R. Dyckhoff, "Contraction-free sequent calculi for intuitionistic logic", Journal of Symbolic Logic, vol. 57, pp. 795--807, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the rule (i) Loop rules may be a cause of severe problems in the proof search. First investigations in elimination of loop rules in intuitionistic calculus were presented by Vorobjov [42] and Orevkov [21] Loop free calculi for the intuitionistic propositional logic were constructed by Dyckhoff [5], Hudelmaier [11] Degtyarev and Voronkov [3] For the propositional modal logic S4 loop free calculus was constructed by Hudelmaier [12] The present paper is a generalization of the papers [30] 32] where loop free infinitary calculus for a fragment of FTL, namely, so called t D sequents (see ....

R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic. --- J. Symbolic Logic 57 (1972), 795--807.

....form. The completeness of the algorithm is proved. 1 Introduction The proof search is a very old subject and many algorithms are known in standard textbooks of logic. But still, much works are being done from the view point of Curry Howard isomorphism and analysis of substructural logics, e.g. [1,8]. But intuitionistic logic is not so intuitive when we consider unprovability. We can hardly accept the unprovability of a formula when we are shown a failure of proof search. A Kripke counter model can explain the unprovability more intuitively. This psychological gap between syntax and ....

....The aim of the present paper is to try to fill this gap. We demonstrate that long normal form proof [4,9] is suit not only for proof search but for counter model generation. Tableaux and sequent calculus with multiple conclusion are standard methods for counter model generation [2] Roy Dyckhoff [1] formulated a contractionfree sequent calculus LJT in which proof search does not loop. In [5] Pinto and Roy Dyckhoff used a variant of LJT and defined a calculus CRIP (Calculus for Refutation of Intuitionistic Propositions) that captures unprovability. They used LJT for proof generation and CRIP ....

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Roy Dyckhoff, Contraction-free sequent calculi for intuitionistic logic, Journal of Symbolic Logic 57(3),795--807, 1992.

.... ( DP96] DP98a] Herbelin calls his calculus LJT, but here we follow Dyckhoff Pinto in calling it MJ, as a calculus intermediate between natural deduction (NJ) and sequent calculus (LJ) This nomenclature also avoids a clash with the calculus here called G4, but elsewhere also called LJT, [Dyc92]) MJ has two kinds of sequent. One looks like the usual kind of sequent; however, only right rules and contraction are applicable to this kind of sequent in backwards proof search. By backwards proof search we mean proof search starting from the root. The other kind of sequent has a formula (on ....

....One can attempt to find a sequent formulation of the logic that terminates when used for backward proof search. An example of this is the contraction free calculus G4 for propositional intuitionistic logic, originating with Vorob ev ( Vor52] Vor58] and rediscovered and expounded by Dyckhoff ([Dyc92]) and by Hudelmaier ( Hud93] These contractionfree calculi are not easily discovered (indeed may not be possible) and so other methods can be useful. Another approach is to place conditions on the sequent calculus to ensure termination of search. It is elegant to be able to build the content of ....

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R. Dyckhoff. Contraction-Free Sequent Calculi for Intuitionistic Logic. Journal of Symbolic Logic, 57(3):795--807, 1992.

....very simple and familiar example the decision problem for classical tautologies. A more realistic example would be decision procedures commonly used such as arith or sup Gamma inf , but those are too complex to treat in these lectures, and the tautology procedure is used in some provers, see [18]. 5.1 Sample Decision Procedure The tableau decision procedure for classical propositional formulas is especially well understood (and still current, see the account in [40] The idea is that given a formula, we try to systematically falsify it by building up a truth assignment to the ....

R. Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. In The Journal of Symbolic Logic, pages Vol.57, Number 3, September 1992.

..... 38 9 Conclusion 40 3 Chapter 1 Introduction Intuitionistic logic has a strong connection to computer science, through its correspondence with type theory and constructive programming. Proof search algorithms and proof generation algorithms are known[1, 2, 3, 9, 11]. But intuitionistic logic is not so widely accepted in practice. One reason is the difficulty of the semantics of the logic. In classical propositional logic, the truth of a formula is determined with the truth values, i.e. with true and false, of atomic formulas in the formula. We can see if the ....

....a seed of a counter model if the input formula is not provable. So we need an algorithm that constructs a counter Kripke model from this deduction tree. This construction is explained in chapter 5. We prove the correctness and completeness of our algorithm in chapter 6. In other proving systems [2, 7, 8], the process of generating a counter model is completely separated from the process of searching the proof. In chapter 7 and 8, we show an implementation of the algorithm and its user interface through internet. We call the system SKIP. It stands for Simple Kripke model Internet Prover and can be ....

R. Dyckhoff: "Contraction-free sequent calculi for intuitionistic logic", The Journal of Symbolic Logic, 57(3), 1992.

....reuse by sharing. The optimized calculus iil is of independent interest. Although the optimized calculus iil is implicit in the work of Russian logician Vorobjev in the 1950 s, iil has been formulated explicitly only in the past several years independently by Lincoln et al. 25] by Dyckhoff [13], and by Hudelmaier [21] Versions of iil are being used in several theorem provers, built by Miller and Hodas (see above) by Paulson, by N. Tennant, and by Dyckhoff, Leslie, and Read. Recent work of Lincoln and Scedrov [24] is concerned with the expressiveness of first order linear logic ....

R. Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 57:795--807, 1992.

....right comma to be a disjunction. The following definition is thus natural: Definition 2.0. 2 We call native disjunction a disjunction for which the following rules are derivable: Gamma Delta; A; B Gamma Delta; A B Gamma Delta; A B Gamma Delta; A; B Dragalin [7] and Dikhoff [8] have suggested to restrict the sole introduction rule of implication of LK to sequents with at most one conclusion. The properties of such calculus depend on the 3 particular rules for disjunction (and also for 9 which we do not consider here) In case disjunction is second order defined as A ....

R. Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. The Journal of Symbolic Logic, 57(3):795--807, 1992.

....one hypothesis. But this restriction gives us a degenerated calculus or brings us back to the symmetrical categorical calculus. ffl A weaker restriction of LK to an intuitionistic calculus consists in restricting only the right introduction rule of implication to a unique conclusion. It is known [6, 7] that this propositional calculus is conservative over intuitionistic propositional logic, we will call it LK 1 . On the other hand, in the first order framework LK 1 is conservative over DIS logic (we already gave the proof of DIS, see proposition 4.2.1) By duality, this restriction extends ....

R. Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. The Journal of Symbolic Logic, 57(3):795--807, 1992.

....Of course, there is still room for further research. The current implementation is not a decision procedure for the propositional intuitionistic logic (which is decidable) since we need multiplicities already in this fragment. For example it would be interesting to integrate some techniques from [3] to get a decision procedure for the propositional fragment of intuitionistic logic. ....

R. Dyckhoff. Contraction-free Sequent Calculi for Intuitionistic Logic. Journal of Symbolic Logic, 57(3):795--807, 1992.

....is num. Error message : 1. Prolog failed The Prolog tactic was not able to prove the subgoal. 4.10.5 Tauto. This tactic, due to C esar Mu noz [70] implements a decision procedure for intuitionistic propositional calculus based on the contraction free sequent calculi LJT of R. Dyckhoff [37]. Note that Tauto succeeds on any instance of an intuitionistic tautological proposition. For instance it succeeds on (x:nat) P:nat Prop)x=O(P x) x=O (P x) while Auto fails. 4.10.6 Intuition. The tactic Intuition takes advantage of the search tree builded by the decision procedure involved ....

Roy Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. The Journal of Symbolic Logic, 57(3), September 1992.

....clauses, which makes it more likely to extend to the general case. In counterpart, some efficiency has to be traded. In this particular propositional setting, we can however refine our bottom up procedure into a decidable one. This procedure can be related to the contraction free sequent system in [4]. Our paper is organized as follows: in section 2, we define the propositional Hereditary Harrop formulas, and illustrate the differences between resolutions for Horn clauses and for Hereditary Harrop formulas. This resolution schema is formally defined in section 3. Section 4 is devoted to ....

....6 : Consider the program [ C oe B) oe A] C D oe B) D. The fixpoint semantics for this program is given by: W 0 T (W 0 ) T 2 (W 0 ) T 3 (W 0 ) D D D A C C D C D B B C D A 5. 3 A decidable refinement An easy consequence of Theorem 3 is the decidability of the phh fragment [2, 4]. Our bottom up resolution schema is however only semi decidable. Indeed, the external promotion rule can always be applied: consider a program P = C[ Delta] The external promotion rule can be applied to successively deeper occurrences of P and Delta, leading to the infinite derivation: ....

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Roy Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 57(3), September 1992.

....which does not contain existentially quantified goals. In particular, this argument shows that the propositional fragment does not require contraction on the right. It is then interesting to pursue the question of whether contraction on the left is required in the propositional case. Dyckhoff [4] has shown that it is possible to use a more intricate proof system in which contraction is not needed at all for any propositional fragment. In our case, we are interested in determining whether the standard rules of LM are contraction free for various propositional fragments. An intriguing ....

.... R (p (p q) q p (p q) R (p (p q) q; p (p q) q q L (p (p q) q q CL However, if we try to omit the contraction, we quickly arrive at an unprovable sequent: 7 p (p q) p (p q) q q L It is interesting to note that Dyckhoff shows in [4] that the formula : p :p) requires contraction in LJ; this formula is essentially the same as the above formula under the transformation of :F to F . We now proceed to show that propositional Horn clauses do not require contraction. We denote by D ; G ; the fragment defined by the ....

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R. Dyckhoff. Contraction-free Sequent Calculi for Intuitionistic Logic. Journal of Symbolic Logic, 57:795--807, 1992.

....1. Classical Logic: 1. LK: Two Sided Gentzen s SC [10] 2: A One Sided Formulation [24] 3: C3 : A SC without Structural Rules 2 2. Intuitionistic Logic: 1. LJ: Gentzen s SC [10] 2. G 3 : A SC without Structural Rules [20] 3. Non standard SC [42] 4. LJT: Contraction free Sequent Calculi [7] 5. other variants [42, 46] 3. Classical Linear Logic: 1. CLL: A Two Sided SC [14] 2. CLV: A Two sided Formulation with Zones 2 3. A One Sided Formulation [6] 4. One sided Implicit Weakening Fragment [45] 5. One sided Formulation without Structural Rules [29] 6. CLL1: One sided ....

....4. Intuitionistic Linear Logic: 1. ILL: A SC [6] 2. JLV: A SC with Four Zones 2 3. An specialised Fragment [12] 6. Minimal Linear Logic: 1. MILL: A SC 2 2. ILV: A SC with Three Zones 2 5. Multiple conclusioned Intuitionistic Logic: 1. A Standard SC [42] 2. MLJT: Multiple conclusioned LJT [7] 3. A variant of multiple conclusioned intuitionistic logic [46] 8. Light Linear Logic: 1. LLL: A One sided Formulation [15] 7. Modal Logic: 1. S4: A SC for Modal Logic [46] 2. S4 S SC for Modal Logic [26] 3. KT S SC for Modal Logic [26] 9. A SC for Logic with Finitely Many Variables ....

Dyckhoff R. Contraction-free Sequent Calculi for Intuitionistic Logic, The Journal of Symbolic Logic, Vol. 57, No. 3, Sept. 1992, pp.795-807 50

....be used effectively with a depth first interpreter because if the implication left rule can be used once, it can be used any number of times, thereby causing the interpreter to loop. Fortunately for this example, a contraction free presentation of propositional intuitionistic logic is given in [6]. That presentation can be expressed directly in this setting by replacing the one formula specifying implication elimination in Figure 5 with the (partial) axiomatization of object level atomic formulas and the five special cases of implication elimination in Figure 6. Executing this linear logic ....

Roy Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. Unpublished report from St. Andrews, Scotland, September 1990.

....35 5.2.14 prop (prop) searches for a proof of the current goal in minimal propositional logic. In particular it provides easy access to the axiom of truth for proving T and to ex falso quodlibet and proofby contradiction. The search mechanism is based on work of Hudelmaier [10, 11] and Dyckhoff [7]. If the search does not succeed, the same is done for intuitionistic propositional logic (by adding ex falso quodlibet assumptions to the context) If it does not succeed again, it does the same for classical propositional logic (by adding stability assumptions to the context) 5.2.15 search ....

Roy Dyckhoff. Contraction--free sequent calculi for intuitionistic logic. The Journal of Symbolic Logic, 57:793--807, 1992.

....on sequents making the hypotheses of each rule of the calculus less than its conclusion. The first order proposition A p OE will be defined by recursion over this well founded relation. The particular sequent calculus 2 we use is that given (independently) by Hudelmaier [6] and Dyckhoff [4]; its implicational part (the most important part) also occurs in the work of Lincoln, Scedrov and Shankar [8] In fact, essentially similar refinements of the sequent calculus for IpC were developed by the Russian school of Proof Theory some time ago see Vorob ev [13] Remark 2 It is perhaps ....

....) are all derivable from the rules in Table 3, and hence that these rules determine the same provable sequents as do those in Table 1. Note that ( is the only rule in Table 3 which fails to have the property that its premisses are structurally simpler than its conclusion. Following Dyckhoff [4] and Hudelmaier [6] we can overcome this defect by replacing ( by 6 0pOE ) 1 ) 3 0p(p OE) 0(OE 1 (OE 2 OE 3 ) 2 ) 0( OE 1 OE 2 ) OE 3 ) 0(OE 1 OE 3 ) OE 2 OE 3 ) 3 ) 0( OE 1 OE 2 ) OE 3 ) 0(OE 2 OE 3 ) OE 1 OE 2 0OE 3 ) 4 ) ....

R. Dyckhoff, Contraction-free Sequent Calculi for Intuitionistic Logic, preprint, University of St.Andrews, 1990.

.... to determine sequent calculi for disproofs in each case, although it is a lot simpler in the classical case than in the intuitionistic one the system given in [9] relies on a careful analysis of the intuitionistic propositional system so that the contraction rule can be eliminated entirely [3], which is a subtle and intricate problem. Whilst both of these approaches suffice for the detection of disprovable formulae, the Negation as Failure rule, as used in logic programs, requires both that a negation as failure connective be introduced, thus making the proof and disproof systems ....

R. Dyckhoff, Contraction-free Sequent Calculi for Intuitionistic Logic, Journal of Symbolic Logic 57:3:795-807, 1992.

.... Her program uses a loop check and tries to construct bottom up a repetition free derivation in a sequent calculus, similarly to the decision procedure already given by Gentzen [3] To avoid a loop check, contraction free sequent calculi were introduced by Hudelmaier [6] and Dyckhoff [1], rediscovering the work of Vorob ev [13] The main idea of their completeness proof is that every derivation in a sequent calculus can be transformed so that every left premise of a left rule for implication (L oe in our notation below) is either an axiom or the conclusion of a right rule (R : ....

....we will use the following property. Lemma 9. If K is a counter model to Gamma; B; Delta ) C, then also to Gamma; AoeB; Delta ) C. ut In the case that the premise of an implication is atomic and is in the left hand side already, we get the following rule(cf. Hudelmaier [6, 7] and Dyckhoff [1]) Gamma; C; Delta ) A L P oe, provided P 2 Gamma; Delta Gamma; P oe C; Delta ) A Corollary 10. L P oe preserves derivability and counter models. Proof. By L oe and Ax, we get that L P oe preserves derivability. Lemma 9 says that L P oe preserves counter models. ut The above section ....

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Roy Dyckhoff. Contraction--free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 57(3):795--807, 1992.

.... of the translations via an intermediate calculus, due to Herbelin in [26] and refined by Dyckhoff and Pinto as the permutation free sequent calculus in [15] Note that this new calculus is called MJ in [15] to avoid confusion between Herbelin s name LJT and Dyckhoff s different calculus LJT in [14]. There are therefore two distinct parts to this work. MJ must be shown to be isomorphic to natural deduction [15] and the reductions must be shown to be normalising with respect to the retraction of LJ into itself via MJ. Note that similar ideas to those of Dyckhoff and Pinto in [17] may also be ....

R. Dyckhoff, `Contraction-free sequent calculi for intuitionistic logic', J. Symbolic Logic 57 (1992) 795--807. 3

.... deductions in NJ (e.g. those in [12] where the strong eliminability of cut, rather than just the admissibility, is required; and, we anticipate, those in [5] and [15] Herbelin called (following [6] his calculus LJT , a name we avoid in case of confusion with that in the first author s [9]: we call its cut free fragment MJ because it is intermediate between [13] Gentzen s cut free LJ and NJ. We apologise to Herbelin for not adopting his nomenclature. We use MJ cut for the extension of MJ with cut rules; similarly, we use NJ cut for the usual natural deduction calculus and ....

Dyckhoff, R.: "Contraction-free sequent calculi for intuitionistic logic", Journal of Symbolic Logic 57 (1992), 795--807.

....Avellone et al., who show its completeness by model theoretic techniques. In our calculus, all the rules of G4 LC 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 IPL; following [21] we call this calculus G4ip. It has the property that root first proof search terminates, thus allowing easy implementation without a loop checker. See [9] for further history of this calculus, ....

....logic 1 Introduction In previous work [9] the author gave a contraction free calculus for zero order intuitionistic logic IPL; following [21] we call this calculus G4ip. It has the property that root first 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 ....

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R. Dyckhoff. Contraction-free sequent calculi for intuitionistic logic, Journal of Symbolic Logic, vol. 57, pp. 795--807, 1992.

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R. Dyckhoff, "Contraction-free sequent calculi for intuitionistic logic", Journal of Symbolic Logic, vol. 57, pp. 795--807, 1992.

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Roy Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 57:795--807, 1992.

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Roy Dyckhoff, Contraction-free sequent calculi for intuitionistic logic, The Journal of Symbolic Logic, vol. 57 (1992), no. 3, pp. 795-807.

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Roy Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 57(3):795--807, September 1992.

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