Tableau System

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Angelo Montanari - One of the best experts on this subject based on the ideXlab platform.

  • One-Pass and Tree-Shaped Tableau Systems for TPTL and TPTLb+Past
    Information & Computation, 2020
    Co-Authors: Luca Geatti, Nicola Gigante, Angelo Montanari, Mark Reynolds
    Abstract:

    Abstract Linear Temporal Logic ( LTL ) is one of the most commonly used formalisms for representing and reasoning about temporal properties of computations. Its application domains range from formal verification to artificial intelligence. Many real-time extensions of LTL have been proposed over the years, including Timed Propositional Temporal Logic ( TPTL ), that makes it possible to constrain the temporal ordering of pairs of events as well as the exact time elapsed between them. The paper focuses on TPTL and Bounded TPTL with Past ( ), a bounded variant of TPTL enriched with past operators, which has been recently introduced to formalise a meaningful class of timeline-based planning problems. allows one to refer to the past while keeping the computational complexity under control: in contrast to the full TPTL with Past ( TPTL+P ), whose satisfiability problem is non-elementary, the satisfiability problem for is -complete. The paper deals with the satisfiability problem for TPTL and by providing an original Tableau System for each of them that suitably generalises Reynolds' one-pass and tree-shaped Tableau for LTL . First, we show how to handle past operators, by devising a one-pass and tree-shaped Tableau System for LTL with Past ( LTL+P ). Then, we adapt it to TPTL and , providing full proofs of the soundness and completeness of the resulting Systems. In particular, completeness is proved by exploiting a novel model-theoretic argument that, compared to the one originally employed for the LTL System, provides a deeper understanding of the crucial role of the prune rule of the System.

  • a sat based encoding of the one pass and tree shaped Tableau System for ltl
    Theorem Proving with Analytic Tableaux and Related Methods, 2019
    Co-Authors: Luca Geatti, Nicola Gigante, Angelo Montanari
    Abstract:

    A new one-pass and tree-shaped Tableau System for Open image in new window satisfiability checking has been recently proposed, where each branch can be explored independently from others and, furthermore, directly corresponds to a potential model of the formula. Despite its simplicity, it proved itself to be effective in practice. In this paper, we provide a SAT-based encoding of such a Tableau System, based on the technique of bounded satisfiability checking. Starting with a single-node Tableau, i.e., depth k of the tree-shaped Tableau equal to zero, we proceed in an incremental fashion. At each iteration, the Tableau rules are encoded in a Boolean formula, representing all branches of the Tableau up to the current depth k. A typical downside of such bounded techniques is the effort needed to understand when to stop incrementing the bound, to guarantee the completeness of the procedure. In contrast, termination and completeness of the proposed algorithm is guaranteed without computing any upper bound to the length of candidate models, thanks to the Boolean encoding of the Open image in new window rule of the original Tableau System. We conclude the paper by describing a tool that implements our procedure, and comparing its performance with other state-of-the-art Open image in new window solvers.

  • TableauX - A SAT-Based Encoding of the One-Pass and Tree-Shaped Tableau System for LTL.
    Lecture Notes in Computer Science, 2019
    Co-Authors: Luca Geatti, Nicola Gigante, Angelo Montanari
    Abstract:

    A new one-pass and tree-shaped Tableau System for Open image in new window satisfiability checking has been recently proposed, where each branch can be explored independently from others and, furthermore, directly corresponds to a potential model of the formula. Despite its simplicity, it proved itself to be effective in practice. In this paper, we provide a SAT-based encoding of such a Tableau System, based on the technique of bounded satisfiability checking. Starting with a single-node Tableau, i.e., depth k of the tree-shaped Tableau equal to zero, we proceed in an incremental fashion. At each iteration, the Tableau rules are encoded in a Boolean formula, representing all branches of the Tableau up to the current depth k. A typical downside of such bounded techniques is the effort needed to understand when to stop incrementing the bound, to guarantee the completeness of the procedure. In contrast, termination and completeness of the proposed algorithm is guaranteed without computing any upper bound to the length of candidate models, thanks to the Boolean encoding of the Open image in new window rule of the original Tableau System. We conclude the paper by describing a tool that implements our procedure, and comparing its performance with other state-of-the-art Open image in new window solvers.

  • LPAR - A One-Pass Tree-Shaped Tableau for LTL+Past.
    2017
    Co-Authors: Nicola Gigante, Angelo Montanari, Mark Reynolds
    Abstract:

    Linear Temporal Logic (LTL) is a de-facto standard formalism for expressing properties of Systems and temporal constraints in formal verification, artificial intelligence, and other areas of computer science. The problem of LTL satisfiability is thus prominently important to check the consistency of these temporal specifications. Although adding past operators to LTL does not increase its expressive power, recently the interest for explicitly handling the past in temporal logics has increased because of the clarity and succinctness that those operators provide. In this work, a recently proposed one-pass tree-shaped Tableau System for LTL is extended to support past operators. The modularity of the required changes provides evidence for the claimed ease of extensibility of this Tableau System.

  • IJCAI - Leviathan: a new LTL satisfiability checking tool based on a one-pass tree-shaped Tableau
    2016
    Co-Authors: Matteo Bertello, Nicola Gigante, Angelo Montanari, Mark Reynolds
    Abstract:

    The paper presents Leviathan, an LTL satisfiability checking tool based on a novel one-pass, treelike Tableau System, which is way simpler than existing solutions. Despite the simplicity of the algorithm, the tool has performance comparable in speed and memory consumption with other tools on a number of standard benchmark sets, and, in various cases, it outperforms the other Tableau-based tools.

Patrick Blackburn - One of the best experts on this subject based on the ideXlab platform.

  • formalizing a seligman style Tableau System for hybrid logic
    International Joint Conference on Automated Reasoning, 2020
    Co-Authors: Asta Halkjaer From, Patrick Blackburn, Jorgen Villadsen
    Abstract:

    Hybrid logic is modal logic enriched with names for worlds. We formalize soundness and completeness proofs for a Seligman-style Tableau System for hybrid logic in the proof assistant Isabelle/HOL. The formalization shows how to lift certain rule restrictions, thereby simplifying the original un-formalized proof. Moreover, the completeness proof we formalize is synthetic which suggests we can extend this work to prove a wider range of results about hybrid logic.

  • Completeness and Termination for a Seligman-style Tableau System
    Journal of Logic and Computation, 2015
    Co-Authors: Patrick Blackburn, Thomas Bolander, Torben Brauner, Klaus Frovin Jørgensen
    Abstract:

    Proof Systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labelling approach lies at the heart of the best known hybrid resolution, natural deduction, and Tableau Systems. But there is another approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was rst introduced and explored by Jerry Seligman in natural deduction [31] and sequent calculus [32] in the 1990s. The purpose of this paper is to introduce a Seligman-style Tableau System, to prove its completeness, and to show how it can be made to terminate. The most obvious feature of Seligman-style Systems is that they work with arbitrary formulas, not just statements prexed by @-operators. They do so by introducing machinery for switching to other proof contexts. We capture this idea in the setting of Tableaus by introducing a rule called GoTo which allows us to \jump to a named world" on a Tableau branch. We rst develop a Seligman-style Tableau System for basic hybrid logic and prove its completeness. We then prove termination of a restricted version of the System without resorting to loop checking, and show that the restrictions do not eect completeness. Both completeness and termination results are proved by explicit translations that transform Tableaus in a standard labelled System into Seligman-style Tableaus and vice-versa.

  • a seligman style Tableau System
    International Conference on Logic Programming, 2013
    Co-Authors: Patrick Blackburn, Thomas Bolander, Torben Brauner, Klaus Frovin Jørgensen
    Abstract:

    Proof Systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labeling approach lies at the heart of most resolution, natural deduction, and Tableau Systems for hybrid logic. But there is another, less well-known approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was first introduced and explored by Jerry Seligman in the setting of natural deduction and sequent calculus in the late 1990s. The purpose of this paper is to introduce a Seligman-style Tableau System.

  • LPAR - A Seligman-Style Tableau System
    Logic for Programming Artificial Intelligence and Reasoning, 2013
    Co-Authors: Patrick Blackburn, Thomas Bolander, Torben Brauner, Klaus Frovin Jørgensen
    Abstract:

    Proof Systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labeling approach lies at the heart of most resolution, natural deduction, and Tableau Systems for hybrid logic. But there is another, less well-known approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was first introduced and explored by Jerry Seligman in the setting of natural deduction and sequent calculus in the late 1990s. The purpose of this paper is to introduce a Seligman-style Tableau System.

Klaus Frovin Jørgensen - One of the best experts on this subject based on the ideXlab platform.

  • Completeness and Termination for a Seligman-style Tableau System
    Journal of Logic and Computation, 2015
    Co-Authors: Patrick Blackburn, Thomas Bolander, Torben Brauner, Klaus Frovin Jørgensen
    Abstract:

    Proof Systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labelling approach lies at the heart of the best known hybrid resolution, natural deduction, and Tableau Systems. But there is another approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was rst introduced and explored by Jerry Seligman in natural deduction [31] and sequent calculus [32] in the 1990s. The purpose of this paper is to introduce a Seligman-style Tableau System, to prove its completeness, and to show how it can be made to terminate. The most obvious feature of Seligman-style Systems is that they work with arbitrary formulas, not just statements prexed by @-operators. They do so by introducing machinery for switching to other proof contexts. We capture this idea in the setting of Tableaus by introducing a rule called GoTo which allows us to \jump to a named world" on a Tableau branch. We rst develop a Seligman-style Tableau System for basic hybrid logic and prove its completeness. We then prove termination of a restricted version of the System without resorting to loop checking, and show that the restrictions do not eect completeness. Both completeness and termination results are proved by explicit translations that transform Tableaus in a standard labelled System into Seligman-style Tableaus and vice-versa.

  • a seligman style Tableau System
    International Conference on Logic Programming, 2013
    Co-Authors: Patrick Blackburn, Thomas Bolander, Torben Brauner, Klaus Frovin Jørgensen
    Abstract:

    Proof Systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labeling approach lies at the heart of most resolution, natural deduction, and Tableau Systems for hybrid logic. But there is another, less well-known approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was first introduced and explored by Jerry Seligman in the setting of natural deduction and sequent calculus in the late 1990s. The purpose of this paper is to introduce a Seligman-style Tableau System.

  • LPAR - A Seligman-Style Tableau System
    Logic for Programming Artificial Intelligence and Reasoning, 2013
    Co-Authors: Patrick Blackburn, Thomas Bolander, Torben Brauner, Klaus Frovin Jørgensen
    Abstract:

    Proof Systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labeling approach lies at the heart of most resolution, natural deduction, and Tableau Systems for hybrid logic. But there is another, less well-known approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was first introduced and explored by Jerry Seligman in the setting of natural deduction and sequent calculus in the late 1990s. The purpose of this paper is to introduce a Seligman-style Tableau System.

Torben Brauner - One of the best experts on this subject based on the ideXlab platform.

  • Many-Valued Hybrid Logic
    2020
    Co-Authors: Jens Ulrik Hansen, Thomas Bolander, Torben Brauner
    Abstract:

    In this paper we define a many-valued semantics for hybrid logic and we give a sound and complete Tableau System which is proof-theoretically well-behaved, in particular, it gives rise to a decision procedure for the logic. This shows that many-valued hybrid logics is a natural enterprise and opens up the way for future applications.

  • Advances in Modal Logic - Many-valued hybrid logic.
    2020
    Co-Authors: Jens Ulrik Hansen, Thomas Bolander, Torben Brauner
    Abstract:

    In this paper we define a many-valued semantics for hybrid logic and we give a sound and complete Tableau System which is proof theoretically well-behaved, in particular, it gives rise to a decision procedure for the logic. This shows that many-valued hybrid logics is a natural enterprise and opens up the way for future applications.

  • Completeness and Termination for a Seligman-style Tableau System
    Journal of Logic and Computation, 2015
    Co-Authors: Patrick Blackburn, Thomas Bolander, Torben Brauner, Klaus Frovin Jørgensen
    Abstract:

    Proof Systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labelling approach lies at the heart of the best known hybrid resolution, natural deduction, and Tableau Systems. But there is another approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was rst introduced and explored by Jerry Seligman in natural deduction [31] and sequent calculus [32] in the 1990s. The purpose of this paper is to introduce a Seligman-style Tableau System, to prove its completeness, and to show how it can be made to terminate. The most obvious feature of Seligman-style Systems is that they work with arbitrary formulas, not just statements prexed by @-operators. They do so by introducing machinery for switching to other proof contexts. We capture this idea in the setting of Tableaus by introducing a rule called GoTo which allows us to \jump to a named world" on a Tableau branch. We rst develop a Seligman-style Tableau System for basic hybrid logic and prove its completeness. We then prove termination of a restricted version of the System without resorting to loop checking, and show that the restrictions do not eect completeness. Both completeness and termination results are proved by explicit translations that transform Tableaus in a standard labelled System into Seligman-style Tableaus and vice-versa.

  • a seligman style Tableau System
    International Conference on Logic Programming, 2013
    Co-Authors: Patrick Blackburn, Thomas Bolander, Torben Brauner, Klaus Frovin Jørgensen
    Abstract:

    Proof Systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labeling approach lies at the heart of most resolution, natural deduction, and Tableau Systems for hybrid logic. But there is another, less well-known approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was first introduced and explored by Jerry Seligman in the setting of natural deduction and sequent calculus in the late 1990s. The purpose of this paper is to introduce a Seligman-style Tableau System.

  • LPAR - A Seligman-Style Tableau System
    Logic for Programming Artificial Intelligence and Reasoning, 2013
    Co-Authors: Patrick Blackburn, Thomas Bolander, Torben Brauner, Klaus Frovin Jørgensen
    Abstract:

    Proof Systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labeling approach lies at the heart of most resolution, natural deduction, and Tableau Systems for hybrid logic. But there is another, less well-known approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was first introduced and explored by Jerry Seligman in the setting of natural deduction and sequent calculus in the late 1990s. The purpose of this paper is to introduce a Seligman-style Tableau System.

Jorgen Villadsen - One of the best experts on this subject based on the ideXlab platform.

  • formalizing a seligman style Tableau System for hybrid logic
    International Joint Conference on Automated Reasoning, 2020
    Co-Authors: Asta Halkjaer From, Patrick Blackburn, Jorgen Villadsen
    Abstract:

    Hybrid logic is modal logic enriched with names for worlds. We formalize soundness and completeness proofs for a Seligman-style Tableau System for hybrid logic in the proof assistant Isabelle/HOL. The formalization shows how to lift certain rule restrictions, thereby simplifying the original un-formalized proof. Moreover, the completeness proof we formalize is synthetic which suggests we can extend this work to prove a wider range of results about hybrid logic.