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Hierarchy of LA Grammar

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Foundations of Computational Linguistics

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Abstract

In this chapter, different kinds of LA Grammar are defined via natural restrictions of its rule system. Then these kinds of grammar are characterized in terms of their generative capacity and computational complexity.

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Notes

  1. 1.

    In automata theory, the recursive languages are defined as those of which each expression may be recognized by at least one Turing machine in finitely many steps (‘halts on all inputs’, Hopcroft and Ullman 1979, p. 151). The PS Grammar hierarchy does not provide a formal characterization of the recursive languages – in contrast to the regular, context-free, context-sensitive, and recursively enumerable languages, which have both an automata-theoretic and a PS Grammar definition.

  2. 2.

    CoL, Theorems 1 and 2, pp. 134f.

  3. 3.

    This proof was provided by Dana Scott.

  4. 4.

    I.e., if ϱ maps the category (X seg c ), representing the surface, into the category (1).

  5. 5.

    CoL, Theorem 5, p. 142.

  6. 6.

    Hopcroft and Ullman (1979), Theorem 9.8, p. 228. CS stands for the class of context-sensitive languages and REC for the class of recursive languages.

  7. 7.

    This finite constant will vary between different grammars.

  8. 8.

    CoL, Theorem 4, p. 138.

    A context-free C LAG (cf LAG for short) consists only of rules with the form

    •  r i : (aX) (b)⇒(αX) rp i , with a,b∈C and α∈C+

    This restriction on the ss- and ss′-categories corresponds to the working of a PDA which may write not just one symbol but a sequence of symbols into the stack (Hopcroft and Ullman 1979, Chap. 5.2). The following two assertions have to be proven:

    1. For each PDA M, a cf LAG σ may be constructed such that L(M)=L(σ).

    This implies \(\mathrm{CF} \subseteq \mathcal{C}_{\mathit{cf}}\).

    2. For each cf LAG σ, a PDA M may be constructed such that L(σ)=L(M).

    This implies \(\mathcal{C}_{\mathit{cf}} \subseteq \mathrm{CF}\).

    In showing 1 and 2, one has to take into consideration that a cf LAG uses rules while a PDA uses states, which are not entirely equivalent. Thus, it is necessary to provide a constructive procedure to convert states into rules and rules into states – a cumbersome, but not particularly difficult task.

    Note with respect to 1 that ε-moves are forbidden in cf LAGs, but not in PDAs (Hopcroft and Ullman 1979, p. 24). However, there exists for each context-free language a PDA working without ε-moves (Harrison 1978, Theorem 5.5.1) and for an ε-free PDA a cf LAG may be constructed.

  9. 9.

    CoL, Theorem 3, p. 138.

  10. 10.

    Another possibility for modifying the generative capacity of LA Grammar consists – at least theoretically – in changing clause 4 of the algebraic definition 10.2.1. For example, if the categorial operations had been defined as arbitrary partial recursive functions, then LA Grammar would generate exactly the recursively enumerable languages. This would amount to an increase of generative capacity, making LA Grammar weakly equivalent to PS Grammar. Alternatively, if the categorial operations had been defined as arbitrary primitive recursive functions, then it could be shown by analogy to Theorem 2 that the resulting kind of LA Grammar generates exactly the primitive recursive languages. This would amount to a decrease in generative capacity compared to the standard definition.

    Using alternative definitions of clause 4 in 10.2.1 is not a good method to obtain different subclasses of LA Grammar, however. First, alternating the categorial operations between partial recursive, total recursive, and primitive recursive functions is a very crude method. Secondly, the resulting language classes are much too big to be of practical interest: even though the primitive recursive functions are a proper subset of the total recursive functions, the primitive recursive functions properly contain the whole class of context-sensitive languages. Third, the categorial operations have been defined as total recursive functions in 10.2.1 for good reason, ensuring that basic LA Grammar has the highest generative capacity possible while still being decidable.

  11. 11.

    For reasons of simplicity, only syntactic causes of ambiguity are considered here. Lexical ambiguities arising from multiple analyses of words have so far been largely ignored in formal language theory, but are unavoidable in the LA Grammatical analysis of natural language. The possible impact of lexical ambiguity on complexity is discussed in CoL, pp. 157f. and 248f.

  12. 12.

    An LA Grammar is lexically ambiguous if its lexicon contains at least two analyzed words with the same surface. A nonlinguistic example of a lexical ambiguity is propositional calculus , e.g., (x∨y∨z)∧(…)…, whereby the propositional variables x, y, z, etc., may be analyzed lexically as ambiguous between [x (1)] and [x (0)], [y (1)] and [y (0)], etc. Thereby [x (1)] is taken to represent a true proposition x, and [x (0)] a false one.

    While syntactic ambiguities arise in the rule-based derivation of more than one new sentence start, lexical ambiguities are caused by additional readings of the next word. Syntactic and lexical ambiguities can also occur at the same time in an LA Grammar. Furthermore, syntactic ambiguities can be reformulated into lexical ambiguities and vice versa (TCS, pp. 303f.)

  13. 13.

    Abstract automata consist of such components as a read/write-head, a write-protected input tape, a certain number of working tapes, the movement of the read/write-head on a certain tape from one cell to another, the reading or deleting of the content in a cell, etc. Classic examples of abstract automata are Turing machines (TMs), linearly bounded automata (LBAs), pushdown automata (PDAs), and finite state automata (FSAa). There is a multitude of additional abstract automata, each defined for the purpose of proving various special complexity, equivalence, and computability properties.

  14. 14.

    In PS Grammar, the notions deterministic and nondeterministic have no counterparts.

  15. 15.

    Rabin and Scott (1959).

  16. 16.

    Hopcroft and Ullman (1979), p. 164, Theorem 7.3.

  17. 17.

    Hopcroft and Ullman (1979), p. 113.

  18. 18.

    Strictly speaking, unambiguous C LAGs have a complexity even better (i.e., lower) than deterministic linear time, namely real time.

  19. 19.

    A C1 LAG for a k b k c m d m∪a k b m c m d k is defined in 11.5.2, for L square and \(\mathsf{L}^{\mathsf{k}}_{\mathit{hast}}\) in Stubert (1993), pp. 16 and 12, for a k b k c k d k e k in CoL, p. 233, for a k b m c k⋅m in TCS, p. 296, and for \(\mathsf{a}^{\mathsf{2}^{\mathsf{i}}}\) in 11.5.1. A C1 LAG for \(\mathsf{a}^{\mathsf{i!}}\) is sketched in TCS, p. 296, footnote 13.

  20. 20.

    Hopcroft and Ullman (1979) present the canonical context-sensitive PS Grammar of \(a^{\mathsf{2}^{\mathsf{i}}}\) on p. 224, and a version as an unrestricted PS Grammar on p. 220.

  21. 21.

    Cf. Hopcroft and Ullman (1979), pp. 99–103.

  22. 22.

    An explicit derivation is given in CoL, pp. 154f.

  23. 23.

    A C2 LAG for WW R is defined in 11.5.4, for \(\mathsf{L}^{\infty}_{\mathit{hast}}\) in Stubert 1993, p. 16, for WW in 11.5.6, for WWW in CoL, p. 215, for W k≥3 in CoL, p. 216, and for \(\mathsf{W}_{1}\mathsf{W}_{2}\mathsf{W}_{1}^{\mathsf{R}}\mathsf{W}_{2}^{\mathsf{R}}\) in 11.5.7.

  24. 24.

    A C3 LAG for L no is defined in 12.3.3, for HCFL in Stubert (1993), p. 16, for SubsetSum in 11.5.8, and for SAT in TCS, footnote 19.

References

  • Earley, J. (1970) “An Efficient Context-Free Parsing Algorithm,” Communications of the ACM 2:94, reprinted in B. Grosz, K. Sparck Jones, and B.L. Webber (eds.), 1986

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  • Harrison, M. (1978) Introduction to Formal Language Theory, Reading: Addison-Wesley

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  • Hopcroft, J.E., and J.D. Ullman (1979) Introduction to Automata Theory, Languages, and Computation, Reading: Addison-Wesley

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  • Rabin, M.O., and D. Scott. (1959) “Finite Automata and Their Decision Problems,” IBM Journal of Research 3.2:115–125

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  • Stubert, B. (1993) “Einordnung der Familie der C-Sprachen zwischen die kontextfreien und die kontextsensitiven Sprachen,” CLUE-betreute Studienarbeit der Informatik, Friedrich Alexander Universität Erlangen Nürnberg

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Authors and Affiliations

  1. Abteilung für Computerlinguistik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany

    Roland Hausser

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Exercises

Exercises

Section 11.1

  1. 1.

    Explain the notions of total recursive function and partial recursive function.

  2. 2.

    What is the formal characterization of the class of recursive languages in the hierarchy of PS Grammar?

  3. 3.

    What is the generative capacity of LA Grammar in its basic unrestricted form?

  4. 4.

    Explain the proofs in 11.1.2 and 11.1.3.

  5. 5.

    Describe the difference in generative capacity of unrestricted PS Grammar and unrestricted LA Grammar.

Section 11.2

  1. 1.

    What are possible structural restrictions on LA Grammar?

  2. 2.

    Explain the grammar classes of A LAGs, B LAGs, and C LAGs.

  3. 3.

    What is the difference between a constant and a nonconstant categorial operation?

  4. 4.

    Describe the reconstruction of the PS Grammar hierarchy in LA Grammar.

  5. 5.

    Is there a class of LA Grammar that generates the recursively enumerable languages?

  6. 6.

    Is there a class of PS Grammar that generates the recursive languages?

  7. 7.

    Is there a class of PS Grammar that generates the C languages?

Section 11.3

  1. 1.

    Explain the notions of ±global ambiguities. What is their relation to the notions of ±deterministic derivations in automata theory and the notion of ambiguity in PS Grammar.

  2. 2.

    What determines the number of rule applications in an LA derivation?

  3. 3.

    Describe three different relations between the input conditions of LA Grammar rules.

  4. 4.

    What is the definition of ambiguous and unambiguous LA Grammars?

  5. 5.

    Explain the notions of ±recursive ambiguity in LA Grammar.

  6. 6.

    Define the rule packages of a −recursively ambiguous LA Grammar with seven rules such that it is maximally ambiguous. How many readings are derived by this grammar after 4, 5, 6, 7, 8, and 9 combination steps, respectively?

Section 11.4

  1. 1.

    Explain the complexity parameters of LA Grammar.

  2. 2.

    How is the elementary operation of C LAGs defined?

  3. 3.

    Compare Earley’s primitive operation for computing the complexity of his algorithm for context-free PS Grammars with that of the C LAGs.

  4. 4.

    How does Earley compute the amount and number parameters of context-free PS Grammar?

  5. 5.

    Name four notions of complexity in automata theory and explain them.

  6. 6.

    Explain the relation between the deterministic and nondeterministic versions of abstract automata.

  7. 7.

    Explain the application of the notions DTIME and NTIME to the C LAGs. What is the alternative?

Section 11.5

  1. 1.

    Describe the subhierarchy of the C1, C2, and C3 LAGs. What is the connection between ambiguity and complexity?

  2. 2.

    Define a C1 LAG for a k b k c m d m∪a k b m c k d m, and explain how this grammar works. Is it ambiguous? What is its complexity?

  3. 3.

    Explain the single return principle.

  4. 4.

    Compare the handling of WW and WW R in PS Grammar and LA Grammar.

  5. 5.

    Describe the well-formed expressions of SubsetSum, and explain why this language is inherently complex.

  6. 6.

    Explain the hierarchy of LA Grammar.

  7. 7.

    Describe the relation between PS Grammar and the nativist theory of language, on the one hand, and between LA Grammar and the Slim theory, on the other. Would it be possible to use a PS Grammar as the syntactic component of the Slim theory? Would it be possible to use an LA Grammar as the syntactic component of the nativist theory?

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Hausser, R. (2014). Hierarchy of LA Grammar. In: Foundations of Computational Linguistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41431-2_11

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  • DOI: https://doi.org/10.1007/978-3-642-41431-2_11

  • Publisher Name: Springer, Berlin, Heidelberg

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  • Online ISBN: 978-3-642-41431-2

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