Pub Date : 2023-01-28DOI: https://dl.acm.org/doi/10.1145/3572837
James Baxter, Ana Cavalcanti, Maciej Gazda, Robert M. Hierons
The existing testing theories for CSP cater for verification of interaction patterns (traces) and deadlocks, but not time. We address here refinement and testing based on a dialect of CSP, called tock-CSP, which can capture discrete time properties. This version of CSP has been of widespread interest for decades; recently, it has been given a denotational semantics, and model checking has become possible using a well established tool. Here, we first equip tock-CSP with a novel semantics for testing, which distinguishes input and output events: the standard models of (tock-)CSP do not differentiate them, but for testing this is essential. We then present a new testing theory for timewise refinement, based on novel definitions of test and test execution. Finally, we reconcile refinement and testing by relating timed ioco testing and refinement in tock-CSP with inputs and outputs. With these results, this paper provides, for the first time, a systematic theory that allows both timed testing and timed refinement to be expressed. An important practical consequence is that this ensures that the notion of correctness used by developers guarantees that tests pass when applied to a correct system and, in addition, faults identified during testing correspond to development mistakes.
{"title":"Testing using CSP Models: Time, Inputs, and Outputs","authors":"James Baxter, Ana Cavalcanti, Maciej Gazda, Robert M. Hierons","doi":"https://dl.acm.org/doi/10.1145/3572837","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3572837","url":null,"abstract":"<p>The existing testing theories for CSP cater for verification of interaction patterns (traces) and deadlocks, but not time. We address here refinement and testing based on a dialect of CSP, called <i>tock</i>-CSP, which can capture discrete time properties. This version of CSP has been of widespread interest for decades; recently, it has been given a denotational semantics, and model checking has become possible using a well established tool. Here, we first equip <i>tock</i>-CSP with a novel semantics for testing, which distinguishes input and output events: the standard models of (<i>tock</i>-)CSP do not differentiate them, but for testing this is essential. We then present a new testing theory for timewise refinement, based on novel definitions of test and test execution. Finally, we reconcile refinement and testing by relating timed ioco testing and refinement in <i>tock</i>-CSP with inputs and outputs. With these results, this paper provides, for the first time, a systematic theory that allows both timed testing and timed refinement to be expressed. An important practical consequence is that this ensures that the notion of correctness used by developers guarantees that tests pass when applied to a correct system and, in addition, faults identified during testing correspond to development mistakes.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"88 1-2","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508210","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 : 2023-01-27DOI: https://dl.acm.org/doi/10.1145/3565286
Olaf Beyersdorff, Joshua Blinkhorn, Meena Mahajan, Tomáš Peitl
We provide a tight characterisation of proof size in resolution for quantified Boolean formulas (QBF) via circuit complexity. Such a characterisation was previously obtained for a hierarchy of QBF Frege systems [16], but leaving open the most important case of QBF resolution. Different from the Frege case, our characterisation uses a new version of decision lists as its circuit model, which is stronger than the CNFs the system works with. Our decision list model is well suited to compute countermodels for QBFs. Our characterisation works for both Q-Resolution and QU-Resolution.
Using our characterisation, we obtain a size-width relation for QBF resolution in the spirit of the celebrated result for propositional resolution [4]. However, our result is not just a replication of the propositional relation—intriguingly ruled out for QBF in previous research [12]—but shows a different dependence between size, width, and quantifier complexity. An essential ingredient is an improved relation between the size and width of term decision lists; this may be of independent interest.
We demonstrate that our new technique elegantly reproves known QBF hardness results and unifies previous lower-bound techniques in the QBF domain.
{"title":"Hardness Characterisations and Size-width Lower Bounds for QBF Resolution","authors":"Olaf Beyersdorff, Joshua Blinkhorn, Meena Mahajan, Tomáš Peitl","doi":"https://dl.acm.org/doi/10.1145/3565286","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3565286","url":null,"abstract":"<p>We provide a <i>tight characterisation of proof size in resolution for quantified Boolean formulas (QBF) via circuit complexity.</i> Such a characterisation was previously obtained for a hierarchy of QBF Frege systems [16], but leaving open the most important case of QBF resolution. Different from the Frege case, our characterisation uses a new version of decision lists as its circuit model, which is stronger than the CNFs the system works with. Our decision list model is well suited to compute countermodels for QBFs. Our characterisation works for both <i>Q-Resolution and QU-Resolution.</i></p><p>Using our characterisation, we obtain a <i>size-width relation for QBF resolution</i> in the spirit of the celebrated result for propositional resolution [4]. However, our result is not just a replication of the propositional relation—intriguingly ruled out for QBF in previous research [12]—but shows a different dependence between size, width, and quantifier complexity. An essential ingredient is an improved relation between the size and width of term decision lists; this may be of independent interest. </p><p>We demonstrate that <i>our new technique elegantly reproves known QBF hardness results</i> and unifies previous lower-bound techniques in the QBF domain.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"36 7","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508189","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 : 2023-01-27DOI: https://dl.acm.org/doi/10.1145/3572907
Theofanis Aravanis
Parikh proposed his relevance-sensitive axiom to remedy the weakness of the classical AGM paradigm in addressing relevant change. An insufficiency of Parikh’s criterion, however, is its dependency on the contingent beliefs of a belief set to be revised, since the former only constrains the revision process of splittable theories (i.e., theories that can be divided in mutually disjoint compartments). The case of arbitrary non-splittable belief sets remains out of the scope of Parikh’s approach. On that premise, we generalize Parikh’s criterion, introducing (both axiomatically and semantically) a new notion of relevance, which we call relevance at the sentential level. We show that the proposed notion of relevance is universal (as it is applicable to arbitrary belief sets) and acts in a more refined way as compared to Parikh’s proposal; as we illustrate, this latter feature of relevance at the sentential level potentially leads to a significant drop in the computational resources required for implementing belief revision. Furthermore, we prove that Dalal’s popular revision operator respects, to a certain extent, relevance at the sentential level. Last but not least, the tight relation between local and relevance-sensitive revision is pointed out.
{"title":"Generalizing Parikh’s Criterion for Relevance-Sensitive Belief Revision","authors":"Theofanis Aravanis","doi":"https://dl.acm.org/doi/10.1145/3572907","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3572907","url":null,"abstract":"<p>Parikh proposed his relevance-sensitive axiom to remedy the weakness of the classical AGM paradigm in addressing relevant change. An insufficiency of Parikh’s criterion, however, is its dependency on the contingent beliefs of a belief set to be revised, since the former only constrains the revision process of splittable theories (i.e., theories that can be divided in mutually disjoint compartments). The case of arbitrary non-splittable belief sets remains out of the scope of Parikh’s approach. On that premise, we generalize Parikh’s criterion, introducing (both axiomatically and semantically) a new notion of relevance, which we call <i>relevance at the sentential level</i>. We show that the proposed notion of relevance is universal (as it is applicable to arbitrary belief sets) and acts in a more refined way as compared to Parikh’s proposal; as we illustrate, this latter feature of relevance at the sentential level potentially leads to a significant drop in the computational resources required for implementing belief revision. Furthermore, we prove that Dalal’s popular revision operator respects, to a certain extent, relevance at the sentential level. Last but not least, the tight relation between local and relevance-sensitive revision is pointed out.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"38 6","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508175","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 : 2023-01-27DOI: https://dl.acm.org/doi/10.1145/3568955
Yisong Wang, Thomas Eiter, Yuanlin Zhang, Fangzhen Lin
In this article, we consider Answer Set Programming (ASP). It is a declarative problem solving paradigm that can be used to encode a problem as a logic program whose answer sets correspond to the solutions of the problem. It has been widely applied in various domains in AI and beyond. Given that answer sets are supposed to yield solutions to the original problem, the question of “why a set of atoms is an answer set” becomes important for both semantics understanding and program debugging. It has been well investigated for normal logic programs. However, for the class of disjunctive logic programs, which is a substantial extension of that of normal logic programs, this question has not been addressed much. In this article, we propose a notion of reduct for disjunctive logic programs and show how it can provide answers to the aforementioned question. First, we show that for each answer set, its reduct provides a resolution proof for each atom in it. We then further consider minimal sets of rules that will be sufficient to provide resolution proofs for sets of atoms. Such sets of rules will be called witnesses and are the focus of this article. We study complexity issues of computing various witnesses and provide algorithms for computing them. In particular, we show that the problem is tractable for normal and headcycle-free disjunctive logic programs, but intractable for general disjunctive logic programs. We also conducted some experiments and found that for many well-known ASP and SAT benchmarks, computing a minimal witness for an atom of an answer set is often feasible.
{"title":"Witnesses for Answer Sets of Logic Programs","authors":"Yisong Wang, Thomas Eiter, Yuanlin Zhang, Fangzhen Lin","doi":"https://dl.acm.org/doi/10.1145/3568955","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3568955","url":null,"abstract":"<p>In this article, we consider Answer Set Programming (ASP). It is a declarative problem solving paradigm that can be used to encode a problem as a logic program whose answer sets correspond to the solutions of the problem. It has been widely applied in various domains in AI and beyond. Given that answer sets are supposed to yield solutions to the original problem, the question of “why a set of atoms is an answer set” becomes important for both semantics understanding and program debugging. It has been well investigated for normal logic programs. However, for the class of disjunctive logic programs, which is a substantial extension of that of normal logic programs, this question has not been addressed much. In this article, we propose a notion of reduct for disjunctive logic programs and show how it can provide answers to the aforementioned question. First, we show that for each answer set, its reduct provides a resolution proof for each atom in it. We then further consider minimal sets of rules that will be sufficient to provide resolution proofs for sets of atoms. Such sets of rules will be called witnesses and are the focus of this article. We study complexity issues of computing various witnesses and provide algorithms for computing them. In particular, we show that the problem is tractable for normal and headcycle-free disjunctive logic programs, but intractable for general disjunctive logic programs. We also conducted some experiments and found that for many well-known ASP and SAT benchmarks, computing a minimal witness for an atom of an answer set is often feasible.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"40 6","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508146","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 : 2023-01-23DOI: https://dl.acm.org/doi/10.1145/3565365
Dylan Bellier, Massimo Benerecetti, Dario Della Monica, Fabio Mogavero
An extension of QPTL is considered where functional dependencies among the quantified variables can be restricted in such a way that their current values are independent of the future values of the other variables. This restriction is tightly connected to the notion of behavioral strategies in game-theory and allows the resulting logic to naturally express game-theoretic concepts. Inspired by the work on logics of dependence and independence, we provide a new compositional semantics for QPTL that allows for expressing such functional dependencies among variables. The fragment where only restricted quantifications are considered, called behavioral quantifications, allows for linear-time properties that are satisfiable if and only if they are realisable in the Pnueli-Rosner sense. This fragment can be decided, for both model checking and satisfiability, in 2Exp Time and is expressively equivalent to QPTL, though significantly less succinct.
{"title":"Good-for-Game QPTL: An Alternating Hodges Semantics","authors":"Dylan Bellier, Massimo Benerecetti, Dario Della Monica, Fabio Mogavero","doi":"https://dl.acm.org/doi/10.1145/3565365","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3565365","url":null,"abstract":"<p>An extension of <sans-serif>QPTL</sans-serif> is considered where <i>functional dependencies</i> among the quantified variables can be restricted in such a way that their current values are <i>independent of the future</i> values of the other variables. This restriction is tightly connected to the notion of <i>behavioral strategies</i> in game-theory and allows the resulting logic to naturally express game-theoretic concepts. Inspired by the work on logics of dependence and independence, we provide a new compositional semantics for <sans-serif>QPTL</sans-serif> that allows for expressing such functional dependencies among variables. The fragment where only restricted quantifications are considered, called <i>behavioral quantifications</i>, allows for linear-time properties that are satisfiable if and only if they are realisable in the Pnueli-Rosner sense. This fragment can be decided, for both <i>model checking</i> and <i>satisfiability</i>, in 2<span>Exp Time</span> and is expressively equivalent to <sans-serif>QPTL</sans-serif>, though significantly less succinct.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"103 1-2","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508219","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 show that the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences exactly match the complexity measures of clause width and depth needed to refute the corresponding graph isomorphism formula in propositional narrow resolution. Using this connection, we obtain upper and lower bounds for refuting graph isomorphism formulas in (normal) resolution. In particular, we show that if k is the minimum number of variables needed to distinguish two graphs with n vertices each, then there is an nO(k) resolution refutation size upper bound for the corresponding isomorphism formula, as well as lower bounds of 2k-1 and k for the treelike resolution size and resolution clause space for this formula. We also show a (normal) resolution size lower bound of exp (Ω (k2/n)) for the case of colored graphs with constant color class sizes. Applying these results, we prove the first exponential lower bound for graph isomorphism formulas in the proof system SRC-1, a system that extends resolution with a global symmetry rule, thereby answering an open question posed by Schweitzer and Seebach.
{"title":"Number of Variables for Graph Differentiation and the Resolution of Graph Isomorphism Formulas","authors":"J. Torán, Florian Wörz","doi":"10.1145/3580478","DOIUrl":"https://doi.org/10.1145/3580478","url":null,"abstract":"We show that the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences exactly match the complexity measures of clause width and depth needed to refute the corresponding graph isomorphism formula in propositional narrow resolution. Using this connection, we obtain upper and lower bounds for refuting graph isomorphism formulas in (normal) resolution. In particular, we show that if k is the minimum number of variables needed to distinguish two graphs with n vertices each, then there is an nO(k) resolution refutation size upper bound for the corresponding isomorphism formula, as well as lower bounds of 2k-1 and k for the treelike resolution size and resolution clause space for this formula. We also show a (normal) resolution size lower bound of exp (Ω (k2/n)) for the case of colored graphs with constant color class sizes. Applying these results, we prove the first exponential lower bound for graph isomorphism formulas in the proof system SRC-1, a system that extends resolution with a global symmetry rule, thereby answering an open question posed by Schweitzer and Seebach.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 25"},"PeriodicalIF":0.5,"publicationDate":"2023-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43704088","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 : 2023-01-20DOI: https://dl.acm.org/doi/10.1145/3549075
Md. Aquil Khan, Mohua Banerjee, Sibsankar Panda
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, Mohua Banerjee, Sibsankar Panda","doi":"https://dl.acm.org/doi/10.1145/3549075","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3549075","url":null,"abstract":"<p>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.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"36 11","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508188","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 prove that the combinatorial Weisfeiler-Leman algorithm of dimension $(3k+4)$ is a complete isomorphism test for the class of all graphs of rank width at most $k$. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width. It was known that isomorphism of graphs of rank width $k$ is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time $n^{f(k)}$ for a non-elementary function $f$. Our result yields an isomorphism test for graphs of rank width $k$ running in time $n^{O(k)}$. Another consequence of our result is the first polynomial time canonisation algorithm for graphs of bounded rank width. Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.
{"title":"Canonisation and Definability for Graphs of Bounded Rank Width","authors":"Martin Grohe, Daniel Neuen","doi":"10.1145/3568025","DOIUrl":"https://doi.org/10.1145/3568025","url":null,"abstract":"We prove that the combinatorial Weisfeiler-Leman algorithm of dimension $(3k+4)$ is a complete isomorphism test for the class of all graphs of rank width at most $k$. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width. It was known that isomorphism of graphs of rank width $k$ is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time $n^{f(k)}$ for a non-elementary function $f$. Our result yields an isomorphism test for graphs of rank width $k$ running in time $n^{O(k)}$. Another consequence of our result is the first polynomial time canonisation algorithm for graphs of bounded rank width. Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"207 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135394055","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 : 2023-01-18DOI: https://dl.acm.org/doi/10.1145/3565363
Yuval Filmus, Meena Mahajan, Gaurav Sood, Marc Vinyals
We study the MaxSAT Resolution (MaxRes) rule in the context of certifying unsatisfiability. We show that it can be exponentially more powerful than tree-like resolution, and when augmented with weakening (the system MaxResW), p-simulates tree-like resolution. In devising a lower bound technique specific to MaxRes (and not merely inheriting lower bounds from Res), we define a new proof system called the SubCubeSums proof system. This system, which p-simulates MaxResW, can be viewed as a special case of the semi-algebraic Sherali–Adams proof system. In expressivity, it is the integral restriction of conical juntas studied in the contexts of communication complexity and extension complexity. We show that it is not simulated by Res. Using a proof technique qualitatively different from the lower bounds that MaxResW inherits from Res, we show that Tseitin contradictions on expander graphs are hard to refute in SubCubeSums. We also establish a lower bound technique via lifting: for formulas requiring large degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums.
{"title":"MaxSAT Resolution and Subcube Sums","authors":"Yuval Filmus, Meena Mahajan, Gaurav Sood, Marc Vinyals","doi":"https://dl.acm.org/doi/10.1145/3565363","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3565363","url":null,"abstract":"<p>We study the MaxSAT Resolution (MaxRes) rule in the context of certifying unsatisfiability. We show that it can be exponentially more powerful than tree-like resolution, and when augmented with weakening (the system MaxResW), <i>p</i>-simulates tree-like resolution. In devising a lower bound technique specific to MaxRes (and not merely inheriting lower bounds from Res), we define a new proof system called the SubCubeSums proof system. This system, which <i>p</i>-simulates MaxResW, can be viewed as a special case of the semi-algebraic Sherali–Adams proof system. In expressivity, it is the integral restriction of conical juntas studied in the contexts of communication complexity and extension complexity. We show that it is not simulated by Res. Using a proof technique qualitatively different from the lower bounds that MaxResW inherits from Res, we show that Tseitin contradictions on expander graphs are hard to refute in SubCubeSums. We also establish a lower bound technique via lifting: for formulas requiring large degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"37 12","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508180","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 : 2023-01-18DOI: https://dl.acm.org/doi/10.1145/3565364
Bahar Aameri, Michael Grüninger
Within knowledge representation in artificial intelligence, a first-order ontology is a theory in first-order logic that axiomatizes the concepts in some domain. Ontology verification is concerned with the relationship between the intended models of an ontology and the models of the axiomatization of the ontology. In particular, we want to characterize the models of an ontology up to isomorphism and determine whether or not these models are equivalent to the intended models of the ontology. Unfortunately, it can be quite difficult to characterize the models of an ontology up to isomorphism. In the first half of this article, we review the different metalogical relationships between first-order theories and identify which relationship is needed for ontology verification. In particular, we will demonstrate that the notion of logical synonymy is needed to specify a representation theorem for the class of models of one first-order ontology with respect to another. In the second half of the article, we discuss the notion of reducible theories and show we can specify representation theorems by which models are constructed by amalgamating models of the constituent ontologies.
{"title":"Reducible Theories and Amalgamations of Models","authors":"Bahar Aameri, Michael Grüninger","doi":"https://dl.acm.org/doi/10.1145/3565364","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3565364","url":null,"abstract":"<p>Within knowledge representation in artificial intelligence, a first-order ontology is a theory in first-order logic that axiomatizes the concepts in some domain. Ontology verification is concerned with the relationship between the intended models of an ontology and the models of the axiomatization of the ontology. In particular, we want to characterize the models of an ontology up to isomorphism and determine whether or not these models are equivalent to the intended models of the ontology. Unfortunately, it can be quite difficult to characterize the models of an ontology up to isomorphism. In the first half of this article, we review the different metalogical relationships between first-order theories and identify which relationship is needed for ontology verification. In particular, we will demonstrate that the notion of logical synonymy is needed to specify a representation theorem for the class of models of one first-order ontology with respect to another. In the second half of the article, we discuss the notion of reducible theories and show we can specify representation theorems by which models are constructed by amalgamating models of the constituent ontologies.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"38 2","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508179","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}