We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical logic, intuitionistic logic, and multiplicative-additive linear logic are derived as fragments of LKU by increasing the sensitivity of specialized structural rules to polarity information. We develop a unified, streamlined framework for proving cut-elimination in the various fragments. Furthermore, each sublogic can interact with other fragments through cut. We also consider the possibility of introducing classical-linear hybrid logics.
{"title":"A Unified Sequent Calculus for Focused Proofs","authors":"Chuck C. Liang, D. Miller","doi":"10.1109/LICS.2009.47","DOIUrl":"https://doi.org/10.1109/LICS.2009.47","url":null,"abstract":"We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical logic, intuitionistic logic, and multiplicative-additive linear logic are derived as fragments of LKU by increasing the sensitivity of specialized structural rules to polarity information. We develop a unified, streamlined framework for proving cut-elimination in the various fragments. Furthermore, each sublogic can interact with other fragments through cut. We also consider the possibility of introducing classical-linear hybrid logics.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126815197","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 graph reachability problem as a language over an infinite alphabet. Namely, we view a word of even lengtha0 b0 ... an b_n over an infinite alphabet as a directed graph with the symbols that appear in a0 b0 ... an bn as the vertices and (a0, b0),...,(an, bn) as the edges. We prove that for any positive integer k, k pebbles are sufficient for recognizing the existence of a path of length 2^k-1 from the vertex a0 to the vertex bn, but are not sufficient for recognizing the existence of a path of length 2^{k+1} - 2 from the vertex a0 to the vertex bn. Based on this result, we establish a number of relations among some classes of languages over infinite alphabets.
{"title":"Graph Reachability and Pebble Automata over Infinite Alphabets","authors":"Tony Tan","doi":"10.1145/2499937.2499940","DOIUrl":"https://doi.org/10.1145/2499937.2499940","url":null,"abstract":"We study the graph reachability problem as a language over an infinite alphabet. Namely, we view a word of even lengtha0 b0 ... an b_n over an infinite alphabet as a directed graph with the symbols that appear in a0 b0 ... an bn as the vertices and (a0, b0),...,(an, bn) as the edges. We prove that for any positive integer k, k pebbles are sufficient for recognizing the existence of a path of length 2^k-1 from the vertex a0 to the vertex bn, but are not sufficient for recognizing the existence of a path of length 2^{k+1} - 2 from the vertex a0 to the vertex bn. Based on this result, we establish a number of relations among some classes of languages over infinite alphabets.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122377971","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}
In modal logics, graded (world) modalities have been deeply investigated as a useful framework for generalizing standard existential and universal modalities in such a way that they can express statements about a given number of immediately accessible worlds. These modalities have been recently investigated with respect to the mu-calculus, which have provided succinctness, without affecting the satisfiability of the extended logic, i.e., it remains solvable in ExpTime. A natural question that arises is how logics that allow reasoning about paths could be affected by considering graded path modalities. In this paper, we investigate this question in the case of the branching-time temporal logic CTL (GCTL, for short). We prove that, although GCTL is more expressive than CTL, the satisfiability problem for GCTL remains solvable in ExpTime. This result is obtained by exploiting an automata-theoretic approach. In particular, we introduce the class of partitioning alternating Büchi tree automata and show that the emptiness problem for them is ExpTime-Complete. The satisfiability result turns even more interesting as we show that GCTL is exponentially more succinct than graded mu-calculus.
{"title":"Graded Computation Tree Logic","authors":"A. Bianco, F. Mogavero, A. Murano","doi":"10.1145/2287718.2287725","DOIUrl":"https://doi.org/10.1145/2287718.2287725","url":null,"abstract":"In modal logics, graded (world) modalities have been deeply investigated as a useful framework for generalizing standard existential and universal modalities in such a way that they can express statements about a given number of immediately accessible worlds. These modalities have been recently investigated with respect to the mu-calculus, which have provided succinctness, without affecting the satisfiability of the extended logic, i.e., it remains solvable in ExpTime. A natural question that arises is how logics that allow reasoning about paths could be affected by considering graded path modalities. In this paper, we investigate this question in the case of the branching-time temporal logic CTL (GCTL, for short). We prove that, although GCTL is more expressive than CTL, the satisfiability problem for GCTL remains solvable in ExpTime. This result is obtained by exploiting an automata-theoretic approach. In particular, we introduce the class of partitioning alternating Büchi tree automata and show that the emptiness problem for them is ExpTime-Complete. The satisfiability result turns even more interesting as we show that GCTL is exponentially more succinct than graded mu-calculus.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122183615","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}
A. Dawar, Martin Grohe, Bjarki Holm, Bastian Laubner
We introduce extensions of first-order logic (FO) and fixed-point logic (FP) with operators that compute the rank of a definable matrix. These operators are generalizations of the counting operations in FP+C (i.e. fixed-point logic with counting) that allow us to count the dimension of a definable vector space, rather than just count the cardinality of a definable set. The logics we define have data complexity contained in polynomial time and all known examples of polynomial time queries that are not definable in FP+C are definable in FP+rk, the extension of FP with rank operators. For each prime number p and each positive integer n, we have rank operators rk_p for determining the rank of a matrix over the finite field GF_p defined by a formula over n-tuples. We compare the expressive power of the logics obtained by varying the values p and n can take. In particular, we show that increasing the arity of the operators yields an infinite hierarchy of expressive power. The rank operators are surprisingly expressive, even in the absence of fixed-point operators. We show that FO+rk_p can define deterministic and symmetric transitive closure. This allows us to show that, on ordered structures, FO+rk_p captures the complexity class MOD_pL, for all prime values of p.
{"title":"Logics with Rank Operators","authors":"A. Dawar, Martin Grohe, Bjarki Holm, Bastian Laubner","doi":"10.1109/LICS.2009.24","DOIUrl":"https://doi.org/10.1109/LICS.2009.24","url":null,"abstract":"We introduce extensions of first-order logic (FO) and fixed-point logic (FP) with operators that compute the rank of a definable matrix. These operators are generalizations of the counting operations in FP+C (i.e. fixed-point logic with counting) that allow us to count the dimension of a definable vector space, rather than just count the cardinality of a definable set. The logics we define have data complexity contained in polynomial time and all known examples of polynomial time queries that are not definable in FP+C are definable in FP+rk, the extension of FP with rank operators. For each prime number p and each positive integer n, we have rank operators rk_p for determining the rank of a matrix over the finite field GF_p defined by a formula over n-tuples. We compare the expressive power of the logics obtained by varying the values p and n can take. In particular, we show that increasing the arity of the operators yields an infinite hierarchy of expressive power. The rank operators are surprisingly expressive, even in the absence of fixed-point operators. We show that FO+rk_p can define deterministic and symmetric transitive closure. This allows us to show that, on ordered structures, FO+rk_p captures the complexity class MOD_pL, for all prime values of p.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130409368","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 characterise type structurally the termination observational congruence of Abadi and Cardelli’s S8 calculus. Pitts’ operational reasoning approach for polymorphic lambda calculi is enhanced with subtyping and primitive covariant object types. Labelling each object with a bound ordinal of terminating method invocations and regarding !-bounded as unlabelled reduction we achieve a crucial object unwinding lemma. Value and term bisimilarities are suitably defined with novel type structural operators and (type-, relation- and value- environment) bindings. We prove term bisimilarity complete and sound with respect to observational congruence, postulated as the largest, substitutive, compatible, (termination) adequate and (type) subsumptive, well typed term relation.
{"title":"Fully Abstract Logical Bisimilarity for a Polymorphic Object Calculus","authors":"Luís Dominguez","doi":"10.1109/LICS.2009.49","DOIUrl":"https://doi.org/10.1109/LICS.2009.49","url":null,"abstract":"We characterise type structurally the termination observational congruence of Abadi and Cardelli’s S8 calculus. Pitts’ operational reasoning approach for polymorphic lambda calculi is enhanced with subtyping and primitive covariant object types. Labelling each object with a bound ordinal of terminating method invocations and regarding !-bounded as unlabelled reduction we achieve a crucial object unwinding lemma. Value and term bisimilarities are suitably defined with novel type structural operators and (type-, relation- and value- environment) bindings. We prove term bisimilarity complete and sound with respect to observational congruence, postulated as the largest, substitutive, compatible, (termination) adequate and (type) subsumptive, well typed term relation.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114729679","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}
Jesper Bengtson, Magnus Johansson, J. Parrow, B. Victor
A psi-calculus is an extension of the pi-calculus with nominal data types for data structures and for logical assertions representing facts about data. These can be transmitted between processes and their names can be statically scoped using the standard pi-calculus mechanism to allow for scope migrations. Other proposed extensions of the pi-calculus can be formulated as psi-calculi; examples include the applied pi-calculus, the spi-calculus, the fusion calculus, the concurrent constraint pi-calculus, and calculi with polyadic communication channels or pattern matching. Psi-calculi can be even more general, for example by allowing structured channels, higher-order formalisms such as the lambda calculus for data structures, and a predicate logic for assertions. Our labelled operational semantics and definition of bisimulation is straightforward, without a structural congruence. We establish minimal requirements on the nominal data and logic in order to prove general algebraic properties of psi-calculi. The proofs have been checked in the interactive proof checker Isabelle. We are the first to formulate a truly compositional labelled operational semantics for calculi of this calibre. Expressiveness and therefore modelling convenience significantly exceeds that of other formalisms, while the purity of the semantics is on par with the original pi-calculus.
{"title":"Psi-calculi: Mobile Processes, Nominal Data, and Logic","authors":"Jesper Bengtson, Magnus Johansson, J. Parrow, B. Victor","doi":"10.1109/LICS.2009.20","DOIUrl":"https://doi.org/10.1109/LICS.2009.20","url":null,"abstract":"A psi-calculus is an extension of the pi-calculus with nominal data types for data structures and for logical assertions representing facts about data. These can be transmitted between processes and their names can be statically scoped using the standard pi-calculus mechanism to allow for scope migrations. Other proposed extensions of the pi-calculus can be formulated as psi-calculi; examples include the applied pi-calculus, the spi-calculus, the fusion calculus, the concurrent constraint pi-calculus, and calculi with polyadic communication channels or pattern matching. Psi-calculi can be even more general, for example by allowing structured channels, higher-order formalisms such as the lambda calculus for data structures, and a predicate logic for assertions. Our labelled operational semantics and definition of bisimulation is straightforward, without a structural congruence. We establish minimal requirements on the nominal data and logic in order to prove general algebraic properties of psi-calculi. The proofs have been checked in the interactive proof checker Isabelle. We are the first to formulate a truly compositional labelled operational semantics for calculi of this calibre. Expressiveness and therefore modelling convenience significantly exceeds that of other formalisms, while the purity of the semantics is on par with the original pi-calculus.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129299817","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 : 2009-08-11DOI: 10.1007/978-3-642-15025-8_14
Yijia Chen, J. Flum
{"title":"A Logic for PTIME and a Parameterized Halting Problem","authors":"Yijia Chen, J. Flum","doi":"10.1007/978-3-642-15025-8_14","DOIUrl":"https://doi.org/10.1007/978-3-642-15025-8_14","url":null,"abstract":"","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126467804","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}
Kristoffer Arnsfelt Hansen, M. Koucký, Peter Bro Miltersen
We exhibit a deterministic concurrent reachability game PURGATORY$_n$ with $n$ non-terminal positions and a binary choice for both players in every position so that any positional strategy for Player 1 achieving the value of the game within given $epsilon
{"title":"Winning Concurrent Reachability Games Requires Doubly-Exponential Patience","authors":"Kristoffer Arnsfelt Hansen, M. Koucký, Peter Bro Miltersen","doi":"10.1109/LICS.2009.44","DOIUrl":"https://doi.org/10.1109/LICS.2009.44","url":null,"abstract":"We exhibit a deterministic concurrent reachability game PURGATORY$_n$ with $n$ non-terminal positions and a binary choice for both players in every position so that any positional strategy for Player 1 achieving the value of the game within given $epsilon","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121716676","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 a substructural logic with ordered, linear, and persistent propositions and then endow a fragment with a committed choice forward-chaining operational interpretation. Exploiting higher-order terms in this metalanguage, we specify the operational semantics of a number of object language features, such as call-by-value, call-by-name, call-by-need, mutable store, parallelism, communication, exceptions and continuations. The specifications exhibit a high degree of uniformity and modularity that allows us to analyze the structural properties required for each feature in isolation. Our substructural framework thereby provides a new methodology for language specification that synthesizes structural operational semantics, abstract machines, and logical approaches.
{"title":"Substructural Operational Semantics as Ordered Logic Programming","authors":"F. Pfenning, R. Simmons","doi":"10.1109/LICS.2009.8","DOIUrl":"https://doi.org/10.1109/LICS.2009.8","url":null,"abstract":"We describe a substructural logic with ordered, linear, and persistent propositions and then endow a fragment with a committed choice forward-chaining operational interpretation. Exploiting higher-order terms in this metalanguage, we specify the operational semantics of a number of object language features, such as call-by-value, call-by-name, call-by-need, mutable store, parallelism, communication, exceptions and continuations. The specifications exhibit a high degree of uniformity and modularity that allows us to analyze the structural properties required for each feature in isolation. Our substructural framework thereby provides a new methodology for language specification that synthesizes structural operational semantics, abstract machines, and logical approaches.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131286483","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 every finitely satisfiable two-variable first-order formula with two equivalence relations has a model of size at most triply exponential with respect to its length. Thus the finite satisfiability problem for two-variable logic over the class of structures with two equivalence relations is decidable in nondeterministic triply exponential time. We also show that replacing one of the equivalence relations in the considered class of structures by a relation which is only required to be transitive leads to undecidability. This sharpens the earlier result that two-variable logic is undecidable over the class of structures with two transitive relations.
{"title":"On Finite Satisfiability of Two-Variable First-Order Logic with Equivalence Relations","authors":"Emanuel Kieronski, Lidia Tendera","doi":"10.1109/LICS.2009.39","DOIUrl":"https://doi.org/10.1109/LICS.2009.39","url":null,"abstract":"We show that every finitely satisfiable two-variable first-order formula with two equivalence relations has a model of size at most triply exponential with respect to its length. Thus the finite satisfiability problem for two-variable logic over the class of structures with two equivalence relations is decidable in nondeterministic triply exponential time. We also show that replacing one of the equivalence relations in the considered class of structures by a relation which is only required to be transitive leads to undecidability. This sharpens the earlier result that two-variable logic is undecidable over the class of structures with two transitive relations.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128141002","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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