A helpful result for proving inherent ambiguity

Ogden, William F.

doi:10.1007/bf01694004

Cited by 134 publications

(55 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The Ogden Lemma [7] is a well-know generalization to the case of pushdown automata of the pumping lemma on finite automata. It can be expressed on paths of the graph as follows.…”

Section: Proofsmentioning

confidence: 99%

Zigzags in Turing Machines

Gajardo

Guillon

2010

Computer Science – Theory and Applications

View full text Add to dashboard Cite

Abstract. We study one-head machines through symbolic and topological dynamics. In particular, a subshift is associated to the system, and we are interested in its complexity in terms of realtime recognition. We emphasize the class of one-head machines whose subshift can be recognized by a deterministic pushdown automaton. We prove that this class corresponds to particular restrictions on the head movement, and to equicontinuity in associated dynamical systems.Keywords: Turing machines, discrete dynamical systems, subshifts, formal languages.We study the dynamics of a system consisting in a finite automaton (the head) that can write and move over an infinite tape, like a Turing machine. We use the approach of symbolic and topological dynamics. Our interest is to understand its properties and limitations, and how dynamical properties are related to computational complexity. This approach was initiated by Kůrka in [1] with two different topologies: one focused on the machine head, and the other on the tape. The first approach was further developed in [2,3]. More recently, in [4,5], a third kind of dynamical system was associated to Turing machines, taking advantage of the following specificity: changes happen only in the head position whilst the rest of the configuration remains unaltered. The whole evolution can therefore be described by the sequence of states taken by the head and the symbols that it reads. This observation actually yields a factor map between Kůrka's first dynamical system and a one-sided subshift.In [4], it has been proved that machines with a sofic subshift correspond to machines whose head makes only bounded cycles. We prove here a similar characterization of machines with a shift that can be recognized by a deterministic pushdown automaton. Moreover, we establish links between these two properties and equicontinuity in all three spaces.In the first section, we recall the definitions and fundamental results. The second section is devoted to defining the different dynamical systems associated to one-head machines, and to stating basic results about equicontinuity within ⋆

show abstract

“…The Ogden Lemma [7] is a well-know generalization to the case of pushdown automata of the pumping lemma on finite automata. It can be expressed on paths of the graph as follows.…”

Section: Proofsmentioning

confidence: 99%

Zigzags in Turing Machines

Gajardo

Guillon

2010

Computer Science – Theory and Applications

View full text Add to dashboard Cite

show abstract

“…is linear and inherently ambiguous (using Ogden's iteration lemma [17] (see also Chapter 6 in [9]) it can be deduced that one always has two different derivations for the words of the form a n b n c n ). Thus there exist inherently ambiguous linear languages.…”

Section: Tullio Ceccherini-silbersteinmentioning

confidence: 99%

“…Note that there are context-free languages that cannot be generated by unambiguous grammars: these are called inherently ambiguous languages. In our setting, the languageis linear and inherently ambiguous (using Ogden's iteration lemma [17] (see also Chapter 6 in [9]) it can be deduced that one always has two different derivations for the words of the form a n b n c n ). Thus there exist inherently ambiguous linear languages.…”

mentioning

confidence: 99%

Growth and ergodicity of context-free languages II: The linear case

Ceccherini‐Silberstein

2006

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. A language L over a finite alphabet Σ is called growth-sensitive if forbidding any non-empty set F of subwords yields a sub-language L F whose exponential growth rate is smaller than that of L. Say that a context-free grammar (and associated language) is ergodic if its dependency di-graph is strongly connected. It is known that regular and unambiguous non-linear context-free languages which are ergodic are growth-sensitive. In this note it is shown that ergodic unambiguous linear languags are growth-sensitive, closing the gap that remained open. IntroductionLet L be a language over the alphabet Σ, that is, a subset of the free monoid Σ * of all finite words over Σ. We write ε for the empty word and Σ + = Σ * \ {ε}. For a word w ∈ Σ * , its length (number of letters) is denoted by |w|. The growth rate of L ⊆ Σ * is the numberAlso recall that L has exponential growth if γ(L) > 1 and it has sub-exponential growth otherwise, namely, if γ(L) = 1. A language L over Σ is growth-sensitive iffor any non-empty F ⊂ Σ * consisting of subwords of elements of L, where L F = {w ∈ L : no v ∈ F is a subword of w}.A context-free grammar is a quadruple C = (V, Σ, P, S), where V is a finite set of variables, disjoint from the finite alphabet Σ, the variable S is the start symbol, and P ⊂ V × (V ∪ Σ) * is a finite set of production rules. We write T u or (T u) ∈ P if (T, u) ∈ P. For v, w ∈ (V ∪ Σ) * , we write v =⇒ w if v = v 1 T v 2 and w = v 1 uv 2 , whereReceived by the editors November 4, 2004.2000 Mathematics Subject Classification. Primary 68Q45; Secondary 05A16, 05C20, 68Q42. Key words and phrases. Context-free grammar, linear language, dependency di-graph, ergodicity, ambiguity, growth, higher block languages, automaton, bilateral automaton. 605License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use TULLIO CECCHERINI-SILBERSTEINA context-free language is a language generated by a context-free grammar. If in addition the production rules T u are such that u contains at most one variable in V, then the grammar and the corresponding language are termed linear. In particular, if u ∈ Σ * ∪ Σ * V (resp. Σ * ∪ VΣ * ), then the language is left (resp. right) linear. In both cases, language and grammar are also called regular.A general context-free grammar C is called unambiguous if for any w ∈ L(C) there is a unique rightmost derivation S * =⇒ w. A context-free language is unambiguous if it is generated by some unambiguous grammar. Note that there are context-free languages that cannot be generated by unambiguous grammars: these are called inherently ambiguous languages. In our setting, the languageis linear and inherently ambiguous (using Ogden's iteration lemma [17] (see also Chapter 6 in [9]) it can be deduced that one always has two different derivations for the words of the form a n b n c n ). Thus there exist inherently ambiguous linear languages.We shall always assume to have a reduced grammar C, that is, each variable is used in some rightmost derivation of a wor...

show abstract

“…En effet, la classe des bilangages réguliers est stable par transcription. Ce résultat n'est pas surprenant, car le principal outil dont on dispose pour démontrer sur des exemples l'ambiguïté inhérente de langage algébrique est le lemme d'Ogden [6] qui précisément permet de construire pour un mot du langage plusieurs arbres syntaxiques associés à ce mot et de structures différentes.…”

Section: Application Aux Langages Algébriques Ayant Une Ambiguïté Inhunclassified

Grammaires parenthésées et bilangages réguliers

Marchand¹

1980

RAIRO. Inform. théor.

View full text Add to dashboard Cite

A helpful result for proving inherent ambiguity

Cited by 134 publications

References 1 publication

Zigzags in Turing Machines

Zigzags in Turing Machines

Growth and ergodicity of context-free languages II: The linear case

Grammaires parenthésées et bilangages réguliers

Contact Info

Product

Resources

About