This paper is an investigation of the matching problem for term equations s = t where s contains context variables and first-order variables, and both terms s and t are given using some kind of compressed representation. The main result is a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. NP-completeness is obtained when the terms are represented using the more general formalism of singleton tree grammars. As an ingredient of this proof, we also show that the special case of first-order matching with singleton tree grammars is decidable in polynomial time.
{"title":"Context Matching for Compressed Terms","authors":"Adrià Gascón, Guillem Godoy, M. Schmidt-Schauß","doi":"10.1109/LICS.2008.17","DOIUrl":"https://doi.org/10.1109/LICS.2008.17","url":null,"abstract":"This paper is an investigation of the matching problem for term equations s = t where s contains context variables and first-order variables, and both terms s and t are given using some kind of compressed representation. The main result is a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. NP-completeness is obtained when the terms are represented using the more general formalism of singleton tree grammars. As an ingredient of this proof, we also show that the special case of first-order matching with singleton tree grammars is decidable in polynomial time.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"30 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":"126877910","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 ancient Euclidean algorithm computes the greatest common divisor gcd(m, n) of two natural numbers from (or relative to) the remainder operation rem, which is assumed as primitive; it requires no more than 2 log(min(m, n)) applications of the remainder operation to compute gcd(m, n) (for m, n ges 2), and it is not known to be optimal: Conjecture: for every algorithm a which computes on Nopf from rem the greatest common divisor function, there is a constant r > 0 such that for infinitely many pairs a ges b ges 1, calpha(a, b) ges rlog2(a), where calpha(m,n) counts the number of calls to "the remainder oracle" required by a for the computation of gcd(m, n). The conjecture claims a logarithmic lower bound for all algorithms which compute gcd(m, n) from the remainder operation, not just those expressed by a specific class of computation models. In this lecture the author develops an approach to the theory of algorithms in the style of abstract model theory which makes it possible to make precise and (on occasion) prove the existence of non-trivial, absolute lower bounds for a wide variety of problems and specified primitives, including many of the results in the bibliography.
古老的欧几里得算法从(或相对于)余数运算rem计算两个自然数的最大公约数gcd(m, n)，假设它是原始的;计算gcd(m, n)(对于m, n ges 2)需要不超过2 log(min(m, n))次的余数运算，并且不知道它是否最优。对于每一个从最大公约数函数rem计算Nopf的算法a，存在一个常数r > 0，使得对于无限多对a ges b ges 1, calpha(a, b) ges rlog2(a)，其中calpha(m,n)计算计算gcd(m, n)所需的“余数oracle”的调用次数。该猜想为所有从余数运算计算gcd(m, n)的算法提供了一个对数下界。而不仅仅是由一类特定的计算模型所表达的。在这个讲座中，作者以抽象模型理论的风格发展了一种算法理论的方法，这种方法可以精确地(有时)证明各种各样的问题和特定原语的非平凡绝对下界的存在，包括参考书目中的许多结果。
{"title":"The Axiomatic Derivation of Absolute Lower Bounds","authors":"Y. Moschovakis","doi":"10.1109/LICS.2008.52","DOIUrl":"https://doi.org/10.1109/LICS.2008.52","url":null,"abstract":"The ancient Euclidean algorithm computes the greatest common divisor gcd(m, n) of two natural numbers from (or relative to) the remainder operation rem, which is assumed as primitive; it requires no more than 2 log(min(m, n)) applications of the remainder operation to compute gcd(m, n) (for m, n ges 2), and it is not known to be optimal: Conjecture: for every algorithm a which computes on Nopf from rem the greatest common divisor function, there is a constant r > 0 such that for infinitely many pairs a ges b ges 1, calpha(a, b) ges rlog2(a), where calpha(m,n) counts the number of calls to \"the remainder oracle\" required by a for the computation of gcd(m, n). The conjecture claims a logarithmic lower bound for all algorithms which compute gcd(m, n) from the remainder operation, not just those expressed by a specific class of computation models. In this lecture the author develops an approach to the theory of algorithms in the style of abstract model theory which makes it possible to make precise and (on occasion) prove the existence of non-trivial, absolute lower bounds for a wide variety of problems and specified primitives, including many of the results in the bibliography.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"10 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":"130811576","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 an extension of the piI-calculus with formal sums of terms. A study of the properties of this sum reveals that its neutral element can be used to make assumptions about the behaviour of the environment of a process. Furthermore, the formal sum appears as a fundamental construct that can be used to decompose both internal and external choice. From these observations, we derive an enriched calculus that enjoys a confluent reduction which preserves the testing semantics of processes. This system is shown to be strongly normalising for terms without replication, and the study of its normal forms provides fully abstract trace semantics for testing of piI processes.
{"title":"An Algebraic Process Calculus","authors":"E. Beffara","doi":"10.1109/LICS.2008.40","DOIUrl":"https://doi.org/10.1109/LICS.2008.40","url":null,"abstract":"We present an extension of the piI-calculus with formal sums of terms. A study of the properties of this sum reveals that its neutral element can be used to make assumptions about the behaviour of the environment of a process. Furthermore, the formal sum appears as a fundamental construct that can be used to decompose both internal and external choice. From these observations, we derive an enriched calculus that enjoys a confluent reduction which preserves the testing semantics of processes. This system is shown to be strongly normalising for terms without replication, and the study of its normal forms provides fully abstract trace semantics for testing of piI processes.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"99 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":"117223194","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 study of constraint satisfaction problems definable in various fragments of Datalog has recently gained considerable importance. We consider constraint satisfaction problems that are definable in the smallest natural recursive fragment of Datalog - monadic linear Datalog with at most one EDB per rule. We give combinatorial and algebraic characterisations of such problems, in terms of caterpillar dualities and lattice operations, respectively. We then apply our results to study graph H-colouring problems.
{"title":"Caterpillar Duality for Constraint Satisfaction Problems","authors":"C. Carvalho, V. Dalmau, A. Krokhin","doi":"10.1109/LICS.2008.19","DOIUrl":"https://doi.org/10.1109/LICS.2008.19","url":null,"abstract":"The study of constraint satisfaction problems definable in various fragments of Datalog has recently gained considerable importance. We consider constraint satisfaction problems that are definable in the smallest natural recursive fragment of Datalog - monadic linear Datalog with at most one EDB per rule. We give combinatorial and algebraic characterisations of such problems, in terms of caterpillar dualities and lattice operations, respectively. We then apply our results to study graph H-colouring problems.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"135 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":"134634956","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}
Individual-based models are a relatively new approach to modelling dynamical systems of interacting entities, for example molecules in a biological cell. Although they are computationally expensive, they have the capability of modelling systems more realistically than traditional state-variable models. We give a formal definition of individual-based models, which includes state-variable models as a special case. We examine the questions of when state-variable models are sufficient for accurate modelling of a system, and when individual-based models are necessary. We define notions of abstraction and approximation, and give sufficient conditions that imply that an individual-based model can be approximated by a deterministic state-variable model. We also give negative results: examples of individual-based models that cannot be approximated by any state-variable model.
{"title":"A Logical Characterization of Individual-Based Models","authors":"J. Lynch","doi":"10.1109/LICS.2008.27","DOIUrl":"https://doi.org/10.1109/LICS.2008.27","url":null,"abstract":"Individual-based models are a relatively new approach to modelling dynamical systems of interacting entities, for example molecules in a biological cell. Although they are computationally expensive, they have the capability of modelling systems more realistically than traditional state-variable models. We give a formal definition of individual-based models, which includes state-variable models as a special case. We examine the questions of when state-variable models are sufficient for accurate modelling of a system, and when individual-based models are necessary. We define notions of abstraction and approximation, and give sufficient conditions that imply that an individual-based model can be approximated by a deterministic state-variable model. We also give negative results: examples of individual-based models that cannot be approximated by any state-variable model.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"30 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":"133393021","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}
Let B be a finite, core relational structure and let A be the algebra associated to B, i.e. whose terms are the operations on the universe of B that preserve the relations of B. We show that if A generates a so-called arithmetical variety then CSP(B), the constraint satisfaction problem associated to B, is solvable in Logspace; in fact notCSP(B) is expressible in symmetric Datalog. In particular, we obtain that notCSP(B) is expressible in Datalog and the relations of B are invariant under a Maltsev operation then notCSP(B) is in symmetric Datalog.
{"title":"Maltsev + Datalog --≫ Symmetric Datalog","authors":"Víctor Dalmau, Benoît Larose","doi":"10.1109/LICS.2008.14","DOIUrl":"https://doi.org/10.1109/LICS.2008.14","url":null,"abstract":"Let B be a finite, core relational structure and let A be the algebra associated to B, i.e. whose terms are the operations on the universe of B that preserve the relations of B. We show that if A generates a so-called arithmetical variety then CSP(B), the constraint satisfaction problem associated to B, is solvable in Logspace; in fact notCSP(B) is expressible in symmetric Datalog. In particular, we obtain that notCSP(B) is expressible in Datalog and the relations of B are invariant under a Maltsev operation then notCSP(B) is in symmetric Datalog.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"110 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":"127983183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a systematic procedure to transform large classes of (Hilbert) axioms into equivalent inference rules in sequent and hypersequent calculi. This allows for the automated generation of analytic calculi for a wide range of prepositional nonclassical logics including intermediate, fuzzy and substructural logics. Our work encompasses many existing results, allows for the definition of new calculi and contains a uniform semantic proof of cut-elimination for hypersequent calculi.
{"title":"From Axioms to Analytic Rules in Nonclassical Logics","authors":"A. Ciabattoni, Nikolaos Galatos, K. Terui","doi":"10.1109/LICS.2008.39","DOIUrl":"https://doi.org/10.1109/LICS.2008.39","url":null,"abstract":"We introduce a systematic procedure to transform large classes of (Hilbert) axioms into equivalent inference rules in sequent and hypersequent calculi. This allows for the automated generation of analytic calculi for a wide range of prepositional nonclassical logics including intermediate, fuzzy and substructural logics. Our work encompasses many existing results, allows for the definition of new calculi and contains a uniform semantic proof of cut-elimination for hypersequent calculi.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"18 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":"114415417","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 the asymptotic complexity of uniformly generated (expressible in first-order (FO) logic) prepositional tautologies for the nullstellensatz proof system (NS) as well as for polynomial calculus, (PC) has four distinct types of asymptotic behavior over fields of finite characteristic. More precisely, based on some highly non-trivial work by Krajicek, we show that for each prime p there exists a function l(n) G isin Omega(log(n)) for NS and l(n) G Omega (log(log(n)) for PC, such that the prepositional translation of any FO formula (that fails in all finite models), has degree proof complexity over fields of characteristic p, that behave in 4 mutually distinct ways: (i) The degree complexity is bound by a constant. (ii) The degree complexity is at least l(n) for all values of n. (iii) The degree complexity is at least l(n) except in a finite number of regular subsequences of infinite size, where the degree is constant. (iv) The degree complexity fluctuates in a very particular way with the degree complexity taking different constant values on an infinite number of regular subsequences each of infinite size. We leave it as an open question whether the classification remains valid for l[n) isin nOmega(1) or even for I (n) isin Omega(n). Finally, we show that for any non-empty proper subset A sube {(i), (ii), (iii), (iv)} the decision problem of whether a given input FO formula Psi has type belonging to A - is undecidable.
我们证明了nullstellensatz证明系统(NS)和多项式微积分(PC)的一致生成(一阶逻辑可表示)介词重言式的渐近复杂性在有限特征域上具有四种不同类型的渐近行为。更准确地说，基于Krajicek的一些高度非平凡的工作，我们证明了对于每个素数p存在一个函数l(n) G isin (log(n))对于NS和l(n) G (log(log(n))对于PC，使得任何FO公式的前移(在所有有限模型中都失败)在特征p域上具有程度证明复杂性，表现为4种相互不同的方式:(i)程度复杂性由一个常数约束。(ii)对于所有n值，复杂度度至少为l(n)。(iii)复杂度度至少为l(n)，除非在有限数量的无限大小的正则子序列中，复杂度度是恒定的。(iv)复杂度以一种非常特殊的方式波动，复杂度在无限数量的无限大小的正则子序列上取不同的常数值。对于I (n) isin(1)，甚至对于I (n) isin (n)，分类是否仍然有效，我们将其作为一个开放的问题。最后，我们证明了对于任意非空固有子集A子{(i)， (ii)， (iii)， (iv)}，给定输入FO公式Psi是否具有属于A -类型的决策问题是不确定的。
{"title":"On the Asymptotic Nullstellensatz and Polynomial Calculus Proof Complexity","authors":"Søren Riis","doi":"10.1109/LICS.2008.30","DOIUrl":"https://doi.org/10.1109/LICS.2008.30","url":null,"abstract":"We show that the asymptotic complexity of uniformly generated (expressible in first-order (FO) logic) prepositional tautologies for the nullstellensatz proof system (NS) as well as for polynomial calculus, (PC) has four distinct types of asymptotic behavior over fields of finite characteristic. More precisely, based on some highly non-trivial work by Krajicek, we show that for each prime p there exists a function l(n) G isin Omega(log(n)) for NS and l(n) G Omega (log(log(n)) for PC, such that the prepositional translation of any FO formula (that fails in all finite models), has degree proof complexity over fields of characteristic p, that behave in 4 mutually distinct ways: (i) The degree complexity is bound by a constant. (ii) The degree complexity is at least l(n) for all values of n. (iii) The degree complexity is at least l(n) except in a finite number of regular subsequences of infinite size, where the degree is constant. (iv) The degree complexity fluctuates in a very particular way with the degree complexity taking different constant values on an infinite number of regular subsequences each of infinite size. We leave it as an open question whether the classification remains valid for l[n) isin nOmega(1) or even for I (n) isin Omega(n). Finally, we show that for any non-empty proper subset A sube {(i), (ii), (iii), (iv)} the decision problem of whether a given input FO formula Psi has type belonging to A - is undecidable.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"39 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":"124380139","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 class of structures satisfies the extension preservation theorem if, on this class, every first order sentence is preserved under extension iff it is equivalent to an existential sentence. We consider different acyclicity notions for hypergraphs (gamma, beta and alpha-acyclicity and also acyclicity on hypergraph quotients) and estimate their influence on the validity of the extension preservation theorem on classes of finite structures. More precisely, we prove that gamma-acyclic classes satisfy the extension preservation theorem, whereas beta-acyclic classes do not. We also extend the validity of the extension preservation theorem for a generalization of gamma-acyclicity that we call gamma-acyclic k-quotient. To achieve this, we make a reduction from finite structures to their k-quotients and we use combinatorial arguments on gamma-acyclic hypergraphs.
{"title":"Hypergraph Acyclicity and Extension Preservation Theorems","authors":"David Duris","doi":"10.1109/LICS.2008.12","DOIUrl":"https://doi.org/10.1109/LICS.2008.12","url":null,"abstract":"A class of structures satisfies the extension preservation theorem if, on this class, every first order sentence is preserved under extension iff it is equivalent to an existential sentence. We consider different acyclicity notions for hypergraphs (gamma, beta and alpha-acyclicity and also acyclicity on hypergraph quotients) and estimate their influence on the validity of the extension preservation theorem on classes of finite structures. More precisely, we prove that gamma-acyclic classes satisfy the extension preservation theorem, whereas beta-acyclic classes do not. We also extend the validity of the extension preservation theorem for a generalization of gamma-acyclicity that we call gamma-acyclic k-quotient. To achieve this, we make a reduction from finite structures to their k-quotients and we use combinatorial arguments on gamma-acyclic hypergraphs.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"45 16 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":"132136813","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}
Variable binding is a prevalent feature of the syntax and proof theory of many logical systems. In this paper, we define a programming language that provides intrinsic support for both representing and computing with binding. This language is extracted as the Curry-Howard interpretation of a focused sequent calculus with two kinds of implication, of opposite polarity. The representational arrow extends systems of definitional reflection with a notion of scoped inference rules, which are used to represent binding. On the other hand, the usual computational arrow classifies recursive functions defined by pattern-matching. Unlike many previous approaches, both kinds of implication are connectives in a single logic, which serves as a rich logical framework capable of representing inference rules that mix binding and computation.
{"title":"Focusing on Binding and Computation","authors":"Daniel R. Licata, N. Zeilberger, R. Harper","doi":"10.1109/LICS.2008.48","DOIUrl":"https://doi.org/10.1109/LICS.2008.48","url":null,"abstract":"Variable binding is a prevalent feature of the syntax and proof theory of many logical systems. In this paper, we define a programming language that provides intrinsic support for both representing and computing with binding. This language is extracted as the Curry-Howard interpretation of a focused sequent calculus with two kinds of implication, of opposite polarity. The representational arrow extends systems of definitional reflection with a notion of scoped inference rules, which are used to represent binding. On the other hand, the usual computational arrow classifies recursive functions defined by pattern-matching. Unlike many previous approaches, both kinds of implication are connectives in a single logic, which serves as a rich logical framework capable of representing inference rules that mix binding and computation.","PeriodicalId":298300,"journal":{"name":"2008 23rd Annual IEEE Symposium on Logic in Computer Science","volume":"50 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":"115422087","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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