Linear Logic is based on the analogy between algebraic linearity (i.e. commutation with sums and with products with scalars) and the computer science linearity (i.e. calling inputs only once). Keeping on this analogy, Ehrhard and Regnier introduced Differential Linear Logic(DiLL) --- an extension of Multiplicative Exponential Linear Logic with differential constructions. In this setting, promotion (the logical exponentiation) can be approximated by a sum of promotion-free proofs f DiLL via Taylor expansion. We present a constructive way to revert Taylor expansion. Precisely, we define merging reduction --- a rewriting system which merges a finite sum of DiLL proofs into a proof with promotion whenever the sum is an approximation of the Taylor expansion of this proof. We prove that this algorithm is sound, complete and can be run in non-deterministic polynomial time.
线性逻辑是基于代数线性(即与和和与标量乘积的交换)和计算机科学线性(即只调用一次输入)之间的类比。在这种类比的基础上，Ehrhard和Regnier引入了微分线性逻辑(Differential Linear Logic, DiLL)——一种具有微分结构的乘法指数线性逻辑的扩展。在这种情况下，提升(逻辑幂)可以近似为通过泰勒展开的DiLL的无提升证明的总和。我们提出了一种建设性的恢复泰勒展开的方法。确切地说，我们定义了归并约简——一个重写系统，它将一个有限的DiLL证明和合并成一个有提升的证明，只要这个和是这个证明的泰勒展开式的近似值。我们证明了该算法是健全的、完备的，并且可以在不确定的多项式时间内运行。
{"title":"The Inverse Taylor Expansion Problem in Linear Logic","authors":"Michele Pagani, C. Tasson","doi":"10.1109/LICS.2009.35","DOIUrl":"https://doi.org/10.1109/LICS.2009.35","url":null,"abstract":"Linear Logic is based on the analogy between algebraic linearity (i.e. commutation with sums and with products with scalars) and the computer science linearity (i.e. calling inputs only once). Keeping on this analogy, Ehrhard and Regnier introduced Differential Linear Logic(DiLL) --- an extension of Multiplicative Exponential Linear Logic with differential constructions. In this setting, promotion (the logical exponentiation) can be approximated by a sum of promotion-free proofs f DiLL via Taylor expansion. We present a constructive way to revert Taylor expansion. Precisely, we define merging reduction --- a rewriting system which merges a finite sum of DiLL proofs into a proof with promotion whenever the sum is an approximation of the Taylor expansion of this proof. We prove that this algorithm is sound, complete and can be run in non-deterministic polynomial time.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"35 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":"115505973","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(2:16)2011
Derek Dreyer, Amal Ahmed, L. Birkedal
We show how to reason about "step-indexed" logical relations in an abstract way, avoiding the tedious, error-prone, and proof-obscuring step-index arithmetic that seems superficially to be an essential element of the method. Specifically, we define a logic LSLR, which is inspired by Plotkin and Abadi's logic for parametricity, but also supports recursively defined relations by means of the modal"later" operator from Appel et al.'s "very modal model" paper. We encode in LSLR a logical relation for reasoning(in-)equationally about programs in call-by-value System F extended with recursive types. Using this logical relation, we derive a useful set of rules with which we can prove contextual (in-)equivalences without mentioning step indices.
{"title":"Logical Step-Indexed Logical Relations","authors":"Derek Dreyer, Amal Ahmed, L. Birkedal","doi":"10.2168/LMCS-7(2:16)2011","DOIUrl":"https://doi.org/10.2168/LMCS-7(2:16)2011","url":null,"abstract":"We show how to reason about \"step-indexed\" logical relations in an abstract way, avoiding the tedious, error-prone, and proof-obscuring step-index arithmetic that seems superficially to be an essential element of the method. Specifically, we define a logic LSLR, which is inspired by Plotkin and Abadi's logic for parametricity, but also supports recursively defined relations by means of the modal\"later\" operator from Appel et al.'s \"very modal model\" paper. We encode in LSLR a logical relation for reasoning(in-)equationally about programs in call-by-value System F extended with recursive types. Using this logical relation, we derive a useful set of rules with which we can prove contextual (in-)equivalences without mentioning step indices.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"66 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":"115856471","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-8(3:19)2012
M. Bojanczyk, Howard Straubing, I. Walukiewicz
We use the recently developed theory of forest algebras to find algebraic characterizations of the languages of unranked trees and forests definable in various logics. These include the temporal logics {CTL} and { EF}, and first-order logic over the ancestor relation. While the characterizations are in general non-effective, we are able to use them to formulate necessary conditions for definability and provide new proofs that a number of languages are not definable in these logics.
{"title":"Wreath Products of Forest Algebras, with Applications to Tree Logics","authors":"M. Bojanczyk, Howard Straubing, I. Walukiewicz","doi":"10.2168/LMCS-8(3:19)2012","DOIUrl":"https://doi.org/10.2168/LMCS-8(3:19)2012","url":null,"abstract":"We use the recently developed theory of forest algebras to find algebraic characterizations of the languages of unranked trees and forests definable in various logics. These include the temporal logics {CTL} and { EF}, and first-order logic over the ancestor relation. While the characterizations are in general non-effective, we are able to use them to formulate necessary conditions for definability and provide new proofs that a number of languages are not definable in these logics.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"142 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":"123198990","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}
Several dynamical systems, such as deterministic automata and labelled transition systems, can be described as coalgebras of so-called Kripke polynomial functors, built up from constants and identities, using product, coproduct and powerset. Locally finite Kripke polynomial coalgebras can be characterized up to bisimulation by a specification language that generalizes Kleene’s regular expressions for finite automata. In this paper, we equip this specification language with an axiomatization and prove it sound and complete with respect to bisimulation, using a purely coalgebraic argument. We demonstrate the usefulness of our framework by providing a finite equational system for (non-)deterministic finite automata, la-belled transition systems with explicit termination, and automata on guarded strings.
{"title":"An Algebra for Kripke Polynomial Coalgebras","authors":"M. Bonsangue, J. Rutten, Alexandra Silva","doi":"10.1109/LICS.2009.18","DOIUrl":"https://doi.org/10.1109/LICS.2009.18","url":null,"abstract":"Several dynamical systems, such as deterministic automata and labelled transition systems, can be described as coalgebras of so-called Kripke polynomial functors, built up from constants and identities, using product, coproduct and powerset. Locally finite Kripke polynomial coalgebras can be characterized up to bisimulation by a specification language that generalizes Kleene’s regular expressions for finite automata. In this paper, we equip this specification language with an axiomatization and prove it sound and complete with respect to bisimulation, using a purely coalgebraic argument. We demonstrate the usefulness of our framework by providing a finite equational system for (non-)deterministic finite automata, la-belled transition systems with explicit termination, and automata on guarded strings.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"21 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":"123682533","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 a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the problem CCSP(Gamma), the constraint satisfaction problem with global cardinality constraints that allows only relations from the set Gamma. The main result of this paper characterizes sets Gamma that give rise to problems solvable in polynomial time, and states that the remaining such problems are NP-complete.
{"title":"The Complexity of Global Cardinality Constraints","authors":"A. Bulatov, D. Marx","doi":"10.2168/LMCS-6(4:4)2010","DOIUrl":"https://doi.org/10.2168/LMCS-6(4:4)2010","url":null,"abstract":"In a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the problem CCSP(Gamma), the constraint satisfaction problem with global cardinality constraints that allows only relations from the set Gamma. The main result of this paper characterizes sets Gamma that give rise to problems solvable in polynomial time, and states that the remaining such problems are NP-complete.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"41 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":"121688823","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 longstanding open problem is whether there exists a model of the untyped lambda calculus in the category CPO of complete partial orderings and Scott continuous functions, whose theory is exactly the least lambda-theory lambda-beta or the least extensional lambda-theory lambda-beta-eta. In this paper we analyze the class of reflexive Scott domains, the models of lambda-calculus living in the category of Scott domains (a full subcategory of CPO). The following are the main results of the paper: (i) Extensional reflexive Scott domains are not complete for the beta-eta-calculus, i.e., there are equations not in lambda-beta-eta which hold in all extensional reflexive Scott domains.(ii) The order theory of an extensional reflexive Scott domain is never recursively enumerable. These results have been obtained by isolating among the reflexive Scott domains a class of webbed models arising from Scott's information systems, called iweb-models. The class of iweb-models includes all extensional reflexive Scott domains, all preordered coherent models and all filter models living in CPO. Based on a fine-grained study of an ``effective'' version of Scott's information systems, we have shown that there are equations not in lambda-beta (resp. lambda-beta-eta) which hold in all (extensional) iweb-models.
{"title":"Reflexive Scott Domains are Not Complete for the Extensional Lambda Calculus","authors":"Alberto Carraro, A. Salibra","doi":"10.1109/LICS.2009.22","DOIUrl":"https://doi.org/10.1109/LICS.2009.22","url":null,"abstract":"A longstanding open problem is whether there exists a model of the untyped lambda calculus in the category CPO of complete partial orderings and Scott continuous functions, whose theory is exactly the least lambda-theory lambda-beta or the least extensional lambda-theory lambda-beta-eta. In this paper we analyze the class of reflexive Scott domains, the models of lambda-calculus living in the category of Scott domains (a full subcategory of CPO). The following are the main results of the paper: (i) Extensional reflexive Scott domains are not complete for the beta-eta-calculus, i.e., there are equations not in lambda-beta-eta which hold in all extensional reflexive Scott domains.(ii) The order theory of an extensional reflexive Scott domain is never recursively enumerable. These results have been obtained by isolating among the reflexive Scott domains a class of webbed models arising from Scott's information systems, called iweb-models. The class of iweb-models includes all extensional reflexive Scott domains, all preordered coherent models and all filter models living in CPO. Based on a fine-grained study of an ``effective'' version of Scott's information systems, we have shown that there are equations not in lambda-beta (resp. lambda-beta-eta) which hold in all (extensional) iweb-models.","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":"129813973","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 provide in this article two characterisation results, describing exactly which terms verify the dinaturality diagram, in Church-style system F and in Curry-style system F. The proof techniques we use here are purely syntactic, giving in particular a direct construction of the two terms generated by the dinaturality diagram. But the origin of these techniques lies in fact directly on the analysis of system F through game semantics. Thus, this article provides an example of backward engineering, where powerful syntactic results can be extracted from a semantic analysis.
{"title":"Dinatural Terms in System F","authors":"J. D. Lataillade","doi":"10.1109/LICS.2009.30","DOIUrl":"https://doi.org/10.1109/LICS.2009.30","url":null,"abstract":"We provide in this article two characterisation results, describing exactly which terms verify the dinaturality diagram, in Church-style system F and in Curry-style system F. The proof techniques we use here are purely syntactic, giving in particular a direct construction of the two terms generated by the dinaturality diagram. But the origin of these techniques lies in fact directly on the analysis of system F through game semantics. Thus, this article provides an example of backward engineering, where powerful syntactic results can be extracted from a semantic analysis.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"85 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":"126236235","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 solve the longstanding open problems of the blow-up involved in the translations (when possible) of a nondeterministic B"uchi word automaton (NBW) to a nondeterministic co-B"uchi word automaton (NCW)and to a deterministic co-B"uchi word automaton (DCW). For the NBW to NCW translation, the currently known upper bound is $2^{O(n log n)}$ and the lower bound is $1.5n$. We improve the upper bound to $n2^n$ and describe a matching lower bound of$2^{Omega(n)}$. For the NBW to DCW translation, the currently known upper bound is $2^{O(n log n)}$. We improve it to $2^{O(n)}$, which is asymptotically tight. Both of our upper-bound constructions are based on a simple subset construction, do not involve intermediate automata with richer acceptance conditions, and can be implemented symbolically. We point to numerous applications of the new constructions. In particular, they imply a simple subset-construction based translation(when possible) of LTL to deterministic B"uchi word automata.
{"title":"Co-ing Büchi Made Tight and Useful","authors":"Udi Boker, O. Kupferman","doi":"10.1109/LICS.2009.32","DOIUrl":"https://doi.org/10.1109/LICS.2009.32","url":null,"abstract":"We solve the longstanding open problems of the blow-up involved in the translations (when possible) of a nondeterministic B\"uchi word automaton (NBW) to a nondeterministic co-B\"uchi word automaton (NCW)and to a deterministic co-B\"uchi word automaton (DCW). For the NBW to NCW translation, the currently known upper bound is $2^{O(n log n)}$ and the lower bound is $1.5n$. We improve the upper bound to $n2^n$ and describe a matching lower bound of$2^{Omega(n)}$. For the NBW to DCW translation, the currently known upper bound is $2^{O(n log n)}$. We improve it to $2^{O(n)}$, which is asymptotically tight. Both of our upper-bound constructions are based on a simple subset construction, do not involve intermediate automata with richer acceptance conditions, and can be implemented symbolically. We point to numerous applications of the new constructions. In particular, they imply a simple subset-construction based translation(when possible) of LTL to deterministic B\"uchi word automata.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"69 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":"125629530","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(2:13)2011
Michele Basaldella, C. Faggian
We prove that is possible to extend Girard's Ludics so as to have repetitions (hence exponentials), and still have the results on semantical types which characterize Ludics in the panorama of Game Semantics. The results are obtained by using less structure than in the original paper; this has an interest on its own, and we hope that it will open the way to applying the approach of Ludics to a larger domain.
{"title":"Ludics with Repetitions (Exponentials, Interactive Types and Completeness)","authors":"Michele Basaldella, C. Faggian","doi":"10.2168/LMCS-7(2:13)2011","DOIUrl":"https://doi.org/10.2168/LMCS-7(2:13)2011","url":null,"abstract":"We prove that is possible to extend Girard's Ludics so as to have repetitions (hence exponentials), and still have the results on semantical types which characterize Ludics in the panorama of Game Semantics. The results are obtained by using less structure than in the original paper; this has an interest on its own, and we hope that it will open the way to applying the approach of Ludics to a larger domain.","PeriodicalId":415902,"journal":{"name":"2009 24th Annual IEEE Symposium on Logic In Computer Science","volume":"52 17","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120839473","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 associate a statistical vector to a trace and a geometrical embedding to a Markov Decision Process, based on a distance on words, and study basic Membership and Equivalence problems. The Membership problem for a trace textit{w} and a Markov Decision Process textit{S} decides if there exists a strategy on textit{S} which generates with high probability traces close to textit{w}. We prove that Membership of a trace is emph{testable} and Equivalence of MDPs is polynomial time approximable. For Probabilistic Automata, Membership is not testable, and approximate Equivalence is undecidable. We give a class of properties, based on results concerning the structure of the tail sigma-field of a finite Markov chain, which characterizes equivalent Markov Decision Processes in this context.
{"title":"Statistic Analysis for Probabilistic Processes","authors":"M. D. Rougemont, M. Tracol","doi":"10.1109/LICS.2009.36","DOIUrl":"https://doi.org/10.1109/LICS.2009.36","url":null,"abstract":"We associate a statistical vector to a trace and a geometrical embedding to a Markov Decision Process, based on a distance on words, and study basic Membership and Equivalence problems. The Membership problem for a trace textit{w} and a Markov Decision Process textit{S} decides if there exists a strategy on textit{S} which generates with high probability traces close to textit{w}. We prove that Membership of a trace is emph{testable} and Equivalence of MDPs is polynomial time approximable. For Probabilistic Automata, Membership is not testable, and approximate Equivalence is undecidable. We give a class of properties, based on results concerning the structure of the tail sigma-field of a finite Markov chain, which characterizes equivalent Markov Decision Processes in this context.","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":"130011496","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：