Three Basic Concepts Languages Grammars Automata

Languages Alphabet: a finite and nonempty set of symbols  = {a, b} String: finite sequence of symbols from  w = abaaa : empty string *: the set of all strings on  (+ = *  {})

Languages Language: a subset L of * Sentence: a string in L

Languages Example 1:  = {a, b} * = {, a, b, aa, ab, ba, aaa, ...} L1 = {a, aa, aab} (finite language) L2 = {anbn | n  0} = {, ab, aabb, ...}

Languages Language concatenation: L1L2 = {xy | xL1, yL2} Ln = L L ... L (n times) L0 = {}

Languages Example 2: L = {anbn | n  0}

Languages Example 2: L = {anbn | n  0} L2 = {anbnambm | n  0, m  0}

Languages Star-closure: L* = L0  L1  L2 ... Positive closure:

Grammars A grammar for a natural language tells us whether a particular sentence is well-formed or not. <sentence>  <noun-phrase><predicate> <noun-phrase>  <article><noun> <predicate>  <verb> <article>  a | the <noun>  boy | dog <verb>  runs | walks

Grammars Formal grammar: G = (V, T, S, P) V: finite set of variables T: finite set of terminal symbols SV: start variable P: finite set of productions

Grammars Productions: x  y x(VT)+ y(VT)* w = uxv derives z = uyv w  z w1 * wn (w1  w2  ...  wn | w1 = wn) w1 + wn

Grammars Generated language: G = (V, T, S, P) L(G) = {wT* | S * w} Derivation: S  w1  w2  ...  wn  wL(G) Sentential forms: S, w1, w2, ..., wn (containing variables)

Grammars Example 3: G = ({S}, {a, b}, S, P) P: S  aSb S   S  aSb  aaSbb  aabb aabb: sentence aaSbb: sentential form

Grammars Example 3: G = ({S}, {a, b}, S, P) P: S  aSb S  

Grammars Example 3: G = ({S}, {a, b}, S, P) P: S  aSb S   L(G) = {anbn | n  0}

Grammars Example 4: G1 = ({A, S}, {a, b}, S, P1) P1: S  aAb |  A  aAb | 

Grammars Example 4: G1 = ({A, S}, {a, b}, S, P1) P1: S  aAb |  A  aAb |  L(G1) = {anbn | n  0} G and G1 are equivalent

Grammars Example 5: G2 = ({S}, {a, b}, S, P2) P2: S  SS S   S  bSa S  aSb S  bSa

Grammars Example 5: G2 = ({S}, {a, b}, S, P2) P2: S  SS S   S  bSa S  aSb S  bSa L(G2) = {w | na(w) = nb(w)}

Automata An abstract model of digital computer: Input file Control unit Storage Output

Automata Input file: is divided into squares. Input is a string over a given alphabet. Each input square holds a symbol. The symbols are read from left to right, one at a time. The end of the input string can be detected.

Automata Storage: consists of an unlimited number of cells. Each cell can hold a symbol from an alphabet (which can be different from the input alphabet). The contents of the storage cells can be read and changed.

Automata Control unit: has a finite number of internal states. Can be in any one of the internal states. Can change state in some defined manner.

current state  input symbol  storage info  next state Automata Transition function: current state  input symbol  storage info  next state Output may be produced Info in the storage may be changed Configuration: current state  input symbol  storage info Move: current configuration  next configuration

Automata General types of automata: Accepter: yes/no output Transducer: string of symbols as output Deterministic: single move Non-deterministic: multiple moves

Homework Exercises: 4, 5, 6, 8, 9, 12, 15, 17 of Section 1.2 - Linz’s book. Reading: Section 1.3 - Linz’s book. Presentation: Section 2.4 - Linz’s book (procedures mark and reduce).