Pub Date : 2023-01-05DOI: https://dl.acm.org/doi/10.1145/3572836
Michael Benedikt, Stanislav Kikot, Piotr Ostropolski-Nalewaja, Miguel Romero
A query Q is monotonically determined over a set of views V if Q can be expressed as a monotonic function of the view image. In the case of relational algebra views and queries, monotonic determinacy coincides with rewritability as a union of conjunctive queries, and it is decidable in important special cases, such as for CQ views and queries [11, 30]. We investigate the situation for views and queries in the recursive query language Datalog. We give both positive and negative results about the ability to decide monotonic determinacy, and also about the co-incidence of monotonic determinacy with Datalog rewritability.
{"title":"On monotonic determinacy and rewritability for recursive queries and views","authors":"Michael Benedikt, Stanislav Kikot, Piotr Ostropolski-Nalewaja, Miguel Romero","doi":"https://dl.acm.org/doi/10.1145/3572836","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3572836","url":null,"abstract":"<p>A query <i>Q</i> is monotonically determined over a set of views <b>V</b> if <i>Q</i> can be expressed as a monotonic function of the view image. In the case of relational algebra views and queries, monotonic determinacy coincides with rewritability as a union of conjunctive queries, and it is decidable in important special cases, such as for CQ views and queries [11, 30]. We investigate the situation for views and queries in the recursive query language Datalog. We give both positive and negative results about the ability to decide monotonic determinacy, and also about the co-incidence of monotonic determinacy with Datalog rewritability.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508164","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 study the expressivity and complexity of two modal logics interpreted on finite forests and equipped with standard modalities to reason on submodels. The logic (mathsf {ML} (,operatorname{raisebox {-2pt}{rule {1.2pt}{2.1ex}}},) ) extends the modal logic K with the composition operator (operatorname{raisebox {-2pt}{rule {1.2pt}{2.1ex}}} ) from ambient logic, whereas (mathsf {ML} (mathbin {ast }) ) features the separating conjunction (mathbin {ast } ) from separation logic. Both operators are second-order in nature. We show that (mathsf {ML} (,operatorname{raisebox {-2pt}{rule {1.2pt}{2.1ex}}},) ) is as expressive as the graded modal logic (mathsf {GML} ) (on trees) whereas (mathsf {ML} (mathbin {ast }) ) is strictly less expressive than (mathsf {GML} ). Moreover, we establish that the satisfiability problem is Tower-complete for (mathsf {ML} (mathbin {ast }) ), whereas it is (only) AExp(_{text{textsc {Pol}}} )-complete for (mathsf {ML} (,operatorname{raisebox {-2pt}{rule {1.2pt}{2.1ex}}},) ), a result which is surprising given their relative expressivity. As by-products, we solve open problems related to sister logics such as static ambient logic and modal separation logic.
{"title":"On Composing Finite Forests with Modal Logics","authors":"Bartosz Bednarczyk, Stéphane Demri, Raul Fervari, Alessio Mansutti","doi":"https://dl.acm.org/doi/10.1145/3569954","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3569954","url":null,"abstract":"<p>We study the expressivity and complexity of two modal logics interpreted on finite forests and equipped with standard modalities to reason on submodels. The logic (mathsf {ML} (,operatorname{raisebox {-2pt}{rule {1.2pt}{2.1ex}}},) ) extends the modal logic K with the composition operator (operatorname{raisebox {-2pt}{rule {1.2pt}{2.1ex}}} ) from ambient logic, whereas (mathsf {ML} (mathbin {ast }) ) features the separating conjunction (mathbin {ast } ) from separation logic. Both operators are second-order in nature. We show that (mathsf {ML} (,operatorname{raisebox {-2pt}{rule {1.2pt}{2.1ex}}},) ) is as expressive as the graded modal logic (mathsf {GML} ) (on trees) whereas (mathsf {ML} (mathbin {ast }) ) is strictly less expressive than (mathsf {GML} ). Moreover, we establish that the satisfiability problem is <span>Tower</span>-complete for (mathsf {ML} (mathbin {ast }) ), whereas it is (only) <span>AExp</span>(_{text{textsc {Pol}}} )-complete for (mathsf {ML} (,operatorname{raisebox {-2pt}{rule {1.2pt}{2.1ex}}},) ), a result which is surprising given their relative expressivity. As by-products, we solve open problems related to sister logics such as static ambient logic and modal separation logic.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508182","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 study the expressivity and complexity of two modal logics interpreted on finite forests and equipped with standard modalities to reason on submodels. The logic (mathsf {ML} ({color{black}{{vert!!vert!vert}}})) extends the modal logic K with the composition operator ({color{black}{{vert!!vert!vert}}}) from ambient logic whereas (mathsf {ML} (mathbin {ast })) features the separating conjunction (mathbin {ast }) from separation logic. Both operators are second-order in nature. We show that (mathsf {ML} ({color{black}{{vert!!vert!vert}}})) is as expressive as the graded modal logic (mathsf {GML}) (on trees) whereas (mathsf {ML} (mathbin {ast })) is strictly less expressive than (mathsf {GML}) . Moreover, we establish that the satisfiability problem is Tower-complete for (mathsf {ML} (mathbin {ast })) , whereas it is (only) AExpPol-complete for (mathsf {ML} ({color{black}{{vert!!vert!vert}}})) , a result that is surprising given their relative expressivity. As by-products, we solve open problems related to sister logics such as static ambient logic and modal separation logic.
{"title":"On Composing Finite Forests with Modal Logics","authors":"Bartosz Bednarczyk, Stephane Demri, Raul Fervari, Alessio Mansutti","doi":"10.1145/3569954","DOIUrl":"https://doi.org/10.1145/3569954","url":null,"abstract":"We study the expressivity and complexity of two modal logics interpreted on finite forests and equipped with standard modalities to reason on submodels. The logic (mathsf {ML} ({color{black}{{vert!!vert!vert}}})) extends the modal logic K with the composition operator ({color{black}{{vert!!vert!vert}}}) from ambient logic whereas (mathsf {ML} (mathbin {ast })) features the separating conjunction (mathbin {ast }) from separation logic. Both operators are second-order in nature. We show that (mathsf {ML} ({color{black}{{vert!!vert!vert}}})) is as expressive as the graded modal logic (mathsf {GML}) (on trees) whereas (mathsf {ML} (mathbin {ast })) is strictly less expressive than (mathsf {GML}) . Moreover, we establish that the satisfiability problem is Tower-complete for (mathsf {ML} (mathbin {ast })) , whereas it is (only) AExpPol-complete for (mathsf {ML} ({color{black}{{vert!!vert!vert}}})) , a result that is surprising given their relative expressivity. As by-products, we solve open problems related to sister logics such as static ambient logic and modal separation logic.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49303322","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}
Although various dynamic or temporal logics have been proposed to verify quantum protocols and systems, these two viewpoints have not been studied comprehensively enough. We propose Linear Temporal Quantum Logic (LTQL), a linear temporal extension of quantum logic with a quantum implication, and extend it to Dynamic Linear Temporal Quantum Logic (DLTQL). This logic has temporal operators to express transitions by unitary operators (quantum gates) and dynamic ones to express those by projections (projective measurement). We then prove some logical properties of the relationship between these two transitions expressed by LTQL and DLTQL. A drawback in applying LTQL to the verification of quantum protocols is that these logics cannot express the future operator in linear temporal logic. We propose a way to mitigate this drawback by using a translation from (D)LTQL to Linear Temporal Modal Logic (LTML) and a simulation. This translation reduces the satisfiability problem of (D)LTQL formulas to that of LTML with the classical semantics over quantum states.
{"title":"Semantic Analysis of a Linear Temporal Extension of Quantum Logic and Its Dynamic Aspect","authors":"Tsubasa Takagi","doi":"10.1145/3576926","DOIUrl":"https://doi.org/10.1145/3576926","url":null,"abstract":"Although various dynamic or temporal logics have been proposed to verify quantum protocols and systems, these two viewpoints have not been studied comprehensively enough. We propose Linear Temporal Quantum Logic (LTQL), a linear temporal extension of quantum logic with a quantum implication, and extend it to Dynamic Linear Temporal Quantum Logic (DLTQL). This logic has temporal operators to express transitions by unitary operators (quantum gates) and dynamic ones to express those by projections (projective measurement). We then prove some logical properties of the relationship between these two transitions expressed by LTQL and DLTQL. A drawback in applying LTQL to the verification of quantum protocols is that these logics cannot express the future operator in linear temporal logic. We propose a way to mitigate this drawback by using a translation from (D)LTQL to Linear Temporal Modal Logic (LTML) and a simulation. This translation reduces the satisfiability problem of (D)LTQL formulas to that of LTML with the classical semantics over quantum states.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42324414","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}
James Baxter, Ana Cavalcanti, Maciej Gazda, R. 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, R. Hierons","doi":"10.1145/3572837","DOIUrl":"https://doi.org/10.1145/3572837","url":null,"abstract":"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.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44799242","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}
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":"T. Aravanis","doi":"10.1145/3572907","DOIUrl":"https://doi.org/10.1145/3572907","url":null,"abstract":"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.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41308455","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}
Satisfiability Modulo Theories (SMT) refers to the problem of deciding the satisfiability of a formula with respect to certain background first-order theories. In this article, we focus on Satisfiablity Modulo Integer Arithmetic, which is referred to as SMT(IA), including both linear and non-linear integer arithmetic theories. Dominant approaches to SMT rely on calling a CDCL-based SAT solver, either in a lazy or eager flavour. Local search, a competitive approach to solving combinatorial problems including SAT, however, has not been well studied for SMT. We develop the first local-search algorithm for SMT(IA) by directly operating on variables, breaking through the traditional framework. We propose a local-search framework by considering the distinctions between Boolean and integer variables. Moreover, we design a novel operator and scoring functions tailored for integer arithmetic, as well as a two-level operation selection heuristic. Putting these together, we develop a local search SMT(IA) solver called LocalSMT. Experiments are carried out to evaluate LocalSMT on benchmark sets from SMT-LIB. The results show that LocalSMT is competitive and complementary with state-of-the-art SMT solvers, and performs particularly well on those formulae with only integer variables. A simple sequential portfolio with Z3 improves the state-of-the-art on satisfiable benchmark sets from SMT-LIB.
{"title":"Local Search For Satisfiability Modulo Integer Arithmetic Theories","authors":"Shaowei Cai, Bohan Li, Xindi Zhang","doi":"10.1145/3597495","DOIUrl":"https://doi.org/10.1145/3597495","url":null,"abstract":"Satisfiability Modulo Theories (SMT) refers to the problem of deciding the satisfiability of a formula with respect to certain background first-order theories. In this article, we focus on Satisfiablity Modulo Integer Arithmetic, which is referred to as SMT(IA), including both linear and non-linear integer arithmetic theories. Dominant approaches to SMT rely on calling a CDCL-based SAT solver, either in a lazy or eager flavour. Local search, a competitive approach to solving combinatorial problems including SAT, however, has not been well studied for SMT. We develop the first local-search algorithm for SMT(IA) by directly operating on variables, breaking through the traditional framework. We propose a local-search framework by considering the distinctions between Boolean and integer variables. Moreover, we design a novel operator and scoring functions tailored for integer arithmetic, as well as a two-level operation selection heuristic. Putting these together, we develop a local search SMT(IA) solver called LocalSMT. Experiments are carried out to evaluate LocalSMT on benchmark sets from SMT-LIB. The results show that LocalSMT is competitive and complementary with state-of-the-art SMT solvers, and performs particularly well on those formulae with only integer variables. A simple sequential portfolio with Z3 improves the state-of-the-art on satisfiable benchmark sets from SMT-LIB.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45805729","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}
S. Ghilezan, J. Pantović, I. Prokić, A. Scalas, N. Yoshida
Session subtyping is a cornerstone of refinement of communicating processes: a process implementing a session type (i.e., a communication protocol) T can be safely used whenever a process implementing one of its supertypes T′ is expected, in any context, without introducing deadlocks nor other communication errors. As a consequence, whenever T ≤ T′ holds, it is safe to replace an implementation of T′ with an implementation of the subtype T, which may allow for more optimised communication patterns. We present the first formalisation of the precise subtyping relation for asynchronous multiparty sessions. We show that our subtyping relation is sound (i.e., guarantees safe process replacement, as outlined above) and also complete: any extension of the relation is unsound. To achieve our results, we develop a novel session decomposition technique, from full session types (including internal/external choices) into single input/output session trees (without choices). We cover multiparty sessions with asynchronous interaction, where messages are transmitted via FIFO queues (as in the TCP/IP protocol), and prove that our subtyping is both operationally and denotationally precise. Our session decomposition technique expresses the subtyping relation as a composition of refinement relations between single input/output trees, and providing a simple reasoning principle for asynchronous message optimisations.
{"title":"Precise Subtyping for Asynchronous Multiparty Sessions","authors":"S. Ghilezan, J. Pantović, I. Prokić, A. Scalas, N. Yoshida","doi":"10.1145/3568422","DOIUrl":"https://doi.org/10.1145/3568422","url":null,"abstract":"Session subtyping is a cornerstone of refinement of communicating processes: a process implementing a session type (i.e., a communication protocol) T can be safely used whenever a process implementing one of its supertypes T′ is expected, in any context, without introducing deadlocks nor other communication errors. As a consequence, whenever T ≤ T′ holds, it is safe to replace an implementation of T′ with an implementation of the subtype T, which may allow for more optimised communication patterns. We present the first formalisation of the precise subtyping relation for asynchronous multiparty sessions. We show that our subtyping relation is sound (i.e., guarantees safe process replacement, as outlined above) and also complete: any extension of the relation is unsound. To achieve our results, we develop a novel session decomposition technique, from full session types (including internal/external choices) into single input/output session trees (without choices). We cover multiparty sessions with asynchronous interaction, where messages are transmitted via FIFO queues (as in the TCP/IP protocol), and prove that our subtyping is both operationally and denotationally precise. Our session decomposition technique expresses the subtyping relation as a composition of refinement relations between single input/output trees, and providing a simple reasoning principle for asynchronous message optimisations.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49126937","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 : 2022-10-20DOI: https://dl.acm.org/doi/10.1145/3529535
Luca Aceto, Valentina Castiglioni, Wan Fokkink, Anna Ingólfsdóttir, Bas 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":"Luca Aceto, Valentina Castiglioni, Wan Fokkink, Anna Ingólfsdóttir, Bas Luttik","doi":"https://dl.acm.org/doi/10.1145/3529535","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3529535","url":null,"abstract":"<p>Bergstra and Klop have shown that <i>bisimilarity</i> has a <i>finite</i> equational axiomatisation over ACP/CCS extended with the binary <i>left</i> and <i>communication merge</i> operators. Moller proved that auxiliary operators are <i>necessary</i> to obtain a finite axiomatisation of bisimilarity over CCS, and Aceto et al. showed that this remains true when <i>Hennessy’s merge</i> is added to that language. These results raise the question of whether there is <i>one</i> auxiliary <i>binary</i> 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.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508185","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 : 2022-10-20DOI: https://dl.acm.org/doi/10.1145/3545114
Adam Case, Christopher P. Porter
In this article, we study several aspects of the intersections of algorithmically random closed sets. First, we answer a question of Cenzer and Weber, showing that the operation of intersecting relatively random closed sets (random with respect to certain underlying measures induced by Bernoulli measures on the space of codes of closed sets), which preserves randomness, can be inverted: a random closed set of the appropriate type can be obtained as the intersection of two relatively random closed sets. We then extend the Cenzer/Weber analysis to the intersection of multiple random closed sets, identifying the Bernoulli measures with respect to which the intersection of relatively random closed sets can be non-empty. We lastly apply our analysis to provide a characterization of the effective Hausdorff dimension of sequences in terms of the degree of intersectability of random closed sets that contain them.
{"title":"The Intersection of Algorithmically Random Closed Sets and Effective Dimension","authors":"Adam Case, Christopher P. Porter","doi":"https://dl.acm.org/doi/10.1145/3545114","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3545114","url":null,"abstract":"<p>In this article, we study several aspects of the intersections of algorithmically random closed sets. First, we answer a question of Cenzer and Weber, showing that the operation of intersecting relatively random closed sets (random with respect to certain underlying measures induced by Bernoulli measures on the space of codes of closed sets), which preserves randomness, can be inverted: a random closed set of the appropriate type can be obtained as the intersection of two relatively random closed sets. We then extend the Cenzer/Weber analysis to the intersection of multiple random closed sets, identifying the Bernoulli measures with respect to which the intersection of relatively random closed sets can be non-empty. We lastly apply our analysis to provide a characterization of the effective Hausdorff dimension of sequences in terms of the degree of intersectability of random closed sets that contain them.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508163","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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