Ehrenfeucht-Fraisse games are a useful technique for proving inexpressibility results in first-order logic. Strategies for a few basic games (on long paths, set-powerset structures and random graphs, to name a few) can be used as a building blocks for strategies in more complicated games. In this talk, I will discuss a few general methods for combining strategies. Applications include results on the expressive power of successor-invariant logic and k-variable logic.
{"title":"Combining Ehrenfeucht-Fraïssé Games","authors":"B. Rossman","doi":"10.1109/LICS.2009.50","DOIUrl":"https://doi.org/10.1109/LICS.2009.50","url":null,"abstract":"Ehrenfeucht-Fraisse games are a useful technique for proving inexpressibility results in first-order logic. Strategies for a few basic games (on long paths, set-powerset structures and random graphs, to name a few) can be used as a building blocks for strategies in more complicated games. In this talk, I will discuss a few general methods for combining strategies. Applications include results on the expressive power of successor-invariant logic and k-variable logic.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"23 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":"133344083","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 physics, Feynman diagrams are used to reason about quantum processes. Similar diagrams can also be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of topological quantum field theory and quantum computation, it became clear that diagrammatic reasoning takes advantage of an extensive network of interlocking analogies between physics, topology, logic and computation. These analogies can be made precise using the formalism of symmetric monoidal closed categories. But symmetric monoidal categories are just the n=1, k=3 entry of a hypothesized "periodic table" of k-tuply monoidal n-categories. This raises the question of how these analogies extend. An important clue comes from the way symmetric monoidal closed 2-categories describe rewrite rules in the lambda calculus and multiplicative intuitionistic linear logic. This talk is based on work in progress with Paul-Andre Mellies and Mike Stay.
{"title":"Computation and the Periodic Table","authors":"J. Baez","doi":"10.1109/LICS.2009.43","DOIUrl":"https://doi.org/10.1109/LICS.2009.43","url":null,"abstract":"In physics, Feynman diagrams are used to reason about quantum processes. Similar diagrams can also be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of topological quantum field theory and quantum computation, it became clear that diagrammatic reasoning takes advantage of an extensive network of interlocking analogies between physics, topology, logic and computation. These analogies can be made precise using the formalism of symmetric monoidal closed categories. But symmetric monoidal categories are just the n=1, k=3 entry of a hypothesized \"periodic table\" of k-tuply monoidal n-categories. This raises the question of how these analogies extend. An important clue comes from the way symmetric monoidal closed 2-categories describe rewrite rules in the lambda calculus and multiplicative intuitionistic linear logic. This talk is based on work in progress with Paul-Andre Mellies and Mike Stay.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"77 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":"126074978","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}
After informally reviewing the main concepts from game semantics and placing the development of the field in a historical context we examine its main applications. We focus in particular on finite state model checking, higher order model checking and more recent developments in hardware design.
{"title":"Applications of Game Semantics: From Program Analysis to Hardware Synthesis","authors":"D. Ghica","doi":"10.1109/LICS.2009.26","DOIUrl":"https://doi.org/10.1109/LICS.2009.26","url":null,"abstract":"After informally reviewing the main concepts from game semantics and placing the development of the field in a historical context we examine its main applications. We focus in particular on finite state model checking, higher order model checking and more recent developments in hardware design.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"140 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":"113968847","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 complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B. This may be seen as a natural generalisation of the non-uniform quantified constraint satisfaction problem QCSP(B). We introduce surjective hyper-endomorphisms and use them in proving a Galois connection that characterises definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem is either in Logspace, is NP-complete, is coNP-complete or is Pspace-complete.
{"title":"The Complexity of Positive First-order Logic without Equality","authors":"B. Martin, Jos Martin","doi":"10.1145/2071368.2071373","DOIUrl":"https://doi.org/10.1145/2071368.2071373","url":null,"abstract":"We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B. This may be seen as a natural generalisation of the non-uniform quantified constraint satisfaction problem QCSP(B). We introduce surjective hyper-endomorphisms and use them in proving a Galois connection that characterises definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem is either in Logspace, is NP-complete, is coNP-complete or is Pspace-complete.","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":"128965038","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 this paper we introduce ``clipping,'' a new method of syntactic approximation which is motivated by and works in conjunction with a sound and decidable denotational model for a given programming language. Like slicing, clipping reduces the size of the source code in preparation for automatic verification; but unlike slicing it is an imprecise but computationally inexpensive algorithm which does not require a whole-program analysis. The technique of clipping can be framed into an iterated refinement cycle to arbitrarily improve its precision. We first present this rather simple idea intuitively with some examples, then work out the technical details in the case of an Algol-like programming language and a decidable approximation of its game-semantic model inspired by Hankin and Malacaria's ``lax functor'' approach. We conclude by presenting an experimental model checking tool based on these ideas and some toy programs.
{"title":"Clipping: A Semantics-Directed Syntactic Approximation","authors":"D. Ghica, Adam Bakewell","doi":"10.1109/LICS.2009.25","DOIUrl":"https://doi.org/10.1109/LICS.2009.25","url":null,"abstract":"In this paper we introduce ``clipping,'' a new method of syntactic approximation which is motivated by and works in conjunction with a sound and decidable denotational model for a given programming language. Like slicing, clipping reduces the size of the source code in preparation for automatic verification; but unlike slicing it is an imprecise but computationally inexpensive algorithm which does not require a whole-program analysis. The technique of clipping can be framed into an iterated refinement cycle to arbitrarily improve its precision. We first present this rather simple idea intuitively with some examples, then work out the technical details in the case of an Algol-like programming language and a decidable approximation of its game-semantic model inspired by Hankin and Malacaria's ``lax functor'' approach. We conclude by presenting an experimental model checking tool based on these ideas and some toy programs.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"164 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":"121514215","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 consider the standard model of finite two-person zero-sum stochastic games with signals. We are interested in the existence of almost-surely winning or positively winning strategies, under reachability, safety, Buchi or co-Buchi winning objectives. We prove two qualitative determinacy results. First, in a reachability game either player $1$ can achieve almost-surely the reachability objective, or player 2 can ensure surely the complementary safety objective, or both players have positively winning strategies. Second, in a Buchi game if player 1 cannot achieve almost-surely the Buchi objective, then player 2 can ensure positively the complementary co-Buchi objective. We prove that players only need strategies with finite-memory, whose sizes range from no memory at all to doubly-exponential number of states, with matching lower bounds. Together with the qualitative determinacy results, we also provide fix-point algorithms for deciding which player has an almost-surely winning or a positively winning strategy and for computing the finite memory strategy. Complexity ranges from EXPTIME to 2-EXPTIME with matching lower bounds, and better complexity can be achieved for some special cases where one of the players is better informed than her opponent.
{"title":"Qualitative Determinacy and Decidability of Stochastic Games with Signals","authors":"N. Bertrand, B. Genest, H. Gimbert","doi":"10.1145/3107926","DOIUrl":"https://doi.org/10.1145/3107926","url":null,"abstract":"We consider the standard model of finite two-person zero-sum stochastic games with signals. We are interested in the existence of almost-surely winning or positively winning strategies, under reachability, safety, Buchi or co-Buchi winning objectives. We prove two qualitative determinacy results. First, in a reachability game either player $1$ can achieve almost-surely the reachability objective, or player 2 can ensure surely the complementary safety objective, or both players have positively winning strategies. Second, in a Buchi game if player 1 cannot achieve almost-surely the Buchi objective, then player 2 can ensure positively the complementary co-Buchi objective. We prove that players only need strategies with finite-memory, whose sizes range from no memory at all to doubly-exponential number of states, with matching lower bounds. Together with the qualitative determinacy results, we also provide fix-point algorithms for deciding which player has an almost-surely winning or a positively winning strategy and for computing the finite memory strategy. Complexity ranges from EXPTIME to 2-EXPTIME with matching lower bounds, and better complexity can be achieved for some special cases where one of the players is better informed than her opponent.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"2005 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":"125618258","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}
Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in first-order propositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterize definable strategies, that is strategies which actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task, which requires to combine advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model, by the means of generators and relations: those strategies can be generated from a finite set of atomic strategies and the equality between strategies admits a finite axiomatization, this equational structure corresponding to a polarized variation of the notion of bialgebra. This work thus bridges algebra and denotational semantics in order to reveal the structure of dependencies induced by first-order quantifiers, and lays the foundations for a mechanized analysis of causality in programming languages.
{"title":"The Structure of First-Order Causality","authors":"S. Mimram","doi":"10.1109/LICS.2009.19","DOIUrl":"https://doi.org/10.1109/LICS.2009.19","url":null,"abstract":"Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in first-order propositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterize definable strategies, that is strategies which actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task, which requires to combine advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model, by the means of generators and relations: those strategies can be generated from a finite set of atomic strategies and the equality between strategies admits a finite axiomatization, this equational structure corresponding to a polarized variation of the notion of bialgebra. This work thus bridges algebra and denotational semantics in order to reveal the structure of dependencies induced by first-order quantifiers, and lays the foundations for a mechanized analysis of causality in programming languages.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"74 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":"134242827","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: 10.2168/LMCS-7(1:12)2011
Taolue Chen, Tingting Han, J. Katoen, A. Mereacre
We study the following problem: given a continuous-time Markov chain (CTMC) C, and a linear real-time property provided as a deterministic timed automaton (DTA) A, what is the probability of the set of paths of C that are accepted by A (C satisfies A)? It is shown that this set of paths is measurable and computing its probability can be reduced to computing the reachability probability in a piecewise deterministic Markov process (PDP). The reachability probability is characterized as the least solution of a system of integral equations and is shown to be approximated by solving a system of partial differential equations. For the special case of single-clock DTA, the system of integral equations can be transformed into a system of linear equations where the coefficients are solutions of ordinary differential equations.
{"title":"Quantitative Model Checking of Continuous-Time Markov Chains Against Timed Automata Specifications","authors":"Taolue Chen, Tingting Han, J. Katoen, A. Mereacre","doi":"10.2168/LMCS-7(1:12)2011","DOIUrl":"https://doi.org/10.2168/LMCS-7(1:12)2011","url":null,"abstract":"We study the following problem: given a continuous-time Markov chain (CTMC) C, and a linear real-time property provided as a deterministic timed automaton (DTA) A, what is the probability of the set of paths of C that are accepted by A (C satisfies A)? It is shown that this set of paths is measurable and computing its probability can be reduced to computing the reachability probability in a piecewise deterministic Markov process (PDP). The reachability probability is characterized as the least solution of a system of integral equations and is shown to be approximated by solving a system of partial differential equations. For the special case of single-clock DTA, the system of integral equations can be transformed into a system of linear equations where the coefficients are solutions of ordinary differential equations.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"40 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":"116497421","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}
Automata on infinite words (ω-automata) have wide applications in formal language theory as well as in modeling and verifying reactive systems. Complementation of ω-automata is a crucial instrument in many these applications, and hence there have been great interests in determining the state complexity of the complementation problem. However, obtaining nontrivial lower bounds has been difficult. For the complementation of Rabin automata, a significant gap exists between the state-of-the-art lower bound 2^Ω(NlgN) and upper bound 2^O(kNlgN), where k, the number of Rabin pairs, can be as large as 2N. In this paper we introduce multidimensional rankings to the full automata technique. Using the improved technique we establish an almost tight lower bound for the complementation of Rabin automata. We also show that the same lower bound holds for the determinization of Rabin automata.
{"title":"An Improved Lower Bound for the Complementation of Rabin Automata","authors":"Yang Cai, Ting Zhang, Haifeng Luo","doi":"10.1109/LICS.2009.13","DOIUrl":"https://doi.org/10.1109/LICS.2009.13","url":null,"abstract":"Automata on infinite words (ω-automata) have wide applications in formal language theory as well as in modeling and verifying reactive systems. Complementation of ω-automata is a crucial instrument in many these applications, and hence there have been great interests in determining the state complexity of the complementation problem. However, obtaining nontrivial lower bounds has been difficult. For the complementation of Rabin automata, a significant gap exists between the state-of-the-art lower bound 2^Ω(NlgN) and upper bound 2^O(kNlgN), where k, the number of Rabin pairs, can be as large as 2N. In this paper we introduce multidimensional rankings to the full automata technique. Using the improved technique we establish an almost tight lower bound for the complementation of Rabin automata. We also show that the same lower bound holds for the determinization of Rabin automata.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"14 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":"125518601","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}
What is reachability in higher-order functional programs? We formulate reachability as a decision problem in the setting of the prototypical functional language PCF, and show that even in the recursion-free fragment generated from a finite base type, several versions of the reachability problem are undecidable from order 4 onwards, and several other versions are reducible to each other. We characterise a version of the reachability problem in terms of a new class of tree automata introduced by Stirling at FoSSaCS 2009, called Alternating Dependency Tree Automata (ADTA). As a corollary, we prove that the ADTA non-emptiness problem is undecidable, thus resolving an open problem raised by Stirling. However, by restricting to contexts constructible from a finite set of variable names, we show that the corresponding solution set of a given instance of the reachability problem is regular. Hence the relativised reachability problem is decidable.
{"title":"Functional Reachability","authors":"C. Ong, N. Tzevelekos","doi":"10.1109/LICS.2009.48","DOIUrl":"https://doi.org/10.1109/LICS.2009.48","url":null,"abstract":"What is reachability in higher-order functional programs? We formulate reachability as a decision problem in the setting of the prototypical functional language PCF, and show that even in the recursion-free fragment generated from a finite base type, several versions of the reachability problem are undecidable from order 4 onwards, and several other versions are reducible to each other. We characterise a version of the reachability problem in terms of a new class of tree automata introduced by Stirling at FoSSaCS 2009, called Alternating Dependency Tree Automata (ADTA). As a corollary, we prove that the ADTA non-emptiness problem is undecidable, thus resolving an open problem raised by Stirling. However, by restricting to contexts constructible from a finite set of variable names, we show that the corresponding solution set of a given instance of the reachability problem is regular. Hence the relativised reachability problem is decidable.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"25 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":"125072981","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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