Some Context-Free Languages and Their Associated Dynamical Systems
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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]. -
Probabilistic Grammars and Their Applications This Discretion to Pursue Political and Economic Ends
Probabilistic Grammars and their Applications this discretion to pursue political and economic ends. and the Law; Monetary Policy; Multinational Cor- Most experiences, however, suggest the limited power porations; Regulation, Economic Theory of; Regu- of privatization in changing the modes of governance lation: Working Conditions; Smith, Adam (1723–90); which are prevalent in each country’s large private Socialism; Venture Capital companies. Further, those countries which have chosen the mass (voucher) privatization route have done so largely out of necessity and face ongoing efficiency problems as a result. In the UK, a country Bibliography whose privatization policies are often referred to as a Armijo L 1998 Balance sheet or ballot box? Incentives to benchmark, ‘control [of privatized companies] is not privatize in emerging democracies. In: Oxhorn P, Starr P exerted in the forms of threats of take-over or (eds.) The Problematic Relationship between Economic and bankruptcy; nor has it for the most part come from Political Liberalization. Lynne Rienner, Boulder, CO Bishop M, Kay J, Mayer C 1994 Introduction: privatization in direct investor intervention’ (Bishop et al. 1994, p. 11). After the steep rise experienced in the immediate performance. In: Bishop M, Kay J, Mayer C (eds.) Pri ati- zation and Economic Performance. Oxford University Press, aftermath of privatizations, the slow but constant Oxford, UK decline in the number of small shareholders highlights Boubakri N, Cosset J-C 1998 The financial and operating the difficulties in sustaining people’s capitalism in the performance of newly privatized firms: evidence from develop- longer run. In Italy, for example, privatization was ing countries. -
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. -
Grammars and Normal Forms
Grammars and Normal Forms Read K & S 3.7. Recognizing Context-Free Languages Two notions of recognition: (1) Say yes or no, just like with FSMs (2) Say yes or no, AND if yes, describe the structure a + b * c Now it's time to worry about extracting structure (and doing so efficiently). Optimizing Context-Free Languages For regular languages: Computation = operation of FSMs. So, Optimization = Operations on FSMs: Conversion to deterministic FSMs Minimization of FSMs For context-free languages: Computation = operation of parsers. So, Optimization = Operations on languages Operations on grammars Parser design Before We Start: Operations on Grammars There are lots of ways to transform grammars so that they are more useful for a particular purpose. the basic idea: 1. Apply transformation 1 to G to get of undesirable property 1. Show that the language generated by G is unchanged. 2. Apply transformation 2 to G to get rid of undesirable property 2. Show that the language generated by G is unchanged AND that undesirable property 1 has not been reintroduced. 3. Continue until the grammar is in the desired form. Examples: • Getting rid of ε rules (nullable rules) • Getting rid of sets of rules with a common initial terminal, e.g., • A → aB, A → aC become A → aD, D → B | C • Conversion to normal forms Lecture Notes 16 Grammars and Normal Forms 1 Normal Forms If you want to design algorithms, it is often useful to have a limited number of input forms that you have to deal with. Normal forms are designed to do just that. -
CS351 Pumping Lemma, Chomsky Normal Form Chomsky Normal
CS351 Pumping Lemma, Chomsky Normal Form Chomsky Normal Form When working with a context-free grammar a simple and useful form is called the Chomsky Normal Form (CNF). A CFG in CNF is one where every rule is of the form: A ! BC A ! a Where a is any terminal and A,B, and C are any variables, except that B and C may not be the start variable. Note that we have two and only two variables on the right hand side of the rule, with the exception that the rule S!ε is permitted where S is the start variable. Theorem: Any context free language may be generated by a context-free grammar in Chomsky normal form. To show how to make this conversion, we will need to do three things: 1. Eliminate all ε rules of the form A!ε 2. Eliminate all unit rules of the form A!B 3. Convert remaining rules into rules of the form A!BC Proof: 1. First add a new start symbol S0 and the rule S0 ! S, where S was the original start symbol. This guarantees that the start symbol doesn’t occur on the right hand side of a rule. 2. Remove all ε rules. Remove a rule A!ε where A is not the start symbol For each occurrence of A on the right-hand side of a rule, add a new rule with that occurrence of A deleted. Ex: R!uAv becomes R!uv This must be done for each occurrence of A, so the rule: R!uAvAw becomes R! uvAw and R! uAvw and R!uvw This step must be repeated until all ε rules are removed, not including the start. -
LING83600: Context-Free Grammars
LING83600: Context-free grammars Kyle Gorman 1 Introduction Context-free grammars (or CFGs) are a formalization of what linguists call “phrase structure gram- mars”. The formalism is originally due to Chomsky (1956) though it has been independently discovered several times. The insight underlying CFGs is the notion of constituency. Most linguists believe that human languages cannot be even weakly described by CFGs (e.g., Schieber 1985). And in formal work, context-free grammars have long been superseded by “trans- formational” approaches which introduce movement operations (in some camps), or by “general- ized phrase structure” approaches which add more complex phrase-building operations (in other camps). However, unlike more expressive formalisms, there exist relatively-efficient cubic-time parsing algorithms for CFGs. Nearly all programming languages are described by—and parsed using—a CFG,1 and CFG grammars of human languages are widely used as a model of syntactic structure in natural language processing and understanding tasks. Bibliographic note This handout covers the formal definition of context-free grammars; the Cocke-Younger-Kasami (CYK) parsing algorithm will be covered in a separate assignment. The definitions here are loosely based on chapter 11 of Jurafsky & Martin, who in turn draw from Hopcroft and Ullman 1979. 2 Definitions 2.1 Context-free grammars A context-free grammar G is a four-tuple N; Σ; R; S such that: • N is a set of non-terminal symbols, corresponding to phrase markers in a syntactic tree. • Σ is a set of terminal symbols, corresponding to words (i.e., X0s) in a syntactic tree. • R is a set of production rules. -
The Logic of Categorial Grammars: Lecture Notes Christian Retoré
The Logic of Categorial Grammars: Lecture Notes Christian Retoré To cite this version: Christian Retoré. The Logic of Categorial Grammars: Lecture Notes. RR-5703, INRIA. 2005, pp.105. inria-00070313 HAL Id: inria-00070313 https://hal.inria.fr/inria-00070313 Submitted on 19 May 2006 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. INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE The Logic of Categorial Grammars Lecture Notes Christian Retoré N° 5703 Septembre 2005 Thème SYM apport de recherche ISRN INRIA/RR--5703--FR+ENG ISSN 0249-6399 The Logic of Categorial Grammars Lecture Notes Christian Retoré ∗ Thème SYM — Systèmes symboliques Projet Signes Rapport de recherche n° 5703 — Septembre 2005 —105 pages Abstract: These lecture notes present categorial grammars as deductive systems, and include detailed proofs of their main properties. The first chapter deals with Ajdukiewicz and Bar-Hillel categorial grammars (AB grammars), their relation to context-free grammars and their learning algorithms. The second chapter is devoted to the Lambek calculus as a deductive system; the weak equivalence with context free grammars is proved; we also define the mapping from a syntactic analysis to a higher-order logical formula, which describes the seman- tics of the parsed sentence. -
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. -
Appendix A: Hierarchy of Grammar Formalisms
Appendix A: Hierarchy of Grammar Formalisms The following figure recalls the language hierarchy that we have developed in the course of the book. '' $$ '' $$ '' $$ ' ' $ $ CFG, 1-MCFG, PDA n n L1 = {a b | n ≥ 0} & % TAG, LIG, CCG, tree-local MCTAG, EPDA n n n ∗ L2 = {a b c | n ≥ 0}, L3 = {ww | w ∈{a, b} } & % 2-LCFRS, 2-MCFG, simple 2-RCG & % 3-LCFRS, 3-MCFG, simple 3-RCG n n n n n ∗ L4 = {a1 a2 a3 a4 a5 | n ≥ 0}, L5 = {www | w ∈{a, b} } & % ... LCFRS, MCFG, simple RCG, MG, set-local MCTAG, finite-copying LFG & % Thread Automata (TA) & % PMCFG 2n L6 = {a | n ≥ 0} & % RCG, simple LMG (= PTIME) & % mildly context-sensitive L. Kallmeyer, Parsing Beyond Context-Free Grammars, Cognitive Technologies, DOI 10.1007/978-3-642-14846-0, c Springer-Verlag Berlin Heidelberg 2010 216 Appendix A For each class the different formalisms and automata that generate/accept exactly the string languages contained in this class are listed. Furthermore, examples of typical languages for this class are added, i.e., of languages that belong to this class while not belonging to the next smaller class in our hier- archy. The inclusions are all proper inclusions, except for the relation between LCFRS and Thread Automata (TA). Here, we do not know whether the in- clusion is a proper one. It is possible that both devices yield the same class of languages. Appendix B: List of Acronyms The following table lists all acronyms that occur in this book. (2,2)-BRCG Binary bottom-up non-erasing RCG with at most two vari- ables per left-hand side argument 2-SA Two-Stack Automaton -
Using Contextual Representations to Efficiently Learn Context-Free
JournalofMachineLearningResearch11(2010)2707-2744 Submitted 12/09; Revised 9/10; Published 10/10 Using Contextual Representations to Efficiently Learn Context-Free Languages Alexander Clark [email protected] Department of Computer Science, Royal Holloway, University of London Egham, Surrey, TW20 0EX United Kingdom Remi´ Eyraud [email protected] Amaury Habrard [email protected] Laboratoire d’Informatique Fondamentale de Marseille CNRS UMR 6166, Aix-Marseille Universite´ 39, rue Fred´ eric´ Joliot-Curie, 13453 Marseille cedex 13, France Editor: Fernando Pereira Abstract We present a polynomial update time algorithm for the inductive inference of a large class of context-free languages using the paradigm of positive data and a membership oracle. We achieve this result by moving to a novel representation, called Contextual Binary Feature Grammars (CBFGs), which are capable of representing richly structured context-free languages as well as some context sensitive languages. These representations explicitly model the lattice structure of the distribution of a set of substrings and can be inferred using a generalisation of distributional learning. This formalism is an attempt to bridge the gap between simple learnable classes and the sorts of highly expressive representations necessary for linguistic representation: it allows the learnability of a large class of context-free languages, that includes all regular languages and those context-free languages that satisfy two simple constraints. The formalism and the algorithm seem well suited to natural language and in particular to the modeling of first language acquisition. Pre- liminary experimental results confirm the effectiveness of this approach. Keywords: grammatical inference, context-free language, positive data only, membership queries 1. -
1 from N-Grams to Trees in Lindenmayer Systems Diego Gabriel Krivochen [email protected] Abstract: in This Paper We
From n-grams to trees in Lindenmayer systems Diego Gabriel Krivochen [email protected] Abstract: In this paper we present two approaches to Lindenmayer systems: the rule-based (or ‘generative’) approach, which focuses on L-systems as Thue rewriting systems and a constraint-based (or ‘model- theoretic’) approach, in which rules are abandoned in favour of conditions over allowable expressions in the language (Pullum, 2019). We will argue that it is possible, for at least a subset of Lsystems and the languages they generate, to map string admissibility conditions (the 'Three Laws') to local tree admissibility conditions (cf. Rogers, 1997). This is equivalent to defining a model for those languages. We will work out how to construct structure assuming only superficial constraints on expressions, and define a set of constraints that well-formed expressions of specific L-languages must satisfy. We will see that L-systems that other methods distinguish turn out to satisfy the same model. 1. Introduction The most popular approach to structure building in contemporary generative grammar is undoubtedly based on set theory: Merge creates sets of syntactic objects (Chomsky, 1995, 2020; Epstein et al., 2015; Collins, 2017), such that Merge(A, B) results in the set {A, B}; in some versions where A always projects a phrasal label, the result is {A, {A, B}}. However, the primacy of set-theory in generative grammar has not been uncontested: syntactic representations can also be expressed in terms of graphs rather than sets, with varying results in terms of empirical adequacy and theoretical consistency. Graphs are sets of nodes and edges; more specifically, a graph is a pair G = (V, E), where V is a set of vertices (also called nodes) and E is a set of edges; v ∈ V is a vertex, and e ∈ E is an edge. -
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.