CS 4410
Automata, Computability, and
Formal Language
Dr. Xuejun Liang
Spring 2021
1
Chapter 11: A Hierarchy of
Formal Languages and Automata
1. Recursive and Recursively Enumerable Languages
• Languages That Are Not Recursively Enumerable
• A Language That Is Not Recursively Enumerable
• A Language That Is Recursively Enumerable But Not
Recursive
2. Unrestricted Grammars
3. Context-Sensitive Grammars and Languages
• Context-Sensitive Languages and Linear Bounded
Automata
• Relation Between Recursive and Context-Sensitive
Languages
4. The Chomsky Hierarchy
2
Learning Objectives
At the conclusion of the chapter, the student will be able to:
• Explain the difference between recursive and recursively enumerable
languages
• Describe the type of productions in an unrestricted grammar
• Identify the types of languages generated by unrestricted grammars
• Describe the type of productions in a context sensitive grammar
• Give a sequence of derivations to generate a string using the
productions in a context sensitive grammar
• Identify the types of languages generated by context-sensitive grammars
• Construct a context-sensitive grammar to generate a particular language
• Describe the structure and components of the Chomsky hierarchy
A Hierarchy of Formal Languages and Automata
Recursively enumerable language
Unrestricted Grammars, Turing Machine
Context-Sensitive Languages
Context-Sensitive grammars
Linear Bounded Automata
Context-free language
Context-free grammar, NDPA
Regular language, Regular grammar,
Regular expression, NFA, DFA
4
Recursive and Recursively Enumerable
Languages
• A language L is recursively enumerable if there exists a
Turing machine that accepts it (as we have previously stated,
rejected strings cause the machine to either not halt or halt
in a nonfinal state)
• A language L is recursive if there exists a Turing machine that
accepts it and is guaranteed to halt on every valid input
string
• In other words, a language is recursive if and only if there
exists a membership algorithm for it
Languages That Are Not Recursively
Enumerable
• Theorem 11.2 states that, for any nonempty alphabet,
there exist languages not recursively enumerable
• One proof involves a technique called diagonalization,
which can be used to show that, in a sense, there are
fewer Turing Machines than there are languages
• More explicitly, Theorem 11.3 describes the existence of a
recursively enumerable language whose complement is
not recursively enumerable
• Furthermore, Theorem 11.5 concludes that the family of
recursive languages is a proper subset of the family of
recursively enumerable languages
Theorem 11.1: Let S be an infinite countable
set. Then its power set 2S is not countable
Let S = {s1, s2, s3, ...}. Then any element of 2S can be represented by
a sequence of 0’s and 1’s. For examples:
1 2 3 4 5 6 7 8 9
the set {s2, s3, s6} =
the set {s1, s3, s5} =
Now, suppose that 2S were countable
and 2S ={t1, t2, t3, …}
Diagonalization
Pick t = 0011…
Then
A contradiction!
So, 2S is not countable
Unrestricted Grammars
• An unrestricted grammar has essentially no restrictions on
the form of its productions:
• Any variables and terminals on the left side, in any order
• Any variables and terminals on the right side, in any order
• The only restriction is that is not allowed as the left side of a
production
• A sample unrestricted grammar has productions
S S1 B
S1 aS1b
bB bbbB
aS1b aa
B
Unrestricted Grammars and Recursively
Enumerable Languages
• Theorem 11.6: Any language generated by an unrestricted
grammar is recursively enumerable
• Theorem 11.7: For every recursively enumerable language L,
there exists an unrestricted grammar G that generates L
• These two theorems establish the result that unrestricted
grammars generate exactly the family of recursively
enumerable languages, the largest family of languages that
can be generated or recognized algorithmically
Context-Sensitive Grammars
• In a context-sensitive grammar, the only restriction is that, for
any production, length of the right side is at least as large as
the length of the left side
• Example 11.2 introduces a sample context-sensitive grammar
with productions
S abc | aAbc Derive the string aabbcc
Ab bA
S aAbc
Ac Bbcc abAc
bB Bb abBbcc
aB aa | aaA aBbbcc
aabbcc
Characteristics of Context-Sensitive Grammars
• An important characteristic of context-sensitive grammars
is that they are noncontracting, in the sense that in any
derivation, the length of successive sentential forms can
never decrease
• These grammars are called context-sensitive because it is
possible to specify that variables may only be replaced in
certain contexts
• For instance, in the grammar of Example 11.2, variable A
can only be replaced if it is followed by either b or c
Ab bA
Ac Bbcc
Context-Sensitive Languages
• A language L is context-sensitive if there is a context-
sensitive grammar G, such that either L = L(G) or L = L(G)
{}
• The empty string is included, because by definition, a
context-sensitive grammar can never generate a language
containing the empty string
• As a result, it can be concluded that the family of context-
free languages is a subset of the family of context-sensitive
languages
• The language { anbncn: n ≥ 1 } is context-sensitive, since it is
generated by the grammar in Example 11.2
Context-Sensitive Languages and Linear
Bounded Automata
• Theorem 11.8 states that, for every context-sensitive
language L not including , there is a linear bounded
automaton that recognizes L
• Theorem 11.9 states that, if a language L is accepted by a
linear bounded automaton M, then there is a context-
sensitive grammar that generates L
• These two theorems establish the result that context-
sensitive grammars generate exactly the family of languages
accepted by linear bounded automata, the context-sensitive
languages
Relationship Between Recursive and Context-
Sensitive Languages
• Theorem 11.10 states that every context-sensitive language is
recursive
• Theorem 11.11 maintains that some recursive languages are
not context-sensitive
• These two theorems help establish a hierarchical relationship
among the various classes of automata and languages:
• Linear bounded automata are less powerful than Turing
machines
• Linear bounded automata are more powerful than pushdown
automata
The Chomsky Hierarchy
• The linguist Noam Chomsky summarized the relationship
between language families by classifying them into four
language types, type 0 to type 3
• This classification, which became known as the Chomsky
Hierarchy, is illustrated as below
Recursively enumerable language
Context-Sensitive Languages
Context-free language
Regular language
An Extended Hierarchy
• We have studied additional language families and their relationships to those
in the Chomsky Hierarchy
• By including deterministic context-free languages and recursive languages,
we obtain the extended hierarchy as below
Recursive languages
Turing machine that
halts on any inputs.
Deterministic
context-free languages
DPDA
A Closer Look at the Family of Context-
Free Languages
The following figure illustrates the relationships among various
subsets of the family of context-free languages: regular (LREG), linear
(LLIN), deterministic context-free (LDCF), and nondeterministic
context-free (LCF)
Linear languages
Linear grammars
