Generalized Tableau Systems for Intemediate Propositional Logics
13 May 1997-pp 43-61
TL;DR: Some sufficient conditions from which, given a well formed formula H, the search for instances of an intermediate propositional logic L can be restricted to a suitable finite set of formulae related to H are studied.
Abstract: Given an intermediate propositional logic L (obtained by adding to intuitionistic logic INT a single axiom-scheme), a pseudo tableau system for L can be given starting from any intuitionistic tableau system and adding a rule which allows to insert in any line of a proof table suitable T-signed instances of the axiom-scheme. In this paper we study some sufficient conditions from which, given a well formed formula H, the search for these instances can be restricted to a suitable finite set of formulae related to H. We illustrate our techniques by means of some known logics, namely, the logic D of Dummett, the logics PR k (k≥1) of Nagata, the logics FIN m (m≥1), the logics G n (n≥1) of Gabbay and de Jongh, and the logic KP of Kreisel and Putnam
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TL;DR: The method generates a set of tableau inference rules that can then be used to reason within the logic and guarantees that the generated rules form a calculus which is sound and constructively complete.
Abstract: This paper presents a method for synthesising sound and complete tableau calculi. Given a specification of the formal semantics of a logic, the method generates a set of tableau inference rules that can then be used to reason within the logic. The method guarantees that the generated rules form a calculus which is sound and constructively complete. If the logic can be shown to admit finite filtration with respect to a well-defined first-order semantics then adding a general blocking mechanism provides a terminating tableau calculus. The process of generating tableau rules can be completely automated and produces, together with the blocking mechanism, an automated procedure for generating tableau decision procedures. For illustration we show the workability of the approach for a description logic with transitive roles and propositional intuitionistic logic.
27 citations
Cites methods from "Generalized Tableau Systems for Int..."
...frequently used (see, for example, [18, 13, 21, 29, 14, 23]; some recent work is [20, 6]). Tableau calculi have been developed and are being used for non-classical logics such as intuitionistic logic [18, 3], conditional logic [2], logics of metric and topology [27] and hybrid logics [37, 10, 15]. Rather than developing tableau calculi one by one for individual logics, it is possible to develop tableau c...
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TL;DR: In this article, the authors present a method for synthesizing sound and complete tableau calculi, given a specification of the formal semantics of a logic, and a set of tableau inference rules that can then be used to reason within the logic.
Abstract: This paper presents a method for synthesising sound and complete tableau calculi. Given a specification of the formal semantics of a logic, the method generates a set of tableau inference rules that can then be used to reason within the logic. The method guarantees that the generated rules form a calculus which is sound and constructively complete. If the logic can be shown to admit finite filtration with respect to a well-defined first-order semantics then adding a general blocking mechanism provides a terminating tableau calculus. The process of generating tableau rules can be completely automated and produces, together with the blocking mechanism, an automated procedure for generating tableau decision procedures. For illustration we show the workability of the approach for a description logic with transitive roles and propositional intuitionistic logic.
24 citations
TL;DR: In this article, a tableau calculus with a multiple premise rule and optimizations is presented to decide propositional Dummett logic, and the resulting implementation outperforms the state of the art graph-based procedure.
10 citations
TL;DR: This work develops a complementary perspective on the generalised subformula properties in terms of logical embeddings and yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory.
Abstract: A sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics (specifically: those with a cut-free hypersequent calculus). We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory.
3 citations
Cites methods from "Generalized Tableau Systems for Int..."
...[1] present a semantic-based method and so-called selection functions to establish that certain intermediate logics are—in our terminology—variable-axiomatisations and (a variant of) formula-axiomatisations of intuitionistic logic....
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01 Jan 2001
TL;DR: This note reports on some experiments, using a handful of standard automated reasoning tools, for exploring Steinitz-Rademacher polyhedra, which are models of a certain first-order theory of incidence structures.
Abstract: This note reports on some experiments, using a handful of standard automated reasoning tools, for exploring Steinitz-Rademacher polyhedra, which are models of a certain first-order theory of incidence structures. This theory and its models, even simple ones, presents significant, geometrically fascinating challenges for automated reasoning tools are.
3 citations
Cites background or methods from "Generalized Tableau Systems for Int..."
...Integration with a genuine higher-order automatic theorem prover, such as LEO-II [5] and Satallax [3], seems necessary....
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...2The answer, known as Steinitz’s theorem [3], says that a directed graph g is isomorphic to the 1-skeleton of a real convex three-dimensional polyhedron iff g is planar and three-connected....
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...Sledgehammer’s performance on higher-order problems is unimpressive, and given the inherent difficulty of performing higher-order reasoning using first-order theorem provers, the way forward is to integrate Sledgehammer with higher-order automatic theorem provers, such as LEO-II [5] and Satallax [3]....
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References
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TL;DR: An equivalent calculus is described, exploiting the Dershowtiz-Manna theorem on multiset orderings, where the problem no longer arises: this gives a simple but effective decision procedure for IPC.
Abstract: ?0. Prologue. 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.
322 citations
Book•
01 Jan 1981
TL;DR: In this article, the Kripke, Beth and Topological Interpretations for HPC are presented, as well as three intermediate logics for Propositional Calculus and three intermediate Logics for Formulas in One Variable.
Abstract: Logical Systems and Semantics.- Introducing HPC.- The Kripke, Beth and Topological Interpretations for HPC.- Heyting's Propositional Calculus and Extensions.- Three Intermediate Logics.- Formulas in One Variable.- Propositional Connectives.- The Interpolation Theorem.- Second Order Propositional Calculus.- Modified Kripke Interpretation.- Theories in HPC 1.- Theories in HPC 2.- Completeness of HPC with Respect to RE and Post Structures.- Undecidability Results.- Decidability Results.
210 citations
123 citations