Linda Buisman

Bio: Linda Buisman is an academic researcher from Australian National University. The author has contributed to research in topics: Cut-elimination theorem & Noncommutative logic. The author has an hindex of 2, co-authored 2 publications receiving 39 citations.

A Cut-free Sequent Calculus for Bi-Intuitionistic Logic: Extended Version

Abstract: Bi-intuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Bi-intuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent ``cut-free'' sequent calculus for BiInt has recently been shown by Uustalu to fail cut-elimination. We present a new cut-free sequent calculus for BiInt, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between implication and its dual, similarly to future and past modalities in tense logic. Our calculus handles this interaction using extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of failed derivation trees. Our simple termination argument allows our calculus to be used for automated deduction, although this is not its main purpose.

A Cut-Free Sequent Calculus for Bi-intuitionistic Logic

Abstract: Bi-intuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Bi-intuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent "cut-free" sequent calculus for BiInt has recently been shown by Uustalu to fail cut-elimination. We present a new cut-free sequent calculus for BiInt , and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between implication and its dual, similarly to future and past modalities in tense logic. Our calculus handles this interaction using extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of failed derivation trees. Our simple termination argument allows our calculus to be used for automated deduction, although this is not its main purpose.

Automated Reasoning with Analytic Tableaux and Related Methods: International Conference, TABLEAUX 2000 St Andrews, Scotland, UK, July 3-7, 2000 Proceedings

Abstract: Invited Lectures.- Tableau Algorithms for Description Logics.- Modality and Databases.- Local Symmetries in Propositional Logic.- Comparison.- Design and Results of TANCS-2000 Non-classical (Modal) Systems Comparison.- Consistency Testing: The RACE Experience.- Benchmark Analysis with FaCT.- MSPASS: Modal Reasoning by Translation and First-Order Resolution.- TANCS-2000 Results for DLP.- Evaluating *SAT on TANCS 2000 Benchmarks.- Research Papers.- A Labelled Tableau Calculus for Nonmonotonic (Cumulative) Consequence Relations.- A Tableau System for Godel-Dummett Logic Based on a Hypersequent Calculus.- An Analytic Calculus for Quantified Propositional Godel Logic.- A Tableau Method for Inconsistency-Adaptive Logics.- A Tableau Calculus for Integrating First-Order and Elementary Set Theory Reasoning.- Hypertableau and Path-Hypertableau Calculi for some Families of Intermediate Logics.- Variants of First-Order Modal Logics.- Complexity of Simple Dependent Bimodal Logics.- Properties of Embeddings from Int to S4.- Term-Modal Logics.- A Subset-Matching Size-Bounded Cache for Satisfiability in Modal Logics.- Dual Intuitionistic Logic Revisited.- Model Sets in a Nonconstructive Logic of Partial Terms with Definite Descriptions.- Search Space Compression in Connection Tableau Calculi Using Disjunctive Constraints.- Matrix-Based Inductive Theorem Proving.- Monotonic Preorders for Free Variable Tableaux.- The Mosaic Method for Temporal Logics.- Sequent-Like Tableau Systems with the Analytic Superformula Property for the Modal Logics KB, KDB, K5, KD5.- A Tableau Calculus for Equilibrium Entailment.- Towards Tableau-Based Decision Procedures for Non-Well-Founded Fragments of Set Theory.- Tableau Calculus for Only Knowing and Knowing At Most.- A Tableau-Like Representation Framework for Efficient Proof Reconstruction.- The Semantic Tableaux Version of the Second Incompleteness Theorem Extends Almost to Robinson's Arithmetic Q.- System Descriptions.- Redundancy-Free Lemmatization in the Automated Model-Elimination Theorem Prover AI-SETHEO.- E-SETHEO: An Automated3 Theorem Prover.

Constructive negation, implication, and co-implication

Abstract: In this paper, a family of paraconsistent propositional logics with constructive negation, constructive implication, and constructive co-implication is introduced. Although some fragments of these ...

Modal logics with Belnapian truth values

Abstract: Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification (support of truth) and the other for falsification (support of falsity). Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to twist structures over modal algebras. Sound and complete tableau calculi are presented as well. Moreover, a number of constructive non-modal logics with strong negation are faithfully embedded into BS4, into its three-valued extension B3S4, or into temporal BS4, BtS4. These logics include David Nelson's three-valued logic N3, the four-valued logic N4 bottom, the connexive logic C, and several extensions of bi-intuitionistic logic by strong ...

Combining Derivations and Refutations for Cut-free Completeness in Bi-intuitionistic Logic

Abstract: Bi-intuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent ‘cut-free’ sequent calculus has recently been shown to fail cut-elimination. We present a new cut-free sequent calculus for bi-intuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose.