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Regulated Grammars and Automata Alexander Meduna Brno University of Technology, Faculty of Information Technology BoΛ‡ zetΛ‡ echova 2, 612 00 Brno, Czech Republic http://www.fit.vutbr.cz/∼meduna Prepared in cooperation with Petr Zemek based on Alexander Meduna and Petr Zemek Regulated Grammars and Automata Springer, New York, pp. 694, 2014, ISBN 978-1-4939-0368-9 http://www.fit.vutbr.cz/ ˜ meduna/books/rga Supported by IT4I Centre of Excellence CZ.1.05/1.1.00/02.0070.

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Regulated Grammars and AutomataPrepared in cooperation with Petr Zemek based on
Alexander Meduna and Petr Zemek Regulated Grammars and Automata Springer, New York, pp. 694, 2014, ISBN 978-1-4939-0368-9 http://www.fit.vutbr.cz/˜meduna/books/rga
Supported by IT4I Centre of Excellence CZ.1.05/1.1.00/02.0070.
Outline
Part I: An Introduction to the Book Basic Idea General Info Contents
Part II: A Sample: One-Sided Random Context Grammars Basic Concept Definitions and Examples Generative Power Normal Forms Reduction Applications
Regulated Grammars and Automata 3 / 22
Basic Idea β€’ a grammar or an automaton based upon a finite set of
rules R
A context-free grammar with the set of rules R:
R : S β†’ ABC A β†’ aA B β†’ bB C β†’ cC A β†’ a B β†’ b C β†’ c
1 2 3 4
5 6 7
S β‡’ ABC [1] β‡’ aABC [2] β‡’ aAbBC [3] β‡’ aAbBcC [4] β‡’ aabBcC [5] β‡’ aabbcC [6] β‡’ aabbcc [7]
L(G) = {
Regulated Grammars and Automata 4 / 22
Basic Idea β€’ a grammar or an automaton based upon a finite set of
rules R β€’ a regulation over R
Example
A context-free grammar with the set of rules R:
R : 1 : S β†’ ABC 2 : A β†’ aA 3 : B β†’ bB 4 : C β†’ cC 5 : A β†’ a 6 : B β†’ b 7 : C β†’ c
1 2 3 4
5 6 7
S β‡’ ABC [1] β‡’ aABC [2] β‡’ aAbBC [3] β‡’ aAbBcC [4] β‡’ aabBcC [5] β‡’ aabbcC [6] β‡’ aabbcc [7]
L(G) = {
Regulated Grammars and Automata 4 / 22
Basic Idea β€’ a grammar or an automaton based upon a finite set of
rules R β€’ a regulation over R
Example
A context-free grammar with the set of rules R:
R : 1 : S β†’ ABC 2 : A β†’ aA 3 : B β†’ bB 4 : C β†’ cC 5 : A β†’ a 6 : B β†’ b 7 : C β†’ c
1 2 3 4
5 6 7
S β‡’ ABC [1]
β‡’ aABC [2] β‡’ aAbBC [3] β‡’ aAbBcC [4] β‡’ aabBcC [5] β‡’ aabbcC [6] β‡’ aabbcc [7]
L(G) = {
Regulated Grammars and Automata 4 / 22
Basic Idea β€’ a grammar or an automaton based upon a finite set of
rules R β€’ a regulation over R
Example
A context-free grammar with the set of rules R:
R : 1 : S β†’ ABC 2 : A β†’ aA 3 : B β†’ bB 4 : C β†’ cC 5 : A β†’ a 6 : B β†’ b 7 : C β†’ c
1 2 3 4
β‡’ aAbBC [3] β‡’ aAbBcC [4] β‡’ aabBcC [5] β‡’ aabbcC [6] β‡’ aabbcc [7]
L(G) = {
Regulated Grammars and Automata 4 / 22
Basic Idea β€’ a grammar or an automaton based upon a finite set of
rules R β€’ a regulation over R
Example
A context-free grammar with the set of rules R:
R : 1 : S β†’ ABC 2 : A β†’ aA 3 : B β†’ bB 4 : C β†’ cC 5 : A β†’ a 6 : B β†’ b 7 : C β†’ c
1 2 3 4
β‡’ aAbBcC [4] β‡’ aabBcC [5] β‡’ aabbcC [6] β‡’ aabbcc [7]
L(G) = {
Regulated Grammars and Automata 4 / 22
Basic Idea β€’ a grammar or an automaton based upon a finite set of
rules R β€’ a regulation over R
Example
A context-free grammar with the set of rules R:
R : 1 : S β†’ ABC 2 : A β†’ aA 3 : B β†’ bB 4 : C β†’ cC 5 : A β†’ a 6 : B β†’ b 7 : C β†’ c
1 2 3 4
β‡’ aabBcC [5] β‡’ aabbcC [6] β‡’ aabbcc [7]
L(G) = {
Regulated Grammars and Automata 4 / 22
Basic Idea β€’ a grammar or an automaton based upon a finite set of
rules R β€’ a regulation over R
Example
A context-free grammar with the set of rules R:
R : 1 : S β†’ ABC 2 : A β†’ aA 3 : B β†’ bB 4 : C β†’ cC 5 : A β†’ a 6 : B β†’ b 7 : C β†’ c
1 2 3 4
5 6 7
S β‡’ ABC [1] β‡’ aABC [2] β‡’ aAbBC [3] β‡’ aAbBcC [4] β‡’ aabBcC [5]
β‡’ aabbcC [6] β‡’ aabbcc [7]
Regulated Grammars and Automata 4 / 22
Basic Idea β€’ a grammar or an automaton based upon a finite set of
rules R β€’ a regulation over R
Example
A context-free grammar with the set of rules R:
R : 1 : S β†’ ABC 2 : A β†’ aA 3 : B β†’ bB 4 : C β†’ cC 5 : A β†’ a 6 : B β†’ b 7 : C β†’ c
1 2 3 4
5 6 7
S β‡’ ABC [1] β‡’ aABC [2] β‡’ aAbBC [3] β‡’ aAbBcC [4] β‡’ aabBcC [5] β‡’ aabbcC [6]
β‡’ aabbcc [7]
Regulated Grammars and Automata 4 / 22
Basic Idea β€’ a grammar or an automaton based upon a finite set of
rules R β€’ a regulation over R
Example
A context-free grammar with the set of rules R:
R : 1 : S β†’ ABC 2 : A β†’ aA 3 : B β†’ bB 4 : C β†’ cC 5 : A β†’ a 6 : B β†’ b 7 : C β†’ c
1 2 3 4
5 6 7
S β‡’ ABC [1] β‡’ aABC [2] β‡’ aAbBC [3] β‡’ aAbBcC [4] β‡’ aabBcC [5] β‡’ aabbcC [6] β‡’ aabbcc [7]
L(G) = {
Regulated Grammars and Automata 4 / 22
Basic Idea β€’ a grammar or an automaton based upon a finite set of
rules R β€’ a regulation over R
Example
A context-free grammar with the set of rules R:
R : 1 : S β†’ ABC 2 : A β†’ aA 3 : B β†’ bB 4 : C β†’ cC 5 : A β†’ a 6 : B β†’ b 7 : C β†’ c
1 2 3 4
5 6 7
S β‡’ ABC [1] β‡’ aABC [2] β‡’ aAbBC [3] β‡’ aAbBcC [4] β‡’ aabBcC [5] β‡’ aabbcC [6] β‡’ aabbcc [7]
L(G) = {
General Info
Alexander Meduna and Petr Zemek Regulated Grammars and Automata Springer, New York, pp. 694, 2014, ISBN 978-1-4939-0368-9 http://www.fit.vutbr.cz/˜meduna/books/rga
Motivation and Subject β€’ an important trend in formal language theory β€’ since 1990, no book has been published on the subject
although many papers have discussed it
Purpose β€’ theoretical: to summarize key results on the subject β€’ practical: to demonstrate applications of regulated
grammars and automata
Alexander Meduna and Petr Zemek Regulated Grammars and Automata Springer, New York, pp. 694, 2014, ISBN 978-1-4939-0368-9 http://www.fit.vutbr.cz/˜meduna/books/rga
Motivation and Subject β€’ an important trend in formal language theory β€’ since 1990, no book has been published on the subject
although many papers have discussed it
Purpose β€’ theoretical: to summarize key results on the subject β€’ practical: to demonstrate applications of regulated
grammars and automata
Regulated Grammars and Automata 6 / 22
General Info (Continued)
Regulated Grammars and Automata 6 / 22
General Info (Continued)
Book Audience β€’ computer scientists: professionals, professors, Ph.D. students β€’ mathematicians β€’ linguists
Regulated Grammars and Automata 7 / 22
General Info (Continued)
Book Audience β€’ computer scientists: professionals, professors, Ph.D. students β€’ mathematicians β€’ linguists
Regulated Grammars and Automata 7 / 22
Contents (1/4)
Part I Introduction and Terminology 1 Introduction 2 Mathematical Background 3 Rudiments of Formal Language Theory
Part II Regulated Grammars: Fundamentals 4 Context-Based Grammatical Regulation 5 Rule-Based Grammatical Regulation
Regulated Grammars and Automata 8 / 22
Contents (1/4)
Part I Introduction and Terminology 1 Introduction 2 Mathematical Background 3 Rudiments of Formal Language Theory
Part II Regulated Grammars: Fundamentals 4 Context-Based Grammatical Regulation 5 Rule-Based Grammatical Regulation
Regulated Grammars and Automata 8 / 22
Contents (2/4)
Part III Regulated Grammars: Special Topics 6 One-Sided Versions of Random Context Grammars 7 On Erasing Rules and Their Elimination 8 Extension of Languages Resulting from Regulated
Grammars 9 Sequential Rewriting over Word Monoids
Part IV Regulated Grammars: Parallelism 10 Regulated ET0L Grammars 11 Uniform Regulated Rewriting in Parallel 12 Parallel Rewriting over Word Monoids
Regulated Grammars and Automata 9 / 22
Contents (2/4)
Part III Regulated Grammars: Special Topics 6 One-Sided Versions of Random Context Grammars 7 On Erasing Rules and Their Elimination 8 Extension of Languages Resulting from Regulated
Grammars 9 Sequential Rewriting over Word Monoids
Part IV Regulated Grammars: Parallelism 10 Regulated ET0L Grammars 11 Uniform Regulated Rewriting in Parallel 12 Parallel Rewriting over Word Monoids
Regulated Grammars and Automata 9 / 22
Contents (3/4)
Part V Regulated Grammar Systems 13 Regulated Multigenerative Grammar Systems 14 Controlled Pure Grammar Systems
Part VI Regulated Automata 15 Self-Regulating Automata 16 Automata Regulated by Control Languages
Part VII Related Unregulated Automata 17 Jumping Finite Automata 18 Deep Pushdown Automata
Regulated Grammars and Automata 10 / 22
Contents (3/4)
Part V Regulated Grammar Systems 13 Regulated Multigenerative Grammar Systems 14 Controlled Pure Grammar Systems
Part VI Regulated Automata 15 Self-Regulating Automata 16 Automata Regulated by Control Languages
Part VII Related Unregulated Automata 17 Jumping Finite Automata 18 Deep Pushdown Automata
Regulated Grammars and Automata 10 / 22
Contents (3/4)
Part V Regulated Grammar Systems 13 Regulated Multigenerative Grammar Systems 14 Controlled Pure Grammar Systems
Part VI Regulated Automata 15 Self-Regulating Automata 16 Automata Regulated by Control Languages
Part VII Related Unregulated Automata 17 Jumping Finite Automata 18 Deep Pushdown Automata
Regulated Grammars and Automata 10 / 22
Contents (4/4)
Part IX Conclusion 21 Concluding Remarks 22 Summary
Regulated Grammars and Automata 11 / 22
Contents (4/4)
Part IX Conclusion 21 Concluding Remarks 22 Summary
Regulated Grammars and Automata 11 / 22
Part II: A Sample: One-Sided Random Context Grammars
Random Context Grammars:Basic Concept
β†βˆ’βˆ’βˆ’. . . . . . A βˆ’βˆ’βˆ’β†’. . . . . .
) ∈ P
bBcECbAcB
Random Context Grammars:Basic Concept
β†βˆ’βˆ’βˆ’. . . . . . A βˆ’βˆ’βˆ’β†’. . . . . .
) ∈ P
bBcECbAcB
Random Context Grammars:Basic Concept
β†βˆ’βˆ’βˆ’. . . . . . A βˆ’βˆ’βˆ’β†’. . . . . .
) ∈ P
bBcECbAcB
Random Context Grammars:Basic Concept
β†βˆ’βˆ’βˆ’. . . . . . A βˆ’βˆ’βˆ’β†’. . . . . .
) ∈ P
Random Context Grammars:Basic Concept
β†βˆ’βˆ’βˆ’. . . . . . A βˆ’βˆ’βˆ’β†’. . . . . .
) ∈ P
One-Sided RCGs: Basic Concept
β€’ a variant of random context grammars β€’ (Aβ†’ x ,U,W ) ∈ P
β€’ P = PL βˆͺ PR
) ∈ PL
bBcECbAcD
One-Sided RCGs: Basic Concept
β€’ a variant of random context grammars β€’ (Aβ†’ x ,U,W ) ∈ P β€’ P = PL βˆͺ PR
β†βˆ’. . .A
) ∈ PL
bBcECbAcD
One-Sided RCGs: Basic Concept
β€’ a variant of random context grammars β€’ (Aβ†’ x ,U,W ) ∈ PL
β€’ P = PL βˆͺ PR
) ∈ PL
bBcECbAcD
One-Sided RCGs: Basic Concept
β€’ a variant of random context grammars β€’ (Aβ†’ x ,U,W ) ∈ PR
β€’ P = PL βˆͺ PR
) ∈ PL
bBcECbAcD
One-Sided RCGs: Basic Concept
β€’ a variant of random context grammars β€’ (Aβ†’ x ,U,W ) ∈ PR
β€’ P = PL βˆͺ PR
) ∈ PL
bBcECbAcD
One-Sided RCGs: Basic Concept
β€’ a variant of random context grammars β€’ (Aβ†’ x ,U,W ) ∈ PR
β€’ P = PL βˆͺ PR
) ∈ PL
One-Sided RCGs: Basic Concept
β€’ a variant of random context grammars β€’ (Aβ†’ x ,U,W ) ∈ PR
β€’ P = PL βˆͺ PR
) ∈ PL
Definitions
Definition
G = ( N, T ,PL,PR , S
) where
β€’ N is an alphabet of nonterminals; β€’ T is an alphabet of terminals (N ∩ T = βˆ…); β€’ PL and PR are two finite sets of rules of the form
(A→ x ,U,W )
where A ∈ N, x ∈ (N βˆͺ T )βˆ—, and U,W βŠ† N; β€’ S ∈ N is the starting nonterminal.
Definition
If (Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that |x | β‰₯ 1, then G is propagating.
Regulated Grammars and Automata 14 / 22
Definitions
Definition
G = ( N, T ,PL,PR , S
) where
β€’ N is an alphabet of nonterminals; β€’ T is an alphabet of terminals (N ∩ T = βˆ…); β€’ PL and PR are two finite sets of rules of the form
(A→ x ,U,W )
where A ∈ N, x ∈ (N βˆͺ T )βˆ—, and U,W βŠ† N; β€’ S ∈ N is the starting nonterminal.
Definition
If (Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that |x | β‰₯ 1, then G is propagating.
Regulated Grammars and Automata 14 / 22
Definitions (Continued)
uAv β‡’ uxv
or
(Aβ†’ x ,U,W ) ∈ PR ,U βŠ† alph(v), and W ∩ alph(v) = βˆ…
Note: alph(y) denotes the set of all symbols appearing in string y
Definition
L(G) = {
Definitions (Continued)
uAv β‡’ uxv
or
(Aβ†’ x ,U,W ) ∈ PR ,U βŠ† alph(v), and W ∩ alph(v) = βˆ…
Note: alph(y) denotes the set of all symbols appearing in string y
Definition
L(G) = {
w ∈ T βˆ— : S β‡’βˆ— w }
whereβ‡’βˆ— is the reflexive-transitive closure ofβ‡’. Regulated Grammars and Automata 15 / 22
Example
Example
G = ( {S,A,B, A, B}, {a,b,c},PL,PR , S
) where PL contains
(B β†’ B, {A}, βˆ…) (B β†’ Ξ΅, βˆ…, {A, A})
and PR contains
(Aβ†’ Ξ΅, {B}, βˆ…)
Example (Continued)
Example
PL : (S β†’ AB, βˆ…, βˆ…) (B β†’ bBc, {A}, βˆ…) (B β†’ B, {A}, βˆ…) (B β†’ Ξ΅, βˆ…, {A, A})
PR : (Aβ†’ aA, {B}, βˆ…) (Aβ†’ A, {B}, βˆ…) (Aβ†’ Ξ΅, {B}, βˆ…)
S β‡’ AB [(S β†’ AB, βˆ…, βˆ…)]
β‡’ aAB [(Aβ†’ aA, {B}, βˆ…)]
β‡’ aAbBc [(B β†’ bBc, {A}, βˆ…)]
β‡’ aAbBc [(Aβ†’ A, {B}, βˆ…)]
β‡’ anbnBcn [(Aβ†’ Ξ΅, {B}, βˆ…)]
L(G) = {
Example (Continued)
Example
PL : (S β†’ AB, βˆ…, βˆ…) (B β†’ bBc, {A}, βˆ…) (B β†’ B, {A}, βˆ…) (B β†’ Ξ΅, βˆ…, {A, A})
PR : (Aβ†’ aA, {B}, βˆ…) (Aβ†’ A, {B}, βˆ…) (Aβ†’ Ξ΅, {B}, βˆ…)
S β‡’ AB [(S β†’ AB, βˆ…, βˆ…)]
β‡’ aAB [(Aβ†’ aA, {B}, βˆ…)]
β‡’ aAbBc [(B β†’ bBc, {A}, βˆ…)]
β‡’ aAbBc [(Aβ†’ A, {B}, βˆ…)]
β‡’ anbnBcn [(Aβ†’ Ξ΅, {B}, βˆ…)]
L(G) = {
Example (Continued)
Example
PL : (S β†’ AB, βˆ…, βˆ…) (B β†’ bBc, {A}, βˆ…) (B β†’ B, {A}, βˆ…) (B β†’ Ξ΅, βˆ…, {A, A})
PR : (Aβ†’ aA, {B}, βˆ…) (Aβ†’ A, {B}, βˆ…) (Aβ†’ Ξ΅, {B}, βˆ…)
S β‡’ AB [(S β†’ AB, βˆ…, βˆ…)]
β‡’ aAB [(Aβ†’ aA, {B}, βˆ…)]
β‡’ aAbBc [(B β†’ bBc, {A}, βˆ…)]
β‡’ aAbBc [(Aβ†’ A, {B}, βˆ…)]
β‡’ anbnBcn [(Aβ†’ Ξ΅, {B}, βˆ…)]
L(G) = {
Example (Continued)
Example
PL : (S β†’ AB, βˆ…, βˆ…) (B β†’ bBc, {A}, βˆ…) (B β†’ B, {A}, βˆ…) (B β†’ Ξ΅, βˆ…, {A, A})
PR : (Aβ†’ aA, {B}, βˆ…) (Aβ†’ A, {B}, βˆ…) (Aβ†’ Ξ΅, {B}, βˆ…)
S β‡’ AB [(S β†’ AB, βˆ…, βˆ…)]
β‡’ aAB [(Aβ†’ aA, {B}, βˆ…)]
β‡’ aAbBc [(B β†’ bBc, {A}, βˆ…)]
β‡’ aAbBc [(Aβ†’ A, {B}, βˆ…)]
β‡’ anbnBcn [(Aβ†’ Ξ΅, {B}, βˆ…)]
L(G) = {
Example (Continued)
Example
PL : (S β†’ AB, βˆ…, βˆ…) (B β†’ bBc, {A}, βˆ…) (B β†’ B, {A}, βˆ…) (B β†’ Ξ΅, βˆ…, {A, A})
PR : (Aβ†’ aA, {B}, βˆ…) (Aβ†’ A, {B}, βˆ…) (Aβ†’ Ξ΅, {B}, βˆ…)
S β‡’ AB [(S β†’ AB, βˆ…, βˆ…)]
β‡’ aAB [(Aβ†’ aA, {B}, βˆ…)]
β‡’ aAbBc [(B β†’ bBc, {A}, βˆ…)]
β‡’ aAbBc [(Aβ†’ A, {B}, βˆ…)]
β‡’ aAbBc [(B β†’ B, {A}, βˆ…)]
L(G) = {
Example (Continued)
Example
PL : (S β†’ AB, βˆ…, βˆ…) (B β†’ bBc, {A}, βˆ…) (B β†’ B, {A}, βˆ…) (B β†’ Ξ΅, βˆ…, {A, A})
PR : (Aβ†’ aA, {B}, βˆ…) (Aβ†’ A, {B}, βˆ…) (Aβ†’ Ξ΅, {B}, βˆ…)
S β‡’ AB [(S β†’ AB, βˆ…, βˆ…)]
β‡’ aAB [(Aβ†’ aA, {B}, βˆ…)]
β‡’ aAbBc [(B β†’ bBc, {A}, βˆ…)]
β‡’ aAbBc [(Aβ†’ A, {B}, βˆ…)]
β‡’ anbnBcn [(Aβ†’ Ξ΅, {B}, βˆ…)]
L(G) = {
Example (Continued)
Example
PL : (S β†’ AB, βˆ…, βˆ…) (B β†’ bBc, {A}, βˆ…) (B β†’ B, {A}, βˆ…) (B β†’ Ξ΅, βˆ…, {A, A})
PR : (Aβ†’ aA, {B}, βˆ…) (Aβ†’ A, {B}, βˆ…) (Aβ†’ Ξ΅, {B}, βˆ…)
S β‡’ AB [(S β†’ AB, βˆ…, βˆ…)]
β‡’ aAB [(Aβ†’ aA, {B}, βˆ…)]
β‡’ aAbBc [(B β†’ bBc, {A}, βˆ…)]
β‡’ aAbBc [(Aβ†’ A, {B}, βˆ…)]
β‡’ anbnBcn [(Aβ†’ Ξ΅, {B}, βˆ…)]
L(G) = {
Example (Continued)
Example
PL : (S β†’ AB, βˆ…, βˆ…) (B β†’ bBc, {A}, βˆ…) (B β†’ B, {A}, βˆ…) (B β†’ Ξ΅, βˆ…, {A, A})
PR : (Aβ†’ aA, {B}, βˆ…) (Aβ†’ A, {B}, βˆ…) (Aβ†’ Ξ΅, {B}, βˆ…)
S β‡’ AB [(S β†’ AB, βˆ…, βˆ…)]
β‡’ aAB [(Aβ†’ aA, {B}, βˆ…)]
β‡’ aAbBc [(B β†’ bBc, {A}, βˆ…)]
β‡’ aAbBc [(Aβ†’ A, {B}, βˆ…)]
β‡’ anbnBcn [(Aβ†’ Ξ΅, {B}, βˆ…)]
L(G) = {
Example (Continued)
Example
PL : (S β†’ AB, βˆ…, βˆ…) (B β†’ bBc, {A}, βˆ…) (B β†’ B, {A}, βˆ…) (B β†’ Ξ΅, βˆ…, {A, A})
PR : (Aβ†’ aA, {B}, βˆ…) (Aβ†’ A, {B}, βˆ…) (Aβ†’ Ξ΅, {B}, βˆ…)
S β‡’ AB [(S β†’ AB, βˆ…, βˆ…)]
β‡’ aAB [(Aβ†’ aA, {B}, βˆ…)]
β‡’ aAbBc [(B β†’ bBc, {A}, βˆ…)]
β‡’ aAbBc [(Aβ†’ A, {B}, βˆ…)]
β‡’ anbnBcn [(Aβ†’ Ξ΅, {B}, βˆ…)]
L(G) = {
Generative Power
RE
CS
CF
RE the family of recursively enumerable languages CS the family of context-sensitive languages CF the family of context-free languages
Regulated Grammars and Automata 18 / 22
Generative Power
RE = ORC
ORC the language family generated by one-sided random context grammars
ORCβˆ’Ξ΅ the language family generated by propagating one-sided random context grammars
Regulated Grammars and Automata 18 / 22
Generative Power
RC the language family generated by random context grammars
RCβˆ’Ξ΅ the language family generated by propagating random context grammars
Regulated Grammars and Automata 18 / 22
Normal Forms
Results Structure
For any one-sided random context grammar, there is an equivalent one-sided random context grammar satisfying one of the following normal forms.
Normal Form I
Normal Form III
(Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that x ∈ NN βˆͺ T βˆͺ {Ξ΅}
Normal Form IV
(Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that U = βˆ… or W = βˆ…
Regulated Grammars and Automata 19 / 22
Normal Forms
Results Structure
For any one-sided random context grammar, there is an equivalent one-sided random context grammar satisfying one of the following normal forms.
Normal Form I
Normal Form III
(Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that x ∈ NN βˆͺ T βˆͺ {Ξ΅}
Normal Form IV
(Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that U = βˆ… or W = βˆ…
Regulated Grammars and Automata 19 / 22
Normal Forms
Results Structure
For any one-sided random context grammar, there is an equivalent one-sided random context grammar satisfying one of the following normal forms.
Normal Form I
Normal Form III
(Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that x ∈ NN βˆͺ T βˆͺ {Ξ΅}
Normal Form IV
(Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that U = βˆ… or W = βˆ…
Regulated Grammars and Automata 19 / 22
Normal Forms
Results Structure
For any one-sided random context grammar, there is an equivalent one-sided random context grammar satisfying one of the following normal forms.
Normal Form I
Normal Form III
(Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that x ∈ NN βˆͺ T βˆͺ {Ξ΅}
Normal Form IV
(Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that U = βˆ… or W = βˆ…
Regulated Grammars and Automata 19 / 22
Normal Forms
Results Structure
For any one-sided random context grammar, there is an equivalent one-sided random context grammar satisfying one of the following normal forms.
Normal Form I
Normal Form III
(Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that x ∈ NN βˆͺ T βˆͺ {Ξ΅}
Normal Form IV
(Aβ†’ x ,U,W ) ∈ PL βˆͺ PR implies that U = βˆ… or W = βˆ…
Regulated Grammars and Automata 19 / 22
Reduction
Theorem
Any one-sided random context grammar can be converted to an equivalent one having no more than 10 nonterminals.
β€’ with respect to the number of right random context nonterminals
Definition
If (Aβ†’ x ,U,W ) ∈ PR , then A is a right random context nonterminal.
Theorem
Any one-sided random context grammar can be converted to an equivalent one having no more than 2 right random context nonterminals.
Regulated Grammars and Automata 20 / 22
Reduction
Theorem
Any one-sided random context grammar can be converted to an equivalent one having no more than 10 nonterminals.
β€’ with respect to the number of right random context nonterminals
Definition
If (Aβ†’ x ,U,W ) ∈ PR , then A is a right random context nonterminal.
Theorem
Any one-sided random context grammar can be converted to an equivalent one having no more than 2 right random context nonterminals.
Regulated Grammars and Automata 20 / 22
Reduction
Theorem
Any one-sided random context grammar can be converted to an equivalent one having no more than 10 nonterminals.
β€’ with respect to the number of right random context nonterminals
Definition
If (Aβ†’ x ,U,W ) ∈ PR , then A is a right random context nonterminal.
Theorem
Any one-sided random context grammar can be converted to an equivalent one having no more than 2 right random context nonterminals.
Regulated Grammars and Automata 20 / 22
Reduction (Continued)
β€’ with respect to the number of right random context rules
Definition
If p ∈ PR , then p is a right random context rule.
Theorem
Any one-sided random context grammar can be converted to an equivalent one having no more than 2 right random context rules.
Regulated Grammars and Automata 21 / 22
Reduction (Continued)
β€’ with respect to the number of right random context rules
Definition
If p ∈ PR , then p is a right random context rule.
Theorem
Any one-sided random context grammar can be converted to an equivalent one having no more than 2 right random context rules.
Regulated Grammars and Automata 21 / 22
Applications
General Application Area β€’ information processing based on the existence or absence
of some information parts
Specific Scientific Disciplines β€’ genetics: modification of genetic codes in which some
prescribed sequences of nucleotides occur while some others do not
β€’ linguistics: generation or verification of texts that contain no forbidding passages, such as vulgarisms or classified information
β€’ computer science: syntax analysis of complicated non-context-free structures during language translation
Regulated Grammars and Automata 22 / 22
Discussion
Basic Idea
General Info
Basic Concept

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