Mathematical Foundations of Computer Science Chapter 3: Regular Languages and Regular Grammars

Languages  A language (over an alphabet Σ) is any subset of the set of all possible strings over Σ. The set of all possible strings is written as Σ*.  Example: Σ = {a, b, c} Σ = {a, b, c} Σ* = {, a, b, c, ab, ac, ba, bc, ca, aaa, …} Σ* = {, a, b, c, ab, ac, ba, bc, ca, aaa, …} one language might be the set of strings of length less than or equal to 2. one language might be the set of strings of length less than or equal to 2. L = {, a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc}

Regular Languages  A regular language (over an alphabet Σ) is any language for which there exists a finite automaton that recognizes it.

Mathematical Models of Computation  This course studies a variety of mathematical models corresponding to notions of computation.  The finite automaton was our first example.  The finite automaton is an example of an automaton model.  There are other models as well.

Mathematical Models of Computation  Another important model is that of a grammar.  We will shortly look at regular grammars.  But first, a digression:

Regular Expressions  A regular expression is a mathematical model for describing a particular type of language.  Regular expressions are kind of like arithmetic expressions.  The regular expression is defined recursively.

Regular Expressions  Given an alphabet Σ , λ and a  Σ are all regular expressions. , λ and a  Σ are all regular expressions. If r 1 and r 2 are regular expressions, then so are r 1 + r 2, r 1  r 2, r 1 * and (r 1 ). If r 1 and r 2 are regular expressions, then so are r 1 + r 2, r 1  r 2, r 1 * and (r 1 ). Note: we usually write r 1  r 2 as r 1 r 2.Note: we usually write r 1  r 2 as r 1 r 2. These are the only things that are regular expressions. These are the only things that are regular expressions. empty set empty string

Regular Expressions  Meaning:  represents the empty language  represents the empty language λ represents the language {λ} λ represents the language {λ} a represents the language {a} a represents the language {a} r 1 + r 2 represents the language L(r 1 )  L(r 2 ) r 1 + r 2 represents the language L(r 1 )  L(r 2 ) r 1  r 2 represents L(r 1 ) L(r 2 ) r 1  r 2 represents L(r 1 ) L(r 2 ) r 1 * represents (L(r 1 ))* r 1 * represents (L(r 1 ))*

Regular Expressions  Example 1: What does a*(a + b) represent? What does a*(a + b) represent? It represents zero or more a's followed by either an a or a b. It represents zero or more a's followed by either an a or a b. {a, b, aa, ab, aaa, aab, aaaa, aaab …} {a, b, aa, ab, aaa, aab, aaaa, aaab …}

Regular Expressions  Example 2: What does (a + b)*(a + bb) represent? What does (a + b)*(a + bb) represent? It represents zero or more symbols, each of which can be an a or a b, followed by either a or bb. It represents zero or more symbols, each of which can be an a or a b, followed by either a or bb. {a, bb, aa, abb, ba, bbb, aaa, aabb, aba, abbb, baa, babb, bba, bbbb, …} {a, bb, aa, abb, ba, bbb, aaa, aabb, aba, abbb, baa, babb, bba, bbbb, …}

Regular Expressions  Example 3: What does (aa)*(bb)*b represent? What does (aa)*(bb)*b represent? All strings over {a, b} that start with an even number of a's which are then followed by an odd number of b's. All strings over {a, b} that start with an even number of a's which are then followed by an odd number of b's. It's important to understand the underlying meaning of a regular expression. It's important to understand the underlying meaning of a regular expression.

Regular Expressions  Example 4: Find a regular expression for strings of 0's and 1's which have at least one pair of consecutive 0's. Find a regular expression for strings of 0's and 1's which have at least one pair of consecutive 0's. Each such string must have a 00 somewhere in it. Each such string must have a 00 somewhere in it. It could have any string in front of it and any string after it, as long as it's there!!! It could have any string in front of it and any string after it, as long as it's there!!! Any string is represented by (0 + 1)* Any string is represented by (0 + 1)* Answer: (0 + 1)*00(0 + 1)* Answer: (0 + 1)*00(0 + 1)*

Regular Expressions  Example: Find a regular expression for strings of 0's and 1's which have no pairs of consecutive 0's. Find a regular expression for strings of 0's and 1's which have no pairs of consecutive 0's. It's a repetition of strings that are either 1's or, if a substring begins with 0, it must be followed by at least one 1.It's a repetition of strings that are either 1's or, if a substring begins with 0, it must be followed by at least one 1. ( *)*( *)* or equivalently, (1 + 01)*or equivalently, (1 + 01)* But such strings can't end in a 0.But such strings can't end in a 0.

Regular Expressions  Example: Find a regular expression for strings of 0's and 1's which have no pairs of consecutive 0's. Find a regular expression for strings of 0's and 1's which have no pairs of consecutive 0's. ( *)*( *)* (1 + 01)*(1 + 01)* But such strings can't end in a 0.But such strings can't end in a 0. So we add (0 + λ) to the end to allow for this.So we add (0 + λ) to the end to allow for this. (1 + 01)* (0 + λ)(1 + 01)* (0 + λ) This is only one of many possible answers. This is only one of many possible answers.

Regular Expressions  Why are they called regular expressions?  Because, as it turns out, the set of languages they describe is that of the regular languages.  That means that regular expressions are just another model for the same thing as finite automata.

Regular Expressions  Homework: Chapter 3, Section 1 Chapter 3, Section 1 Problems 1-11, 17, 18Problems 1-11, 17, 18

Regular Expressions and Regular Languages  As we have said, regular expressions and finite automata are really different ways of expressing the same thing.  Let's see why.  Given a regular expression, how can we build an equivalent finite automaton?  (We won't bother going the other way, although it can be done.)

Regular Expressions and Regular Languages  Clearly there are simple finite automata corresponding to the simple regular expressions:  λ λ a λ a Note that each of these has an initial state and one accepting state.

Regular Expressions and Regular Languages  On the previous slide, we saw that the simplest regular expressions can be represented by a finite automaton with an initial state (duh!) and one isolated accepting state:

Regular Expressions and Regular Languages  We can build more complex automata for more complex regular expressions using this model:

Regular Expressions and Regular Languages  Here's how we build an nfa for r 1 + r 2 : λ λλ λ r1r1r1r1 r2r2r2r2 r 1 + r 2

Regular Expressions and Regular Languages  Here's how we build an nfa for r 1 r 2 : r1r1r1r1 r2r2r2r2 λ λ λ r1 r2r1 r2r1 r2r1 r2

Regular Expressions and Regular Languages  Here's how we build an nfa for (r 1 )*: λλ λ λ r1r1r1r1 (r 1 )* λ Note: the last state added is not in book. For safety, I do it to have only one arc going into the final state.

Building an nfa from a regular expression  Example: Consider the regular expression (a + bb)(a+b)*(bb) Consider the regular expression (a + bb)(a+b)*(bb) a b b λ λ λ λ λ λ a b λ λ λ λ λ λ λ λ b b sometimes we just get tired and take an obvious shortcut

Building regular expression from a finite automaton  The book goes on to show that it works the other way around as well: we can find a corresponding regular expression for any finite automaton.  It's fairly easy in some cases and you can "just do it."  However, it's generally complicated and not worth the bother studying.  You are not responsible for this material

Building regular expression from a finite automaton  The above automaton clearly corresponds to a*(a+b)c* a, b c a

Regular Expressions and nfa's  Homework: Chapter 3, Section 2 Chapter 3, Section 2 Problems 1-5Problems 1-5

Regular Grammars  Review: A grammar is a quadruple G = (V, T, S, P) where V is a finite set of variables V is a finite set of variables T is a finite set of symbols, called terminals T is a finite set of symbols, called terminals S is in V and is called the start symbol S is in V and is called the start symbol P is a finite set of productions, which are rules of the form α → β P is a finite set of productions, which are rules of the form α → β where α and β are strings consisting of terminals and variables.where α and β are strings consisting of terminals and variables.

Regular Grammars  A grammar is said to be right-linear if every production in P is of the form A → xB or A → xB or A → x A → x where A and B are variables (perhaps the same, perhaps the start symbol S) in V where A and B are variables (perhaps the same, perhaps the start symbol S) in V and x is any string of terminal symbols (including the empty string λ) and x is any string of terminal symbols (including the empty string λ)

Regular Grammars  An alternate (and better) definition of a right- linear grammar says that every production in P is of the form A → aB or A → aB or A → a or A → a or S → λ (to allow λ to be in the language) S → λ (to allow λ to be in the language) where A and B are variables (perhaps the same, but B can't be S) in V where A and B are variables (perhaps the same, but B can't be S) in V and a is any terminal symbol and a is any terminal symbol

Regular Grammars  The reason I prefer the second definition (although I accept the first one that happens to be used in the book) is It's easier to work with in proving things. It's easier to work with in proving things. It's the much more common definition. It's the much more common definition.

Regular Grammars  A grammar is said to be left-linear if every production in P is of the form A → Bx or A → Bx or A → x A → x where A and B are variables (perhaps the same, perhaps the start symbol S) in V where A and B are variables (perhaps the same, perhaps the start symbol S) in V and x is any string of terminal symbols (including the empty string λ) and x is any string of terminal symbols (including the empty string λ)

Regular Grammars  The alternate definition of a left-linear grammar says that every production in P is of the form A → Ba or A → Ba or A → a or A → a or S → λ S → λ where A and B are variables (perhaps the same, but B can't be S) in V where A and B are variables (perhaps the same, but B can't be S) in V and a is any terminal symbol and a is any terminal symbol

Regular Grammars  Any left-linear or right-linear grammar is called a regular grammar.

Regular Grammars  For brevity, we often write a set of productions such as A → x 1 A → x 1 A → x 2 A → x 2 A → x 3 A → x 3  As A → x 1 | x 2 | x 3 A → x 1 | x 2 | x 3

Regular Grammars  A derivation in grammar G is any sequence of strings in V and T, connected with connected with starting with S and ending with a string containing no variables starting with S and ending with a string containing no variables where each subsequent string is obtained by applying a production in P is called a derivation. where each subsequent string is obtained by applying a production in P is called a derivation.  S  x 1  x 2  x 3 ...  x n abbreviated as:  S x n  *

Regular Grammars  S  x 1  x 2  x 3 ...  x n  abbreviated as:  S x n  We say that x n is a sentence of the language generated by G, L(G).  We say that the other x's are sentential forms.  *

Regular Grammars  L(G) = {w | w  T* and S x n }  We call L(G) the language generated by G  L(G) is the set of all sentences over grammar G  *

Example 1  S → abS | a is an example of a right-linear grammar.  Can you figure out what language it generates?  L = {w  {a,b}* | w contains alternating a's and b's, begins with an a, and ends with a b}  {a}  L((ab)*a)

Example 2  S → Aab A → Aab | aB B → a is an example of a left-linear grammar.  Can you figure out what language it generates?  L = {w  {a,b}* | w is aa followed by at least one set of alternating ab's}  L(aaab(ab)*)

Example 3  Consider the grammar S → A A → aB | λ B → Ab  This grammar is NOT regular.  No "mixing and matching" left- and right- recursive productions.

Regular Grammars and nfa's  It's not hard to show that regular grammars generate and nfa's accept the same class of languages: the regular languages!  It's a long proof, where we must show that any finite automaton has a corresponding left- or right-linear grammar, any finite automaton has a corresponding left- or right-linear grammar, and any regular grammar has a corresponding nfa. and any regular grammar has a corresponding nfa.  We won't bother with the details.

Regular Grammars and nfa's  We get a feel for this by example. Let S → aA A → abS | b Let S → aA A → abS | b SA a b b a

Regular Grammars and Regular Expressions  Example: L(aab*a)  We can easily construct a regular language for this expression: S → aA S → aA A → aB A → aB B → bB B → bB B → a B → a

Regular Languages regular expressions regular grammars finite automata

Regular Languages  Homework: Chapter 3, Section 3 Chapter 3, Section 3 Problems Problems