The Chomsky Hierarchy
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The Structure of Index Sets and Reduced Indexed Grammars Informatique Théorique Et Applications, Tome 24, No 1 (1990), P
INFORMATIQUE THÉORIQUE ET APPLICATIONS R. PARCHMANN J. DUSKE The structure of index sets and reduced indexed grammars Informatique théorique et applications, tome 24, no 1 (1990), p. 89-104 <http://www.numdam.org/item?id=ITA_1990__24_1_89_0> © AFCET, 1990, tous droits réservés. L’accès aux archives de la revue « Informatique théorique et applications » im- plique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Informatique théorique et Applications/Theoretical Informaties and Applications (vol. 24, n° 1, 1990, p. 89 à 104) THE STRUCTURE OF INDEX SETS AND REDUCED INDEXED GRAMMARS (*) by R. PARCHMANN (*) and J. DUSKE (*) Communicated by J. BERSTEL Abstract. - The set of index words attached to a variable in dérivations of indexed grammars is investigated. Using the regularity of these sets it is possible to transform an mdexed grammar in a reducedfrom and to describe the structure ofleft sentential forms of an indexed grammar. Résumé. - On étudie Vensemble des mots d'index d'une variable dans les dérivations d'une grammaire d'index. La rationalité de ces ensembles peut être utilisée pour transformer une gram- maire d'index en forme réduite, et pour décrire la structure des mots apparaissant dans les dérivations gauches d'une grammaire d'index. 1. INTRODUCTION In this paper we will further investigate indexed grammars and languages introduced by Aho [1] as an extension of context-free grammars and lan- guages. -
The Mathematics of Syntactic Structure: Trees and Their Logics
Computational Linguistics Volume 26, Number 2 The Mathematics of Syntactic Structure: Trees and their Logics Hans-Peter Kolb and Uwe MÈonnich (editors) (Eberhard-Karls-UniversitÈat Tubingen) È Berlin: Mouton de Gruyter (Studies in Generative Grammar, edited by Jan Koster and Henk van Riemsdijk, volume 44), 1999, 347 pp; hardbound, ISBN 3-11-016273-3, $127.75, DM 198.00 Reviewed by Gerald Penn Bell Laboratories, Lucent Technologies Regular languages correspond exactly to those languages that can be recognized by a ®nite-state automaton. Add a stack to that automaton, and one obtains the context- free languages, and so on. Probably all of us learned at some point in our univer- sity studies about the Chomsky hierarchy of formal languages and the duality be- tween the form of their rewriting systems and the automata or computational re- sources necessary to recognize them. What is perhaps less well known is that yet another way of characterizing formal languages is provided by mathematical logic, namely in terms of the kind and number of variables, quanti®ers, and operators that a logical language requires in order to de®ne another formal language. This mode of characterization, which is subsumed by an area of research called descrip- tive complexity theory, is at once more declarative than an automaton or rewriting system, more ¯exible in terms of the primitive relations or concepts that it can pro- vide resort to, and less wedded to the tacit, not at all unproblematic, assumption that the right way to view any language, including natural language, is as a set of strings. -
Genome Grammars
Genome Grammars Genome Sequences and Formal Languages Andreas de Vries Version: June 17, 2011 Wir sind aus Staub und Fantasie Andreas Bourani, Nur in meinem Kopf (2011) Contents 1 Genetics 5 1.1 Cell physiology ............................. 5 1.2 Amino acids and proteins ....................... 6 1.2.1 Geometry of peptide bonds .................. 8 1.2.2 Protein structure ......................... 10 1.3 Nucleic acids ............................... 12 1.4 DNA replication ............................. 14 1.5 Flow of information for cell growth .................. 16 1.5.1 The genetic code ......................... 18 1.5.2 Open reading frames, coding regions, and genes ...... 19 2 Formal languages 23 2.1 The Chomsky hierarchy ......................... 25 2.2 Regular languages ............................ 28 2.2.1 Regular expressions ....................... 30 2.3 Context-free languages ......................... 31 2.3.1 Linear languages ........................ 33 2.4 Context-sensitive languages ...................... 34 2.4.1 Indexed languages ....................... 35 2.5 Languages and machines ........................ 38 3 Grammar of DNA Strings 42 3.1 Searl’s approach to DNA language .................. 42 3.2 Gene regulation and inadequacy of context-free grammars ..... 45 3.3 DNA splicing rule ............................ 46 A Mathematical Foundations 49 A.1 Notation ................................. 49 A.2 Sets .................................... 49 A.3 Maps ................................... 52 A.4 Algebra ................................. -
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. -
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. -
Hierarchy and Interpretability in Neural Models of Language Processing ILLC Dissertation Series DS-2020-06
Hierarchy and interpretability in neural models of language processing ILLC Dissertation Series DS-2020-06 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Science Park 107 1098 XG Amsterdam phone: +31-20-525 6051 e-mail: [email protected] homepage: http://www.illc.uva.nl/ The investigations were supported by the Netherlands Organization for Scientific Research (NWO), through a Gravitation Grant 024.001.006 to the Language in Interaction Consortium. Copyright © 2019 by Dieuwke Hupkes Publisher: Boekengilde Printed and bound by printenbind.nl ISBN: 90{6402{222-1 Hierarchy and interpretability in neural models of language processing Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. ir. K.I.J. Maex ten overstaan van een door het College voor Promoties ingestelde commissie, in het openbaar te verdedigen op woensdag 17 juni 2020, te 13 uur door Dieuwke Hupkes geboren te Wageningen Promotiecommisie Promotores: Dr. W.H. Zuidema Universiteit van Amsterdam Prof. Dr. L.W.M. Bod Universiteit van Amsterdam Overige leden: Dr. A. Bisazza Rijksuniversiteit Groningen Dr. R. Fern´andezRovira Universiteit van Amsterdam Prof. Dr. M. van Lambalgen Universiteit van Amsterdam Prof. Dr. P. Monaghan Lancaster University Prof. Dr. K. Sima'an Universiteit van Amsterdam Faculteit der Natuurwetenschappen, Wiskunde en Informatica to my parents Aukje and Michiel v Contents Acknowledgments xiii 1 Introduction 1 1.1 My original plan . .1 1.2 Neural networks as explanatory models . .2 1.2.1 Architectural similarity . .3 1.2.2 Behavioural similarity . -
The Complexity of Narrow Syntax : Minimalism, Representational
Erschienen in: Measuring Grammatical Complexity / Frederick J. Newmeyer ... (Hrsg.). - Oxford : Oxford University Press, 2014. - S. 128-147. - ISBN 978-0-19-968530-1 The complexity of narrow syntax: Minimalism, representational economy, and simplest Merge ANDREAS TROTZKE AND JAN-WOUTER ZWART 7.1 Introduction The issue of linguistic complexity has recently received much attention by linguists working within typological-functional frameworks (e.g. Miestamo, Sinnemäki, and Karlsson 2008; Sampson, Gil, and Trudgill 2009). In formal linguistics, the most prominent measure of linguistic complexity is the Chomsky hierarchy of formal languages (Chomsky 1956), including the distinction between a finite-state grammar (FSG) and more complicated types of phrase-structure grammar (PSG). This dis- tinction has played a crucial role in the recent biolinguistic literature on recursive complexity (Sauerland and Trotzke 2011). In this chapter, we consider the question of formal complexity measurement within linguistic minimalism (cf. also Biberauer et al., this volume, chapter 6; Progovac, this volume, chapter 5) and argue that our minimalist approach to complexity of derivations and representations shows simi- larities with that of alternative theoretical perspectives represented in this volume (Culicover, this volume, chapter 8; Jackendoff and Wittenberg, this volume, chapter 4). In particular, we agree that information structure properties should not be encoded in narrow syntax as features triggering movement, suggesting that the relevant information is established at the interfaces. Also, we argue for a minimalist model of grammar in which complexity arises out of the cyclic interaction of subderivations, a model we take to be compatible with Construction Grammar approaches. We claim that this model allows one to revisit the question of the formal complexity of a generative grammar, rephrasing it such that a different answer to the question of formal complexity is forthcoming depending on whether we consider the grammar as a whole, or just narrow syntax. -
Fundamental Methodological Issues of Syntactic Pattern Recognition
Pattern Anal Applic (2014) 17:465–480 DOI 10.1007/s10044-013-0322-1 ORIGINAL ARTICLE Fundamental methodological issues of syntactic pattern recognition Mariusz Flasin´ski • Janusz Jurek Received: 22 February 2012 / Accepted: 30 January 2013 / Published online: 9 March 2013 Ó The Author(s) 2013. This article is published with open access at Springerlink.com Abstract Fundamental open problems, which are fron- Syntactic pattern recognition prevails over ‘‘standard’’ tiers of syntactic pattern recognition are discussed in the pattern recognition approaches (probabilistic, discriminant paper. Methodological considerations on crucial issues in function-based, NN, etc.) when patterns considered can be areas of string and graph grammar-based syntactic methods characterized better with structural features than vectors of are made. As a result, recommendations concerning an features. What is more, using this approach not only can we enhancement of context-free grammars as well as con- make a classification (in a sense of ascribing a pattern to a structing parsable and inducible classes of graph grammars pre-defined category), but also a (structural) interpretation are formulated. of an unknown pattern. Therefore, for structurally-oriented recognition problems such as: character recognition, speech Keyword Syntactic pattern recognition Á recognition, scene analysis, chemical and biological struc- Formal language Á Graph grammar tures analysis, texture analysis, fingerprint recognition, geophysics, a syntactic approach has been applied suc- cessfully since its beginning in the early 1960s for the next 1 Introduction two decades. A rapid development of syntactic methods has slowed down since 1990s and the experts in this area (see Representing a pattern as a structure of the form of string, e.g. -
What Was Wrong with the Chomsky Hierarchy?
What Was Wrong with the Chomsky Hierarchy? Makoto Kanazawa Hosei University Chomsky Hierarchy of Formal Languages recursively enumerable context- sensitive context- free regular Enduring impact of Chomsky’s work on theoretical computer science. 15 pages devoted to the Chomsky hierarchy. Hopcroft and Ullman 1979 No mention of the Chomsky hierarchy. 450 INDEX The same with the latest edition of Carmichael, R. D., 444 CNF-formula, 302 Cartesian product, 6, 46 Co-Turing-recognizableHopcroft, language, 209 Motwani, and Ullman (2006). CD-ROM, 349 Cobham, Alan, 444 Certificate, 293 Coefficient, 183 CFG, see Context-free grammar Coin-flip step, 396 CFL, see Context-free language Complement operation, 4 Chaitin, Gregory J., 264 Completed rule, 140 Chandra, Ashok, 444 Complexity class Characteristic sequence, 206 ASPACE(f(n)),410 Checkers, game of, 348 ATIME(t(n)),410 Chernoff bound, 398 BPP,397 Chess, game of, 348 coNL,354 Chinese remainder theorem, 401 coNP,297 Chomsky normal form, 108–111, 158, EXPSPACE,368 198, 291 EXPTIME,336 Chomsky, Noam, 444 IP,417 Church, Alonzo, 3, 183, 255 L,349 Church–Turing thesis, 183–184, 281 NC,430 CIRCUIT-SAT,386 NL,349 Circuit-satisfiability problem, 386 NP,292–298 CIRCUIT-VALUE,432 NPSPACE,336 Circular definition, 65 NSPACE(f(n)),332 Clause, 302 NTIME(f(n)),295 Clique, 28, 296 P,284–291,297–298 CLIQUE,296 PH,414 Sipser 2013Closed under, 45 PSPACE,336 Closure under complementation RP,403 context-free languages, non-, 154 SPACE(f(n)),332 deterministic context-free TIME(f(n)),279 languages, 133 ZPP,439 P,322 Complexity -
Iterated Stack Automata and Complexity Classes
INFORMATION AND COMPUTATION 95, 2 1-75 ( t 99 1) Iterated Stack Automata and Complexity Classes JOOST ENCELFRIET Department of Computer Science, Leiden University. P.O. Box 9.512, 2300 RA Leiden, The Netherlands An iterated pushdown is a pushdown of pushdowns of . of pushdowns. An iterated exponential function is 2 to the 2 to the to the 2 to some polynomial. The main result presented here is that the nondeterministic 2-way and multi-head iterated pushdown automata characterize the deterministic iterated exponential time complexity classes. This is proved by investigating both nondeterministic and alternating auxiliary iterated pushdown automata, for which similar characteriza- tion results are given. In particular it is shown that alternation corresponds to one more iteration of pushdowns. These results are applied to the l-way iterated pushdown automata: (1) they form a proper hierarchy with respect to the number of iterations, and (2) their emptiness problem is complete in deterministic iterated exponential time. Similar results are given for iterated stack (checking stack, non- erasing stack, nested stack, checking stack-pushdown) automata. ? 1991 Academic Press. Inc INTRODUCTION It is well known that several types of 2-way and multi-head pushdown automata and stack automata have the same power as certain time or space bounded Turing machines; see, e.g., Chapter 14 of (Hopcroft and Ullman, 1979), or Sections 13 and 20.2 of (Wagner and Wechsung, 1986). For the deterministic and nondeterministic case such characterizations were given by Fischer (1969) for checking stack automata, by Hopcroft and Ullman (1967b) for nonerasing stack automata, by Cook (1971) for auxiliary pushdown and 2-way stack automata, by Ibarra (1971) for auxiliary (nonerasing and erasing) stack automata, by Beeri (1975) for 2-way and auxiliary nested stack automata, and by van Leeuwen (1976) for auxiliary checking stack-pushdown automata. -
Lecture 5 Mildly Context-Sensitive Languages
Lecture 5 Mildly Context-Sensitive Languages Last modified 2016/07/08 Multiple context-free grammars In the previous lecture, we proved the equivalence between TAG and LIG. In fact, there is yet another grammar formalism, namely the head grammar, that is equivalent to these two formalisms. A head grammar is a special kind of multiple context-free grammar (MCFG), so we introduce the latter formalism first. The MCFG is a natural generalization of the “bottom-up” view of the CFG. The standard, “top-down” view of the CFG takes a rule A X ::: X ! 1 n as a permission to rewrite A into X1 ::: Xn. In contrast, the bottom-up view of the CFG interprets the same rule not as a rewriting instruction, but as an implication, which says: A ∗ x ::: x if X ∗ x X ∗ x : ) 1 n 1 ) 1 ^ · · · ^ n ) n In fact, to define the language L(G) = w Σ S w of a CFG G, there is no f 2 ∗ j )∗ g need to define the derivation relation which holds between strings of terminals )∗ and nonterminals. All you need is an inductive definition of the subrelation of this relation that holds between single nonterminals and strings of terminals: ∗ (N Σ ∗): ) \ × To express that nonterminal A and terminal string x stand in this relation (i.e., A x), we may write A(x), treating nonterminal A as a unary predicate on )∗ 5–1 terminal strings. Then the bottom-up interpretation of the CFG rule can be written in the form of a Horn clause: A(x ::: x ) X (x );:::; X (x ): 1 n 1 1 n n A context-free grammar now becomes a Horn clause program consisting of rules of the above form.