Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types. Number 7 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for QLL is represented as a Gentzen style calculus. In the rules presented in Fig. 1, Gamma represents a finite (possibly empty) sequence of hypotheses, and Delta represents a finite (possibly empty) sequence of assertions. The rules are multi succedent versions of the intuitionistic system [GLT89], plus two Lax Logic specific rules, fl L and fl R. of QLL, then it is derivable without using the Cut rule. Proof See Appendix A. Intuitively, the theory QLL may be regarded as a variant of QLL such that: QLL : fl false is the class of Kripke constraint structures validating ....

J-Y. Girard, Y. LaFont, and P. Taylor. Proofs and Types. Number 7 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....and a lot more reduction rules so this requires a proof. Theorem 17 (Strong Normalization) If A E : t #u then there is a normal form G such that E G . 22 Proof. By Theorem 11, A E : t #u implies jAj C jEj : jtj. By the Strong Normalization theorem for Curry typed calculus [7] there is a Curry normal form F such that jEj F . We can now apply the first Simulation theorem (Theorem 15) to obtain a term G such that E As jGj = F is a normal form it has no fi redexes, but in G some other reductions may be applicable. Because of the Church Rosser theorem, only using ....

Jean-Yves Girard, Paul Taylor, and Yves Lafont. Proofs and Types. Number 7 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....proofs of propositions from assumptions, and reduction to proof normalization. The basic example of the correspondence relates the simply typed lambda calculus with function, pair and disjoint union types to intutionisitic propositional logic with implication, conjunction and disjunction [GLT89] Whilst it may be true that almost no realistic programming language corresponds accurately to anything which might plausibly be called a logic (because of the presence of general recursion, if nothing else) logic and proof theory can still provide helpful insights into the design of ....

....to ensure that normal deductions satisfy the subformula property, for example. These occur when the system contains elimination rules which have a minor premiss (Girard calls this a parasitic formula and refers to the necessity for these extra reduction the shame of natural deduction [GLT89] In general, when we have such a rule, we want to be able to commute the last rule in the derivation of the minor premiss down past the rule, or to move the application of a rule to the conclusion of the elimination up past the elimination rule into to the derivation of the minor premiss. The ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Number 7 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

.... [2, 21] in which the basic judgments do not relate phrases directly to their interpretations, but rather to the types of their interpretations, and the actual interpretations are extracted from derivations by virtue of some version of the Curry Howard isomorphism between propositions and types [6, 11, 14, 15]. Categorial grammar attempts to merge the above two views by identifying grammatical categories and semantic types. However, semantic types are too coarse grained to make all the necessary syntactic distinctions. Types must be specialized: for example, in the directed Lambek calculus [19] the ....

....# to #)andv has type #, the application of u to v has type #. The abstraction rule (4) states that, if by assuming that an arbitrary x has type # one can conclude that u has type #, then the abstraction #x.u has type ###. The assumption x : # is said to be discharged by the rule. As is well known [11, 15, 33], the pairing between types and terms given by the above rules can also be seen as a pairing between formulas and terms describing their proofs. With the type constructor # interpreted as implication, the provable formulas are then just those provable in the implicational fragment of ....

[Article contains additional citation context not shown here]

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Number 7 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, England, 1989.

....proofs 65 of propositions from assumptions, and reduction to proof normalization. The basic example of the correspondence relates the simply typed lambda calculus with function, pair and disjoint union types to intutionisitic propositional logic with implication, conjunction and disjunction [GLT89] Whilst it may be true that almost no realistic programming language corresponds accurately to anything which might plausibly be called a logic (because of the presence of general recursion, if nothing else) logic and proof theory can still provide helpful insights into the design of ....

....to ensure that normal deductions satisfy the subformula property, for example. These occur when the system contains elimination rules which have a minor premiss (Girard calls this a parasitic formula and refers to the necessity for these extra reduction the shame of natural deduction [GLT89] In general, when we have such a rule, we want to be able to commute the last rule in the derivation of the minor premiss down past the rule, or to move the application of a rule to the conclusion of the elimination up past the elimination rule into to the derivation of the minor premiss. The ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Number 7 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....for QLL is represented as a Gentzen style calculus. In the rules presented in Fig. 1, Gamma represents a finite (possibly empty) sequence of hypotheses, and Delta represents a finite (possibly empty) sequence of assertions. The rules are multi succedent versions of the intuitionistic system [GLT89], plus two Lax Logic specific rules, fl L and fl R. Theorem 2.8 (Cut Elimination) If the sequent Delta is derivable using the proof system of QLL, then it is derivable without using the Cut rule. Proof See Appendix A. 3 System QLL Intuitively, the theory QLL may be regarded as a ....

J-Y. Girard, Y. LaFont, and P. Taylor. Proofs and Types. Number 7 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....applied to the concrete CLP program. Such information is called intentional information. But how do we extract this information Intuitionistic logic possesses a well known property known as the Curry Howard isomorphism between natural deduction proofs and terms of the lambda calculus (see e.g. GLT89] That is to say, for a particular (natural deduction) proof of a sequent in intuitionistic logic, there exists a unique lambda term (up to variable renaming) The logical rules of natural deduction can be seen as rules governing typing judgements on lambda terms, e.g. Gamma p : M Gamma q ....

J-Y. Girard, Y. LaFont, and P. Taylor. Proofs and Types. Number 7 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

.... somewhat out of date) coverage of both proof theory and semantics of LL is provided in the book by Anne Troelstra [151] well written and easy to read; above all the only book on LL to date) and a good book on proof theoretical issues and their computational interpretation is Proofs and Types [73]. AWWW page about LL is maintained by Lincoln, http: www.csl.sri.com linear sri csl ll.html. Available from ftp.cs.ualberta.ca: pub TechReports TR93 18, file TR93 18.ps.gz or TR93 18.ps.Z. Also in Bulletin of the IGPL 2(1) March 1994. Comments are most welcome. z Last revision September ....

....In the presence of non logical (proper) axiom, e.g. the predicate definitions in a logic program, an analogous property states that Cut can be pushed all the way up to the leaves of the proof tree where one of its premises is a proper axiom. Cut elimination for LL has been proved by Girard [62, 73] by largely the same method as Gentzen s Hauptsatz which proves it for CL. Cut elimination is important both for reducing the space of proofs (only cut free proofs will be searched for, see Section 3) and because the process of cut elimination (proof normalization) can be regarded as computation ....

J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Number 7 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1988.

No context found.

Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types. Number 7 in Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC