Arnaud Carayol, M. Hague, A. Meyer, C. Ong, O. Serre
In this paper we consider parity games defined by higher-order pushdown automata. These automata generalise pushdown automata by the use of higher-order stacks, which are nested "stack of stacks" structures. Representing higher-order stacks as well-bracketed words in the usual way, we show that the winning regions of these games are regular sets of words. Moreover a finite automaton recognising this region can be effectively computed. A novelty of our work are abstract pushdown processes which can be seen as (ordinary) pushdown automata but with an infinite stack alphabet. We use the device to give a uniform presentation of our results.From our main result on winning regions of parity games we derive a solution to the Modal Mu-Calculus Global Model-Checking Problem for higher-order pushdown graphs as well as for ranked trees generated by higher-order safe recursion schemes.
{"title":"Winning Regions of Higher-Order Pushdown Games","authors":"Arnaud Carayol, M. Hague, A. Meyer, C. Ong, O. Serre","doi":"10.1109/LICS.2008.41","DOIUrl":"https://doi.org/10.1109/LICS.2008.41","url":null,"abstract":"In this paper we consider parity games defined by higher-order pushdown automata. These automata generalise pushdown automata by the use of higher-order stacks, which are nested \"stack of stacks\" structures. Representing higher-order stacks as well-bracketed words in the usual way, we show that the winning regions of these games are regular sets of words. Moreover a finite automaton recognising this region can be effectively computed. A novelty of our work are abstract pushdown processes which can be seen as (ordinary) pushdown automata but with an infinite stack alphabet. We use the device to give a uniform presentation of our results.From our main result on winning regions of parity games we derive a solution to the Modal Mu-Calculus Global Model-Checking Problem for higher-order pushdown graphs as well as for ranked trees generated by higher-order safe recursion schemes.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121366265","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 present a syntax for MALL (multiplicative additive linear logic without units) proof nets which refines Girard's one. It is also based on the use of monomial weights for identifying additive components (slices). Our generalization gives the possibility of representing a kind of sharing of nodes which does not exist in Girard's nets. This sharing leads to the definition of a strong cut elimination procedure for MALL. We give a correctness criterion which is proved to be stable by reduction and to give a sequentialization theorem with respect to the sequent calculus. Sequentialization is proved by showing that an expansion procedure allows us to unfold any of our proof nets into a Girard proof net.
{"title":"Cut Elimination for Monomial MALL Proof Nets","authors":"Olivier Laurent, Roberto Maieli","doi":"10.1109/LICS.2008.31","DOIUrl":"https://doi.org/10.1109/LICS.2008.31","url":null,"abstract":"We present a syntax for MALL (multiplicative additive linear logic without units) proof nets which refines Girard's one. It is also based on the use of monomial weights for identifying additive components (slices). Our generalization gives the possibility of representing a kind of sharing of nodes which does not exist in Girard's nets. This sharing leads to the definition of a strong cut elimination procedure for MALL. We give a correctness criterion which is proved to be stable by reduction and to give a sequentialization theorem with respect to the sequent calculus. Sequentialization is proved by showing that an expansion procedure allows us to unfold any of our proof nets into a Girard proof net.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115529335","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}
Certain principles are fundamental to operational semantics, regardless of the languages or idioms involved. Such principles include rule-based definitions and proof techniques for congruence results. We formulate these principles in the general context of categorical logic. From this general formulation we recover precise results for particular language idioms by interpreting the logic in particular categories. For instance, results for first-order calculi, such as CCS, arise from considering the general results in the category of sets. Results for languages involving substitution and name generation, such as the pi-calculus, arise from considering the general results in categories of sheaves and group actions. As an extended example, we develop a tyft/tyxt-like rule format for open bisimulation in the pi-calculus.
{"title":"General Structural Operational Semantics through Categorical Logic","authors":"S. Staton","doi":"10.1109/LICS.2008.43","DOIUrl":"https://doi.org/10.1109/LICS.2008.43","url":null,"abstract":"Certain principles are fundamental to operational semantics, regardless of the languages or idioms involved. Such principles include rule-based definitions and proof techniques for congruence results. We formulate these principles in the general context of categorical logic. From this general formulation we recover precise results for particular language idioms by interpreting the logic in particular categories. For instance, results for first-order calculi, such as CCS, arise from considering the general results in the category of sets. Results for languages involving substitution and name generation, such as the pi-calculus, arise from considering the general results in categories of sheaves and group actions. As an extended example, we develop a tyft/tyxt-like rule format for open bisimulation in the pi-calculus.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"435 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123009869","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}
Pub Date : 2008-06-24DOI: 10.2168/LMCS-11(3:9)2015
Hubie Chen, Florent R. Madelaine, B. Martin
We study two containment problems related to the quantified constraint satisfaction problem (QCSP). Firstly, we give a combinatorial condition on finite structures A and B that is necessary and sufficient to render QCSP(A) a subset of QCSP(B). The required condition is the existence of a positive integer r such that there is a surjective homomorphism from the power structure A^r to B. We note that this condition is already necessary to guarantee containment of the Pi_2 restriction of QCSP, that is Pi_2-CSP(A) a subset of Pi_2-CSP(B). Since we are able to give an effective bound on such an r, we provide a decision procedure for the model containment problem with non-deterministic double-exponential time complexity. Secondly, we prove that the entailment problem for quantified conjunctive-positive first-order logic is decidable. That is, given two sentences phi and psi of first-order logic with no instances of negation or disjunction, we give an algorithm that determines whether "phi implies psi" is true in all structures (models). Our result is in some sense tight, since we show that the entailment problem for positive first-order logic (i.e. quantified conjunctive-positive logic plus disjunction) is undecidable.
{"title":"Quantified Constraints and Containment Problems","authors":"Hubie Chen, Florent R. Madelaine, B. Martin","doi":"10.2168/LMCS-11(3:9)2015","DOIUrl":"https://doi.org/10.2168/LMCS-11(3:9)2015","url":null,"abstract":"We study two containment problems related to the quantified constraint satisfaction problem (QCSP). Firstly, we give a combinatorial condition on finite structures A and B that is necessary and sufficient to render QCSP(A) a subset of QCSP(B). The required condition is the existence of a positive integer r such that there is a surjective homomorphism from the power structure A^r to B. We note that this condition is already necessary to guarantee containment of the Pi_2 restriction of QCSP, that is Pi_2-CSP(A) a subset of Pi_2-CSP(B). Since we are able to give an effective bound on such an r, we provide a decision procedure for the model containment problem with non-deterministic double-exponential time complexity. Secondly, we prove that the entailment problem for quantified conjunctive-positive first-order logic is decidable. That is, given two sentences phi and psi of first-order logic with no instances of negation or disjunction, we give an algorithm that determines whether \"phi implies psi\" is true in all structures (models). Our result is in some sense tight, since we show that the entailment problem for positive first-order logic (i.e. quantified conjunctive-positive logic plus disjunction) is undecidable.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126808689","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}
C. Baier, N. Bertrand, P. Bouyer, Thomas Brihaye, Marcus Größer
In this paper, we define two relaxed semantics (one based on probabilities and the other one based on the topological notion of largeness) for LTL over infinite runs of timed automata which rule out unlikely sequences of events. We prove that these two semantics match in the framework of single-clock timed automata (and only in that framework), and prove that the corresponding relaxed model-checking problems are PSPACE-Complete. Moreover, we prove that the probabilistic non-Zenoness can be decided for single-clocktimed automata in NLOGSPACE.
{"title":"Almost-Sure Model Checking of Infinite Paths in One-Clock Timed Automata","authors":"C. Baier, N. Bertrand, P. Bouyer, Thomas Brihaye, Marcus Größer","doi":"10.1109/LICS.2008.25","DOIUrl":"https://doi.org/10.1109/LICS.2008.25","url":null,"abstract":"In this paper, we define two relaxed semantics (one based on probabilities and the other one based on the topological notion of largeness) for LTL over infinite runs of timed automata which rule out unlikely sequences of events. We prove that these two semantics match in the framework of single-clock timed automata (and only in that framework), and prove that the corresponding relaxed model-checking problems are PSPACE-Complete. Moreover, we prove that the probabilistic non-Zenoness can be decided for single-clocktimed automata in NLOGSPACE.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131830066","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}
Kleene algebra with tests (KAT) is an equational system for program verification that combines Kleene algebra (KA), or the algebra of regular expressions, with Boolean algebra. It can model basic programming and verification constructs such as conditional tests, while loops, and Hoare triples, thus providing a relatively simple equational approach to program equivalence and partial correctness. In this paper we show how KAT can be used to give a rigorous equational treatment of control constructs involving nonlocal transfer of control such as unconditional jumps, loop statements with multi-level breaks, and exception handlers. We develop a compositional semantics and a complete equational axiomatization. The approach has some novel technical features, including a treatment of multi-level break statements that is reminiscent of de Bruijn indices in the variable-free lambda calculus. We illustrate the use of the system by giving a purely calculational proof that every deterministic flowchart is equivalent to a loop program with multi-level breaks.
{"title":"Nonlocal Flow of Control and Kleene Algebra with Tests","authors":"D. Kozen","doi":"10.1109/LICS.2008.32","DOIUrl":"https://doi.org/10.1109/LICS.2008.32","url":null,"abstract":"Kleene algebra with tests (KAT) is an equational system for program verification that combines Kleene algebra (KA), or the algebra of regular expressions, with Boolean algebra. It can model basic programming and verification constructs such as conditional tests, while loops, and Hoare triples, thus providing a relatively simple equational approach to program equivalence and partial correctness. In this paper we show how KAT can be used to give a rigorous equational treatment of control constructs involving nonlocal transfer of control such as unconditional jumps, loop statements with multi-level breaks, and exception handlers. We develop a compositional semantics and a complete equational axiomatization. The approach has some novel technical features, including a treatment of multi-level break statements that is reminiscent of de Bruijn indices in the variable-free lambda calculus. We illustrate the use of the system by giving a purely calculational proof that every deterministic flowchart is equivalent to a loop program with multi-level breaks.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133662604","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}
Superdeduction is a formalism closely related to deduction modulo which permits to enrich a deduction system - especially a first-order one such as natural deduction or sequent calculus - with new inference rules automatically computed from the presentation of a theory. We give a natural encoding from every functional pure type system (PTS) into superdeduction by defining an appropriate first-order theory. We prove that this translation is correct and conservative, showing a correspondence between valid typing judgments in the PTS and provable sequents in the corresponding superdeductive system. As a byproduct, we also introduce the superdeductive sequent calculus for intuitionistic logic, which was until now only defined for classical logic. We show its equivalence with the superdeductive natural deduction. This implies that superdeduction can be easily used as a logical framework. These results lead to a better understanding of the implementation and the automation of proof search for PTS, as well as to more cooperation between proof assistants.
{"title":"A First-Order Representation of Pure Type Systems Using Superdeduction","authors":"Guillaume Burel","doi":"10.1109/LICS.2008.22","DOIUrl":"https://doi.org/10.1109/LICS.2008.22","url":null,"abstract":"Superdeduction is a formalism closely related to deduction modulo which permits to enrich a deduction system - especially a first-order one such as natural deduction or sequent calculus - with new inference rules automatically computed from the presentation of a theory. We give a natural encoding from every functional pure type system (PTS) into superdeduction by defining an appropriate first-order theory. We prove that this translation is correct and conservative, showing a correspondence between valid typing judgments in the PTS and provable sequents in the corresponding superdeductive system. As a byproduct, we also introduce the superdeductive sequent calculus for intuitionistic logic, which was until now only defined for classical logic. We show its equivalence with the superdeductive natural deduction. This implies that superdeduction can be easily used as a logical framework. These results lead to a better understanding of the implementation and the automation of proof search for PTS, as well as to more cooperation between proof assistants.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129169458","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 show that reachability and termination for lossy channel systems is exactly at level Fomegaomega in the fast-growing hierarchy of recursive functions, the first level that dominates all multiply-recursive functions.
{"title":"The Ordinal Recursive Complexity of Lossy Channel Systems","authors":"P. Chambart, P. Schnoebelen","doi":"10.1109/LICS.2008.47","DOIUrl":"https://doi.org/10.1109/LICS.2008.47","url":null,"abstract":"We show that reachability and termination for lossy channel systems is exactly at level Fomegaomega in the fast-growing hierarchy of recursive functions, the first level that dominates all multiply-recursive functions.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130427543","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 classes of Buchi and Rabin automatic structures. For Buchi (Rabin) automatic structures their domains consist of infinite strings (trees), and the basic relations, including the equality relation, and graphs of operations are recognized by Buchi (Rabin) automata. A Buchi (Rabin) automatic structure is injective if different infinite strings (trees) represent different elements of the structure. The first part of the paper is devoted to understanding the automata- theoretic content of the well-known Lowenheim-Skolem theorem in model theory. We provide automata-theoretic versions of Lowenheim-Skolem theorem for Rabin and Buchi automatic structures. In the second part, we address the following two well-known open problems in the theory of automatic structures: Does every Buchi automatic structure have an injective Buchi presentation? Does every Rabin automatic structure have an injective Rabin presentation? We provide examples of Buchi structures without injective Buchi and Rabin presentations. To answer these questions we introduce Borel structures and use some of the basic properties of Borel sets and isomorphisms. Finally, in the last part of the paper we study the isomorphism problem for Buchi automatic structures.
{"title":"From Automatic Structures to Borel Structures","authors":"G. Hjorth, B. Khoussainov, A. Montalbán, A. Nies","doi":"10.1109/LICS.2008.28","DOIUrl":"https://doi.org/10.1109/LICS.2008.28","url":null,"abstract":"We study the classes of Buchi and Rabin automatic structures. For Buchi (Rabin) automatic structures their domains consist of infinite strings (trees), and the basic relations, including the equality relation, and graphs of operations are recognized by Buchi (Rabin) automata. A Buchi (Rabin) automatic structure is injective if different infinite strings (trees) represent different elements of the structure. The first part of the paper is devoted to understanding the automata- theoretic content of the well-known Lowenheim-Skolem theorem in model theory. We provide automata-theoretic versions of Lowenheim-Skolem theorem for Rabin and Buchi automatic structures. In the second part, we address the following two well-known open problems in the theory of automatic structures: Does every Buchi automatic structure have an injective Buchi presentation? Does every Rabin automatic structure have an injective Rabin presentation? We provide examples of Buchi structures without injective Buchi and Rabin presentations. To answer these questions we introduce Borel structures and use some of the basic properties of Borel sets and isomorphisms. Finally, in the last part of the paper we study the isomorphism problem for Buchi automatic structures.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124940924","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}
The paper develops a mathematical theory in the spirit of categorical algebra that provides a model theory for second-order and dependently-sorted syntax. The theory embodies notions such as alpha-equivalence, variable binding, capture-avoiding simultaneous substitution, term metavariable, meta-substitution, mono and multi sorting, and sort dependency. As a matter of illustration, a model is used to extract a second-order syntactic theory, which is thus guaranteed to be correct by construction.
{"title":"Second-Order and Dependently-Sorted Abstract Syntax","authors":"M. Fiore","doi":"10.1109/LICS.2008.38","DOIUrl":"https://doi.org/10.1109/LICS.2008.38","url":null,"abstract":"The paper develops a mathematical theory in the spirit of categorical algebra that provides a model theory for second-order and dependently-sorted syntax. The theory embodies notions such as alpha-equivalence, variable binding, capture-avoiding simultaneous substitution, term metavariable, meta-substitution, mono and multi sorting, and sort dependency. As a matter of illustration, a model is used to extract a second-order syntactic theory, which is thus guaranteed to be correct by construction.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122842045","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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