Production Rules in TOC

In Theory of Computation (TOC), production rules are fundamental components of formal grammars used to define languages. These rules describe how non-terminal symbols can be replaced with combinations of terminal and non-terminal symbols, enabling the generation of strings in a language.  As we know, every grammar generates a specific language. Each grammar holds some rules, which are called Production Rules. Strings are generated by using production rules. 

Basic Format of a Production Rule

A → α

  • A is a non-terminal symbol.
  • α is a string that may contain both terminal and non-terminal symbols.

Production Rules Representation Methods

Let’s suppose the following production rule, where S can either be replaced by aSb or by c

S → aSb | c

Here are the various methods that represent he same results as the above production rule

Method 1:

You can represent the rules as separate lines, clearly showing both alternatives:

S → aSb
S → c

Explanation: This format breaks down the rules into two distinct production rules.

Method 2:

If you prefer to break down the recursion into smaller steps, you could introduce an intermediate non-terminal, say T:

S → aT
T → Sb | c

Explanation: The rule S → aT initiates the recursion, and T → Sb | c handles the recursive structure or termination.

Method 3:

If you want to make it more generalized or add another level of abstraction:

S → aXb
X → Sb | c

Explanation: The non-terminal X is introduced to handle the recursive part, and it eventually leads to the termination with c.

Production Rules Examples in TOC

Based on the Chomsky hierarchy, production rules are categorized into different types of grammars, depending on the form and constraints of the rules. The types of production rules correspond to different levels of the hierarchy, each with increasing complexity and expressive power.

Example 1:

S → aSb | ε

Explanation:

  • The rule S → aSb means that S can be replaced by an a, followed by another S, and then a b.
  • The rule S → ε allows S to be replaced by the empty string, effectively stopping the recursion.
  • By applying these rules, strings with equal numbers of a’s and b’s are generated, such as ab, aabb, aaabbb.
  • All possible strings are { ε, ab, aabb, aaabbb, aaaabbbb, … }

Example 2:

S → a | b

Explanation:

  • S → a: The non-terminal S can be replaced by the terminal symbol a.
  • S → b: The non-terminal S can also be replaced by the terminal symbol b.
  • These rules generate the strings a and b.
  • The grammar simply allows S to produce either the character a or b, without any recursion or further complexity.

Example 3:

S → aSb | c

Explanation:

  • S → aSb: The non-terminal S can be replaced by a, followed by another S, and then b.
  • S → c: The non-terminal S can be replaced by the terminal symbol c, ending the recursion.
  • This grammar generates strings with a structure like a^n c b^n, where n is the number of a’s and b’s.
  • Examples of strings generated include c, acb, aacbb, aaacbbb, etc., where the number of a’s and b’s is equal, with c in the middle.

Example 4:

S → abS | aa 

Explanation:

  • S can be replaced by abS, meaning the string starts with ab and then recursively applies S.
  • Alternatively, S can be replaced by aa.
  • This grammar generates strings like aa, abaa, ababaa, etc.

Example 5:

S → aA | b | c
A → xxA | xx

Explanation:

  • S → b produces the string b, and S → c produces the string c.
  • S → aA generates strings starting with a, followed by what A generates.
  • A → xx generates the string xx, while A → xxA recursively generates multiple xx pairs.

This example generates strings like b, c, axx, axxxx, axxxxxxx, axxxxxxxxx, …

Example 6

S → aSb | ab

Explanation:

  • S → aSb: The non-terminal S is replaced by a, followed by another S, and then b.
  • S → ab: This is the base case where S is replaced by ab, which stops the recursion.

All possible strings: { ab, aabb, aaabbb, aaaabbbb, … }

Example 7

S → aSbb | aa

Explanation:

  • S → aSbb: S is replaced by a, followed by another S, and then two b’s.
  • S → aa: The recursion stops when S is replaced by aa.

All possible strings: { aa, aaabb, aaaabbbb, aaabbbbbb, … }

Example 8

S → aS | bS | c

Explanation:

  • S → aS: This rule means S can be replaced by a followed by another S.
  • S → bS: Similarly, S can also be replaced by b followed by another S.
  • S → c: This is the base case where S is replaced by c, ending the recursion.

All possible strings: { c, ac, bc, aac, aabc, abbbc, … }

Example 9

S → aSb | c | aSbb

Explanation:

  • S → aSb: This rule adds an a at the beginning and a b at the end, with another S in between.
  • S → c: S can be replaced by c, which is the base case.
  • S → aSbb: This rule starts with a, then applies S, and appends two b’s after it.

All possible strings: { c, acb, aacbbb, … }

Example 10

S → aA | bB
A → Aa | aa
B → Bb | bb

Explanation:

  • S → aA and S → bB: S can be replaced by either aA or bB.
  • A → Aa: A can be replaced by A followed by a, allowing A to generate multiple a’s.
  • A → aa: The recursion for A stops with two a’s.
  • B → Bb: B can be replaced by B followed by b, generating multiple b’s.
  • B → bb: The recursion for B stops with two b’s.

All possible strings: { aa, bb, aaa, bbb,  … }

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