Pub Date : 2022-07-26DOI: 10.48550/arXiv.2207.12953
J. Keiren, R. Cleaveland
This paper revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus in order to develop proof techniques that permit the seamless inclusion of new features in this logic. Our approach relies on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reason about the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableau method is sound and complete for timed transition systems.
{"title":"Extensible Proof Systems for Infinite-State Systems","authors":"J. Keiren, R. Cleaveland","doi":"10.48550/arXiv.2207.12953","DOIUrl":"https://doi.org/10.48550/arXiv.2207.12953","url":null,"abstract":"This paper revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus in order to develop proof techniques that permit the seamless inclusion of new features in this logic. Our approach relies on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reason about the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableau method is sound and complete for timed transition systems.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46662838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-22DOI: https://dl.acm.org/doi/10.1145/3506702
Erfan Khaniki
The proof system Res (PCd,R) is a natural extension of the Resolution proof system that instead of disjunctions of literals operates with disjunctions of degree d multivariate polynomials over a ring R with Boolean variables. Proving super-polynomial lower bounds for the size of Res(PC1,R)-refutations of Conjunctive normal forms (CNFs) is one of the important problems in propositional proof complexity. The existence of such lower bounds is even open for Res(PC1,𝔽) when 𝔽 is a finite field, such as 𝔽2. In this article, we investigate Res(PCd,R) and tree-like Res(PCd,R) and prove size-width relations for them when R is a finite ring. As an application, we prove new lower bounds and reprove some known lower bounds for every finite field 𝔽 as follows:
(1)
We prove almost quadratic lower bounds for Res(PCd,𝔽)-refutations for every fixed d. The new lower bounds are for the following CNFs:
(a)
Mod q Tseitin formulas (char(𝔽)≠ q) and Flow formulas,
(b)
Random k-CNFs with linearly many clauses.
(2)
We also prove super-polynomial (more than nk for any fixed k) and also exponential (2nϵ for an ϵ > 0) lower bounds for tree-like Res(PCd,𝔽)-refutations based on how big d is with respect to n for the following CNFs:
(a)
Mod q Tseitin formulas (char(𝔽)≠ q) and Flow formulas,
(b)
Random k-CNFs of suitable densities,
(c)
Pigeonhole principle and Counting mod q principle.
The lower bounds for the dag-like systems are the first nontrivial lower bounds for these systems, including the case d=1. The lower bounds for the tree-like systems were known for the case d=1 (except for the Counting mod q principle, in which
{"title":"On Proof Complexity of Resolution over Polynomial Calculus","authors":"Erfan Khaniki","doi":"https://dl.acm.org/doi/10.1145/3506702","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3506702","url":null,"abstract":"<p>The proof system <sans-serif>Res (PC</sans-serif><sub><i>d,R</i></sub>) is a natural extension of the Resolution proof system that instead of disjunctions of literals operates with disjunctions of degree <i>d</i> multivariate polynomials over a ring <i>R</i> with Boolean variables. Proving super-polynomial lower bounds for the size of <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub>1,<i>R</i></sub>)-refutations of Conjunctive normal forms (CNFs) is one of the important problems in propositional proof complexity. The existence of such lower bounds is even open for <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub>1,𝔽</sub>) when 𝔽 is a finite field, such as 𝔽<sub>2</sub>. In this article, we investigate <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub><i>d,R</i></sub>) and tree-like <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub><i>d,R</i></sub>) and prove size-width relations for them when <i>R</i> is a finite ring. As an application, we prove new lower bounds and reprove some known lower bounds for every finite field 𝔽 as follows:\u0000<p><table border=\"0\" list-type=\"ordered\" width=\"95%\"><tr><td valign=\"top\"><p>(1)</p></td><td colspan=\"5\" valign=\"top\"><p>We prove almost quadratic lower bounds for <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub><i>d</i></sub>,𝔽)-refutations for every fixed <i>d</i>. The new lower bounds are for the following CNFs:</p><p><table border=\"0\" list-type=\"ordered\" width=\"95%\"><tr><td valign=\"top\"><p>(a)</p></td><td colspan=\"5\" valign=\"top\"><p>Mod <i>q</i> Tseitin formulas (<i>char</i>(𝔽)≠ <i>q</i>) and Flow formulas,</p></td></tr><tr><td valign=\"top\"><p>(b)</p></td><td colspan=\"5\" valign=\"top\"><p>Random <i>k</i>-CNFs with linearly many clauses.</p></td></tr></table></p></td></tr><tr><td valign=\"top\"><p>(2)</p></td><td colspan=\"5\" valign=\"top\"><p>We also prove super-polynomial (more than <i>n</i><sup><i>k</i></sup> for any fixed <i>k</i>) and also exponential (2<i><sup>nϵ</sup></i> for an ϵ > 0) lower bounds for tree-like <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub><i>d</i>,𝔽</sub>)-refutations based on how big <i>d</i> is with respect to <i>n</i> for the following CNFs:</p><p><table border=\"0\" list-type=\"ordered\" width=\"95%\"><tr><td valign=\"top\"><p>(a)</p></td><td colspan=\"5\" valign=\"top\"><p>Mod <i>q</i> Tseitin formulas (<i>char</i>(𝔽)≠ <i>q</i>) and Flow formulas,</p></td></tr><tr><td valign=\"top\"><p>(b)</p></td><td colspan=\"5\" valign=\"top\"><p>Random <i>k</i>-CNFs of suitable densities,</p></td></tr><tr><td valign=\"top\"><p>(c)</p></td><td colspan=\"5\" valign=\"top\"><p>Pigeonhole principle and Counting mod <i>q</i> principle.</p></td></tr></table></p></td></tr></table></p> The lower bounds for the dag-like systems are the first nontrivial lower bounds for these systems, including the case <i>d</i>=1. The lower bounds for the tree-like systems were known for the case <i>d</i>=1 (except for the Counting mod <i>q</i> principle, in which","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138542040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The article discusses temporal information systems (TISs) that add the dimension of time to complete or incomplete information systems. Through TISs, one can accommodate the possibility of domains or attribute values for objects changing with time or the availability of currently missing information with time. Different patterns of flow of information give different TISs. The corresponding logics with sound and complete axiomatization are presented.
{"title":"Logics for Temporal Information Systems in Rough Set Theory","authors":"Md. Aquil Khan, M. Banerjee, Sibsankar Panda","doi":"10.1145/3549075","DOIUrl":"https://doi.org/10.1145/3549075","url":null,"abstract":"The article discusses temporal information systems (TISs) that add the dimension of time to complete or incomplete information systems. Through TISs, one can accommodate the possibility of domains or attribute values for objects changing with time or the availability of currently missing information with time. Different patterns of flow of information give different TISs. The corresponding logics with sound and complete axiomatization are presented.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49318217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Simon Doherty, Sadegh Dalvandi, Brijesh Dongol, H. Wehrheim
In this article, we propose an approach to program verification using an abstract characterisation of weak memory models. Our approach is based on a hierarchical axiom scheme that captures the observational properties of a memory model. In particular, we show that it is possible to prove correctness of a program with respect to a particular axiom scheme, and we show this proof to suffice for any memory model that satisfies the axioms. Our axiom scheme is developed using a characterisation of weakest liberal preconditions for weak memory. This characterisation naturally extends to Hoare logic and Owicki-Gries reasoning by lifting weakest liberal preconditions (defined over read/write events) to the level of programs. We study three memory models (SC, TSO, and RC11-RAR) as example instantiations of the axioms, then we demonstrate the applicability of our reasoning technique on a number of litmus tests. The majority of the proofs in this article are supported by mechanisation within Isabelle/HOL.
{"title":"Unifying Operational Weak Memory Verification: An Axiomatic Approach","authors":"Simon Doherty, Sadegh Dalvandi, Brijesh Dongol, H. Wehrheim","doi":"10.1145/3545117","DOIUrl":"https://doi.org/10.1145/3545117","url":null,"abstract":"In this article, we propose an approach to program verification using an abstract characterisation of weak memory models. Our approach is based on a hierarchical axiom scheme that captures the observational properties of a memory model. In particular, we show that it is possible to prove correctness of a program with respect to a particular axiom scheme, and we show this proof to suffice for any memory model that satisfies the axioms. Our axiom scheme is developed using a characterisation of weakest liberal preconditions for weak memory. This characterisation naturally extends to Hoare logic and Owicki-Gries reasoning by lifting weakest liberal preconditions (defined over read/write events) to the level of programs. We study three memory models (SC, TSO, and RC11-RAR) as example instantiations of the axioms, then we demonstrate the applicability of our reasoning technique on a number of litmus tests. The majority of the proofs in this article are supported by mechanisation within Isabelle/HOL.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41288180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe the categorical semantics for a simply typed variant and a simplified dependently typed variant of Cocon, a contextual modal type theory where the box modality mediates between the weak function space that is used to represent higher-order abstract syntax (HOAS) trees and the strong function space that describes (recursive) computations about them. What makes Cocon different from standard type theories is the presence of first-class contexts and contextual objects to describe syntax trees that are closed with respect to a given context of assumptions. Following M. Hofmann’s work, we use a presheaf model to characterise HOAS trees. Surprisingly, this model already provides the necessary structure to also model Cocon. In particular, we can capture the contextual objects of Cocon using a comonad ♭ that restricts presheaves to their closed elements. This gives a simple semantic characterisation of the invariants of contextual types (e.g. substitution invariance) and identifies Cocon as a type-theoretic syntax of presheaf models. We further extend this characterisation to dependent types using categories with families and show that we can model a fragment of Cocon without recursor in the Fitch-style dependent modal type theory presented by Birkedal et al.
{"title":"A Category Theoretic View of Contextual Types: From Simple Types to Dependent Types","authors":"Jason Z. S. Hu, B. Pientka, Ulrich Schöpp","doi":"10.1145/3545115","DOIUrl":"https://doi.org/10.1145/3545115","url":null,"abstract":"We describe the categorical semantics for a simply typed variant and a simplified dependently typed variant of Cocon, a contextual modal type theory where the box modality mediates between the weak function space that is used to represent higher-order abstract syntax (HOAS) trees and the strong function space that describes (recursive) computations about them. What makes Cocon different from standard type theories is the presence of first-class contexts and contextual objects to describe syntax trees that are closed with respect to a given context of assumptions. Following M. Hofmann’s work, we use a presheaf model to characterise HOAS trees. Surprisingly, this model already provides the necessary structure to also model Cocon. In particular, we can capture the contextual objects of Cocon using a comonad ♭ that restricts presheaves to their closed elements. This gives a simple semantic characterisation of the invariants of contextual types (e.g. substitution invariance) and identifies Cocon as a type-theoretic syntax of presheaf models. We further extend this characterisation to dependent types using categories with families and show that we can model a fragment of Cocon without recursor in the Fitch-style dependent modal type theory presented by Birkedal et al.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45776127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let V be a set of number-theoretical functions. We define a notion of V-realizability for predicate formulas in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Intuitionistic Predicate Calculus is sound with respect to the semantics of V-realizability if and only if some natural conditions for V hold.
{"title":"A Generalized Realizability and Intuitionistic Logic","authors":"A. Y. Konovalov","doi":"10.1145/3565367","DOIUrl":"https://doi.org/10.1145/3565367","url":null,"abstract":"Let V be a set of number-theoretical functions. We define a notion of V-realizability for predicate formulas in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Intuitionistic Predicate Calculus is sound with respect to the semantics of V-realizability if and only if some natural conditions for V hold.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48224824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. However, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph problems that are commonly studied in parameterized algorithmics. By adding the atomic predicates connk(x,y,z_1,..., zk) that hold true in a graph if there exists a path between (the valuations of) x and y after (the valuations of) z1,..., zk have been deleted, we obtain separator logic FO + conn. We show that separator logic can express many interesting problems, such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO + connk the fragment of separator logic that is restricted to connectivity predicates with at most k + 2 variables (that is, at most k deletions), we show that FO + connk + 1 is strictly more expressive than FO + connk for all k ≥ 0. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic FO + DP by adding the atomic predicates disjoint-pathsk[(x1, y1),..., (xk, yk) that evaluate to true if there are internally vertex-disjoint paths between (the valuations of) xi and yi for all 1 ≤ i ≤ k. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again, we show that the fragments FO + DPk that use predicates for at most k disjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic.
{"title":"First-order Logic with Connectivity Operators","authors":"Nicole Schirrmacher, S. Siebertz, Alexandre Vigny","doi":"10.1145/3595922","DOIUrl":"https://doi.org/10.1145/3595922","url":null,"abstract":"First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. However, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph problems that are commonly studied in parameterized algorithmics. By adding the atomic predicates connk(x,y,z_1,..., zk) that hold true in a graph if there exists a path between (the valuations of) x and y after (the valuations of) z1,..., zk have been deleted, we obtain separator logic FO + conn. We show that separator logic can express many interesting problems, such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO + connk the fragment of separator logic that is restricted to connectivity predicates with at most k + 2 variables (that is, at most k deletions), we show that FO + connk + 1 is strictly more expressive than FO + connk for all k ≥ 0. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic FO + DP by adding the atomic predicates disjoint-pathsk[(x1, y1),..., (xk, yk) that evaluate to true if there are internally vertex-disjoint paths between (the valuations of) xi and yi for all 1 ≤ i ≤ k. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again, we show that the fragments FO + DPk that use predicates for at most k disjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48028297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Carvalho, Florent R. Madelaine, B. Martin, Dmitriy Zhuk
Let 𝔸 be an idempotent algebra on a finite domain. By mediating between results of Chen [1] and Zhuk [2], we argue that if 𝔸 satisfies the polynomially generated powers property (PGP) and ℬ is a constraint language invariant under 𝔸 (i.e., in Inv(𝔸)), then QCSP ℬ is in NP. In doing this, we study the special forms of PGP, switchability, and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv}(𝔸) satisfies the exponentially generated powers property (EGP), then QCSP (Inv(𝔸)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture. This result becomes more significant now that the original Chen Conjecture (see [3]) is known to be false [4]. Switchability was introduced by Chen [1] as a generalization of the already-known collapsibility [5]. There, an algebra 𝔸 :=({ 0,1,2};r) was given that is switchable and not collapsible. We prove that, for all finite subsets Δ of Inv (𝔸 A), Pol (Δ) is collapsible. The significance of this is that, for QCSP on finite structures, it is still possible all QCSP tractability (in NP) explained by switchability is already explained by collapsibility. At least, no counterexample is known to this.
{"title":"The Complexity of Quantified Constraints: Collapsibility, Switchability, and the Algebraic Formulation","authors":"C. Carvalho, Florent R. Madelaine, B. Martin, Dmitriy Zhuk","doi":"10.1145/3568397","DOIUrl":"https://doi.org/10.1145/3568397","url":null,"abstract":"Let 𝔸 be an idempotent algebra on a finite domain. By mediating between results of Chen [1] and Zhuk [2], we argue that if 𝔸 satisfies the polynomially generated powers property (PGP) and ℬ is a constraint language invariant under 𝔸 (i.e., in Inv(𝔸)), then QCSP ℬ is in NP. In doing this, we study the special forms of PGP, switchability, and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv}(𝔸) satisfies the exponentially generated powers property (EGP), then QCSP (Inv(𝔸)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture. This result becomes more significant now that the original Chen Conjecture (see [3]) is known to be false [4]. Switchability was introduced by Chen [1] as a generalization of the already-known collapsibility [5]. There, an algebra 𝔸 :=({ 0,1,2};r) was given that is switchable and not collapsible. We prove that, for all finite subsets Δ of Inv (𝔸 A), Pol (Δ) is collapsible. The significance of this is that, for QCSP on finite structures, it is still possible all QCSP tractability (in NP) explained by switchability is already explained by collapsibility. At least, no counterexample is known to this.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46187472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model...
{"title":"Expressiveness and Nash Equilibrium in Iterated Boolean Games","authors":"GutiérrezJulián, HarrensteinPaul, PerelliGiuseppe, WooldridgeMichael","doi":"10.1145/3439900","DOIUrl":"https://doi.org/10.1145/3439900","url":null,"abstract":"We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model...","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3439900","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44371734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is proof-theoretical naturalness: The system consists of all and only the inference rules generated by the single, simple, linear scheme of the recently introduced subatomic logic. Thanks to this regularity, cuts are eliminated via a natural construction. The second reason is that the system generates efficient proofs. Indeed, we show that a certain class of tautologies due to Statman, which cannot have better than exponential cut-free proofs in the sequent calculus, have polynomial cut-free proofs in our system. We achieve this by using the same construction that we use for cut elimination. In summary, by expanding the language of propositional logic, we make its proof theory more regular and generate more proofs, some of which are very efficient. That design is made possible by considering atoms as superpositions of their truth values, which are connected by self-dual, non-commutative connectives. A proof can then be projected via each atom into two proofs, one for each truth value, without a need for cuts. Those projections are semantically natural and are at the heart of the constructions in this article. To accommodate self-dual non-commutativity, we compose proofs in deep inference.
{"title":"A Subatomic Proof System for Decision Trees","authors":"Chris Barrett, Alessio Guglielmi","doi":"10.1145/3545116","DOIUrl":"https://doi.org/10.1145/3545116","url":null,"abstract":"We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is proof-theoretical naturalness: The system consists of all and only the inference rules generated by the single, simple, linear scheme of the recently introduced subatomic logic. Thanks to this regularity, cuts are eliminated via a natural construction. The second reason is that the system generates efficient proofs. Indeed, we show that a certain class of tautologies due to Statman, which cannot have better than exponential cut-free proofs in the sequent calculus, have polynomial cut-free proofs in our system. We achieve this by using the same construction that we use for cut elimination. In summary, by expanding the language of propositional logic, we make its proof theory more regular and generate more proofs, some of which are very efficient. That design is made possible by considering atoms as superpositions of their truth values, which are connected by self-dual, non-commutative connectives. A proof can then be projected via each atom into two proofs, one for each truth value, without a need for cuts. Those projections are semantically natural and are at the heart of the constructions in this article. To accommodate self-dual non-commutativity, we compose proofs in deep inference.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45735718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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