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Bentzen Bruno - One of the best experts on this subject based on the ideXlab platform.

  • A Henkin-style Completeness Proof for the modal logic S5
    2021
    Co-Authors: Bentzen Bruno
    Abstract:

    This paper presents a recent formalization of a Henkin-style Completeness Proof for the propositional modal logic S5 using the Lean theorem prover. The Proof formalized is close to that of Hughes and Cresswell, but the system, based on a different choice of axioms, is better described as a Mendelson system augmented with axiom schemes for K, T, S4, and B, and the necessitation rule as a rule of inference. The language has the false and implication as the only primitive logical connectives and necessity as the only primitive modal operator. The full source code is available online and has been typechecked with Lean 3.4.2

  • A Henkin-style Completeness Proof for the modal logic S5
    'Springer Science and Business Media LLC', 2021
    Co-Authors: Bentzen Bruno
    Abstract:

    This paper presents a recent formalization of a Henkin-style Completeness Proof for the propositional modal logic S5 using the Lean theorem prover. The Proof formalized is close to that of Hughes and Cresswell, but the system, based on a different choice of axioms, is better described as a Mendelson system augmented with axiom schemes for K, T, S4, and B, and the necessitation rule as a rule of inference. The language has the false and implication as the only primitive logical connectives and necessity as the only primitive modal operator. The full source code is available online at https://github.com/bbentzen/mpl/ and has been typechecked with Lean 3.4.2

  • A Henkin-style Completeness Proof for the modal logic S5
    2019
    Co-Authors: Bentzen Bruno
    Abstract:

    This paper presents a recent formalization of a Henkin-style Completeness Proof for the propositional modal logic S5 using the Lean theorem prover. The Proof formalized is close to that of Hughes and Cresswell [9], except that it is given for a system based on a different choice of axioms. Here the Proof is based on a Hilbert-style presentation better described as a Mendelson system augmented with axiom schemes for K, T, S4, and B, and the necessitation rule as rule of inference. The language has the false and implication as the only primitive logical connectives and necessity as the only primitive modal operator. The full source code is available online and has been typechecked with Lean 3.4.1

Ryo Kashima - One of the best experts on this subject based on the ideXlab platform.

  • Completeness Proof by semantic diagrams for transitive closure of accessibility relation
    Advances in Modal Logic, 2010
    Co-Authors: Ryo Kashima
    Abstract:

    We treat the smallest normal modal propositional logic with two modal operators 2 and 2+. While 2 is interpreted in Kripke models by the accessibility relation R, 2+ is interpreted by the transitive closure of R. Intuitively the formula 2+φ means the infinite conjunction 2φ ∧ 22φ ∧ 222φ ∧ · · · . There is a Hilbert style axiomatization of this logic (a characteristic axiom is 2φ ∧ 2+(φ → 2φ) → 2+φ, called “induction axiom”), and its Completeness with respect to finite models was shown by the canonical model method. This paper gives an alternative Proof of this Completeness. We use the method of “semantic diagram”, which is a variant of semantic tableaux, as follows. Given an unprovable formula φ, we first make a small model (consisting of one world that forces φ to be false); then we add worlds step by step using the Hilbert system as an oracle, and finally we get a finite countermodel for φ. The point is how to handle 2+ in this construction.

Ben C Moszkowski - One of the best experts on this subject based on the ideXlab platform.

  • a complete axiom system for propositional interval temporal logic with infinite time
    Logical Methods in Computer Science, 2012
    Co-Authors: Ben C Moszkowski
    Abstract:

    Interval Temporal Logic (ITL) is an established temporal formalism for reasoning about time periods. For over 25 years, it has been applied in a number of ways and several ITL variants, axiom systems and tools have been investigated. We solve the longstanding open problem of finding a complete axiom system for basic quantifier-free propositional ITL (PITL) with infinite time for analysing nonterminating computational systems. Our Completeness Proof uses a reduction to Completeness for PITL with finite time and conventional propositional linear-time temporal logic. Unlike Completeness Proofs of equally expressive logics with nonelementary computational complexity, our semantic approach does not use tableaux, subformula closures or explicit deductions involving encodings of omega automata and nontrivial techniques for complementing them. We believe that our result also provides evidence of the naturalness of interval-based reasoning.

  • a hierarchical Completeness Proof for propositional interval temporal logic with finite time
    Journal of Applied Non-Classical Logics, 2004
    Co-Authors: Ben C Moszkowski
    Abstract:

    We present a Completeness Proof for Propositional Interval Temporal Logic (PITL) with finite time which avoids certain difficulties of conventional methods. It is more gradated than previous efforts since we progressively reduce reasoning within the original logic to simpler reasoning in sublogics. Furthermore, our approach benefits from being less constructive since it is able to invoke certain theorems about regular languages over finite words without the need to explicitly describe the associated intricate Proofs. A modified version of regular expressions called Fusion Expressions is used as part of an intermediate logic called Fusion Logic. Both have the same expressiveness as PITL but are lower-level notations which play an important role in the hierarchical structure of the overall Completeness Proof. In particular, showing Completeness for PITL is reduced to showing Completeness for Fusion Logic. This in turn is shown to hold relative to Completeness for conventional linear-time temporal logic with...

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.

Waldmann Clara - One of the best experts on this subject based on the ideXlab platform.

  • Existence and Complexity of Approximate Equilibria in Weighted Congestion Games
    2020
    Co-Authors: Christodoulou George, Gairing Martin, Giannakopoulos Yiannis, Poças Diogo, Waldmann Clara
    Abstract:

    We study the existence of approximate pure Nash equilibria ($\alpha$-PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree $d$. Previously it was known that $d$-approximate equilibria always exist, while nonexistence was established only for small constants, namely for $1.153$-PNE. We improve significantly upon this gap, proving that such games in general do not have $\tilde{\Theta}(\sqrt{d})$-approximate PNE, which provides the first super-constant lower bound. Furthermore, we provide a black-box gap-introducing method of combining such nonexistence results with a specific circuit gadget, in order to derive NP-Completeness of the decision version of the problem. In particular, deploying this technique we are able to show that deciding whether a weighted congestion game has an $\tilde{O}(\sqrt{d})$-PNE is NP-complete. Previous hardness results were known only for the special case of exact equilibria and arbitrary cost functions. The circuit gadget is of independent interest and it allows us to also prove hardness for a variety of problems related to the complexity of PNE in congestion games. For example, we demonstrate that the question of existence of $\alpha$-PNE in which a certain set of players plays a specific strategy profile is NP-hard for any $\alpha < 3^{d/2}$, even for unweighted congestion games. Finally, we study the existence of approximate equilibria in weighted congestion games with general (nondecreasing) costs, as a function of the number of players $n$. We show that $n$-PNE always exist, matched by an almost tight nonexistence bound of $\tilde\Theta(n)$ which we can again transform into an NP-Completeness Proof for the decision problem

  • Existence and Complexity of Approximate Equilibria in Weighted Congestion Games
    LIPIcs - Leibniz International Proceedings in Informatics. 47th International Colloquium on Automata Languages and Programming (ICALP 2020), 2020
    Co-Authors: Christodoulou George, Gairing Martin, Giannakopoulos Yiannis, Waldmann Clara
    Abstract:

    We study the existence of approximate pure Nash equilibria (?-PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree d. Previously it was known that d-approximate equilibria always exist, while nonexistence was established only for small constants, namely for 1.153-PNE. We improve significantly upon this gap, proving that such games in general do not have ??(?d)-approximate PNE, which provides the first super-constant lower bound. Furthermore, we provide a black-box gap-introducing method of combining such nonexistence results with a specific circuit gadget, in order to derive NP-Completeness of the decision version of the problem. In particular, deploying this technique we are able to show that deciding whether a weighted congestion game has an O?(?d)-PNE is NP-complete. Previous hardness results were known only for the special case of exact equilibria and arbitrary cost functions. The circuit gadget is of independent interest and it allows us to also prove hardness for a variety of problems related to the complexity of PNE in congestion games. For example, we demonstrate that the question of existence of ?-PNE in which a certain set of players plays a specific strategy profile is NP-hard for any ? < 3^(d/2), even for unweighted congestion games. Finally, we study the existence of approximate equilibria in weighted congestion games with general (nondecreasing) costs, as a function of the number of players n. We show that n-PNE always exist, matched by an almost tight nonexistence bound of ??(n) which we can again transform into an NP-Completeness Proof for the decision problem