The algebraic theory of rational languages has provided powerful decidability results. Among them, one of the most fundamental is the definability of a rational language in the class of aperiodic languages, i.e., languages recognized by finite automata whose transition relation defines an aperiodic congruence. An important corollary of this result is the first-order definability of monadic second-order formulas over finite words.Our goal is to extend these results to rational transductions, i.e. word functions realized by finite transducers. We take an algebraic approach and consider definability problems of rational transductions in a given variety of congruences (or monoids).The strength of the algebraic theory of rational languages relies on the existence of a congruence canonically attached to every language, the syntactic congruence. In a similar spirit, Reutenauer and Schützenberger have defined a canonical device for rational transductions, that we extend to establish our main contribution: an effective characterization of V-transductions, i.e. rational transductions realizable by transducers whose transition relation defines a congruence in a (decidable) variety V. In particular, it provides an algorithm to decide the definability of a rational transduction by an aperiodic finite transducer.Using those results, we show that the FO-definability of a rational transduction is decidable, where FO-definable means definable in a first-order restriction of logical transducers à la Courcelle.Categories and Subject Descriptors F.4.2 [Mathematical Logic and Formal Languages]: Formal Languages
{"title":"First-order definability of rational transductions: An algebraic approach","authors":"E. Filiot, Olivier Gauwin, N. Lhote","doi":"10.1145/2933575.2934520","DOIUrl":"https://doi.org/10.1145/2933575.2934520","url":null,"abstract":"The algebraic theory of rational languages has provided powerful decidability results. Among them, one of the most fundamental is the definability of a rational language in the class of aperiodic languages, i.e., languages recognized by finite automata whose transition relation defines an aperiodic congruence. An important corollary of this result is the first-order definability of monadic second-order formulas over finite words.Our goal is to extend these results to rational transductions, i.e. word functions realized by finite transducers. We take an algebraic approach and consider definability problems of rational transductions in a given variety of congruences (or monoids).The strength of the algebraic theory of rational languages relies on the existence of a congruence canonically attached to every language, the syntactic congruence. In a similar spirit, Reutenauer and Schützenberger have defined a canonical device for rational transductions, that we extend to establish our main contribution: an effective characterization of V-transductions, i.e. rational transductions realizable by transducers whose transition relation defines a congruence in a (decidable) variety V. In particular, it provides an algorithm to decide the definability of a rational transduction by an aperiodic finite transducer.Using those results, we show that the FO-definability of a rational transduction is decidable, where FO-definable means definable in a first-order restriction of logical transducers à la Courcelle.Categories and Subject Descriptors F.4.2 [Mathematical Logic and Formal Languages]: Formal Languages","PeriodicalId":206395,"journal":{"name":"2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130423700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper modifies our previous work in combining classical logic with intuitionistic logic [16], [17] to also include affine linear logic, resulting in a system we call Affine Control Logic. A propositional system with six binary connectives is defined and given a phase space interpretation. Choosing classical, intuitionistic or affine reasoning is entirely dependent on the subformula property. Moreover, the connectives of these logics can mix without restriction. We give a sound and complete sequent calculus that requires novel proof transformations for cut elimination. Compared to linear logic, classical fragments of proofs are better isolated from non-classical fragments. One of our goals is to allow non-classical restrictions to coexist with computational interpretations of classical logic such as found in the λμ calculus. In fact, we show that the transition between different modes of proof, classical, intuitionistic and affine, can be interpreted by delimited control operators. We also discuss how to extend the definition of focused proofs to this logic.
{"title":"Unified Semantics and Proof System for Classical, Intuitionistic and Affine Logics","authors":"Chuck C. Liang","doi":"10.1145/2933575.2933581","DOIUrl":"https://doi.org/10.1145/2933575.2933581","url":null,"abstract":"This paper modifies our previous work in combining classical logic with intuitionistic logic [16], [17] to also include affine linear logic, resulting in a system we call Affine Control Logic. A propositional system with six binary connectives is defined and given a phase space interpretation. Choosing classical, intuitionistic or affine reasoning is entirely dependent on the subformula property. Moreover, the connectives of these logics can mix without restriction. We give a sound and complete sequent calculus that requires novel proof transformations for cut elimination. Compared to linear logic, classical fragments of proofs are better isolated from non-classical fragments. One of our goals is to allow non-classical restrictions to coexist with computational interpretations of classical logic such as found in the λμ calculus. In fact, we show that the transition between different modes of proof, classical, intuitionistic and affine, can be interpreted by delimited control operators. We also discuss how to extend the definition of focused proofs to this logic.","PeriodicalId":206395,"journal":{"name":"2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133642066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Guarded recursion is a form of recursion where recursive calls are guarded by delay modalities. Previous work has shown how guarded recursion is useful for reasoning operationally about programming languages with advanced features including general references, recursive types, countable non-determinism and concurrency.Guarded recursion also offers a way of adding recursion to type theory while maintaining logical consistency. In previous work we initiated a programme of denotational semantics in type theory using guarded recursion, by constructing a computationally adequate model of the language PCF (simply typed lambda calculus with fixed points). This model was intensional in that it could distinguish between computations computing the same result using a different number of fixed point unfoldings.In this work we show how also programming languages with recursive types can be given denotational semantics in type theory with guarded recursion. More precisely, we give a computationally adequate denotational semantics to the language FPC (simply typed lambda calculus extended with recursive types), modelling recursive types using guarded recursive types. The model is inten-sional in the same way as was the case in previous work, but we show how to recover extensionality using a logical relation.All constructions and reasoning in this paper, including proofs of theorems such as soundness and adequacy, are by (informal) reasoning in type theory, often using guarded recursion.Categories and Subject Descriptors F.3.2 [Semantics of Programming Languages]: Denotational semantics; F.4.1 [Mathematical Logic and Formal Languages]: Lambda calculus and related systems
{"title":"Denotational semantics of recursive types in synthetic guarded domain theory","authors":"Rasmus E. Møgelberg, M. Paviotti","doi":"10.1145/2933575.2934516","DOIUrl":"https://doi.org/10.1145/2933575.2934516","url":null,"abstract":"Guarded recursion is a form of recursion where recursive calls are guarded by delay modalities. Previous work has shown how guarded recursion is useful for reasoning operationally about programming languages with advanced features including general references, recursive types, countable non-determinism and concurrency.Guarded recursion also offers a way of adding recursion to type theory while maintaining logical consistency. In previous work we initiated a programme of denotational semantics in type theory using guarded recursion, by constructing a computationally adequate model of the language PCF (simply typed lambda calculus with fixed points). This model was intensional in that it could distinguish between computations computing the same result using a different number of fixed point unfoldings.In this work we show how also programming languages with recursive types can be given denotational semantics in type theory with guarded recursion. More precisely, we give a computationally adequate denotational semantics to the language FPC (simply typed lambda calculus extended with recursive types), modelling recursive types using guarded recursive types. The model is inten-sional in the same way as was the case in previous work, but we show how to recover extensionality using a logical relation.All constructions and reasoning in this paper, including proofs of theorems such as soundness and adequacy, are by (informal) reasoning in type theory, often using guarded recursion.Categories and Subject Descriptors F.3.2 [Semantics of Programming Languages]: Denotational semantics; F.4.1 [Mathematical Logic and Formal Languages]: Lambda calculus and related systems","PeriodicalId":206395,"journal":{"name":"2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115711666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Guilhem Jaber, Gabriel Lewertowski, Pierre-Marie Pédrot, Matthieu Sozeau, Nicolas Tabareau
This paper studies forcing translations of proofs in dependent type theory, through the Curry-Howard correspondence. Based on a call-by-push-value decomposition, we synthesize two simply-typed translations: i) one call-by-value, corresponding to the translation derived from the presheaf construction as studied in a previous paper; ii) one call-by-name, whose intuitions already appear in Krivine and Miquel’s work. Focusing on the call-by-name translation, we adapt it to the dependent case and prove that it is compatible with the definitional equality of our system, thus avoiding coherence problems. This allows us to use any category as forcing conditions, which is out of reach with the call-by-value translation. Our construction also exploits the notion of storage operators in order to interpret dependent elimination for inductive types. This is a novel example of a dependent theory with side-effects, clarifying how dependent elimination for inductive types must be restricted in a non-pure setting. Being implemented as a Coq plugin, this work gives the possibility to formalize easily consistency results, for instance the consistency of the negation of Voevodsky’s univalence axiom.
{"title":"The Definitional Side of the Forcing","authors":"Guilhem Jaber, Gabriel Lewertowski, Pierre-Marie Pédrot, Matthieu Sozeau, Nicolas Tabareau","doi":"10.1145/2933575.2935320","DOIUrl":"https://doi.org/10.1145/2933575.2935320","url":null,"abstract":"This paper studies forcing translations of proofs in dependent type theory, through the Curry-Howard correspondence. Based on a call-by-push-value decomposition, we synthesize two simply-typed translations: i) one call-by-value, corresponding to the translation derived from the presheaf construction as studied in a previous paper; ii) one call-by-name, whose intuitions already appear in Krivine and Miquel’s work. Focusing on the call-by-name translation, we adapt it to the dependent case and prove that it is compatible with the definitional equality of our system, thus avoiding coherence problems. This allows us to use any category as forcing conditions, which is out of reach with the call-by-value translation. Our construction also exploits the notion of storage operators in order to interpret dependent elimination for inductive types. This is a novel example of a dependent theory with side-effects, clarifying how dependent elimination for inductive types must be restricted in a non-pure setting. Being implemented as a Coq plugin, this work gives the possibility to formalize easily consistency results, for instance the consistency of the negation of Voevodsky’s univalence axiom.","PeriodicalId":206395,"journal":{"name":"2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"326 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115872884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the existence of Hanf normal forms for extensions FO(Q) of first-order logic by sets ${mathbf{Q}} subseteq mathcal{P}(mathbb{N})$ of unary counting quantifiers. A formula is in Hanf normal form if it is a Boolean combination of formulas $xi (bar x)$ describing the isomorphism type of a local neighbourhood around its free variables $bar x$ and statements of the form "the number of witnesses y of ψ(y) belongs to (Q+k)" here Q ∈ Q, k ∈ ℕ, and ψ describes the isomorphism type of a local neighbourhood around its unique free variable y.We show that a formula from FO(Q) can be transformed into a formula in Hanf normal form that is equivalent on all structures of degree ⩽ d if, and only if, all counting quantifiers occurring in the formula are ultimately periodic. This transformation can be carried out in worst-case optimal 3-fold exponential time.In particular, this yields an algorithmic version of Nurmonen’s extension of Hanf’s theorem for first-order logic with modulo-counting quantifiers. As an immediate consequence, we obtain that on finite structures of degree ⩽ d, model checking of first-order logic with modulo-counting quantifiers is fixed-parameter tractable.
{"title":"Hanf normal form for first-order logic with unary counting quantifiers","authors":"Lucas Heimberg, D. Kuske, Nicole Schweikardt","doi":"10.1145/2933575.2934571","DOIUrl":"https://doi.org/10.1145/2933575.2934571","url":null,"abstract":"We study the existence of Hanf normal forms for extensions FO(Q) of first-order logic by sets ${mathbf{Q}} subseteq mathcal{P}(mathbb{N})$ of unary counting quantifiers. A formula is in Hanf normal form if it is a Boolean combination of formulas $xi (bar x)$ describing the isomorphism type of a local neighbourhood around its free variables $bar x$ and statements of the form \"the number of witnesses y of ψ(y) belongs to (Q+k)\" here Q ∈ Q, k ∈ ℕ, and ψ describes the isomorphism type of a local neighbourhood around its unique free variable y.We show that a formula from FO(Q) can be transformed into a formula in Hanf normal form that is equivalent on all structures of degree ⩽ d if, and only if, all counting quantifiers occurring in the formula are ultimately periodic. This transformation can be carried out in worst-case optimal 3-fold exponential time.In particular, this yields an algorithmic version of Nurmonen’s extension of Hanf’s theorem for first-order logic with modulo-counting quantifiers. As an immediate consequence, we obtain that on finite structures of degree ⩽ d, model checking of first-order logic with modulo-counting quantifiers is fixed-parameter tractable.","PeriodicalId":206395,"journal":{"name":"2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116281631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is inspired by two works from the early 90s. The first one is by Bogaert and Tison who considered a model of automata on finite ranked trees where one can check equality and disequality constraints between direct subtrees: they proved that this class of automata is closed under Boolean operations and that both the emptiness and the finiteness problem of the accepted language are decidable. The second one is by Niwinski who showed that one can compute the cardinality of any ω-regular language of infinite trees.Here, we generalise the model of automata of Tison and Bogaert to the setting of infinite binary trees. Roughly speaking we consider parity tree automata where some transitions are guarded and can be used only when the two direct sub-trees of the current node are equal/disequal. We show that the resulting class of languages encompasses the one of ω-regular languages of infinite trees while sharing most of its closure properties, in particular it is a Boolean algebra. Our main technical contribution is then to prove that it also enjoys a decidable cardinality problem. In particular, this implies the decidability of the emptiness problem.
{"title":"Automata on Infinite Trees with Equality and Disequality Constraints Between Siblings","authors":"Arnaud Carayol, Christof Löding, O. Serre","doi":"10.1145/2933575.2934504","DOIUrl":"https://doi.org/10.1145/2933575.2934504","url":null,"abstract":"This article is inspired by two works from the early 90s. The first one is by Bogaert and Tison who considered a model of automata on finite ranked trees where one can check equality and disequality constraints between direct subtrees: they proved that this class of automata is closed under Boolean operations and that both the emptiness and the finiteness problem of the accepted language are decidable. The second one is by Niwinski who showed that one can compute the cardinality of any ω-regular language of infinite trees.Here, we generalise the model of automata of Tison and Bogaert to the setting of infinite binary trees. Roughly speaking we consider parity tree automata where some transitions are guarded and can be used only when the two direct sub-trees of the current node are equal/disequal. We show that the resulting class of languages encompasses the one of ω-regular languages of infinite trees while sharing most of its closure properties, in particular it is a Boolean algebra. Our main technical contribution is then to prove that it also enjoys a decidable cardinality problem. In particular, this implies the decidability of the emptiness problem.","PeriodicalId":206395,"journal":{"name":"2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121250424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Quiescent consistency is a notion of correctness for a concurrent object that gives meaning to the object’s behaviours in quiescent states, i.e., states in which none of the object’s operations are being executed. The condition enables greater flexibility in object design by allowing more behaviours to be admitted, which in turn allows the algorithms implementing quiescent consistent objects to be more efficient (when executed in a multithreaded environment).Quiescent consistency of an implementation object is defined in terms of a corresponding abstract specification. This gives rise to two important verification questions: membership (checking whether a behaviour of the implementation is allowed by the specification) and correctness (checking whether all behaviours of the implementation are allowed by the specification). In this paper, we consider the membership and correctness conditions for quiescent consistency, as well as a restricted form that assumes an upper limit on the number of events between two quiescent states. We show that the membership problem for unrestricted quiescent consistency is NP-complete and that the correctness problem is decidable, coNEXPTIME-hard, and in EXPSPACE. For the restricted form, we show that membership is in PTIME, while correctness is PSPACE-complete.
{"title":"Decidability and Complexity for Quiescent Consistency","authors":"Brijesh Dongol, R. Hierons","doi":"10.1145/2933575.2933576","DOIUrl":"https://doi.org/10.1145/2933575.2933576","url":null,"abstract":"Quiescent consistency is a notion of correctness for a concurrent object that gives meaning to the object’s behaviours in quiescent states, i.e., states in which none of the object’s operations are being executed. The condition enables greater flexibility in object design by allowing more behaviours to be admitted, which in turn allows the algorithms implementing quiescent consistent objects to be more efficient (when executed in a multithreaded environment).Quiescent consistency of an implementation object is defined in terms of a corresponding abstract specification. This gives rise to two important verification questions: membership (checking whether a behaviour of the implementation is allowed by the specification) and correctness (checking whether all behaviours of the implementation are allowed by the specification). In this paper, we consider the membership and correctness conditions for quiescent consistency, as well as a restricted form that assumes an upper limit on the number of events between two quiescent states. We show that the membership problem for unrestricted quiescent consistency is NP-complete and that the correctness problem is decidable, coNEXPTIME-hard, and in EXPSPACE. For the restricted form, we show that membership is in PTIME, while correctness is PSPACE-complete.","PeriodicalId":206395,"journal":{"name":"2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127968125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe an interpretation of recursive computation in a symmetric monoidal category with infinite biproducts and cofree commutative comonoids (for instance, the category of free modules over a complete semiring). Such categories play a significant role in "quantitative" models of computation: they bear a canonical complete monoid enrichment, but may not be cpo-enriched, making standard techniques for reasoning about fixed points unavailable. By constructing a bifree algebra for the cofree exponential, we obtain fixed points for morphisms in its co-Kleisli category without requiring any order-theoretic structure. These fixed points corresponding to infinite sums of finitary approximants indexed over the nested finite multisets, each representing a unique call-pattern for computation of the fixed point. We illustrate this construction by using it to give a denotational semantics for PCF with non-deterministic choice and scalar weights from a complete semiring, proving that this is computationally adequate with respect to an operational semantics which evaluates a term by taking a weighted sum of the residues of its terminating reduction paths.
{"title":"Fixed Points In Quantitative Semantics","authors":"J. Laird","doi":"10.1145/2933575.2934569","DOIUrl":"https://doi.org/10.1145/2933575.2934569","url":null,"abstract":"We describe an interpretation of recursive computation in a symmetric monoidal category with infinite biproducts and cofree commutative comonoids (for instance, the category of free modules over a complete semiring). Such categories play a significant role in \"quantitative\" models of computation: they bear a canonical complete monoid enrichment, but may not be cpo-enriched, making standard techniques for reasoning about fixed points unavailable. By constructing a bifree algebra for the cofree exponential, we obtain fixed points for morphisms in its co-Kleisli category without requiring any order-theoretic structure. These fixed points corresponding to infinite sums of finitary approximants indexed over the nested finite multisets, each representing a unique call-pattern for computation of the fixed point. We illustrate this construction by using it to give a denotational semantics for PCF with non-deterministic choice and scalar weights from a complete semiring, proving that this is computationally adequate with respect to an operational semantics which evaluates a term by taking a weighted sum of the residues of its terminating reduction paths.","PeriodicalId":206395,"journal":{"name":"2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129739452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce differential refinement logic (dRℒ), a logic with first-class support for refinement relations on hybrid systems, and a proof calculus for verifying such relations. dRℒ simultaneously solves several seemingly different challenges common in theorem proving for hybrid systems: 1. When hybrid systems are complicated, it is useful to prove properties about simpler and related sub-systems before tackling the system as a whole. 2. Some models of hybrid systems can be implementation-specific. Verification can be aided by abstracting the system down to the core components necessary for safety, but only if the relations between the abstraction and the original system can be guaranteed. 3. One approach to taming the complexities of hybrid systems is to start with a simplified version of the system and iteratively expand it. However, this approach can be costly, since every iteration has to be proved safe from scratch, unless refinement relations can be leveraged in the proof. 4. When proofs become large, it is di cult to maintain a modular or comprehensible proof structure. By using a refinement relation to arrange proofs hierarchically according to the structure of natural subsystems, we can increase the readability and modularity of the resulting proof. dRℒ extends an existing specification and verification language for hybrid systems (differential dynamic logic, dℒ) by adding a refinement relation to directly compare hybrid systems. This paper gives a syntax, semantics, and proof calculus for dRℒ. We demonstrate its usefulness with examples where using refinement results in easier and better-structured proofs.
{"title":"Differential Refinement Logic*","authors":"Sarah M. Loos, André Platzer","doi":"10.1145/2933575.2934555","DOIUrl":"https://doi.org/10.1145/2933575.2934555","url":null,"abstract":"We introduce differential refinement logic (dRℒ), a logic with first-class support for refinement relations on hybrid systems, and a proof calculus for verifying such relations. dRℒ simultaneously solves several seemingly different challenges common in theorem proving for hybrid systems: 1. When hybrid systems are complicated, it is useful to prove properties about simpler and related sub-systems before tackling the system as a whole. 2. Some models of hybrid systems can be implementation-specific. Verification can be aided by abstracting the system down to the core components necessary for safety, but only if the relations between the abstraction and the original system can be guaranteed. 3. One approach to taming the complexities of hybrid systems is to start with a simplified version of the system and iteratively expand it. However, this approach can be costly, since every iteration has to be proved safe from scratch, unless refinement relations can be leveraged in the proof. 4. When proofs become large, it is di cult to maintain a modular or comprehensible proof structure. By using a refinement relation to arrange proofs hierarchically according to the structure of natural subsystems, we can increase the readability and modularity of the resulting proof. dRℒ extends an existing specification and verification language for hybrid systems (differential dynamic logic, dℒ) by adding a refinement relation to directly compare hybrid systems. This paper gives a syntax, semantics, and proof calculus for dRℒ. We demonstrate its usefulness with examples where using refinement results in easier and better-structured proofs.","PeriodicalId":206395,"journal":{"name":"2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114094641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jakub Gajarský, Petr Hliněný, J. Obdržálek, D. Lokshtanov, M. Ramanujan
We study the FO model checking problem of dense graph classes, namely those which are FO-interpretable in some sparse graph classes. Note that if an input dense graph is given together with the corresponding FO interpretation in a sparse graph, one can easily solve the model checking problem using the existing algorithms for sparse graph classes. However, if the assumed interpretation is not given, then the situation is markedly harder.In this paper we give a structural characterization of graph classes which are FO interpretable in graph classes of bounded degree. This characterization allows us to efficiently compute such an interpretation for an input graph. As a consequence, we obtain an FPT algorithm for FO model checking of graph classes FO interpretable in graph classes of bounded degree. The approach we use to obtain these results may also be of independent interest.
{"title":"A New Perspective on FO Model Checking of Dense Graph Classes","authors":"Jakub Gajarský, Petr Hliněný, J. Obdržálek, D. Lokshtanov, M. Ramanujan","doi":"10.1145/2933575.2935314","DOIUrl":"https://doi.org/10.1145/2933575.2935314","url":null,"abstract":"We study the FO model checking problem of dense graph classes, namely those which are FO-interpretable in some sparse graph classes. Note that if an input dense graph is given together with the corresponding FO interpretation in a sparse graph, one can easily solve the model checking problem using the existing algorithms for sparse graph classes. However, if the assumed interpretation is not given, then the situation is markedly harder.In this paper we give a structural characterization of graph classes which are FO interpretable in graph classes of bounded degree. This characterization allows us to efficiently compute such an interpretation for an input graph. As a consequence, we obtain an FPT algorithm for FO model checking of graph classes FO interpretable in graph classes of bounded degree. The approach we use to obtain these results may also be of independent interest.","PeriodicalId":206395,"journal":{"name":"2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123428804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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