FoCL, Chapter 12: LA- and PS-hierarchies in comparison) 191

12. LA- and PS-hierarchies in comparison

12.1 Language classes of LA- and PS-grammar

12.1.1 Complexity degrees of the LA- and PS-hierarchy

LA-grammar PS-grammar

undecidable — recursively enumerable languages

A-languages

exponential B-languages context-sensitive languages

C3-languages

polynomial  C2-languages  context-free languages
linear  C1-languages  regular languages

FoCL, Chapter 12: LA- and PS-hierarchies in comparison) 192

12.2 Subset relations in the two hierarchies

12.2.1 Subset relations in the PS-hierarchy

regular lang. ⊂ context-free lang. ⊂ context-sensitive lang. ⊂ rec. enum. languages 12.2.2 Subset relations in the LA-hierarchy

C1-languages ⊆ C2-languages ⊆ C3-languages ⊆ B-languages ⊂ A-languages

FoCL, Chapter 12: LA- and PS-hierarchies in comparison) 193

12.3 Non-equivalence of the LA- and PS-hierarchy

12.3.1 Languages which are in the same class in PS-grammar, but in different classes in LA-grammar a^{k}b^{k} and WW^{R} are in the same class in PS-grammar (i.e. context-free), but in different classes in LA-grammar: a^{k}b^{k} is a C1-LAG parsing in linear time, while WW^{R} is a C2-LAG parsing in n^{2}. 12.3.2 Languages which are in the same class in LA-grammar, but in different classes in PS-grammar a^{k}b^{k} and a^{k}b^{k}c^{k} are in the same class in LA-grammar (i.e. C1-LAGs), but in different classes in PS-grammar: a^{k}b^{k} is context-free, while a^{k}b^{k}c^{k} is context-sensitive. 12.3.3 Inherent complexity The inherent complexity of a language is based on the number of operations required in the worst case on an abstract machine (e.g. a Turing or register machine). This form of analysis occurs on a very low level corresponding to machine or assembler code. 12.3.4 Class assigned complexity The complexity of artificial and natural languages is usually analyzed at the abstraction level of grammar formalisms, whereby complexity is determined for the grammar type and its language class as a whole.

\begin{matrix}c \\ ◯ \end{matrix}1999 Roland Hausser FoCL, Chapter 12: LA- and PS-hierarchies in comparison) 194

12.3.5 Difference between the two types of complexity Languages which are inherently of high complexity (e.g. 3SAT and SUBSET SUM) are necessarily in a high complexity class (here exponential) in any possible grammar formalism. Languages which are inherently of low complexity (e.g. a^{k}b^{k}c^{k}) may be assigned high or low class complexity, depending on the formalism. 12.3.6 PS-Grammar of L_{no}

S → 1S1 S → 1S S → # S → 0S0 S → 0S

12.3.7 PS-grammar derivation of 10010#101 in L_{no}

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} S S S S S S 1  1 0  0 0 1  1 0  0S. # derivation tree:  1S1 10S01 100S01 1001S101 10010S101 10010#101 generated chains:  states: 1.S1 1.S 0.S0 0.S 0.S0 0.S 1.S1 1.S 0.S0 0.S #.  1S1. 0S0. 0S. 1S1.
\begin{matrix}c \\ ◯ \end{matrix}1999 Roland Hausser

12.3.8 C3-LAG for L_{no}

LX =_{def} {[0 (0)], [1 (1)], [# (#)]} ST_{S} =_{def} {[(seg_{c}) {r_{1}, r_{2}, r_{3}, r_{4}, r_{5}}] }, where seg_{c}, seg_{d} ϵ {0, 1}.

r_{1}: (seg_{c})(seg_{d})  ⇒ ε  {r_{1}, r_{2}, r_{3}, r_{4}, r_{5}}
r_{2}: (seg_{c})(seg_{d})  ⇒ (seg_{d})  {r_{1}, r_{2}, r_{3}, r_{4}, r_{5}}
r_{3}: (X)(seg_{c})  ⇒ (X)  {r_{1} r_{2}, r_{3}, r_{4}, r_{5}}

r_{4}: (X)(seg_{c}) ⇒ (seg_{c} X) {r_{1} r_{2}, r_{3}, r_{4}, r_{5}} r_{5}: (X)(#) ⇒ (X) {r_{6}} r_{6}: (seg_{c} X)(seg_{c}) ⇒ (X) {r_{6}} ST_{F} =_{def} {[ε rp_{6}]}

FoCL, Chapter 12: LA- and PS-hierarchies in comparison) 196

12.4 Comparing the lower LA- and PS-classes

Context-free PS-grammar has been widely used because it provides the greatest amount of generative capacity within the PS-grammar hierarchy while being computationally tractable. 12.4.1 How suitable is context-free grammar for describing natural and programming languages? There is general agreement in linguistics that context-free PS-grammar does not properly fit the structures characteristic of natural language. The same holds for computer science:

It is no secret that context-free grammars are only a first order approximation to the various mechanisms used for specifying the syntax of modern programming languages.

S. Ginsberg 1980, p.7

12.4.2 Conservative extensions of the PS-grammar hierarchy

regular lang. ⊂ context-free lang. ⊂ TAL ⊂ index lang. ⊂ context-sensitive lang. ⊂ r.e. languages

\begin{matrix}c \\ ◯ \end{matrix}1999 Roland Hausser FoCL, Chapter 12: LA- and PS-hierarchies in comparison) 197

12.4.3 Orthogonal relation between C- and cf-languages

[figure: a^{k}b^{k}; a^{k}b^{k}c^{k}; a^{k}b^{k}c^{k}d^{k}e^{k}; a^{2n}; a^{n!}; C1; C2; C3; WW^{R}; WW; WWW; L_{no}; HCFL; 3SAT; SUBSET-SUM; context-free languages]

\begin{matrix}c \\ ◯ \end{matrix}1999 Roland Hausser

FoCL, Chapter 12: LA- and PS-hierarchies in comparison) 198

12.4.4 Orthogonal L_{dcf}, L_{cf}, C_{1}, C_{2}, and C_{3} classifications

[figure: C3; C2; C1; L_{cf}; L_{dcf}; L_{hast}^{k≥2}; 0^{k} 1^{k} ∪ 0^{k} 1^{2k}; L_{hast}^{1}; L_{rabin}; WW; M_{hast}^{∞}; L_{hast}^{∞}; WWR; L_{rose}; M_{hast}^{1}; M_{hast}^{k≥2}; HCFL; Lno; SUBSET-SUM; 3SAT]

FoCL, Chapter 12: LA- and PS-hierarchies in comparison) 199

12.5 Linear complexity of natural language

12.5.1 Why the natural languages are likely to be C-languages In a context-sensitive language which is not a C-language, the category length would have to grow just within the LBA-definition of context-sensitive languages, but grow faster than the pattern-based categorial operations of the C-LAGs would permit. That this type of language should be characteristic for the structure of natural language is highly improbable. 12.5.2 If the natural languages are C-LAGs then the following two questions are equivalent:

(i) How complex are the natural languages? (ii) How ambiguous are the natural languages? This is because the C-LAG subclasses differ solely in their degrees of ambiguity.

12.5.3 Slim-theoretic analysis of ambiguity

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12.5.4 Multiple interpretations of prepositional phrases: a syntactic or a semantic ambiguity?

The man saw the girl with the telescope. Julia ate the apple on the table behind the tree in the garden.

12.5.5 Incorrect analysis of a semantic ambiguity

?^{analysis 1:}  [place to roost]^{contextual referent}^{The osprey is looking for a perch}  ^{The osprey is looking for a perch}^{analysis 2:} [kind of fish]

12.5.6 Correct analysis of a semantic ambiguity

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12.5.7 Incorrect analysis: a recursive pseudo-ambiguity

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 postnominal^{in the garden.}^{in the garden.} adverbial^{in the garden.} postnominal^{behind the tree} adverbial  ^{in the garden.} adverbial^{in the garden.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} postnominal^{3}^{behind the tree} postnominal  ^{in the garden.} adverbial^{in the garden.} postnominal^{behind the tree} adverbial  ^{in the garden.} adverbial^{behind the tree} postnominal^{on the table} postnominal^{on the table} adverbial^{Julia ate the apple}^{number of readings:}  ^{2}  ^{2}  ^{2}^{1}  ^{2}

12.5.8 Correct analysis with semantic doubling

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12.5.9 CoNSyx hypothesis (Complexity of natural language syntax)

The natural languages are contained in the class of C1-languages and parse in linear time. This hypothesis holds as long as no recursive ambiguities are found in natural language.