Structure of Finite Semigroups and Language Equations
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Fast Graph Simplification for Interleaved Dyck-Reachability
Fast Graph Simplification for Interleaved Dyck-Reachability Yuanbo Li Qirun Zhang Thomas Reps Georgia Institute of Technology Georgia Institute of Technology University of Wisconsin-Madison USA USA USA [email protected] [email protected] [email protected] Abstract CCS Concepts: • Mathematics of computing ! Graph ! Many program-analysis problems can be formulated as graph- algorithms; • Theory of computation Program anal- reachability problems. Interleaved Dyck language reachabil- ysis. ity (InterDyck-reachability) is a fundamental framework to Keywords: CFL-Reachability, Static Analysis express a wide variety of program-analysis problems over edge-labeled graphs. The InterDyck language represents ACM Reference Format: an intersection of multiple matched-parenthesis languages Yuanbo Li, Qirun Zhang, and Thomas Reps. 2020. Fast Graph Simpli- fication for Interleaved Dyck-Reachability. In Proceedings of the 41st (i.e., Dyck languages). In practice, program analyses typically ACM SIGPLAN International Conference on Programming Language leverage one Dyck language to achieve context-sensitivity, Design and Implementation (PLDI ’20), June 15–20, 2020, London, and other Dyck languages to model data dependences, such UK. ACM, New York, NY, USA, 14 pages. https://doi.org/10.1145/ as field-sensitivity and pointer references/dereferences. In 3385412.3386021 the ideal case, an InterDyck-reachability framework should model multiple Dyck languages simultaneously. Unfortunately, precise InterDyck-reachability is undecid- 1 Introduction able. Any practical solution must over-approximate the exact The L language-reachability (L-reachability) framework is answer. In the literature, a lot of work has been proposed to a popular model to formulate many program-analysis prob- over-approximate the InterDyck-reachability formulation. lems [14]. -
Grammatical Compression: Compressed Equivalence and Other Problems Alberto Bertoni, Roberto Radicioni
Grammatical compression: compressed equivalence and other problems Alberto Bertoni, Roberto Radicioni To cite this version: Alberto Bertoni, Roberto Radicioni. Grammatical compression: compressed equivalence and other problems. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2010, 12 (4), pp.109- 126. hal-00990458 HAL Id: hal-00990458 https://hal.inria.fr/hal-00990458 Submitted on 13 May 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science DMTCS vol. 12:4, 2010, 109–126 Grammatical compression: compressed equivalence and other problems Alberto Bertoni1y and Roberto Radicioni2z 1Dipartimento di Scienze dell’Informazione, Universita` degli Studi di Milano, Italy 2Dipartimento di Informatica e Comunicazione, Universita` degli Studi dell’Insubria, Italy received 30th October 2009, revised 1st June 2010, accepted 11th October 2010. In this work, we focus our attention to algorithmic solutions for problems where the instances are presented as straight-line programs on a given algebra. In our exposition, we try to survey general results by presenting some meaningful examples; moreover, where possible, we outline the proofs in order to give an insight of the methods and the techniques. We recall some recent results for the problem PosSLP, consisting of deciding if the integer defined by a straight-line program on the ring Z is greater than zero; we discuss some implications in the areas of numerical analysis and strategic games. -
Equidivisible Pseudovarieties of Semigroups
Pr´e-Publica¸c˜oes do Departamento de Matem´atica Universidade de Coimbra Preprint Number 16–05 EQUIDIVISIBLE PSEUDOVARIETIES OF SEMIGROUPS JORGE ALMEIDA AND ALFREDO COSTA Abstract: We give a complete characterization of pseudovarieties of semigroups whose finitely generated relatively free profinite semigroups are equidivisible. Be- sides the pseudovarieties of completely simple semigroups, they are precisely the pseudovarieties that are closed under Mal’cev product on the left by the pseudova- riety of locally trivial semigroups. A further characterization which turns out to be instrumental is as the non-completely simple pseudovarieties that are closed under two-sided Karnofsky-Rhodes expansion. Keywords: semigroup, equidivisible, pseudovariety, Karnofsky-Rhodes expansion, connected expansion, two-sided Cayley graph. AMS Subject Classification (2010): Primary 20M07, 20M05. 1. Introduction A pseudovariety of semigroups is a class of finite semigroups closed un- der taking subsemigroups, homomorphic images and finitary products. In the past few decades, pseudovarieties provided the main framework for the research on finite semigroups, motivated by Eilenberg’s correspondence the- orem between pseudovarieties and varieties of languages. In this context, the finitely generated relatively free profinite semigroups associated to each pseu- dovariety proved to be of fundamental importance. We assume the reader has some familiarity with this background. The books [19, 1] are indicated as supporting references. The paper [2] might also be useful for someone looking for a brief introduction. In this paper we are concerned with equidivisible relatively free profinite semigroups. A semigroup S is equidivisible if for every u,v,x,y ∈ S, the equality uv = xy implies that u = x and v = y, or that there is t ∈ S such that ut = x and v = ty, or such that xt = u and y = tv. -
Visibly Counter Languages and Constant Depth Circuits
Visibly Counter Languages and Constant Depth Circuits Andreas Krebs, Klaus-Jörn Lange, and Michael Ludwig WSI – University of Tübingen Sand 13, 72076 Tübingen, Germany {krebs,lange,ludwigm}@informatik.uni-tuebingen.de Abstract We examine visibly counter languages, which are languages recognized by visibly counter au- tomata (a.k.a. input driven counter automata). We are able to effectively characterize the visibly counter languages in AC0 and show that they are contained in FO[+]. 1998 ACM Subject Classification F.1.1 Models of Computation, F.1.3 Complexity Measures and Classes, F.4.1 Mathematical Logic, F.4.3 Formal Languages Keywords and phrases visibly counter automata, constant depth circuits, AC0, FO[+] Digital Object Identifier 10.4230/LIPIcs.STACS.2015.594 1 Introduction One important topic of complexity theory is the characterization of regular languages contained in constant depth complexity classes [7, 3, 5, 4]. In [4] Barrington et al. showed that the regular sets in AC0 are exactly the languages definable by first order logic using regular predicates. We extend this approach to certain non-regular languages. The visibly pushdown languages (VPL) are a sub-class of the context-free languages containing the regular sets which exhibits many of the decidability and closure properties of the regular languages. Their essential feature is that the use of the pushdown store is purely input-driven: The input alphabet is partitioned into call, return and internal letters. On a call letter, the pushdown automaton (PDA) must push a symbol onto its stack, on a return letter it must pop a symbol, and on an internal letter it cannot access the stack at all. -
The Inclusion Problem of Context-Free Languages: Some Tractable Cases?
The Inclusion Problem of Context-free Languages: Some Tractable Cases? Alberto Bertoni1, Christian Choffrut2, and Roberto Radicioni3 1 Universit`adegli Studi di Milano Dipartimento di Scienze dell’Informazione Via Comelico 39, 20135 Milano, Italy [email protected] 2 L.I.A.F.A. (Laboratoire d’Informatique Algorithmique, Fondements et Applications), Universit´eParis VII, 2 Place Jussieu, 75221 Paris - France [email protected] 3 Universit`adegli Studi dell’Insubria Dipartimento di Informatica e Comunicazione Via Mazzini 5, 21100 Varese, Italy [email protected] Abstract. We study the problem of testing whether a context-free lan- guage is included in a fixed set L0, where L0 is the language of words reducing to the empty word in the monoid defined by a complete string rewrite system. We prove that, if the monoid is cancellative, then our inclusion problem is polynomially reducible to the problem of testing equivalence of straight-line programs in the same monoid. As an appli- cation, we obtain a polynomial time algorithm for testing if a context-free language is included in a Dyck language (the best previous algorithm for this problem was doubly exponential). Introduction In this paper we analyze problems of this kind: given a fixed language L0, to de- cide whether the language LG generated by a context-free grammar is contained in L0. It is known that he class of context-free grammars generating languages L0 for which the problem is decidable is not recursive ([7]). This problem is unde- cidable even for some deterministic context-free language L0, while it turns out to be decidable if L0 is superdeterministic ([6]); in this case a doubly exponential time algorithm has been shown. -
CDM [1Ex]Context-Free Grammars
CDM Context-Free Grammars Klaus Sutner Carnegie Mellon Universality 60-cont-free 2017/12/15 23:17 1 Generating Languages Properties of CFLs Generation vs. Recognition 3 Turing machines can be used to check membership in decidable sets. They can also be used to enumerate semidecidable sets, whence the classical notion of recursively enumerable sets. For languages L ⊆ Σ? there is a similar notion of generation. The idea is to set up a system of simple rules that can be used to derive all words in a particular formal language. These systems are typically highly nondeterministic and it is not clear how to find (efficient) recognition algorithms for the corresponding languages. Noam Chomsky 4 Historically, these ideas go back to work by Chomsky in the 1950s. Chomsky was mostly interested natural languages: the goal is to develop grammars that differentiate between grammatical and and ungrammatical sentences. 1 The cat sat on the mat. 2 The mat on the sat cat. Alas, this turns out to be inordinately difficult, syntax and semantics of natural languages are closely connected and very complicated. But for artificial languages such as programming languages, Chomsky's approach turned out be perfectly suited. Cat-Mat Example 5 Sentence Noun Phrase Verb Phrase Punctuation Determiner Noun Verb Prepositional Phrase . The cat sat Preposition Noun Phrase on Determiner Noun the mat Mat-Cat Example 6 Noun Phrase Noun Phrase Prepositional Phrase Punctuation Determiner Noun Preposition Noun Phrase . The mat on Determiner Adjective Noun the sat cat Killer App: Programming Languages 7 Many programming languages have a block structure like so: begin begin end begin begin end begin end end end Clearly, this is not a regular language and cannot be checked by a finite state machine. -
Semigroup, Monoid and Group Models of Groupoid Identities 1. Introduction
Quasigroups and Related Systems 16 (2008), 25 ¡ 29 Semigroup, monoid and group models of groupoid identities Nick C. Fiala Abstract In this note, we characterize those groupoid identities that have a (nite) non-trivial (semigroup, monoid, group) model. 1. Introduction Denition 1.1. A groupoid consists of a non-empty set equipped with a binary operation, which we simply denote by juxtaposition. A groupoid G is non-trivial if jGj > 1, otherwise it is trivial.A semigroup is a groupoid S that is associative ((xy)z = x(yz) for all x; y; z 2 S). A monoid is a semigroup M possessing a neutral element e 2 M such that ex = xe = x for all x 2 M (the letter e will always denote the neutral element of a monoid). A group is a monoid G such that for all x 2 G there exists an inverse x¡1 such that x¡1x = xx¡1 = e.A quasigroup is a groupoid Q such that for all a; b 2 Q, there exist unique x; y 2 Q such that ax = b and ya = b.A loop is a quasigroup possessing a neutral element. A groupoid term is a product of universally quantied variables. A groupoid identity is an equation, the left-hand side and right-hand side of which are groupoid terms. By the words term and identity, we shall always mean groupoid term and groupoid identity, respectively. The letters s and t will always denote terms. We will say that an identity s = t has a (nite) non-trivial model if there exists a (nite) non-trivial groupoid G such that s = t is valid in G. -
Inverse Monoids, Trees, and Context-Free Languages
transactions of the american mathematical society Volume 335, Number 1, January 1993 INVERSE MONOIDS, TREES, AND CONTEXT-FREE LANGUAGES STUART W. MARGOLIS AND JOHN C. MEAKIN Abstract. This paper is concerned with a study of inverse monoids presented by a set X subject to relations of the form e¡ = f¡, i € I, where e¡ and f¡ are Dyck words, i.e. idempotents of the free inverse monoid on X . Some general results of Stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the Cayley graph of the free group on X . In the finitely presented case the word problem is solved by using Rabin's theorem on the second order monadic logic of the infinite binary tree. Some connections with the theory of rational subsets of the free group and the theory of context-free languages are explored. 1. Introduction We refer the reader to Lallement [5] for basic information about semigroups and connections with automata theory and formal language theory. We shall be concerned in this paper primarily with inverse semigroups and inverse monoids. For the convenience of the reader we briefly summarize the basic notions and results about inverse monoids that we will need in the present paper; many more details and results about inverse monoids may be found in the book of Petrich [9]- An inverse semigroup is a semigroup S with the property that, for each a eS there is a unique element a~x £ S such that a = aa~xa and a~x = a~xaa~x. -
The Dyck Language Edit Distance Problem in Near-Linear Time
The Dyck Language Edit Distance Problem in Near-linear Time Barna Saha Dept. of Computer Science University of Massachusetts, Amherst Amherst, USA Email: [email protected]. Abstract Given a string σ over alphabet Σ and a grammar G defined over the same alphabet, how many minimum number of repairs (insertions, deletions and substitutions) are required to map σ into a valid member of G? The seminal work of Aho and Peterson in 1972 initiated the study of this language edit distance problem providing a dynamic programming algorithm for context free languages that runs in O(jGj2n3) time, where n is the string length and jGj is the grammar size. While later improvements reduced the running time to O(jGjn3), the cubic running time on the input length held a major bottleneck for applying these algorithms to their multitude of applications. In this paper, we study the language edit distance problem for a fundamental context free language, DYCK(s) representing the language of well-balanced parentheses of s different types, that has been piv- otal in the development of formal language theory. We provide the very first near-linear time algorithm to tightly approximate the DYCK(s) language edit distance problem for any arbitrary s.DYCK(s) lan- guage edit distance significantly generalizes the well-studied string edit distance problem, and appears in most applications of language edit distance ranging from data quality in databases, generating auto- mated error-correcting parsers in compiler optimization to structure prediction problems in biological sequences. Its nondeterministic counterpart is known as the hardest context free language. -
Semigroups with Operation-Compatible Green’S Quasiorders
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Essex Research Repository SEMIGROUPS WITH OPERATION-COMPATIBLE GREEN'S QUASIORDERS ZSOFIA´ JUHASZ,´ ALEXEI VERNITSKI Abstract. We call a semigroup on which the Green's quasiorder ≤J (≤L, ≤R) is operation-compatible, a ≤J -compatible (≤L-compatible, ≤R-compatible) semigroup. We study the classes of ≤J -compatible, ≤L-compatible and ≤R-compatible semigroups, using the smallest operation-compatible quasiorders containing Green's quasiorders as a tool. We prove a number of results, including the following. The class of ≤L-compatible (≤R-compatible) semigroups is closed under taking homomorphic images. A regular periodic semigroup is ≤J -compatible if and only if it is a semilattice of simple semi- groups. Every negatively orderable semigroup can be embedded into a negatively orderable ≤J -compatible semigroup. 1. Introduction Green's relations L, R and J are one of the most important tools in studying the structure of semigroups. They can also be viewed from a less common angle: as being defined via quasiorders (or pre- orders), which we shall refer to as Green's quasiorders and denote by ≤L, ≤R and ≤J , respectively. Studying the properties of these qua- siorders is of interest, because of the importance of Green's relations and due to the fact that in a certain sense these associated quasiorders contain `more information' about a semigroup than Green's relations: given only a Green's quasiorder on a semigroup we can reconstruct the corresponding Green's relation, whereas the converse is not true. -
Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 3, March 2009, Pages 1429–1461 S 0002-9947(08)04712-0 Article electronically published on October 20, 2008 REPRESENTATION THEORY OF FINITE SEMIGROUPS, SEMIGROUP RADICALS AND FORMAL LANGUAGE THEORY JORGE ALMEIDA, STUART MARGOLIS, BENJAMIN STEINBERG, AND MIKHAIL VOLKOV Abstract. In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an ar- bitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semi- groups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; and Cern´ˇ y’s conjecture for an important class of automata. 1. Introduction For over 100 years, the theory of linear representations has played a fundamental role in studying finite groups, finite dimensional algebras and Lie algebras as well as other parts of algebra. By way of contrast, the theory of semigroup representations, which was intensively developed during the 1950s and 1960s in classic work such as Clifford [15], Munn [29, 30] and Ponizovsky (see [16, Chapter 5] for an account of this work, as well as [26, 56] for nicer treatments restricting to the case of finite semigroups) has found almost no applications in the theory of finite semigroups. It was pointed out by McAlister in his survey of 1971 [28] that the only paper applying representation theoretic results to finite semigroups was the paper [51] of Rhodes. -
Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language
Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language Andris Ambainis Kaspars Balodis Center for Quantum Computer Science, Faculty Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia of Computing, University of Latvia, Riga, Latvia J¯anis Iraids Kamil Khadiev Center for Quantum Computer Science, Faculty Kazan Federal University, Russia of Computing, University of Latvia, Riga, Latvia Vladislavs Kl,evickis Krišj¯anis Prusis¯ Center for Quantum Computer Science, Faculty Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia of Computing, University of Latvia, Riga, Latvia Yixin Shen Juris Smotrovs Université de Paris, CNRS, IRIF, Center for Quantum Computer Science, Faculty F-75006 Paris, France of Computing, University of Latvia, Riga, Latvia Jevg¯enijs Vihrovs Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia Abstract We study the quantum query complexity of two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most k. We call this the Dyckk,n problem. We prove a √ lower bound of Ω(ck n), showing that the complexity of this problem increases exponentially in k. Here n is the length of the word. When k is a constant, this is interesting as a representative √ example of star-free languages for which a surprising O˜( n) query quantum algorithm was recently constructed by Aaronson et al. [1]. Their proof does not give rise to a general algorithm. When k is √ 0.5k not a constant, Dyckk,n is not context-free.