The class of regular transformations has several equivalent characterizations such as functional MSO transductions, deterministic two-way transducers, streaming string transducers, as well as regular transducer expressions (RTE). For algorithmic applications, it is very common and useful to transform a specification, here, an RTE, to a machine, here, a transducer. In this paper, we give an efficient construction of a two-way reversible transducer (2RFT) equivalent to a given RTE. 2RFTs form a well behaved class of transducers which are deterministic and co-deterministic (hence allows evaluation in linear time w.r.t. the input word), and where composition has only polynomial complexity. As a significant complexity improvement over existing techniques, we give the first elementary procedure for translating RTEs to machines. For full RTE, the constructed 2RFT has size doubly exponential in the size of the expression. If the RTE does not use Hadamard product or chained-star, the constructed 2RFT has size exponential in the size of the RTE.
{"title":"Efficient Construction of Reversible Transducers from Regular Transducer Expressions","authors":"L. Dartois, P. Gastin, R. Govind, S. Krishna","doi":"10.1145/3531130.3533364","DOIUrl":"https://doi.org/10.1145/3531130.3533364","url":null,"abstract":"The class of regular transformations has several equivalent characterizations such as functional MSO transductions, deterministic two-way transducers, streaming string transducers, as well as regular transducer expressions (RTE). For algorithmic applications, it is very common and useful to transform a specification, here, an RTE, to a machine, here, a transducer. In this paper, we give an efficient construction of a two-way reversible transducer (2RFT) equivalent to a given RTE. 2RFTs form a well behaved class of transducers which are deterministic and co-deterministic (hence allows evaluation in linear time w.r.t. the input word), and where composition has only polynomial complexity. As a significant complexity improvement over existing techniques, we give the first elementary procedure for translating RTEs to machines. For full RTE, the constructed 2RFT has size doubly exponential in the size of the expression. If the RTE does not use Hadamard product or chained-star, the constructed 2RFT has size exponential in the size of the RTE.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127495052","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}
Polynomial closure is a standard operator which is applied to a class of regular languages. In this paper, we investigate three restrictions called left (LPol), right (RPol) and mixed polynomial closure (MPol). The first two were known while MPol is new. We look at two decision problems that are defined for every class . Membership takes a regular language as input and asks if it belongs to . Separation takes two regular languages as input and asks if there exists a third language in including the first one and disjoint from the second. We prove that LPol, RPol and MPol preserve the decidability of membership under mild hypotheses on the input class, and the decidability of separation under much stronger hypotheses. We apply these results to natural hierarchies. First, we look at several language theoretic hierarchies that are built by applying LPol, RPol and MPol recursively to a single input class. We prove that these hierarchies can actually be defined using almost exclusively MPol. We also consider quantifier alternation hierarchies for two-variable first-order logic (FO2) and prove that one can climb them using MPol. The result is generic in the sense that it holds for most standard choices of signatures. We use it to prove that for most of these choices, membership is decidable for all levels in the hierarchy. Finally, we prove that separation is decidable for the hierarchy of two-variable first-order logic equipped with only the linear order (FO2(<)).
{"title":"The amazing mixed polynomial closure and its applications to two-variable first-order logic","authors":"Thomas Place","doi":"10.1145/3531130.3532410","DOIUrl":"https://doi.org/10.1145/3531130.3532410","url":null,"abstract":"Polynomial closure is a standard operator which is applied to a class of regular languages. In this paper, we investigate three restrictions called left (LPol), right (RPol) and mixed polynomial closure (MPol). The first two were known while MPol is new. We look at two decision problems that are defined for every class . Membership takes a regular language as input and asks if it belongs to . Separation takes two regular languages as input and asks if there exists a third language in including the first one and disjoint from the second. We prove that LPol, RPol and MPol preserve the decidability of membership under mild hypotheses on the input class, and the decidability of separation under much stronger hypotheses. We apply these results to natural hierarchies. First, we look at several language theoretic hierarchies that are built by applying LPol, RPol and MPol recursively to a single input class. We prove that these hierarchies can actually be defined using almost exclusively MPol. We also consider quantifier alternation hierarchies for two-variable first-order logic (FO2) and prove that one can climb them using MPol. The result is generic in the sense that it holds for most standard choices of signatures. We use it to prove that for most of these choices, membership is decidable for all levels in the hierarchy. Finally, we prove that separation is decidable for the hierarchy of two-variable first-order logic equipped with only the linear order (FO2(<)).","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122420341","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 monoidal streams: a generalization of causal stream functions to monoidal categories. In the same way that streams provide semantics to dataflow programming with pure functions, monoidal streams provide semantics to dataflow programming with theories of processes represented by a symmetric monoidal category. At the same time, monoidal streams form a feedback monoidal category, which can be used to interpret signal flow graphs. As an example, we study a stochastic dataflow language.
{"title":"Monoidal Streams for Dataflow Programming","authors":"Elena Di Lavore, G. Felice, Mario Rom'an","doi":"10.1145/3531130.3533365","DOIUrl":"https://doi.org/10.1145/3531130.3533365","url":null,"abstract":"We introduce monoidal streams: a generalization of causal stream functions to monoidal categories. In the same way that streams provide semantics to dataflow programming with pure functions, monoidal streams provide semantics to dataflow programming with theories of processes represented by a symmetric monoidal category. At the same time, monoidal streams form a feedback monoidal category, which can be used to interpret signal flow graphs. As an example, we study a stochastic dataflow language.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131855481","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}
Jannik Dreier, Jakub Gajarsk'y, Sandra Kiefer, Michal Pilipczuk, Szymon Toruńczyk
We give new decomposition theorems for classes of graphs that can be transduced in first-order logic from classes of sparse graphs — more precisely, from classes of bounded expansion and nowhere dense classes. In both cases, the decomposition takes the form of a single colored rooted tree of bounded depth where, in addition, there can be links between nodes that are not related in the tree. The constraint is that the structure formed by the tree and the links has to be sparse. Using the decomposition theorem for transductions of nowhere dense classes, we show that they admit low-shrubdepth covers of size , where n is the vertex count and ε > 0 is any fixed real. This solves an open problem posed by Gajarský et al. (ACM TOCL ’20) and also by Briański et al. (SIDMA ’21).
{"title":"Treelike Decompositions for Transductions of Sparse Graphs","authors":"Jannik Dreier, Jakub Gajarsk'y, Sandra Kiefer, Michal Pilipczuk, Szymon Toruńczyk","doi":"10.1145/3531130.3533349","DOIUrl":"https://doi.org/10.1145/3531130.3533349","url":null,"abstract":"We give new decomposition theorems for classes of graphs that can be transduced in first-order logic from classes of sparse graphs — more precisely, from classes of bounded expansion and nowhere dense classes. In both cases, the decomposition takes the form of a single colored rooted tree of bounded depth where, in addition, there can be links between nodes that are not related in the tree. The constraint is that the structure formed by the tree and the links has to be sparse. Using the decomposition theorem for transductions of nowhere dense classes, we show that they admit low-shrubdepth covers of size , where n is the vertex count and ε > 0 is any fixed real. This solves an open problem posed by Gajarský et al. (ACM TOCL ’20) and also by Briański et al. (SIDMA ’21).","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132651927","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}
Martin Avanzini, G. Moser, Romain Péchoux, S. Perdrix, Vladimir Zamdzhiev
We introduce a new kind of expectation transformer for a mixed classical-quantum programming language. Our semantic approach relies on a new notion of a cost structure, which we introduce and which can be seen as a specialisation of the Kegelspitzen of Keimel and Plotkin. We show that our weakest precondition analysis is both sound and adequate with respect to the operational semantics of the language. Using the induced expectation transformer, we provide formal analysis methods for the expected cost analysis and expected value analysis of classical-quantum programs. We illustrate the usefulness of our techniques by computing the expected cost of several well-known quantum algorithms and protocols, such as coin tossing, repeat until success, entangled state preparation, and quantum walks.
{"title":"Quantum Expectation Transformers for Cost Analysis","authors":"Martin Avanzini, G. Moser, Romain Péchoux, S. Perdrix, Vladimir Zamdzhiev","doi":"10.1145/3531130.3533332","DOIUrl":"https://doi.org/10.1145/3531130.3533332","url":null,"abstract":"We introduce a new kind of expectation transformer for a mixed classical-quantum programming language. Our semantic approach relies on a new notion of a cost structure, which we introduce and which can be seen as a specialisation of the Kegelspitzen of Keimel and Plotkin. We show that our weakest precondition analysis is both sound and adequate with respect to the operational semantics of the language. Using the induced expectation transformer, we provide formal analysis methods for the expected cost analysis and expected value analysis of classical-quantum programs. We illustrate the usefulness of our techniques by computing the expected cost of several well-known quantum algorithms and protocols, such as coin tossing, repeat until success, entangled state preparation, and quantum walks.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121445728","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 framework of quantitative equational logic has been successfully applied to reason about algebras whose carriers are metric spaces and operations are nonexpansive. We extend this framework in two orthogonal directions: algebras endowed with generalised metric space structures, and operations being nonexpansive up to a lifting. We apply our results to the algebraic axiomatisation of the Łukaszyk–Karmowski distance on probability distributions, which has recently found application in the field of representation learning on Markov processes.
{"title":"Beyond Nonexpansive Operations in Quantitative Algebraic Reasoning","authors":"M. Mio, Ralph Sarkis, Valeria Vignudelli","doi":"10.1145/3531130.3533366","DOIUrl":"https://doi.org/10.1145/3531130.3533366","url":null,"abstract":"The framework of quantitative equational logic has been successfully applied to reason about algebras whose carriers are metric spaces and operations are nonexpansive. We extend this framework in two orthogonal directions: algebras endowed with generalised metric space structures, and operations being nonexpansive up to a lifting. We apply our results to the algebraic axiomatisation of the Łukaszyk–Karmowski distance on probability distributions, which has recently found application in the field of representation learning on Markov processes.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132885082","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 orbit-finite systems of linear equations, in the setting of sets with atoms. Our principal contribution is a decision procedure for solvability of such systems. The procedure works for every field (and even commutative ring) under mild effectiveness assumptions, and reduces a given orbit-finite system to a number of finite ones: exponentially many in general, but polynomially many when the atom dimension of input systems is fixed. Towards obtaining the procedure we push further the theory of vector spaces generated by orbit-finite sets, and show that each such vector space admits an orbit-finite basis. This fundamental property is a key tool in our development, but should be also of wider interest.
{"title":"Solvability of orbit-finite systems of linear equations","authors":"Arka P. Ghosh, Piotr Hofman, S. Lasota","doi":"10.1145/3531130.3533333","DOIUrl":"https://doi.org/10.1145/3531130.3533333","url":null,"abstract":"We study orbit-finite systems of linear equations, in the setting of sets with atoms. Our principal contribution is a decision procedure for solvability of such systems. The procedure works for every field (and even commutative ring) under mild effectiveness assumptions, and reduces a given orbit-finite system to a number of finite ones: exponentially many in general, but polynomially many when the atom dimension of input systems is fixed. Towards obtaining the procedure we push further the theory of vector spaces generated by orbit-finite sets, and show that each such vector space admits an orbit-finite basis. This fundamental property is a key tool in our development, but should be also of wider interest.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123089516","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}
Michael Blondin, Filip Mazowiecki, Philip Offtermatt
Workflow nets are a popular variant of Petri nets that allow for the algorithmic formal analysis of business processes. The central decision problems concerning workflow nets deal with soundness, where the initial and final configurations are specified. Intuitively, soundness states that from every reachable configuration one can reach the final configuration. We settle the widely open complexity of the three main variants of soundness: classical, structural and generalised soundness. The first two are EXPSPACE-complete, and, surprisingly, the latter is PSPACE-complete, thus computationally simpler.
{"title":"The complexity of soundness in workflow nets","authors":"Michael Blondin, Filip Mazowiecki, Philip Offtermatt","doi":"10.1145/3531130.3533341","DOIUrl":"https://doi.org/10.1145/3531130.3533341","url":null,"abstract":"Workflow nets are a popular variant of Petri nets that allow for the algorithmic formal analysis of business processes. The central decision problems concerning workflow nets deal with soundness, where the initial and final configurations are specified. Intuitively, soundness states that from every reachable configuration one can reach the final configuration. We settle the widely open complexity of the three main variants of soundness: classical, structural and generalised soundness. The first two are EXPSPACE-complete, and, surprisingly, the latter is PSPACE-complete, thus computationally simpler.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124481008","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. Gutsfeld, A. Meier, Christoph Ohrem, J. Virtema
In this paper, we study a novel approach to asynchronous hyperproperties by reconsidering the foundations of temporal team semantics. We consider three logics: , and , which are obtained by adding quantification over so-called time evaluation functions controlling the asynchronous progress of traces. We then relate synchronous to our new logics and show how it can be embedded into them. We show that the model checking problem for with Boolean disjunctions is highly undecidable by encoding recurrent computations of non-deterministic 2-counter machines. Finally, we present a translation from to Alternating Asynchronous Büchi Automata and obtain decidability results for the path checking problem as well as restricted variants of the model checking and satisfiability problems.
{"title":"Temporal Team Semantics Revisited","authors":"J. Gutsfeld, A. Meier, Christoph Ohrem, J. Virtema","doi":"10.1145/3531130.3533360","DOIUrl":"https://doi.org/10.1145/3531130.3533360","url":null,"abstract":"In this paper, we study a novel approach to asynchronous hyperproperties by reconsidering the foundations of temporal team semantics. We consider three logics: , and , which are obtained by adding quantification over so-called time evaluation functions controlling the asynchronous progress of traces. We then relate synchronous to our new logics and show how it can be embedded into them. We show that the model checking problem for with Boolean disjunctions is highly undecidable by encoding recurrent computations of non-deterministic 2-counter machines. Finally, we present a translation from to Alternating Asynchronous Büchi Automata and obtain decidability results for the path checking problem as well as restricted variants of the model checking and satisfiability problems.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123498134","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 pebbling comonad, introduced by Abramsky, Dawar and Wang, provides a categorical interpretation for the k-pebble games from finite model theory. The coKleisli category of the pebbling comonad specifies equivalences under different fragments and extensions of infinitary k-variable logic. Moreover, the coalgebras over this pebbling comonad characterise treewidth and correspond to tree decompositions. In this paper we introduce the pebble-relation comonad, which characterises pathwidth and whose coalgebras correspond to path decompositions. We further show that the existence of a coKleisli morphism in this comonad is equivalent to truth preservation in the restricted conjunction fragment of k-variable infinitary logic. We do this using Dalmau’s pebble-relation game and an equivalent all-in-one pebble game. We then provide a similar treatment to the corresponding coKleisli isomorphisms via a bijective version of the all-in-one pebble game with a hidden pebble placement. Finally, we show as a consequence a new Lovász-type theorem relating pathwidth to the restricted conjunction fragment of k-variable infinitary logic with counting quantifiers.
{"title":"The Pebble-Relation Comonad in Finite Model Theory","authors":"Yoàv Montacute, Nihil Shah","doi":"10.1145/3531130.3533335","DOIUrl":"https://doi.org/10.1145/3531130.3533335","url":null,"abstract":"The pebbling comonad, introduced by Abramsky, Dawar and Wang, provides a categorical interpretation for the k-pebble games from finite model theory. The coKleisli category of the pebbling comonad specifies equivalences under different fragments and extensions of infinitary k-variable logic. Moreover, the coalgebras over this pebbling comonad characterise treewidth and correspond to tree decompositions. In this paper we introduce the pebble-relation comonad, which characterises pathwidth and whose coalgebras correspond to path decompositions. We further show that the existence of a coKleisli morphism in this comonad is equivalent to truth preservation in the restricted conjunction fragment of k-variable infinitary logic. We do this using Dalmau’s pebble-relation game and an equivalent all-in-one pebble game. We then provide a similar treatment to the corresponding coKleisli isomorphisms via a bijective version of the all-in-one pebble game with a hidden pebble placement. Finally, we show as a consequence a new Lovász-type theorem relating pathwidth to the restricted conjunction fragment of k-variable infinitary logic with counting quantifiers.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124497138","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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