Argumentation is a well-established formalism dealing with conflicting information by generating and comparing arguments. It has been playing a major role in AI for decades. In logic-based argumentation, we explore the internal structure of an argument. Informally, a set of formulas is the support for a given claim if it is consistent, subset-minimal, and implies the claim. In such a case, the pair of the support and the claim together is called an argument. In this article, we study the propositional variants of the following three computational tasks studied in argumentation: ARG (exists a support for a given claim with respect to a given set of formulas), ARG-Check (is a given set a support for a given claim), and ARG-Rel (similarly as ARG plus requiring an additionally given formula to be contained in the support). ARG-Check is complete for the complexity class DP, and the other two problems are known to be complete for the second level of the polynomial hierarchy (Creignou et al. 2014 and Parson et al., 2003) and, accordingly, are highly intractable. Analyzing the reason for this intractability, we perform a two-dimensional classification: First, we consider all possible propositional fragments of the problem within Schaefer’s framework (STOC 1978) and then study different parameterizations for each of the fragments. We identify a list of reasonable structural parameters (size of the claim, support, knowledge base) that are connected to the aforementioned decision problems. Eventually, we thoroughly draw a fine border of parameterized intractability for each of the problems showing where the problems are fixed-parameter tractable and when this exactly stops. Surprisingly, several cases are of very high intractability (para-NP and beyond).
{"title":"Parameterized Complexity of Logic-based Argumentation in Schaefer’s Framework","authors":"Yasir Mahmood, A. Meier, Johannes Schmidt","doi":"10.1145/3582499","DOIUrl":"https://doi.org/10.1145/3582499","url":null,"abstract":"Argumentation is a well-established formalism dealing with conflicting information by generating and comparing arguments. It has been playing a major role in AI for decades. In logic-based argumentation, we explore the internal structure of an argument. Informally, a set of formulas is the support for a given claim if it is consistent, subset-minimal, and implies the claim. In such a case, the pair of the support and the claim together is called an argument. In this article, we study the propositional variants of the following three computational tasks studied in argumentation: ARG (exists a support for a given claim with respect to a given set of formulas), ARG-Check (is a given set a support for a given claim), and ARG-Rel (similarly as ARG plus requiring an additionally given formula to be contained in the support). ARG-Check is complete for the complexity class DP, and the other two problems are known to be complete for the second level of the polynomial hierarchy (Creignou et al. 2014 and Parson et al., 2003) and, accordingly, are highly intractable. Analyzing the reason for this intractability, we perform a two-dimensional classification: First, we consider all possible propositional fragments of the problem within Schaefer’s framework (STOC 1978) and then study different parameterizations for each of the fragments. We identify a list of reasonable structural parameters (size of the claim, support, knowledge base) that are connected to the aforementioned decision problems. Eventually, we thoroughly draw a fine border of parameterized intractability for each of the problems showing where the problems are fixed-parameter tractable and when this exactly stops. Surprisingly, several cases are of very high intractability (para-NP and beyond).","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 25"},"PeriodicalIF":0.5,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42428743","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}
Jinsheng Chen, G. Greco, A. Palmigiano, A. Tzimoulis
A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e., those calculi that support general and modular proof-strategies for cut elimination) and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analyticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e., those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (a.k.a. unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms, which can be equivalently and algorithmically transformed into analytic structural rules so the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination, and the subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically, i.e., by showing that a (cut-free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far, this proof strategy for syntactic completeness has been implemented on a case-by-case base and not in general. In this article, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut-free derivation can be effectively generated that has a specific shape, referred to as pre-normal form.
{"title":"Syntactic Completeness of Proper Display Calculi","authors":"Jinsheng Chen, G. Greco, A. Palmigiano, A. Tzimoulis","doi":"10.1145/3529255","DOIUrl":"https://doi.org/10.1145/3529255","url":null,"abstract":"A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e., those calculi that support general and modular proof-strategies for cut elimination) and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analyticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e., those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (a.k.a. unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms, which can be equivalently and algorithmically transformed into analytic structural rules so the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination, and the subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically, i.e., by showing that a (cut-free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far, this proof strategy for syntactic completeness has been implemented on a case-by-case base and not in general. In this article, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut-free derivation can be effectively generated that has a specific shape, referred to as pre-normal form.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"23 1","pages":"1 - 46"},"PeriodicalIF":0.5,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49190842","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 introduce and study a natural extension of the Alternating time temporal logic ATL, called Temporal Logic of Coalitional Goal Assignments (TLCGA). It features one new and quite expressive coalitional strategic operator, called the coalitional goal assignment operator ⦉ γ ⦊, where γ is a mapping assigning to each set of players in the game its coalitional goal, formalised by a path formula of the language of TLCGA, i.e., a formula prefixed with a temporal operator X, U, or G, representing a temporalised objective for the respective coalition, describing the property of the plays on which that objective is satisfied. Then, the formula ⦉ γ ⦊ intuitively says that there is a strategy profile Σ for the grand coalition Agt such that for each coalition C, the restriction Σ |C of Σ to C is a collective strategy of C that enforces the satisfaction of its objective γ (C) in all outcome plays enabled by Σ |C. We establish fixpoint characterizations of the temporal goal assignments in a μ-calculus extension of TLCGA, discuss its expressiveness and illustrate it with some examples, prove bisimulation invariance and Hennessy–Milner property for it with respect to a suitably defined notion of bisimulation, construct a sound and complete axiomatic system for TLCGA, and obtain its decidability via finite model property.
{"title":"The Temporal Logic of Coalitional Goal Assignments in Concurrent Multiplayer Games","authors":"S. Enqvist, V. Goranko","doi":"10.1145/3517128","DOIUrl":"https://doi.org/10.1145/3517128","url":null,"abstract":"We introduce and study a natural extension of the Alternating time temporal logic ATL, called Temporal Logic of Coalitional Goal Assignments (TLCGA). It features one new and quite expressive coalitional strategic operator, called the coalitional goal assignment operator ⦉ γ ⦊, where γ is a mapping assigning to each set of players in the game its coalitional goal, formalised by a path formula of the language of TLCGA, i.e., a formula prefixed with a temporal operator X, U, or G, representing a temporalised objective for the respective coalition, describing the property of the plays on which that objective is satisfied. Then, the formula ⦉ γ ⦊ intuitively says that there is a strategy profile Σ for the grand coalition Agt such that for each coalition C, the restriction Σ |C of Σ to C is a collective strategy of C that enforces the satisfaction of its objective γ (C) in all outcome plays enabled by Σ |C. We establish fixpoint characterizations of the temporal goal assignments in a μ-calculus extension of TLCGA, discuss its expressiveness and illustrate it with some examples, prove bisimulation invariance and Hennessy–Milner property for it with respect to a suitably defined notion of bisimulation, construct a sound and complete axiomatic system for TLCGA, and obtain its decidability via finite model property.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"23 1","pages":"1 - 58"},"PeriodicalIF":0.5,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48325977","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}
L. Aceto, Valentina Castiglioni, W. Fokkink, Anna Igolfsdottir, B. Luttik
Bergstra and Klop have shown that bisimilarity has a finite equational axiomatisation over ACP/CCS extended with the binary left and communication merge operators. Moller proved that auxiliary operators are necessary to obtain a finite axiomatisation of bisimilarity over CCS, and Aceto et al. showed that this remains true when Hennessy’s merge is added to that language. These results raise the question of whether there is one auxiliary binary operator whose addition to CCS leads to a finite axiomatisation of bisimilarity. We contribute to answering this question in the simplified setting of the recursion-, relabelling-, and restriction-free fragment of CCS. We formulate three natural assumptions pertaining to the operational semantics of auxiliary operators and their relationship to parallel composition and prove that an auxiliary binary operator facilitating a finite axiomatisation of bisimilarity in the simplified setting cannot satisfy all three assumptions.
{"title":"Are Two Binary Operators Necessary to Obtain a Finite Axiomatisation of Parallel Composition?","authors":"L. Aceto, Valentina Castiglioni, W. Fokkink, Anna Igolfsdottir, B. Luttik","doi":"10.1145/3529535","DOIUrl":"https://doi.org/10.1145/3529535","url":null,"abstract":"Bergstra and Klop have shown that bisimilarity has a finite equational axiomatisation over ACP/CCS extended with the binary left and communication merge operators. Moller proved that auxiliary operators are necessary to obtain a finite axiomatisation of bisimilarity over CCS, and Aceto et al. showed that this remains true when Hennessy’s merge is added to that language. These results raise the question of whether there is one auxiliary binary operator whose addition to CCS leads to a finite axiomatisation of bisimilarity. We contribute to answering this question in the simplified setting of the recursion-, relabelling-, and restriction-free fragment of CCS. We formulate three natural assumptions pertaining to the operational semantics of auxiliary operators and their relationship to parallel composition and prove that an auxiliary binary operator facilitating a finite axiomatisation of bisimilarity in the simplified setting cannot satisfy all three assumptions.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"23 1","pages":"1 - 56"},"PeriodicalIF":0.5,"publicationDate":"2020-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49668375","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}
Property testing algorithms are highly efficient algorithms that come with probabilistic accuracy guarantees. For a property P, the goal is to distinguish inputs that have P from those that are far from having P with high probability correctly, by querying only a small number of local parts of the input. In property testing on graphs, the distance is measured by the number of edge modifications (additions or deletions) that are necessary to transform a graph into one with property P. Much research has focused on the query complexity of such algorithms, i. e., the number of queries the algorithm makes to the input, but in view of applications, the running time of the algorithm is equally relevant. In (Adler, Harwath, STACS 2018), a natural extension of the bounded degree graph model of property testing to relational databases of bounded degree was introduced, and it was shown that on databases of bounded degree and bounded tree-width, every property that is expressible in monadic second-order logic with counting (CMSO) is testable with constant query complexity and sublinear running time. It remains open whether this can be improved to constant running time. In this article we introduce a new model, which is based on the bounded degree model, but the distance measure allows both edge (tuple) modifications and vertex (element) modifications. We show that every property that is testable in the classical model is testable in our model with the same query complexity and running time, but the converse is not true. Our main theorem shows that on databases of bounded degree and bounded tree-width, every property that is expressible in CMSO is testable with constant query complexity and constant running time in the new model. Our proof methods include the semilinearity of the neighborhood histograms of databases having the property and a result by Alon (Proposition 19.10 in Lovász, Large networks and graph limits, 2012) that states that for every bounded degree graph (mathcal {G}) there exists a constant size graph (mathcal {H}) that has a similar neighborhood distribution to (mathcal {G}) . It can be derived from a result in (Benjamini et al., Advances in Mathematics 2010) that hyperfinite hereditary properties are testable with constant query complexity and constant running time in the classical model (and hence in the new model). Using our methods, we give an alternative proof that hyperfinite hereditary properties are testable with constant query complexity and constant running time in the new model. We argue that our model is natural and our meta-theorem showing constant-time CMSO testability supports this.
性能测试算法是具有概率准确性保证的高效算法。对于属性P，目标是通过仅查询输入的少量局部部分，以高概率正确区分具有P的输入和远没有P的输入。在图的属性测试中，距离是通过将图转换为具有属性P的图所需的边缘修改（添加或删除）的数量来衡量的。许多研究都集中在这种算法的查询复杂性上，即算法对输入进行的查询数量，但从应用的角度来看，算法的运行时间同样相关。在（Adler，Harwath，STACS 2018）中，引入了性质测试的有界度图模型对有界度关系数据库的自然扩展，并表明在有界度和有界树宽的数据库上，在具有计数的一元二阶逻辑（CMSO）中表示的每一个性质都是可测试的，其查询复杂度和运行时间不变。是否可以将其改进为恒定的运行时间仍悬而未决。在本文中，我们介绍了一种新的模型，它基于有界度模型，但距离度量允许边（元组）修改和顶点（元素）修改。我们证明了在具有相同查询复杂度和运行时间的情况下，在经典模型中可测试的每个属性在我们的模型中都是可测试的，但反之亦然。我们的主要定理表明，在有界度和有界树宽的数据库上，CMSO中可表达的每一个性质在新模型中都是可测试的，具有恒定的查询复杂度和恒定的运行时间。我们的证明方法包括具有该性质的数据库的邻域直方图的半线性性，以及Alon（Lovász，Large networks and graph limits，2012中的19.10命题）的一个结果，该结果指出，对于每个有界度图。从（Benjamini et al.，Advances in Mathematics 2010）中的一个结果可以得出，在经典模型中（因此在新模型中），超有限遗传性质可以在恒定的查询复杂度和恒定的运行时间下进行测试。使用我们的方法，我们给出了一个替代的证明，即在新模型中，超有限遗传属性在恒定的查询复杂度和恒定的运行时间下是可测试的。我们认为我们的模型是自然的，并且我们的元定理显示了恒定时间CMSO可测试性支持这一点。
{"title":"Faster Property Testers in a Variation of the Bounded Degree Model","authors":"Isolde Adler, Polly Fahey","doi":"10.1145/3584948","DOIUrl":"https://doi.org/10.1145/3584948","url":null,"abstract":"Property testing algorithms are highly efficient algorithms that come with probabilistic accuracy guarantees. For a property P, the goal is to distinguish inputs that have P from those that are far from having P with high probability correctly, by querying only a small number of local parts of the input. In property testing on graphs, the distance is measured by the number of edge modifications (additions or deletions) that are necessary to transform a graph into one with property P. Much research has focused on the query complexity of such algorithms, i. e., the number of queries the algorithm makes to the input, but in view of applications, the running time of the algorithm is equally relevant. In (Adler, Harwath, STACS 2018), a natural extension of the bounded degree graph model of property testing to relational databases of bounded degree was introduced, and it was shown that on databases of bounded degree and bounded tree-width, every property that is expressible in monadic second-order logic with counting (CMSO) is testable with constant query complexity and sublinear running time. It remains open whether this can be improved to constant running time. In this article we introduce a new model, which is based on the bounded degree model, but the distance measure allows both edge (tuple) modifications and vertex (element) modifications. We show that every property that is testable in the classical model is testable in our model with the same query complexity and running time, but the converse is not true. Our main theorem shows that on databases of bounded degree and bounded tree-width, every property that is expressible in CMSO is testable with constant query complexity and constant running time in the new model. Our proof methods include the semilinearity of the neighborhood histograms of databases having the property and a result by Alon (Proposition 19.10 in Lovász, Large networks and graph limits, 2012) that states that for every bounded degree graph (mathcal {G}) there exists a constant size graph (mathcal {H}) that has a similar neighborhood distribution to (mathcal {G}) . It can be derived from a result in (Benjamini et al., Advances in Mathematics 2010) that hyperfinite hereditary properties are testable with constant query complexity and constant running time in the classical model (and hence in the new model). Using our methods, we give an alternative proof that hyperfinite hereditary properties are testable with constant query complexity and constant running time in the new model. We argue that our model is natural and our meta-theorem showing constant-time CMSO testability supports this.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 24"},"PeriodicalIF":0.5,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49207202","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}
A. Artale, J. C. Jung, Andrea Mazzullo, A. Ozaki, F. Wolter
The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP reduce potentially hard existence problems to entailment in the underlying logic. Description (and modal) logics with nominals and/or role inclusions do not enjoy the CIP nor the PBDP, but interpolants and explicit definitions have many applications, in particular in concept learning, ontology engineering, and ontology-based data management. In this article we show that, even without Beth and Craig, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as (mathcal {ALCO} ) , (mathcal {ALCH} ) and (mathcal {ALCHOI} ) and corresponding hybrid modal logics. However, living without Beth and Craig makes these problems harder than entailment: the existence problems become 2ExpTime-complete in the presence of an ontology or the universal modality, and coNExpTime-complete otherwise. We also analyze explicit definition existence if all symbols (except the one that is defined) are admitted in the definition. In this case the complexity depends on whether one considers individual or concept names. Finally, we consider the problem of computing interpolants and explicit definitions if they exist and turn the complexity upper bound proof into an algorithm computing them, at least for description logics with role inclusions.
{"title":"Living Without Beth and Craig: Definitions and Interpolants in Description and Modal Logics with Nominals and Role Inclusions","authors":"A. Artale, J. C. Jung, Andrea Mazzullo, A. Ozaki, F. Wolter","doi":"10.1145/3597301","DOIUrl":"https://doi.org/10.1145/3597301","url":null,"abstract":"The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP reduce potentially hard existence problems to entailment in the underlying logic. Description (and modal) logics with nominals and/or role inclusions do not enjoy the CIP nor the PBDP, but interpolants and explicit definitions have many applications, in particular in concept learning, ontology engineering, and ontology-based data management. In this article we show that, even without Beth and Craig, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as (mathcal {ALCO} ) , (mathcal {ALCH} ) and (mathcal {ALCHOI} ) and corresponding hybrid modal logics. However, living without Beth and Craig makes these problems harder than entailment: the existence problems become 2ExpTime-complete in the presence of an ontology or the universal modality, and coNExpTime-complete otherwise. We also analyze explicit definition existence if all symbols (except the one that is defined) are admitted in the definition. In this case the complexity depends on whether one considers individual or concept names. Finally, we consider the problem of computing interpolants and explicit definitions if they exist and turn the complexity upper bound proof into an algorithm computing them, at least for description logics with role inclusions.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41876247","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}
GajarskýJakub, HliněnýPetr, ObdržálekJan, LokshtanovDaniel, S. RamanujanM.
We study the first-order (FO) model checking problem of dense graph classes, namely, those that have FO interpretations in (or are FO transductions of) some sparse graph classes. We give a structur...
{"title":"A New Perspective on FO Model Checking of Dense Graph Classes","authors":"GajarskýJakub, HliněnýPetr, ObdržálekJan, LokshtanovDaniel, S. RamanujanM.","doi":"10.1145/3383206","DOIUrl":"https://doi.org/10.1145/3383206","url":null,"abstract":"We study the first-order (FO) model checking problem of dense graph classes, namely, those that have FO interpretations in (or are FO transductions of) some sparse graph classes. We give a structur...","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"12 1","pages":"1-23"},"PeriodicalIF":0.5,"publicationDate":"2020-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75796783","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}
H. Aamer, B. Bogaerts, D. Surinx, E. Ternovska, J. V. Bussche
The logic of information flows (LIF) is a general framework in which tasks of a procedural nature can be modeled in a declarative, logic-based fashion. The first contribution of this article is to propose semantic and syntactic definitions of inputs and outputs of LIF expressions. We study how the two relate and show that our syntactic definition is optimal in a sense that is made precise. The second contribution is a systematic study of the expressive power of sequential composition in LIF. Our results on composition tie in the results on inputs and outputs and relate LIF to first-order logic (FO) and bounded-variable LIF to bounded- variable FO. This article is the extended version of a paper presented at KR 2020 [2].
{"title":"Inputs, Outputs, and Composition in the Logic of Information Flows","authors":"H. Aamer, B. Bogaerts, D. Surinx, E. Ternovska, J. V. Bussche","doi":"10.1145/3604553","DOIUrl":"https://doi.org/10.1145/3604553","url":null,"abstract":"The logic of information flows (LIF) is a general framework in which tasks of a procedural nature can be modeled in a declarative, logic-based fashion. The first contribution of this article is to propose semantic and syntactic definitions of inputs and outputs of LIF expressions. We study how the two relate and show that our syntactic definition is optimal in a sense that is made precise. The second contribution is a systematic study of the expressive power of sequential composition in LIF. Our results on composition tie in the results on inputs and outputs and relate LIF to first-order logic (FO) and bounded-variable LIF to bounded- variable FO. This article is the extended version of a paper presented at KR 2020 [2].","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 44"},"PeriodicalIF":0.5,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44651658","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 develop a doubly exponential decision procedure for the satisfiability problem of guarded separation logic—a novel fragment of separation logic featuring user-supplied inductive predicates, Boolean connectives, and separating connectives, including restricted (guarded) versions of negation, magic wand, and septraction. Moreover, we show that dropping the guards for any of the preceding connectives leads to an undecidable fragment. We further apply our decision procedure to reason about entailments in the popular symbolic heap fragment of separation logic. In particular, we obtain a doubly exponential decision procedure for entailments between (quantifier-free) symbolic heaps with inductive predicate definitions of bounded treewidth (SLbtw)—one of the most expressive decidable fragments of separation logic. Together with the recently shown 2ExpTime-hardness for entailments in said fragment, we conclude that the entailment problem for SLbtw is 2ExpTime-complete—thereby closing a previously open complexity gap.
{"title":"A Decision Procedure for Guarded Separation Logic Complete Entailment Checking for Separation Logic with Inductive Definitions","authors":"C. Matheja, J. Pagel, Florian Zuleger","doi":"10.1145/3534927","DOIUrl":"https://doi.org/10.1145/3534927","url":null,"abstract":"We develop a doubly exponential decision procedure for the satisfiability problem of guarded separation logic—a novel fragment of separation logic featuring user-supplied inductive predicates, Boolean connectives, and separating connectives, including restricted (guarded) versions of negation, magic wand, and septraction. Moreover, we show that dropping the guards for any of the preceding connectives leads to an undecidable fragment. We further apply our decision procedure to reason about entailments in the popular symbolic heap fragment of separation logic. In particular, we obtain a doubly exponential decision procedure for entailments between (quantifier-free) symbolic heaps with inductive predicate definitions of bounded treewidth (SLbtw)—one of the most expressive decidable fragments of separation logic. Together with the recently shown 2ExpTime-hardness for entailments in said fragment, we conclude that the entailment problem for SLbtw is 2ExpTime-complete—thereby closing a previously open complexity gap.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 76"},"PeriodicalIF":0.5,"publicationDate":"2020-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43143635","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}
Proofs in propositional logic are typically presented as trees of derived formulas or, alternatively, as directed acyclic graphs of derived formulas. This distinction between tree-like vs. dag-like structure is particularly relevant when making quantitative considerations regarding, for example, proof size. Here we analyze a more general type of structural restriction for proofs in rule-based proof systems. In this definition, proofs are directed graphs of derived formulas in which cycles are allowed as long as every formula is derived at least as many times as it is required as a premise. We call such proofs “circular”. We show that, for all sets of standard inference rules with single or multiple conclusions, circular proofs are sound. We start the study of the proof complexity of circular proofs at Circular Resolution, the circular version of Resolution. We immediately see that Circular Resolution is stronger than dag-like Resolution since, as we show, the propositional encoding of the pigeonhole principle has circular Resolution proofs of polynomial size. Furthermore, for derivations of clauses from clauses, we show that Circular Resolution is, surprisingly, equivalent to Sherali-Adams, a proof system for reasoning through polynomial inequalities that has linear programming at its base. As corollaries we get: (1) polynomial-time (LP-based) algorithms that find Circular Resolution proofs of constant width, (2) examples that separate Circular from dag-like Resolution, such as the pigeonhole principle and its variants, and (3) exponentially hard cases for Circular Resolution. Contrary to the case of Circular Resolution, for Frege we show that circular proofs can be converted into tree-like proofs with at most polynomial overhead.
{"title":"Circular (Yet Sound) Proofs in Propositional Logic","authors":"Albert Atserias, Massimo Lauria","doi":"10.1145/3579997","DOIUrl":"https://doi.org/10.1145/3579997","url":null,"abstract":"Proofs in propositional logic are typically presented as trees of derived formulas or, alternatively, as directed acyclic graphs of derived formulas. This distinction between tree-like vs. dag-like structure is particularly relevant when making quantitative considerations regarding, for example, proof size. Here we analyze a more general type of structural restriction for proofs in rule-based proof systems. In this definition, proofs are directed graphs of derived formulas in which cycles are allowed as long as every formula is derived at least as many times as it is required as a premise. We call such proofs “circular”. We show that, for all sets of standard inference rules with single or multiple conclusions, circular proofs are sound. We start the study of the proof complexity of circular proofs at Circular Resolution, the circular version of Resolution. We immediately see that Circular Resolution is stronger than dag-like Resolution since, as we show, the propositional encoding of the pigeonhole principle has circular Resolution proofs of polynomial size. Furthermore, for derivations of clauses from clauses, we show that Circular Resolution is, surprisingly, equivalent to Sherali-Adams, a proof system for reasoning through polynomial inequalities that has linear programming at its base. As corollaries we get: (1) polynomial-time (LP-based) algorithms that find Circular Resolution proofs of constant width, (2) examples that separate Circular from dag-like Resolution, such as the pigeonhole principle and its variants, and (3) exponentially hard cases for Circular Resolution. Contrary to the case of Circular Resolution, for Frege we show that circular proofs can be converted into tree-like proofs with at most polynomial overhead.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 26"},"PeriodicalIF":0.5,"publicationDate":"2018-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41952068","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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