Descriptional complexity of semi-conditional grammars

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Descriptional Complexity of Semi-Conditional Grammars

Tom´ aˇs Masopust

Faculty of Information Technology, Brno University of Technology, Boˇzetˇechova 2, Brno 61266, Czech Republic

Alexander Meduna

Faculty of Information Technology, Brno University of Technology, Boˇzetˇechova 2, Brno 61266, Czech Republic

Abstract

The descriptional complexity of semi-conditional grammars is studied. It is proved that every recursively enumerable language is generated by a semi-conditional grammar of degree (2,1) with no more than seven conditional productions and eight nonterminals.

Key words: formal languages, semi-conditional grammars, descriptional complexity

1991 MSC: 68Q19

1 Introduction

Semi-conditional grammars (see [3,7,8]) are context-free grammars, in which two strings, called a permitting and a forbidding context, are attached to each production. Such a production is applicable if its permitting context occurs in the current sentential form while its forbidding context does not.

Simple semi-conditional grammars represents a straightforward simplification of semi-conditional grammars, in which each production has just one attached string—either a permitting or a forbidding context.

Email addresses: masopust@fit.vutbr.cz (Tom´aˇs Masopust), meduna@fit.vutbr.cz (Alexander Meduna).

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The formal language theory has discussed the descriptional complexity of simple semi-conditional grammars in detail (see [4,6,7,9]). In [7], it is proved that every recursively enumerable language is generated by a (simple) semi- conditional grammar of degree (2,1) with no more than twelve conditional productions and thirteen nonterminals. Later, in [9], this result was improved by demonstrating that every recursively enumerable language is generated by a (simple) semi-conditional grammar of degree (2,1) with no more than ten conditional productions and twelve nonterminals. Finally, this result was improved in [4] by demonstrating that every recursively enumerable language is generated by a (simple) semi-conditional grammar of degree (2,1) with no more than nine conditional productions and ten nonterminals.

This paper discusses the descriptional complexity of semi-conditional grammars because this topic has not been studied at all so far. It demonstrates stronger results about this complexity for them than the above results for simple semi-conditional grammars. Specifically, it proves that every recursively enumerable language is generated by a semi-conditional grammar of degree (2,1) with no more than seven conditional productions and eight nonterminals.

2 Preliminaries and Definitions

This paper assumes that the reader is familiar with the theory of formal languages (see [1,5]). For an alphabetV,V^∗ represents the free monoid generated byV. The unit ofV^∗ is denoted byε. SetV⁺ =V^∗− {ε}. Set sub(w) = {u:u is a substring ofw}.

In [2], it was shown that every recursively enumerable language is generated by a grammar

G= ({S, A, B, C}, T, P ∪ {ABC →ε}, S)

in theGeffert normal form, where P contains context-free productions of the form

S →uSa, where u∈ {A, AB}^∗, a∈T,

S →uSv, where u∈ {A, AB}^∗, v ∈ {BC, C}^∗, S →uv, where u∈ {A, AB}^∗, v ∈ {BC, C}^∗.

In addition, any terminal derivation is of the form S ⇒^∗ w₁w₂w

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by productions from P, wherew₁ ∈ {A, B}^∗, w₂ ∈ {B, C}^∗, w∈T^∗, and w1w2w⇒^∗ w

byABC →ε.

A semi-conditional grammar, G, is a quadruple G= (N, T, P, S), where

• N is a nonterminal alphabet;

• T is a terminal alphabet such that N ∩T =∅;

• S ∈N is the start symbol; and

• P is a finite set of productions of the form (X →α, u, v)

with X ∈N, α∈(N∪T)^∗, and u, v ∈(N∪T)⁺∪ {0}, where 06∈N ∪T is a special symbol.

If u 6= 0 or v 6= 0, then the production (X → α, u, v) ∈ P is said to be conditional. G hasdegree (i, j) if for all productions (X →α, u, v)∈P,u6= 0 implies|u| ≤iand v 6= 0 implies|v| ≤j. Forx∈(N∪T)⁺ andy∈(N∪T)^∗, x directly derives y according to the production (X → α, u, v) ∈ P, denoted by

x⇒y

if x=x₁Xx₂, y =x₁αx₂, for some x₁, x₂ ∈(N ∪T)^∗, and u6= 0 implies that u ∈ sub(x) and v 6= 0 implies that v 6∈ sub(x). As usual, ⇒ is extended to

⇒ⁱ, for i ≥ 0, ⇒⁺, and ⇒^∗. The language generated by a semi-conditional grammar, G, is defined as

L(G) ={w∈T^∗ :S ⇒^∗ w}.

A derivation of the formS ⇒^∗ w withw ∈T^∗ is called aterminal derivation.

3 Main Result

This section presents the main result concerning the descriptional complexity of semi-conditional grammars.

Theorem 1 Every recursively enumerable language is generated by a semi- conditional grammar of degree (2,1) with no more than 7 conditional productions and 8 nonterminals.

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PROOF. Let L be a recursively enumerable language. There is a grammar G⁰ = ({S, A, B, C}, T, P ∪ {ABC → ε}, S) in the Geffert normal form such that L=L(G⁰). Construct the grammar

G= ({S, A, B, C,#, B⁰, C⁰,$}, T, P⁰∪P⁰⁰, S), where

P⁰ ={(X →α,0,0) :X →α∈P}, and P⁰⁰ contains the following seven conditional productions:

(1) (A→$#,0,$), (2) (B →B⁰,#, B⁰), (3) (C →C⁰$,#B⁰, C⁰), (4) (B⁰ →ε, B⁰C⁰,0), (5) (C⁰ →ε,#C⁰,0), (6) (#→ε,#$,0), (7) ($→ε,0,#).

To prove thatL(G⁰)⊆ L(G), consider a derivation S ⇒^∗ wABCw⁰v ⇒ww⁰v in G⁰ by productions from P with only one application of the production ABC → ε, where w, w⁰ ∈ {A, B, C}^∗ and v ∈ T^∗. Then, S ⇒^∗ wABCw⁰v in G by productions fromP⁰. Moreover, by productions 1, 2, 3, 4, 5, 6, 7, 7, we get

wABCw⁰v⇒w$#BCw⁰v

⇒w$#B⁰Cw⁰v

⇒w$#B⁰C⁰$w⁰v

⇒w$#C⁰$w⁰v

⇒w$#$w⁰v

⇒w$$w⁰v

⇒w$w⁰v

⇒ww⁰v.

The inclusion follows by induction.

To prove that L(G⁰) ⊇ L(G), consider a terminal derivation. Let X from {A, B, C}be in a sentential form of this derivation. To eliminate X, there are the following three possibilities:

(1) IfX =A, then there must beC andB (by productions 6 and 3) in some (previous) sentential form;

(2) IfX =B, then there must beC and A(by productions 4 and 3) in some (previous) sentential form;

(3) IfX =C, then there must beA andB (by productions 5 and 3) in some (previous) sentential form.

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In all above cases, there are A, B, and C in some sentential form of the derivation. By productions 1, 2, 3, and 7, there cannot be more than one

#, B⁰, and C⁰ in any sentential form. By productions 3 and 4, #B⁰C⁰ is a substring of a sentential form and there is no terminal symbol between any two nonterminals. Thus, the first part of any terminal derivation in G is of the form

S ⇒^∗ w1ABCw2w⇒³w1$#B⁰C⁰$w2w (1) by productions from P⁰ and productions 1, 2, and 3, where w₁ ∈ {A, B}^∗, w₂ ∈ {B, C}^∗, and w∈T^∗. Next, only production 4 is applicable. Thus,

w1$#B⁰C⁰$w2w⇒w1$#C⁰$w2w.

Besides a possible application of production 2, only production 5 is applicable.

Thus,

w₁$#C⁰$w₂w⇒⁺w⁰₁$#$w⁰₂w

where w₁⁰ ∈ {A, B, B⁰}^∗, w⁰₂ ∈ {B, B⁰, C}^∗. Besides a possible application of production 2, only production 6 is applicable. Thus,

w⁰₁$#$w₂⁰w⇒⁺w⁰⁰₁$$w⁰⁰₂w

where w₁⁰⁰ ∈ {A, B, B⁰}^∗, w⁰⁰₂ ∈ {B, B⁰, C}^∗. Finally, only production 7 is applicable, i.e.,

w⁰⁰₁$$w⁰⁰₂w⇒²w⁰⁰₁w₂⁰⁰w.

Then,

w⁰⁰₁w₂⁰⁰w⇒^∗uvw

by productions 1, 2, 3, or 1, 3, if production 2 has already been applied, where uvw∈ {u₁$#B⁰C⁰$u₂w:u₁ ∈ {A, B}^∗, u₂ ∈ {B, C}^∗}

or uv =ε. Thus, the substring ABC and only this substring was eliminated.

By induction (see (1)), the inclusion holds.

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Acknowledgements

This work was supported by the Czech Grant Agency projects 201/07/0005 and 102/05/H050, FRVˇS grant FR762/2007/G1, and the Czech Ministry of Education under the Research Plan MSM 0021630528.

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