Sciences, St Andrews University, St Andrews, Fife KY16 9SS, Scotland,

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1 A Deterministic Terminating Sequent Calculus for Godel-Dummett logic ROY DYCKHOFF, School of Mathematical and Computational Sciences, St Andrews University, St Andrews, Fife KY16 9SS, Scotland, Abstract We give a short proof-theoretic treatment of a terminating contraction-free calculus G4-C for the zero-order Godel-Dummett logic C. This calculus is a slight variant of a calculus given by Avellone et al, who show its completeness by model-theoretic techniques. In our calculus, all the rules of G4-C are invertible, thus allowing a deterministic proof-search procedure. Keywords: sequent calculus, contraction-free, terminating, Godel-Dummett logic 1 Introduction In previous work [9] the author gave a \contraction-free calculus" for zero-order intuitionistic logic IP; following [21] we call this calculus G4ip. It has the property that root-rst proof search terminates, thus allowing easy implementation without a loop-checker. See [9] for further history of this calculus, developed independently by Hudelmaier [16] and others, and with ideas from Vorob'ev's 1950 work (presented later in [22]). We now call this a \terminating" calculus to distinguish it from other \contraction-free calculi", notably Dragalin's GHPC in [7] and G3i in [21], i.e. those in which the contraction rule is not primitive but admissible. By \terminating" we mean just that every sequence of steps, each of which replaces a sequent by the premisses of a rule whose conclusion matches the sequent, is nite. Dummett [8] described a logic, usually known as C, with semantics based on linearly ordered Kripke frames: it is intuitionistic logic plus the axiom (schema) (AB) _ (BA). Godel had previously considered [13] some nite approximations to this logic, in recognition of which the logic is often, especially in Central Europe, called G, Godel logic. The logic is of interest now mainly for its relationship [15] with multi-valued and fuzzy logics. Sonobe presented [20] a cut-free sequent calculus for zero-order C; this was presented as a tableau calculus and extended to the rst-order case by Corsi [5], [6]. Avellone et al showed [1] how in the zero-order case the calculus could be modied, using some of the ideas from [9] to make a terminating calculus; this has been extended to the rst-order case in [2], except that (of course) not all the rules have the desired termination property. These two papers of Avellone et al use tableau calculi and Kripke models, but conveniently translate the tableau calculus into roughly the sequent calculus below; [1] argues rather vigorously that these model-theoretic. J. of the IGP, Vol. 0 No. 0, pp. 1{ c Oxford University Press

2 2 A Deterministic Terminating Sequent Calculus for Godel-Dummett logic techniques are superior to our proof-theoretic ones. We give here a short proof-theoretic treatment, along the lines of [9], of the zeroorder case. For simplicity we present the argument just in the implicational case: the other zero-order constants cause no extra problems. Extension of the argument to the rst-order case is non-trivial: see [11] and [12]. In what follows P; Q; : : : are meta-variables for atoms p; q; : : :; A; B; ::: are metavariables for formulae; and?; are for multisets of formulae. We deal with two-sided sequents? ). 2 A non-terminating calculus G3-C We begin with a non-terminating calculus G3-C as a basis. The axioms and inference rules for the implicational fragment of G3-C are as follows:?; P ) P; Axiom?; AB ) A;?; B )?; AB ) f?; A i ) B i ; i g? ) In the last rule, contains m > 0 implicational formulae A i B i ; i = 1; :::; m; may also contain some other kinds of formulae. indicates the multiset consisting of just the m implicational formulae of. i is just after removal of the i th formula A i B i. This rule is roughly as given by Sonobe [20]. Note the repetition of AB in the left premiss of and the associated termination problems discussed in e.g. [9]. It is routine to follow the approach in, say, [7] to see that this calculus admits the rules of Contraction and Cut. By Cut we mean either the additive or the multiplicative version: in the presence of Contraction the two are equivalent. Note that the left and right contraction rules must be proved admissible simultaneously. The rule is easily seen to be invertible; the rule R is invertible when the succedent of the conclusion consists only of implicational formulae. This calculus is that of [20], except that Sonobe's calculus is based on sets of formulae: by using multisets we choose not to hide a contraction rule in the notation. Also, rather than having it as primitive we build it into the logical rules. The calculus' implementation needs a loop-checker, because the rule has its left premiss more complex than the conclusion. Admitting Cut, the calculus is easily seen to formalise C, the latter now being taken to be implicational logic plus axioms of the form ((AB)C)(((BA)C)C), a formulation easily seen to be equivalent, in the presence of disjunction, to that using (AB) _ (BA). R 3 A terminating calculus G4-C We say that a sequent? ) is irreducible 1.? contains only atoms P; :::; and atomic implications QB; :::, with no P equalling the body Q of an atomic implication; 2. contains only atoms and implications; 3.? and have no atoms in common. if

3 A Deterministic Terminating Sequent Calculus for Godel-Dummett logic 3 Condition 2 is uninformative until we add the other logical constants to the language. Irreducibility of a sequent is equivalent to the sequent not being an axiom or the conclusion of any rule (of G4-C) other than R 0. The axioms and inference rules for the implicational fragment of G4-C are as follows:?; P ) P; Axiom?; P; B )?; P; P B ) 0?; DB ) CD;?; B )?; (CD)B ) f?; A i ) B i ; i g? ) The R 0 rule has the same conventions as R, except that we only permit its use when the conclusion is irreducible. Theorem 3.1 G4-C is a terminating calculus for zero-order C in which all the rules are invertible. Proof: Since P; P B ) B, (CD)B ) DB and CD; (CD)B ) B are derivable in G3-C, and the latter admits Contraction and Cut, it is easy to verify that these four rules are admissible in G3-C, showing the soundness of G4-C, and (similarly) that 0 and are invertible (w.r.t. G3-C). The irreducibility restriction on use of the R 0 rule forces its invertibility: if the conclusion of an instance of the rule is derivable in G4-C and is irreducible then there is no way it could be derived other than by R 0, and, in contrast to the case with IP where one could analyse any implication in the succedent, here all must be analysed simultaneously. Further, G4-C is terminating, because there is a well-founded measure \weight" on sequents such that every premiss is of lower weight than the conclusion, just as in [9] or the presentation of the same material in [21]. We now show, by induction on the weight of sequents, that derivability in G3-C implies derivability in G4-C. Consider a sequent? ) derivable in G3-C. If the sequent contains either a nested implication (CD)B or both an atomic implication P B and the atom P, then, using invertibility of the rule or (resp.) 0, we may replace it by two (resp. one) G3-C-derivable sequent(s) of lower weight, inductively nd derivations thereof in G4-C and then use or, respectively, 0 to get a derivation of the sequent in G4-C. Thus, w.l.o.g. we may take the sequent to be irreducible. Consider the leftmost branch of its derivation; there must be an R step, otherwise it consists entirely of steps which can never (as we move upwards) transfer anything other than atoms to the succedent and these atoms are, by irreducibility, not in the antecedent, so we never reach an axiom. If we examine closely the premisses of this R step, we see that any atoms accumulated in the conclusion's succedent by all the steps can be ignored. So, w.l.o.g. the last step of the G3-C derivation is by R (note the compliance with the irreducibility restriction), with premisses that are of lower weight and so, by inductive hypothesis, derivable in G4-C. Using R 0, we obtain a derivation of? ) in G4-C. Thus, G4-C is a sound and complete calculus for implicational Godel-Dummett logic and so admits the same structural rules as does G3-C. It is easy to add the other constants, using ideas from [9] to deal with implications with conjunctive or R 0

4 4 A Deterministic Terminating Sequent Calculus for Godel-Dummett logic disjunctive bodies. We treat negation as an abbreviation rather than as a primitive notion, adding also the usual axiom? and the invertible rule allowing weakening with? in the succedent, because otherwise the irreducibility restriction forbids the derivation of, for example, the sequent ) pp;?. 2 4 Example We illustrate the above with a derivation of an instance of the characteristic axiom for implicational G4-C, p; q and r being distinct atoms. For brevity, we write pq for pq, pqr for (pq)r, etc. First, we have: and, second: pr; r; q; p ) q Axiom pr; r; q ) p; pq R0 pr; qr; q ) p; pq 0 r; qr; p ) qp; r Axiom pr; qr; p ) qp; r 0 r; qr; p ) r Axiom qpr; qr; p ) r p; qr ) q; qprr R0 r; qr; p; q ) p Axiom r; qr; p ) qp; q R0 pr; qr; p ) qp; q 0 pr; qr ) qp; pq; r R 0 r; qr ) pq; r Axiom qpr; qr ) pq; r Combining these with further steps we obtain p; qr ) q; qprr qpr; qr ) pq; r r; qpr ) r Axiom qr ) pq; qprr R 0 r ) qprr R0 pqr ) qprr ) pqr(qprr) R0 When p; q and r are replaced by nonatomic formulae or by non-distinct atoms, this derivation must be modied to meet the irreducibility constraints: nevertheless, the completeness theorem guarantees the existence of a derivation in each case. 5 Refutations and Kripke semantics The above calculus is of special interest because of the invertibility of all the rules. Our rst reaction to this was to think that such a situation only occurs in classical logic (and thus that our results were erroneous): however, this view is mistaken. The trivial calculus for IP, in which the axioms are all the sequents? ) A for which A is provable, is complete, admits all the structural rules and, having no primitive rules, has all of them invertible: but it is not classical logic. More generally, systems such as that of Maslov [18] have all rules invertible without being classical. Refutation systems have been considered by several authors: see [14] for a partial survey. We use the word \refutation" in the traditional sense, not in the modern sense popularised by, for example, [4], where a \refutation" of a set S of clauses is a deduction of the empty clause from the set, also called (!) a \proof" of S: in our

5 A Deterministic Terminating Sequent Calculus for Godel-Dummett logic 5 sense, a formula is refutable when it has a counter-model, and in the sense of [4] when, the formula being represented as a set of clauses, it is unsatisable.) Following the approach of [19], we may construct a refutation calculus as follows: the judgments are \antisequents"? 6) ; a derivation of such an antisequent is called a \refutation" of the sequent? ). The rules are? 6) (1)?; P; B 6)?; P; P B 6) (2)?; DB 6) CD;?; (CD)B 6) (3)?; B 6)?; (CD)B 6) (4)?; A 6) B;? 6) AB; (5) in which (1) has the side-condition that? ) is irreducible and just consists of atoms; and in (5), consists of the implicational formulae of (without the requirement that they number more than zero); and in (5) we take? ) AB; to be irreducible. Note that, because the rules of G4-C are all invertible, there is no refutation rule with several premisses, each corresponding to a dierent non-invertible way to expand the proof (as in [19]). One may now argue, as in [19] and [23], that for each sequent one may construct either a derivation or a refutation, thus giving a constructive proof of a completeness theorem from which an implementation could be mechanically extracted (as in [23]). By inspection, we see that each rule of G4-C \preserves refutability" in the sense of Weich [23], i.e. if such a rule has a refutable premiss then the conclusion is refutable. In fact, the refutation rules \are" the rules of G4-C, except that we only consider one premiss at a time. It is also clear that the techniques used in [19] for extracting Kripke counter-models from refutations will construct, in the present case, only counter-models based on linear frames, because each of our refutation rules has at most one premiss: branching frames come from refutations using rules with more than one premiss. [23] discusses a stronger property than \refutability-preserving", namely the property of preserving counter-models: a rule has this property when, if a premiss has M as a counter-model then so does the conclusion. When all the rules have this property, then, one may conjecture, the logic is classical; however, a counterexample to this conjecture is given at the start of this section. Proposition 5.1 Each of the rules of G4-C except for R 0 preserves counter-models. Proof: Consider for example the left premiss?; DB ) CD; of an instance of. Suppose that the Kripke structure K based on the linearly ordered set with root k 0 is a counter-model to this premiss, so k 0 k?a for each A 2?, k 0 k?db but k 0 k6?cd and k 0 k6?a for each A 2. We must show that k 0 k?(cd)b. Since (CD)B is an implication, consider an arbitrary k 1 k 0 with k 1 k?cd: we must show that k 1 k?b. It suces to show, since k 1 k 0 and k 0 k?db, that k 1 k?d. Since k 0 k6?cd, there is k 2 k 0 with k 2 k?c but k 2 k6?d. If k 2 k 1, then we obtain

6 6 A Deterministic Terminating Sequent Calculus for Godel-Dummett logic a contradiction from k 1 k?cd, k 2 k?c and k 2 k6?d; by the linearity of, we infer that k 1 k 2. Since k 2 k?c we obtain that k 1 k?c, whence k 1 k?d. 2 Note that we are only discussing linear counter-models. Our conclusion then is that we see the invertibility of all the rules as giving us directly the \preservation of refutability" for each rule; the distinction in [23] between this and \preservation of counter-models" provides the distinction we initially missed between logic with linear frames and logic with trivial frames (i.e. classical logic). 6 Related work Dummett [8] showed that zero-order C was decidable: in fact, that a formula is satisable i satised in an n+1-node model, where n is the number of atoms occurring in the formula. The sequent calculus of Sonobe, requiring loop-checking, has already been mentioned above. The G4-C calculus presented is from [1], except that the latter 1. uses the non-invertible rule? ) P; B;? )?; P B ) A rather than our invertible rule 0; 2. treats negation as a primitive notion rather than as a dened notion; 3. does not use?; 4. allows 2 m? 1 versions of the R rule rather than our single version R 0, by allowing any non-empty subset of to be used in place of, and thus has much more non-determinism in proof search. 5. makes no irreducibility restriction on the R rule and thus does not have it as an invertible rule. [3] (page 548) gives a translation of formulae, such that a formula is C-derivable i its translation is classically derivable; the translation is cubic in the formula length, measured as the number of subformulae. Weich [24], following our suggestion in a talk at Arhus and using ideas from [17], proves the decidability of C by giving a constructive proof that (for a given sequent) either there exists a derivation or there exists a counter-model; this should lead as in [23] to a Scheme implementation. His algorithm is, in the worst case, apparently more ecient than, but also more complex than, that implicit in G4-C and that in [3]; how they all behave on average cases is not yet clear. 7 Conclusion Our proof of admissibility of the usual structural rules for the terminating sequent calculus G4-C, complete for Godel-Dummett logic, is short and simple; and our calculus improves in various ways on that of [1], mainly by providing the invertibility of all the rules, thus allowing (we believe) for more ecient implementation. More

7 A Deterministic Terminating Sequent Calculus for Godel-Dummett logic 7 interestingly, the invertibility of all rules allows a transparent explanation for the linearity of the frames that underlie the Kripke counter-models. It is not completely clear how to extend the main proof to the rst-order case covered in [2]: appropriate techniques have been developed in [11] for dealing with rst-order extensions of G4ip and have been extended to C in [12]. 8 Acknowledgments We thank both Klaus Weich and Pierangelo Miglioli (and the latter's colleagues) for unpublished papers ([24], [1], [2]) and Klaus Weich and Sara Negri for helpful discussions. Anne Troelstra and Grigori Mints gave us helpful (but contradictory) comments, for which we are also grateful: an anomous referee also made helpful remarks. References [1] A. Avellone, M. Ferrari, P. Miglioli, Duplication-free tableau calculi together with cut-free and contraction-free sequent calculi for the interpolable propositional intermediate logics. Technical Report 210{97, Dipartimento di Scienze dell'informazione, Universita degli Studi di Milano, (Submitted to JC). [2] A. Avellone, M. Ferrari, P. Miglioli, U. Moscato. A tableau calculus and a cut-free sequent calculus for Dummett predicate logic. (Position paper, Tableaux 98, obtainable from H. de Swart, Faculty of Philosophy, Tilburg University.) [3] A. Chagrov, M. Zakharyaschev, \Modal logic", Oxford University Press, [4] C.-. Chang, R.C.-T. ee, \Symbolic logic and mechanical theorem proving", Academic Press, [5] G. Corsi, A cut-free calculus for Dummett's C quantied, Zeitschr. f. math. ogik und Grundlagen d. Math., vol 35, pp. 289{301, [6] G. Corsi, Completeness theorem for Dummett's C quantied and some of its extensions, Studia ogica 51, pp 3176{335, [7] A. G. Dragalin. \Mathematical Intuitionism", Translations of Mathematical Monographs, vol. 67, American Math. Soc., Providence, Rhode Island, [8] M. Dummett, A propositional calculus with denumerable matrix. Journal of Symbolic ogic, vol. 24, pp. 96{107, [9] R. Dyckho. Contraction-free sequent calculi for intuitionistic logic, Journal of Symbolic ogic, vol. 57, pp. 795{807, [10] R. Dyckho. Dragalin's proofs of cut-admissibility for the intuitionistic sequent calculi G3i and G3i 0, Research Report CS/97/8, Computer Science Division, St Andrews University, 1997, available from \ [11] R. Dyckho, S. Negri. Admissibility of structural rules for contraction-free systems of intuitionistic logic. Journal of Symbolic ogic, accepted December [12] R. Dyckho, S. Negri. Admissibility of structural rules for contraction-free systems for extensions of intuitionistic logic. In preparation, [13] K. Godel, \On the intuitionistic propositional calculus", 1932, in Collected Works, Vol. 1, edited by S. Feferman et al, Oxford University Press, [14] V. Goranko. Refutation systems in modal logic, Studia ogica, vol. 53, pp. 299{324, [15] P. Hajek, \ Metamathematics of fuzzy logic", Kluwer Academic Publishers, [16] J. Hudelmaier. Bounds for cut elimination in intuitionistic propositional logic, Archive for Mathematical ogic, vol. 31, pp. 331{354, [17] J. Hudelmaier. An O(n log(n))-space decision procedure for intuitionistc propositional logic, Journal of ogic and Computation, vol. 3, pp. 63{75, 1993.

8 8 A Deterministic Terminating Sequent Calculus for Godel-Dummett logic [18] A. Maslov. Invertible sequential variant of constructive predicate calculus, Studies in Constructive Mathematics and Mathematical ogic, edited by A. O. Slisenko, V. A. Steklov Math. Inst., eningrad, part 1, pp 36{42, [19]. Pinto, R. Dyckho. oop-free construction of counter-models for intuitionistic propositional logic, Symposia Gaussiana, Conf. A, Eds. Behara/Fritsch/intz, Walter de Gruyter & Co, Berlin, pp 225{232, [20] O. Sonobe, A Gentzen-type formulation of some intermediate propositional logics, J. Tsuda College 7, pp 7{13, [21] A. S. Troelstra, H. Schwichtenberg. \Basic Proof Theory", Cambridge University Press, [22] N. N. Vorob'ev. A new algorithm for derivability in the constructive propositional calculus, American Math. Soc. Translations, ser. 2, vol. 94 (1970), pp 37{71. [23] K. Weich. Decision procedures for intuitionistic propositional logic by program extraction, Proceedings of the Tableaux 98 Conference, edited by H. de Swart, NCS 1397, Springer-Verlag, pp 292{306, [24] K. Weich. A bicomplete decision procedure for Dummett's logic C, draft of 31 August, Received March 31, 1999