Frank Pfenning, Structural cut elimination in linear logic, Tech. Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....refinement checking and the cut rule for linear logic entailment) and show that the resulting system is sound and complete with respect to the original refinement checking specification. We carry out the proof by modifying and extending the logical cut elimination proof in earlier work by Pfenning [28]. In the second step we eliminate the subsumption rule and introduce annotations in order to eliminate two critical sources of non determinism present in the previous system. The first source is the non syntax directedness of the subsumption rule. We therefore incorporate the subsumption rule ....

F. Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

....language Elf by Pfenning [Pfe89, Pfe92, Pfe94a] In recent years LF and Elf became more and more popular. Significant problems have been represented in LF and Elf, for example the Church Rosser theorem [Pfe99] and a structural cut elimination theorem for classical, intuitionistic and linear logic [Pfe94c, Pfe94b]. Elf is a logic programming language and not an automated theorem proving system. Consequently, it serves the purpose of representing meta theoretical results, but it does not support their derivation. The calculus of constructions [CH88, PM93] and Martin Lof type theory [ML84, ML84] are type ....

....(lam (lam (app (1 ) 1) lam (lam (1 ) lam 1) Y. Elf verifies our expectation: Y = clo (empty ; clo empty (lam 1) lam (1 ) yes This concludes the section on LF and Elf. For more examples how to use LF type theory and the programming language Elf, we refer the reader to the literature [MP91, Pfe99, Pfe92, Pfe94c, Pfe94b, Pfe95]. 2.3. CALCULUS OF INDUCTIVE CONSTRUCTIONS AND COQ 27 2.3 Calculus of Inductive Constructions and Coq In this chapter we want to present the representation of the language T in a different type theory: the calculus of inductive constructions (CIC) This signature is then represented in Coq. ....

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Frank Pfenning. Structural cut elimination in linear logic. Technical Report CMUCS -94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

....of provability. All type families representing the different deductive systems are either level 0 or level 1, i.e. either non dependent type families, or dependent type families depending only on level 0 type families. level 2 type families are used to represent relations between derivations [Pfe94a] 3 Theorems, Proofs, and LF Because of higher order abstract syntax, LF is a very elegant logical framework. It is also very powerful because it allows the representation of proofs. To motivate this feature, we present in this section the type preservation theorem for Mini ML, and the deduction ....

F. Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS94 -222, Carnegie Mellon University, Department of Computer Science, December 1994.

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Frank Pfenning, Structural cut elimination in linear logic, Tech. Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

No context found.

F. Pfenning. Structural cut elimination in linear logic. Technical Report CMUCS -94-222, Carnegie Mellon University, Department of Computer Science, December 1994.

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F. Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Cepartment of Computer Science, Carnegie Mellon University, December 1994.

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Frank Pfenning, Structural cut elimination in linear logic, Tech. Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

No context found.

Frank Pfenning, Structural cut elimination in linear logic, Technical Report CMUCS -94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

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F. Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Carnegie Mellon University, Department of Computer Science, December 1994.

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Frank Pfenning, Structural cut elimination in linear logic, Tech. Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

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Frank Pfenning, Structural cut elimination in linear logic, Tech. Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

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Frank Pfenning, Structural cut elimination in linear logic, Technical Report CMUCS -94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

....backward calculus 2. contraction) If #, A, A # =# C. Proof. By structural induction on the derivations. The backward sequent calculus enjoys the following substitution properties, often written as explicit cut rules. Theorem 2 (Cut) # 1 , # 2 =# C. # =# C. Proof. See [11, 8]. 3 Controlled Weakening We will now turn to forward reasoning, i.e. with the aim of assembling the conclusion from the premisses. To summarize, the forward direction has the following resource management problems: 4 1. Undetermined contexts: the linear context in and the unrestricted ....

Frank Pfenning. Structural cut elimination in linear logic. Technical Report CMUCS -94-222, Carnegie Mellon University, December 1994.

....in FILL seems closer to its classical counterpart; furthermore, FILL has a clear categorical semantics [10] that we have not yet explored for JILL. Related structural proofs of cut elimination have appeared for intuitionistic and classical logic [22] classical linear logic in an unpublished note [21], and ordered logic [25] but these did not incorporate possibility and related connectives ( #) To our knowledge, the double negation translation from classical into intuitionistic linear logic that can optionally account for MIX is also new in this paper. Lamarche [17] has previously ....

....proofs of cut admissibility or cut elimination in the literature, we find it remarkable that a nested structural induction suffices, requiring no additional restrictions or induction measures. Similar structural proofs for the admissibility of cut have been demonstrated for classical linear logic [21] and ordered logic [25] Sequents have valid interpretations in the natural deduction calculus, which we state as a soundness theorem for the sequent calculus. Theorem 3 (Soundness of =# wrt #) then # C true. C, then # C poss. Proof. Proceed by simultaneous induction on ....

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Frank Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

.... Gamma M # P Delta; Gamma M P This might be proven by using the translations between natural deduction and the sequent calculus as in [Pfe99] together with a cut elimination argument for modal logic in a formulation based on judgments very similar to the one for linear logic in [Pfe94]. At present, we have not verified the details of such a construction. 10 Conclusion We have presented a judgmental reconstruction of the modal logic of necessity and possibility, leading to a clean and simple formulation of natural deduction and associated proof terms. Because the definitions ....

Frank Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

....case here is a cut between two initial sequents with the cut formula as a side formula. Finally, we have applied the ideas in this paper to obtain similar structural cut elimination results for intuitionistic and classical linear logics. These are sketched in [Pfe95] and given in more detail in [Pfe94] They will be the subject of a subsequent paper [Pfe] APPENDIX A. DETAILED ADMISSIBILITY PROOFS FOR CUT In this appendix we give the details of the admissibility of cut for intuitionistic and classical sequent calculi. For each case in the two proofs we show the formalization as an Elf ....

Pfenning, F. (1994), Structural Cut Elimination in Linear Logic," Technical Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University.

.... Gamma M # P # Delta; Gamma M P This might be proven by using the translations between natural deduction and the sequent calculus as in [Pfe99] together with a cut elimination argument for modal logic in a formulation based on judgments very similar to the one for linear logic in [Pfe94]. At present, we have not verified the details of such a construction. 10 Conclusion We have presented a judgmental reconstruction of the modal logic of necessity and possibility, leading to a clean and simple formulation of natural deduction and associated proof terms. Because the definitions of ....

Frank Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

....has been implemented as the logic programming language Elf by Pfenning[Pfe94a] In recent years, LF and Elf have become more popular. Significant problems have been represented in LF, for example, Pfenning proved a structural cut elimination theorem for intuitionistic, classical, and linear logic [Pfe94b] and showed its representation in LF. Because Elf is a programming language and not an automated theorem proving system, it serves the purpose of representing meta theoretical results, but it does not support their derivation. The calculus of constructions [CH88, PM93] and Martin Lof type theory ....

Frank Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

....an equivalent cut free derivation D 0 . Since it operates linearly on the formulas appearing in it, D would be adequately represented by a term pDq in a linear calculus; the same holds for D 0 . This problem was encoded in LF by representing sequents as types and derivations as proof terms [18]. LF is intuitionistic and therefore the linearity of pDq needed to be checked explicitly as a property of pDq, complicating the meta theory to the extent that it became infeasible and only the cut elimination algorithm without the linearity check was implemented. On the other hand, encoding ....

....lapp Psi Sigma MN # B Psi Sigma M # Pix : A:B Psi Sigma N A oa iapp Psi Sigma M N # [N=x]B Figure 2. A Pre canonical Deduction System for LLF, objects context Delta of an intuitionistic sequent Delta Gamma A as a multiset of linear formulas. More recent works [8, 10, 18] prefer to refine it as a pair Gamma; Delta of multisets so that the sequent Gamma; Delta Gamma A is equi provable with ( Gamma; Delta) Gamma A in the traditional formulation. The two components of Gamma; Delta are called the intuitionistic and the linear context, respectively. Our ....

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F. Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, Dec. 1994.

....Here A j B is the strongest possibly equivalence which requires ordered implications in both directions. Our sequent system combines the ideas of a multi zone presentation due to Andreoli [3] with implicit structural rules to permit a proof of cut elimination by structural induction as in [18]. Theorem 4.1 (Admissibility of Cut) i) Cut Omega : If Gamma; Delta C ; Omega C Gamma C and Gamma; Delta; Omega L C Omega R Gamma A then Gamma; Delta C . Delta; Omega L Omega C Omega R Gamma A. ii) Cut Delta : If Gamma; Delta C ; Delta Gamma C and Gamma; Delta L ....

....the result of an appeal to the induction hypothesis. 2 Polakow and Pfenning Our proofs above are constructive and inherently contain a method for translation between sequent derivations in INCLL and natural deductions. This translation could be written out concisely on proof terms (similar to [18]) but this is beyond the scope of this summary. Clearly, the correspondence is very close, but it is not a bijection, because the order in which left rules are applied in a sequent derivation may be irrelevant to the resulting natural deduction. If one wants to establish a bijection, one has to ....

F. Pfenning. Structural cut elimination in linear logic. Technical Report CMUCS -94-222, Carnegie Mellon University, Department of Computer Science, December 1994.

....to establish that an Elf program constitutes a decision procedure. This allows us to make a meta mathematical statement about an object language formalized in Elf simply by exhibiting a checked program in the same framework. Some examples of this approach are: a formalization of linear logic [21], where the linearity of derivations is a decidable property, an implementation of the sequent calculus [22] with a terminating cut elimination procedure, a formulation of refinement types [19] for which the subtype property is decidable, and a representation of Mini ML [11] for which ....

Frank Pfenning. Structural cut elimination in linear logic. Technical Report CMUCS -94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

No context found.

F. Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

No context found.

F. Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

No context found.

F. Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS-94-222, Department of Computer Science, Carnegie Mellon University, December 1994.

No context found.

Frank Pfenning. Structural cut elimination in linear logic. Technical Report CMU-CS94 -222, Department of Computer Science, Carnegie Mellon University, December 1994.

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