{"title":"Reasoning about Petri Nets: A Calculus Based on Resolution and Dynamic Logic","authors":"Bruno Lopes, C. Nalon, E. Haeusler","doi":"10.1145/3441655","DOIUrl":"https://doi.org/10.1145/3441655","url":null,"abstract":"Petri Nets are a widely used formalism to deal with concurrent systems. Dynamic Logics (DLs) are a family of modal logics where each modality corresponds to a program. Petri-PDL is a logical language that combines these two approaches: it is a dynamic logic where programs are replaced by Petri Nets. In this work we present a clausal resolution-based calculus for Petri-PDL. Given a Petri-PDL formula, we show how to obtain its translation into a normal form to which a set of resolution-based inference rules are applied. We show that the resulting calculus is sound, complete, and terminating. Some examples of the application of the method are also given.","PeriodicalId":396362,"journal":{"name":"ACM Trans. Comput. Log.","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132867909","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}
J. Adámek, Liang-Ting Chen, Stefan Milius, Henning Urbat
{"title":"Reiterman's Theorem on Finite Algebras for a Monad","authors":"J. Adámek, Liang-Ting Chen, Stefan Milius, Henning Urbat","doi":"10.1145/3464691","DOIUrl":"https://doi.org/10.1145/3464691","url":null,"abstract":"\u0000 Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e., classes of finite algebras closed under finite products, subalgebras and quotients. In this article, Reiterman’s theorem is generalized to finite Eilenberg-Moore algebras for a monad \u0000 T\u0000 on a category D: we prove that a class of finite\u0000 T\u0000 -algebras is a pseudovariety iff it is presentable by profinite equations. As a key technical tool, we introduce the concept of a profinite monad\u0000 T\u0000 ^\u0000 associated to the monad\u0000 T\u0000 , which gives a categorical view of the construction of the space of profinite terms.\u0000","PeriodicalId":396362,"journal":{"name":"ACM Trans. Comput. Log.","volume":"292 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117327627","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":"The Complexity of Counting Problems over Incomplete Databases","authors":"M. Arenas, Pablo Barcel'o, Mikaël Monet","doi":"10.1145/3461642","DOIUrl":"https://doi.org/10.1145/3461642","url":null,"abstract":"\u0000 We study the complexity of various fundamental counting problems that arise in the context of incomplete databases, i.e., relational databases that can contain unknown values in the form of labeled nulls. Specifically, we assume that the domains of these unknown values are finite and, for a Boolean query \u0000 q\u0000 , we consider the following two problems: Given as input an incomplete database \u0000 D\u0000 , (a) return the number of completions of \u0000 D\u0000 that satisfy \u0000 q\u0000 ; or (b) return the number of valuations of the nulls of \u0000 D\u0000 yielding a completion that satisfies \u0000 q\u0000 . We obtain dichotomies between #P-hardness and polynomial-time computability for these problems when \u0000 q\u0000 is a self-join–free conjunctive query and study the impact on the complexity of the following two restrictions: (1) every null occurs at most once in \u0000 D\u0000 (what is called\u0000 Codd tables\u0000 ); and (2) the domain of each null is the same. Roughly speaking, we show that counting completions is much harder than counting valuations: For instance, while the latter is always in #P, we prove that the former is not in #P under some widely believed theoretical complexity assumption. Moreover, we find that both (1) and (2) can reduce the complexity of our problems. We also study the approximability of these problems and show that, while counting valuations always has a fully polynomial-time randomized approximation scheme (FPRAS), in most cases counting completions does not. Finally, we consider more expressive query languages and situate our problems with respect to known complexity classes.\u0000","PeriodicalId":396362,"journal":{"name":"ACM Trans. Comput. Log.","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117181434","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":"Small circuits and dual weak PHP in the universal theory of p-time algorithms","authors":"J. Krajícek","doi":"10.1145/3446207","DOIUrl":"https://doi.org/10.1145/3446207","url":null,"abstract":"\u0000 We prove, under a computational complexity hypothesis, that it is consistent with the true universal theory of p-time algorithms that a specific p-time function extending\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 bits to\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 bits violates the dual weak pigeonhole principle: Every string\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 equals the value of the function for some\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 . The function is the truth-table function assigning to a circuit the table of the function it computes and the hypothesis is that every language in P has circuits of a fixed polynomial size\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 .\u0000","PeriodicalId":396362,"journal":{"name":"ACM Trans. Comput. Log.","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125874828","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":"Generalized Realizability and Basic Logic","authors":"A. Y. Konovalov","doi":"10.1145/3468856","DOIUrl":"https://doi.org/10.1145/3468856","url":null,"abstract":"\u0000 Let\u0000 V\u0000 be a set of number-theoretical functions. We define a notion of absolute\u0000 V\u0000 -realizability for predicate formulas and sequents in such a way that the indices of functions in\u0000 V\u0000 are used for interpreting the implication and the universal quantifier. In this article, we prove that Basic Predicate Calculus is sound with respect to the semantics of absolute\u0000 V\u0000 -realizability if\u0000 V\u0000 satisfies some natural conditions.\u0000","PeriodicalId":396362,"journal":{"name":"ACM Trans. Comput. Log.","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126168383","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":"Action Logic is Undecidable","authors":"S. Kuznetsov","doi":"10.1145/3445810","DOIUrl":"https://doi.org/10.1145/3445810","url":null,"abstract":"\u0000 Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. One of the operations of this logic is the Kleene star, which is axiomatized by an induction scheme. For a stronger system that uses an\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 -rule instead (infinitary action logic), Buszkowski and Palka (2007) proved\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 -completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by Kozen in 1994. In this article, we show that it is undecidable, more precisely,\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 -complete. We also prove the same undecidability results for all recursively enumerable logics between action logic and infinitary action logic, for fragments of these logics with only one of the two lattice (additive) connectives, and for action logic extended with the law of distributivity.\u0000","PeriodicalId":396362,"journal":{"name":"ACM Trans. Comput. Log.","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130137297","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":"A Formal System for the Universal Quantification of Schematic Variables","authors":"F. Guidi","doi":"10.1145/3470646","DOIUrl":"https://doi.org/10.1145/3470646","url":null,"abstract":"We advocate the use of de Bruijn's universal abstraction $lambda^infty$ for the quantification of schematic variables in the predicative setting and we present a typed $lambda$-calculus featuring the quantifier $lambda^infty$ accompanied by other practically useful constructions like explicit substitutions and expected type annotations. The calculus stands just on two notions, i.e., bound rt-reduction and parametric validity, and has the expressive power of $lambdarightarrow$. Thus, while not aiming at being a logical framework by itself, it does enjoy many desired invariants of logical frameworks including confluence of reduction, strong normalization, preservation of type by reduction, decidability, correctness of types and uniqueness of types up to conversion. This calculus belongs to the $lambdadelta$ family of formal systems, which borrow some features from the pure type systems and some from the languages of the Automath tradition, but stand outside both families. In particular, the calculus includes and evolves two earlier systems of this family. Moreover, a machine-checked specification of its theory is available.","PeriodicalId":396362,"journal":{"name":"ACM Trans. Comput. Log.","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127286903","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":"Arithmetic complexity","authors":"L. Dries, Y. Moschovakis","doi":"10.1145/1459010.1459012","DOIUrl":"https://doi.org/10.1145/1459010.1459012","url":null,"abstract":"We obtain lower bounds on the cost of computing various arithmetic functions and deciding various arithmetic relations from specified primitives. This includes lower bounds for computing the greatest common divisor and deciding coprimeness of two integers, from primitives like addition, subtraction, division with remainder and multiplication. Some of our results are in terms of recursive programs, but they generalize directly to most (plausibly all) algorithms from the specified primitives. Our methods involve some elementary number theory as well as the development of some basic notions and facts about recursive algorithms.","PeriodicalId":396362,"journal":{"name":"ACM Trans. Comput. Log.","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129158910","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":"Complete Abstractions for Checking Language Inclusion","authors":"P. Ganty, Francesco Ranzato, Pedro Valero","doi":"10.1145/3462673","DOIUrl":"https://doi.org/10.1145/3462673","url":null,"abstract":"\u0000 We study the language inclusion problem\u0000 L\u0000 1\u0000 ⊆\u0000 L\u0000 2\u0000 , where\u0000 L\u0000 1\u0000 is regular or context-free. Our approach relies on abstract interpretation and checks whether an overapproximating abstraction of\u0000 L\u0000 1\u0000 , obtained by approximating the Kleene iterates of its least fixpoint characterization, is included in\u0000 L\u0000 2\u0000 . We show that a language inclusion problem is decidable whenever this overapproximating abstraction satisfies a completeness condition (i.e., its loss of precision causes no false alarm) and prevents infinite ascending chains (i.e., it guarantees termination of least fixpoint computations). This overapproximating abstraction of languages can be defined using quasiorder relations on words, where the abstraction gives the language of all the words “greater than or equal to” a given input word for that quasiorder. We put forward a range of such quasiorders that allow us to systematically design decision procedures for different language inclusion problems, such as regular languages into regular languages or into trace sets of one-counter nets, and context-free languages into regular languages. In the case of inclusion between regular languages, some of the induced inclusion checking procedures correspond to well-known state-of-the-art algorithms, like the so-called antichain algorithms. Finally, we provide an equivalent language inclusion checking algorithm based on a greatest fixpoint computation that relies on quotients of languages and, to the best of our knowledge, was not previously known.\u0000","PeriodicalId":396362,"journal":{"name":"ACM Trans. Comput. Log.","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130121886","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 Circuits for Rank Logic","authors":"A. Dawar, G. Wilsenach","doi":"10.4230/LIPIcs.CSL.2018.20","DOIUrl":"https://doi.org/10.4230/LIPIcs.CSL.2018.20","url":null,"abstract":"Fixed-point logic with rank (FPR) is an extension of fixed-point logic with counting (FPC) with operators for computing the rank of a matrix over a finit field. The expressive power of FPR properly extends that of FPC and is contained in P, but it is not known if that containment is proper. We give a circuit characterization for FPR in terms of families of symmetric circuits with rank gates, along the lines of that for FPC given by Anderson and Dawar in 2017. This requires the development of a broad framework of circuits in which the individual gates compute functions that are not symmetric (i.e., invariant under all permutations of their inputs). This framework also necessitates the development of novel techniques to prove the equivalence of circuits and logic. Both the framework and the techniques are of greater generality than the main result.","PeriodicalId":396362,"journal":{"name":"ACM Trans. Comput. Log.","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125825059","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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