=http://www.cs.cmu.edu/~fp/papers/mdorf01.pdf}}~~
	*[[Robert Harper (computer scientist)\|Robert Harper]], Furio Honsell and [[Gordon Plotkin]]. ''A Framework For Defining Logics''. Journal of the ~~Association for Computing Machinery, 40(1):143-184, 1993~~
	*Arnon Avron, Furio Honsell, Ian Mason and Randy Pollack. ''Using typed lambda calculus to implement formal systems on a machine''. Journal of Automated Reasoning, 9:309-354, 1992.
	*Robert Harper. ''An Equational Formulation of LF''. Technical Report, University of Edinburgh, 1988. LFCS report ECS-LFCS-88-67.
	*Robert Harper, Donald Sannella and Andrzej Tarlecki. ~~''Structured Theory Presentations and Logic Representations''. Annals of Pure and Applied Logic, 67(1-3):113-160, 1994.~~
	*Samin Ishtiaq and David Pym. ''A Relevant Analysis of Natural Deduction''. Journal of Logic and Computation 8, 809-838, 1998.
	* Samin Ishtiaq and David Pym. ''Kripke Resource Models of a Dependently-typed, Bunched <~~math~~>~~\lambda~~<~~/math~~>~~-calculus''. Journal of Logic and Computation 12(6), 1061-1104, 2002.~~
	* Per Martin-Löf. "[http://www.hf.uio.no/filosofi/njpl/vol1no1/meaning/meaning.html On the ~~Meanings of the Logical Constants and the Justifications of the Logical Laws]." "[[Nordic Journal of Philosophical Logic]]", 1(1): 11~~-~~60, 1996.~~
	* Bengt Nordström, Kent Petersson, and Jan M. Smith. ''Programming in Martin-Löf's Type Theory''. Oxford University Press, 1990. (The book is out of print, but [http://~~www~~.~~cs.chalmers.se~~/Cs/~~Research~~/~~Logic/book/ a free version~~] ~~has been made available.)~~
	*David Pym. ''A Note on the Proof Theory of the <math>\lambda\Pi</math>-calculus''. Studia Logica 54: 199-230, 1995.
	*David Pym and Lincoln Wallen. ''Proof-search in the <math>\lambda\Pi</math>-calculus''. In: G. Huet and G. Plotkin (eds), Logical Frameworks, Cambridge University Press, 1991.
	*Didier Galmiche and David Pym. ''Proof-search in type-theoretic languages:an introduction''. Theoretical Computer Science 232 (2000) 5-53.
	*Philippa Gardner. ''Representing Logics in Type Theory''. Technical Report, University of Edinburgh, 1992. LFCS report ECS-LFCS-92-227.
	*Gilles Dowek. ''The undecidability of typability in the lambda-pi-calculus''. In M. Bezem, J.F. Groote (Eds.), Typed Lambda Calculi and Applications. Volume 664 of ''Lecture Notes in Computer Science'', 139-~~145, 1993.~~
	*David Pym. ''Proofs, Search and Computation in General Logic''. Ph.D. thesis, University of Edinburgh, 1990.
	*David Pym. ''A Unification Algorithm for the <math>\lambda\Pi</math>-calculus.'' Int. J. of Foundations of Computer Science 3(3), 333-378, 1992.

	~~== External links ==~~
	* [http://www.cs.cmu.edu/~fp/lfs-impl.html Specific Logical Frameworks and Implementations] (a list maintained by [[Frank Pfenning]])

	~~[[Category:Logic in computer science]]~~
	~~[[Category:Type theory]]~~
	~~[[Category:Proof assistants]]~~
	~~[[Category:Dependently typed programming]]~~

134.84.140.54 at 02:09, 24 October 2013

2013-10-24T02:09:56Z

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en>Kirsim at 10:08, 29 June 2012

2012-06-29T10:08:26Z

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en>Cuzkatzimhut: /* top */ removed 15-month old merger template, as per consensus to ''not'' merge.

en>Vieque: Disambiguated: state vector → Quantum state

134.84.140.54 at 02:09, 24 October 2013

en>Kirsim at 10:08, 29 June 2012