Pfenning, F., Structural cut elimination, in: D. Kozen, editor, Proc. LICS'95 (1995), pp. 156--166.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....applicable. We are aware of only four other attempts to formalise proof theory or cut elimination in sequent calculi. The first is the work of Pfenning which formalises the admissibility (weaknormalisation) of cut for classical, intuitionistic and linear logic using the logical framework Elf [21]. Pfenning readily admits this is not a full formalisation [21] but Pfenning s proofs have now been fully formally verified in Twelf by Schurmann; see [23, Section 8.5] Both formalisations make explicit use of properties of the underlying logical framework to obtain certain structural rules like ....

....proof theory or cut elimination in sequent calculi. The first is the work of Pfenning which formalises the admissibility (weaknormalisation) of cut for classical, intuitionistic and linear logic using the logical framework Elf [21] Pfenning readily admits this is not a full formalisation [21], but Pfenning s proofs have now been fully formally verified in Twelf by Schurmann; see [23, Section 8.5] Both formalisations make explicit use of properties of the underlying logical framework to obtain certain structural rules like exchange, weakening and contraction for free . The associated ....

Pfenning, F. Structural cut elimination. In Proc. LICS 94, 1994.

....induction over the cut rank and degree a la Gentzen. In [2] we implemented a shallow embedding of #RA which enabled us to mimic derivations in #RA using Isabelle Pure. But it was impossible to reason about derivations since they existed only as the trace of the particular Isabelle session. In [9], Pfenning has given a formalisation of cut admissibility for traditional sequent calculi for various nonclassical logics using the logical framework Supported by an Australian Research Council Large Grant Supported by an Australian Research Council QEII Fellowship Elf, which is based upon ....

....to be captured as terms, they do not enable us to formalise all aspects of a meta theoretic proof. As Pfenning admits, the Elf formalisation cannot be used for checking the correct use of the induction principles used in the cut admissibility proof, since this requires a deep embedding [9]. The use of such deep embeddings to formalise meta logical results is rare [8, 7] To our knowledge, the only full formalisation of a proof of cut admissibility is that of Schurmann [10] but the calculi used by both Pfenning and Schurmann contain no explicit structural rules, and structural ....

F Pfenning. Structural cut elimination. In Proc. LICS 94, 1994.

....derivation. The sequent calculus [Gen35] de ning judgment = G is depicted in Figure 4 is de ned for the same fragment as the natural deduction calculus from Figure 2. For this example it is not so important what the sequent calculus actually is, but it is important how it is represented in LF [Pfe95] Example 2 (Sequent deductions) Extends from Example 1. hyp : o type; conc : o type; init : hyp G conc G; impR : hyp G 1 conc G 2 ) conc (G 1 G 2 ) impL : conc G 1 (hyp G 2 conc G 3 ) hyp (G 1 G 2 ) conc G 3 ) cut : conc G 1 (hyp G 1 conc G 2 ) ....

Frank Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156{ 166, San Diego, California, June 1995. IEEE Computer Society Press.

....example we use a mapping of natural deduction derivations into derivations of the sequent calculus. Although we present only one connective, the example scales to full rst order logic. The judgment for sequent derivations is = G, its rules and its adequate encoding are depicted in Figure 2 [Pfe95] Hypotheses to the left of the sequent symbol = are encoded using hyp in order to distinguish them from the conclusion to the right which is encoded using conc . Only by representing them as two separate type families we can guarantee the adequacy of the encoding which also relies on the ....

....In our experience many algorithms related to programming languages and logics can be directly represented in T . In the setting of rst order intuitionistic logic, for example, we have encoded an algorithm that transforms sequent derivations with cut into sequent derivations without cut [Pfe95] In the interest of space, we can only give the de nition of the regular world and the types of the two recursive T functions. The sequent calculus for full rst order intuitionistic logic extends Figure 2 with additional left and right rules for the other connectives. Let i be the LF ....

Frank Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156{ 166, San Diego, California, June 1995. IEEE Computer Society Press.

....the derivation above, we felt uncertain that we could use it to find all required derivations, primarily because of the lack of user control over a proof. 3. 3 The Cut Elimination Theorem in Twelf At this point we should refer to the proof of a cut elimination theorem, using Twelf, described in [19]. This uses a rather ingenious representation of sequents; a cut free proof of a traditional Gentzen sequent A # B is represented as the type neg A pos B , and a cut free proof of it as neg A pos B #, as explained in [19] The two rules shown below left are both represented as the ....

.... proof of a cut elimination theorem, using Twelf, described in [19] This uses a rather ingenious representation of sequents; a cut free proof of a traditional Gentzen sequent A # B is represented as the type neg A pos B , and a cut free proof of it as neg A pos B #, as explained in [19]. The two rules shown below left are both represented as the type shown on the right: 9 #, A # # #, A B # # A # A B # (neg A #) neg (A and B) #) In fact, the second rule shown would be directly represented by the type shown, but the first rule shown could be directly ....

F Pfenning. Structural cut elimination. In Proc. Tenth Annual Symposium on Logic in Comp. Sci. , pages 156--166. IEEE Computer Society Press, 1995.

....while a stands for an already defined parameter. To adopt a logical point of view, the term on the left of a subterm relation can be interpreted as universally quantified and the term on the right as existentially quantified. Another example is taken from the representation of first order logic [Pfe95]. We can represent formulas by the type family o. Individuals are described by the type family i. The constructor 8 can be defined as forall: i o) o. We might want to show that A T (which represents [t=x]A)is smaller than forall x:A x (which represents 8x:A) Similarly, we might count the ....

Frank Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156-- 166, San Diego, California, June 1995. IEEE Computer Society Press.

....Sequents are encoded by two separate type families. Hypotheses to the left of the sequent symbol = are labeled with h and encoded using hyp whereas the conclusion to the right is encoded by conc . And again, the representation of the sequent calculus in LF employs hypothetical judgments [Pfe95] The premiss of a derivation which ends in the R rule, for example, is encoded as an LF function of type hyp G 1 conc G 2 . 3 3. THE REGULAR WORLD ASSUMPTION Example 3 (Sequent Calculus) Provability judgment: G hyp : o type conc : o type init ; h : G = G init : hyp G ....

....higher order features and dependent types. In this paper we present a small calculus of functions which is sucient to express interesting algorithms such as the one above, other examples include normalization algorithms based on the Church Rosser theorem [Pfe93] or the cut elimination theorem [Pfe95] As a running example we use Gentzen s result. Lemma 1 (Natural deduction ) sequent calculus) For any set of assumptions and formula G, if D is a natural deduction derivation of G then there exists a sequent derivation of = G. 4 3. THE REGULAR WORLD ASSUMPTION Proof. By structural ....

Frank Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156{ 166, San Diego, California, June 1995. IEEE Computer Society Press.

....although the theorem prover did successfully find the derivation above, we felt uncertain that we would be able to use it to find all required derivations. 3. 3 The Cut Elimination Theorem in Twelf At this point we should refer to the proof of a cut elimination theorem, using Twelf, described in [20]. This uses a rather ingenious representation of sequents; a cut free proof of a traditional Gentzen sequent A # B is represented as the type neg A pos B , and a cut free proof of it as neg A pos B #, as explained in [20] The two rules shown below left are represented as the type ....

.... proof of a cut elimination theorem, using Twelf, described in [20] This uses a rather ingenious representation of sequents; a cut free proof of a traditional Gentzen sequent A # B is represented as the type neg A pos B , and a cut free proof of it as neg A pos B #, as explained in [20]. The two rules shown below left are represented as the type shown on the right: #, A # # #, A B # # A # A B # (neg A #) neg (A and B) #) In fact, the second rule shown would be directly represented by the type shown, but the first rule shown could be directly represented by ....

Frank Pfennig. Structural cut elimination. In D Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156--166. IEEE Computer Society Press, 1995.

....simply not confluent. The critical 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 ....

Pfenning, F. (1995), Structural cut elimination, in Proceedings of the Tenth Annual Symposium on Logic in Computer Science, San Diego, California" (D. Kozen, Ed.), IEEE Comput. Soc., Los Alamitos, CA.

....and Computation 157, 84#141 (2000) Structural Cut Elimination I. Intuitionistic and Classical Logic 1 Frank Pfenning Department of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 3891 We present new variants of known proofs of cut elimination for intuitionistic and classical sequent calculi. In both cases the proofs proceed by three nested structural inductions, avoiding the ....

....They also involve global conditions on occurrences of parameters in sequent derivations. In this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91] Multi sets are avoided altogether in these proofs, and termination measures are replaced by three nested structural inductions. Parameters are treated as variables bound in derivations, thus naturally capturing occurrence conditions. A starting point for the proofs is Kleene s sequent system ....

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Pfenning, F., Structural cut elimination. II. Linear logic, in preparation.

....details of the fine structure of proofs in such a clear manner that many logic presentations employ sequent calculi. The laws governing the structure of proofs, however, are more complicated than the Curry Howard isomorphism for natural deduction might suggest and are still the subject of study [Her95, Pfe95]. We choose natural deduction as our definitional formalism as the purest and most widely applicable. Later we justify the sequent calculus as a calculus of proof search for natural deduction and explicitly relate the two forms of presentation. We begin by introducing natural deduction for ....

Frank Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156--166, San Diego, California, June 1995. IEEE Computer Society Press.

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Pfenning, F., Structural cut elimination, in: D. Kozen, editor, Proc. LICS'95 (1995), pp. 156--166.

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F. Pfenning. Structural cut elimination. In D. Kozen, editor, Proc. LICS'95, pages 156--166, San Diego, CA, 1995. IEEE Computer Society Press.

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Frank Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156-- 166, San Diego, California, June 1995. IEEE Computer Society Press.

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Pfenning, F. Structural Cut Elimination. In Proceedings of the Tenth Annual IEEE Symposium on Logic in Computer Science (San Diego, CA, June 1995), pp. 156--166.

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Pfenning, F. Structural Cut Elimination. In Proceedings of the Tenth Annual IEEE Symposium on Logic in Computer Science (San Diego, CA, June 1995), pp. 156--166.

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Pfenning, F. (1995). Structural Cut Elimination. In 10th Annual IEEE Symposium on Logic in Computer Science (LICS'95).

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Frank Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156-- 166, San Diego, California, June 1995. IEEE Computer Society Press.

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Frank Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156--166, San Diego, California, June 1995. IEEE Computer Society Press.

No context found.

F. Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156--166. Computer Society Press, 1995.

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F. Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156--166. Computer Society Press, 1995.

No context found.

F. Pfenning. Structural Cut-Elimination. In Logic and Computer Science, pages 156--166. IEEE Computer Society, 1995.

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F. Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156--166. Computer Society Press, 1995.

No context found.

F. Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156--166. Computer Society Press, 1995.

No context found.

Frank Pfenning. Structural cut elimination. In D. Kozen, editor, Proceedings of the Tenth Annual Symposium on Logic in Computer Science, pages 156--166, San Diego, California, June 1995. IEEE Computer Society Press.

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