{"title":"Some Methods of Problem Solving in Elementary Geometry","authors":"T. Hales","doi":"10.1109/LICS.2007.43","DOIUrl":"https://doi.org/10.1109/LICS.2007.43","url":null,"abstract":"Many elementary problems in geometry arise as part of the proof of the Kepler conjecture on sphere packings. In the original proof, most of these problems were solved by hand. This article investigates the methods that were used in the original proof and describes a number of other methods that might be used to automate the proofs of these problems. A companion article presents the collection of elementary problems in geometry for which automated proofs are sought. This article is a contribution to the Flyspeck project, which aims to give a complete formal proof of the Kepler conjecture.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"124 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128117354","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}
{"title":"Examining The Fragments of G","authors":"S. Perron","doi":"10.1109/LICS.2007.18","DOIUrl":"https://doi.org/10.1109/LICS.2007.18","url":null,"abstract":"When restricted to proving Sigma<sub>i</sub> <sup>q</sup> formulas, the quantified prepositional proof system G<sub>i</sub>* is closely related to the Sigma<sub>i</sub> <sup>b</sup> theorems of Buss's theory S<sub>2</sub> <sup>i</sup>. Namely, G<sub>i</sub>* has polynomial- size proofs of the translations of theorems of S<sub>2</sub> <sup>i</sup>, and S<sub>2</sub> <sup>i</sup> proves that G<sub>i</sub>* is sound. However, little is known about G* when proving more complex formulas. In this paper, we prove a witnessing theorem for G<sub>i</sub>* similar in style to the KPT witnessing theorem for T<sub>2</sub> <sup>i</sup>. This witnessing theorem is then used to show that S<sub>2</sub> <sup>i</sup> proves G* is sound with respect to prenex Sigma<sub>i+1</sub> <sup>q</sup> formulas. Note that unless the polynomial hierarchy collapses S<sub>2</sub> <sup>i</sup> is the weakest theory in the S<sub>2</sub> <sup>i</sup> hierarchy for which this is true. The witnessing theorem is also used to show that G<sub>1</sub>* is p-equivalent to a quantified version of extended-Frege. This is followed by a proof that Gi p-simulates G*<sub>i+1</sub>. We finish by proving that S<sub>2</sub> can be axiomatized by S<sub>2</sub> <sup>1</sup> plus axioms stating that the cut-free version of G* is sound. All together this shows that the connection between G<sub>i</sub>* and S<sub>2</sub> <sup>i</sup> does not extend to more complex formulas.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"41 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113933402","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}
{"title":"Symmetric Datalog and Constraint Satisfaction Problems in Logspace","authors":"László Egri, B. Larose, Pascal Tesson","doi":"10.1109/LICS.2007.47","DOIUrl":"https://doi.org/10.1109/LICS.2007.47","url":null,"abstract":"We introduce symmetric Datalog, a syntactic restriction of linear Datalog and show that its expressive power is exactly that of restricted symmetric Krom monotone SNP. The deep result of Reingold [17] on the complexity of undirected connectivity suffices to show that symmetric Datalog queries can be evaluated in logarithmic space. We show that for a number of constraint languages Gamma, the complement of the constraint satisfaction problem CSP(Gamma) can be expressed in symmetric Datalog. In particular, we show that if CSP(Gamma) is first-order definable and Lambda is a finite subset of the relational clone generated by Gamma then notCSP(Lambda) is definable in symmetric Datalog. Over the two-element domain and under standard complexity-theoretic assumptions, expressibility of notCSP(Gamma) in symmetric Datalog corresponds exactly to the class of CSPs computable in logarithmic space. Finally, we describe a fairly general subclass of implicational (or 0/1/all) constraints for which the complement of the corresponding CSP is also definable in symmetric Datalog. Our results provide preliminary evidence that symmetric Datalog may be a unifying explanation for families of CSPs lying in L.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117157059","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}
{"title":"Reflections on Finite Model Theory","authors":"Phokion G. Kolaitis","doi":"10.1109/LICS.2007.39","DOIUrl":"https://doi.org/10.1109/LICS.2007.39","url":null,"abstract":"Advances in finite model theory have appeared in LICS proceedings since the very beginning of the LICS Symposium. The goal of this paper is to reflect on finite model theory by highlighting some of its successes, examining obstacles that were encountered, and discussing some open problems that have stubbornly resisted solution.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116241977","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}
{"title":"Strong Normalization as Safe Interaction","authors":"Colin Riba","doi":"10.1109/LICS.2007.46","DOIUrl":"https://doi.org/10.1109/LICS.2007.46","url":null,"abstract":"When enriching the lambda-calculus with rewriting, union types may be needed to type all strongly normalizing terms. However, with rewriting, the elimination rule (orE) of union types may also allow to type non normalizing terms (in which case we say that (orE) is unsafe). This occurs in particular with non-determinism, but also with some confluent systems. It appears that studying the safety of (orE) amounts to the characterization, in a term, of safe interactions between some of its subterms. In this paper, we study the safety of (orE) for an extension of the lambda-calculus with simple rewrite rules. We prove that the union and intersection type discipline without (orE) is complete w.r.t. strong normalization. This allows to show that (orE) is safe if and only if an interpretation of types based on biorthogonals is sound for it. We also discuss two sufficient conditions for the safety of (orE), and study an alternative biorthogonality relation, based on the observation of the least reducibility candidate.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122804913","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}
{"title":"Bialgebraic Operational Semantics and Modal Logic","authors":"Bartek Klin","doi":"10.1109/LICS.2007.13","DOIUrl":"https://doi.org/10.1109/LICS.2007.13","url":null,"abstract":"A novel, general approach is proposed to proving the compositionality of process equivalences on languages defined by structural operational semantics (SOS). The approach, based on modal logic, is inspired by the simple observation that if the set of formulas satisfied by a process can be derived from the corresponding sets for its subprocesses, then the logical equivalence is a congruence. Striving for generality, SOS rules are modeled categorically as bialgebraic distributive laws for some notions of process syntax and behaviour, and modal logics are modeled via coalgebraic polyadic modal logic. Compositionality is proved by providing a suitable notion of behaviour for the logic together with a dual distributive law, reflecting the one modeling the SOS specification. Concretely, the dual laws may appear as SOS-like rules where logical formulas play the role of processes, and their behaviour models logical decomposition over process syntax. The approach can be used either to proving compositionality for specific languages or for defining SOS congruence formats.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123961311","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}
{"title":"Lindstrom theorems for fragments of first-order logic","authors":"B. T. Cate, J. Benthem, J. Väänänen","doi":"10.2168/LMCS-5(3:3)2009","DOIUrl":"https://doi.org/10.2168/LMCS-5(3:3)2009","url":null,"abstract":"Lindstrom theorems characterize logics in terms of model-theoretic conditions such as Compactness and the Lowenheim-Skolem property. Most existing Lindstrom theorems concern extensions of first-order logic. On the other hand, many logics relevant to computer science are fragments or extensions of fragments of first-order logic, e.g., k-variable logics and various modal logics. Finding Lindstrom theorems for these languages can be challenging, as most known techniques rely on coding arguments that seem to require the full expressive power of first-order logic. In this paper, we provide Lindstrom characterizations for a number of fragments of first-order logic. These include the k-variable fragments for k > 2, Tarski's relation algebra, graded modal logic, and the binary guarded fragment. We use two different proof techniques. One is a modification of the original Lindstrom proof. The other involves the modal concepts of bisimulation, tree unraveling, and finite depth. Our results also imply semantic preservation theorems. Characterizing the 2-variable fragment or the full guarded fragment remain open problems.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131485913","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}
{"title":"Pi-Calculus in Logical Form","authors":"M. Bonsangue, A. Kurz","doi":"10.1109/LICS.2007.36","DOIUrl":"https://doi.org/10.1109/LICS.2007.36","url":null,"abstract":"Abramsky's logical formulation of domain theory is extended to encompass the domain theoretic model for pi-calculus processes of Stark and of Fiore, Moggi and Sangiorgi. This is done by defining a logical counterpart of categorical constructions including dynamic name allocation and name exponentiation, and showing that they are dual to standard constructs in functor categories. We show that initial algebras of functors defined in terms of these constructs give rise to a logic that is sound, complete, and characterises bisimilarity. The approach is modular, and we apply it to derive a logical formulation of pi-calculus. The resulting logic is a modal calculus with primitives for input, free output and bound output.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121379678","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}
{"title":"Categorical Combinatorics for Innocent Strategies","authors":"Russell Harmer, M. Hyland, Paul-André Melliès","doi":"10.1109/LICS.2007.14","DOIUrl":"https://doi.org/10.1109/LICS.2007.14","url":null,"abstract":"We show how to construct the category of games and innocent strategies from a more primitive category of games. On that category we define a comonad and monad with the former distributing over the latter. Innocent strategies are the maps in the induced two-sided Kleisli category. Thus the problematic composition of innocent strategies reflects the use of the distributive law. The composition of simple strategies, and the combinatorics of pointers used to give the comonad and monad are themselves described in categorical terms. The notions of view and of legal play arise naturally in the explanation of the distributivity. The category-theoretic perspective provides a clear discipline for the necessary combinatorics.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"184 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114743451","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}
{"title":"Alternation-free modal mu-calculus for data trees","authors":"M. Jurdzinski, R. Lazic","doi":"10.1109/lics.2007.11","DOIUrl":"https://doi.org/10.1109/lics.2007.11","url":null,"abstract":"An alternation-free modal mu-calculus over data trees is introduced and studied. A data tree is an unranked ordered tree whose every node is labelled by a letter from a finite alphabet and an element (\"datum\") from an infinite set. For expressing data-sensitive properties, the calculus is equipped with freeze quantification. A freeze quantifier stores in a register the datum labelling the current tree node, which can then be accessed for equality comparisons deeper in the formula. The main results in the paper are that, for the fragment with forward modal operators and one register, satisfiability over finite data trees is decidable but not primitive recursive, and that for the subfragment consisting of safety formulae, satisfiability over countable data trees is decidable but not elementary. The proofs use alternating tree automata which have registers, and establish correspondences with nondeterministic tree automata which have faulty counters. Allowing backward modal operators or two registers causes undecidability. As consequences, decidability is obtained for two data-sensitive fragments of the XPath query language. The paper shows that, for reasoning about data trees, the forward fragment of the calculus with one register is a powerful alternative to a recently proposed first-order logic with two variables.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130790861","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}