Home

# Contraction-free sequent calculi for intuitionistic logic

Extract

Gentzen's sequent calculus LJ, and its variants such as G3 [21], are (as is well known) convenient as a basis for automating proof search for IPC (intuitionistic propositional calculus). But a problem arises: that of detecting loops, arising from the use (in reverse) of the rule ⊃⇒ for implication introduction on the left. We describe below an equivalent calculus, yet another variant on these systems, where the problem no longer arises: this gives a simple but effective decision procedure for IPC.

The underlying method can be traced back forty years to Vorob′ev [33], [34]. It has been rediscovered recently by several authors (the present author in August 1990, Hudelmaier [18], [19], Paulson [27], and Lincoln et al. [23]). Since the main idea is not plainly apparent in Vorob′ev's work, and there are mathematical applications [28], it is desirable to have a simple proof. We present such a proof, exploiting the Dershowtiz-Manna theorem [4] on multiset orderings.

Consider the task of constructing proofs in Gentzen's sequent calculus LJ of intuitionistic sequents Γ⇒ G, where Γ is a set of assumption formulae and G is a formula (in the language of zero-order logic, using the nullary constant f [absurdity], the unary constant ¬ [negation, with ¬A =defAf] and the binary constants &, ∨, and ⊃ [conjunction, disjunction, and implication respectively]). By the Hauptsatz [15], there is an apparently simple algorithm which breaks up the sequent, growing the proof tree until one reaches axioms (of the form Γ⇒ A where A is in Γ), or can make no further progress and must backtrack or even abandon the search. (Gentzen's argument in fact was to use the subformula property derived from the Hauptsatz to limit the size of the search tree. Došen [5] improves on this argument.)

References
Hide All
[1]Abramsky, Samson, Computational interpretations of linear logic, technical report DOC 90/20, Imperial College, London; Theoretical Computer Science, to appear.
[2]Beeson, Michael, Some applications of Gentzen's proof theory in automated deduction, Extensions of logic programming workshop, 1989 (Schroeder-Heister, P., editor), Lecture Notes in Computer Science, vol. 475, Springer-Verlag, Berlin, 1991, pp. 101156.
[3]Dardzhanta, G. K., Intuitionistic system without contraction, Polish Academy of Sciences, Institute of Philosophy and Sociology, Bulletin of the Section of Logic, vol. 6 (1977), pp. 28.
[4]Dershowitz, Nachum and Manna, Zohar, Proving termination with multiset orderings, Communications of the ACM, vol. 22(1979), pp. 465476; Automata, languages and programming: sixth colloquium (ICALP '79), Lecture Notes in Computer Science, vol. 71, Springer-Verlag, Berlin, 1979, pp. 188-202.
[5]Došen, Kosta, A note on Gentzen's decision procedure for intuitionistic prepositional logic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 33 (1987), pp. 453456.
[6]Dragalin, Albert Grigorevich, Mathematical intuitionism—introduction to proof theory, Translations of Mathematical Monographs, vol. 67, American Mathematical Society, Providence, Rhode Island, 1988.
[7]Dummett, Michael, Elements of intuitionism, Clarendon Press, Oxford, 1977.
[8]Dyckhoff, Roy, Leslie, Neil and Read, Stephen, MALT St Andrews (MacLogic—a proof assistant for first-order logic), Computerised Logic Teaching Bulletin, vol. 2, no. 1, St Andrews University, 1989, pp. 5160.
[9]Felty, Amy, A logic program for transforming sequent proofs to natural deduction proofs, Extensions of logic programming workshop, 1989 (Schroeder-Heister, P., editor), Lecture Notes in Computer Science, vol. 475, Springer-Verlag, Berlin, 1991, pp. 157178.
[10]Fitting, Melvin, Proof methods for modal and intuitionistic logics, Reidel, Dordrecht, 1983.
[11]Fitting, Melvin, First-order modal tableaux, Journal of Automated Reasoning, vol. 4 (1988), pp. 191213.
[12]Fitting, Melvin, First-order logic and automated theorem proving, Springer-Verlag, Berlin, 1990.
[13]Franzén, Torkel, Algorithmic aspects of intuitionistic propositional logic I & II, Technical reports R87010B and R89006, Swedish Institute of Computer Science, Kista, Sweden, 1987 & 1989 (see also [13a]).
[13a]Sahlin, Dan, Franzén, Torkel and Haridi, Seif, An intuitionistic predicate logic theorem prover, Journal of Logic and Computation, to appear.
[14]Gabbay, Dov M., Algorithmic proof with diminishing resources I, Computer science logic: proceedings, Heidelberg, 1990 (Börger, E., Büning, H. Kleine, Richter, M. M. and Schönfeld, W., editors), Lecture Notes in Computer Science, vol. 533, Springer-Verlag, Berlin, 1991, pp. 156173.
[15]Gentzen, Gerhard, The collected papers of Gerhard Gentzen (Szabo, M., editor), North-Holland, Amsterdam, 1969.
[16]Girard, Jean-Yves, Proofs and types, Cambridge University Press, Cambridge, 1989.
[17]Hodas, Joshua and Miller, Dale, Logic programming in a fragment of intuitionistic linear logic, Sixth annual IEEE symposium on Logic in Computer Science: proceedings, Amsterdam, 1991, IEEE Computer Society Press, Los Alamitos, California, 1991, pp. 3242.
[18]Hudelmaier, Jörg, A Prolog program for intuitionistic logic, SNS-Bericht 88-28, Universität Tübingen, Tübingen, 01 1988.
[19]Hudelmaier, Jörg, Bounds for cut elimination in intuitionistic propositional logic, Dissertation, Mathematics Faculty, Universität Tübingen, Tübingen, 1989; also to appear in Archive for Mathematical Logic.
[20]Hudelmaier, Jörg, A decision procedure for intuitionistic propositional logic, submitted to Journal of Logic and Computation.
[21]Kleene, Stephen C., Introduction to metamathematics, North-Holland, Amsterdam, 1964.
[22]Kleene, Stephen C., Permutation of inferences in Gentzen's calculi LK and LJ, Two papers on the predicate calculus, Memoir no. 10, American Mathematical Society, Providence, Rhode Island, 1951, pp. 126.
[23]Lincoln, P., Scedrov, A. and Shankar, N., Linearizing intuitionistic implication, Sixth annual IEEE symposium on Logic in Computer Science: proceedings, Amsterdam, 1991, IEEE Computer Society Press, Los Alamitos, California, 1991. pp. 5162.
[24]Mints, Grigori, Gentzen-type systems and resolution rules. Part I: Propositional logic, COLOG-88:proceedings, Tallinn, 1988 (Martin-Löf, P. and Mints, G., editors), Lecture Notes in Computer Science, vol. 417, Springer-Verlag, Berlin, 1990, pp. 198231.
[25]Mints, Grigori, Proof theory in the USSR 1925-1969, this Journal, vol. 56 (1991), pp. 385424.
[26]Ono, Hiroakira and Komori, Yuichi, Logics without the contraction rule, this Journal, vol. 50 (1985), pp. 169201.
[27]Paulson, Larry, personal communication, concerning Isabelle (Computer Laboratory, Cambridge University), Cambridge, 01 1991.
[28]Pitts, Andrew, On an interpretation of second order quantification in first-order intuitionistic propositional logic, this Journal, vol. 57 (1992), pp. 3352.
[29]Prawitz, Dag, Natural deduction, Almqvist & Wiksell, Stockholm, 1965.
[30]Slaney, John and Girle, Rod, Tableau and sequent calculus method in minimal logic theorem proving, ANU-Fujitsu-joint workshop on logic and computation, Canberra, 06 1990 (unpublished talk).
[31]Tennant, Neil, Computational logic, seminar, A.I. Department Edinburgh, University, Edinburgh, 05 1991.
[32]van Gelder, Allen, Efficient loop detection in Prolog using the tortoise-and-hare technique, Journal of Logic Programming, vol. 4 (1987), pp. 2331.
[33]Vorob'ev, N. N., The derivability problem in the constructive propositional calculus with strong negation, Doklady Akademii Nauk SSSR, vol. 85 (1952), pp. 689692. (Russian)
[34]Vorob'ev, N. N., A new algorithm for derivability in the constructive propositional calculus, Trudy Matematicheskogo Instituta imeni V. A. Steklova, vol. 52 (1958), pp. 193225; English translation in American Mathematical Society Translations, ser. 2, vol. 94 (1970), pp. 37-71.
[35]Wallen, Lincoln, Automated deduction in non-classical logics: efficient matrix proof methods for modal and intuitionistic logics, M.I.T. Press, Cambridge, Massachusetts, 1990.
Recommend this journal

The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic
Who would you like to send this to? *

×

## Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *