So far... A language is a set of strings over an alphabet. We have defined languages by: (i) regular expressions (ii) finite state automata Both (i) and (ii) give us exactly the same class of languages. Languages serve two purposes in computing: (a) communicating instructions or information (b) defining valid communications What about languages outwith this class?

Specifying Non-Regular Languages We have already seen a number of languages that are not regular. In particular, {a n b n : n ≥ 0} the language of matched round brackets arithmetic expressions standard programming languages are not regular. However, these languages are all systematic constructions, and can be clearly and explicitly defined. Consider L = {a n b n : n ≥ 0}: (i)  L (ii) if x  L, then axb  L (iii) nothing else is in L This is a clear and concise specification of L. Can we use it to generate members of L?

Generating Languages Using the previous definition of L, and the notion of string substitution, we can give a generative definition of L. Let X be a new symbol. 1) X -> 2) X -> aXb This definition says that if we have a symbol X, we can replace it by the empty string, or by aXb. We now define L to be all strings over {a,b} formed by starting with X and applying rules 1) and 2) until we get a string with no X's. Example: X => aXb => aaXbb => aabb X => X => aXb => aaXbb => aaaXbbb => aaabbb

Grammar Formalising the previous notion of a generative definition based on string substitution, we get: A grammar is a 4-tuple, G = (N, T, S, P), where N is a finite alphabet called the non-terminals; T is a finite alphabet, called the terminals; N  T =  ; S  N is the start symbol; and P is a finite set of productions of the form , where  (N  T) +,  has at least one member from N, and   (N  T)* Thus the previous example is a grammar where N = {X} T = {a, b} S = X P = { X ->, X -> aXb} so G = ({X}, {a,b}, {X}, {X ->, X -> aXb})

Definitions and Notation Let G = (N,T,S,P) be a grammar. If s, t, x, y, u and v are strings s.t. s = xuy, t = xvy, and (u -> v )  P then s directly derives t., written s => t. If there is a sequence of strings s 0, s 1,..., s n s.t. s 0 => s 1 =>... => s n-1 => s n, then s 0 derives s n, written s 0 =>* s n. A sentential form of G is a string w  (N  T)* s.t. S =>* w. A sentence of G is a sentential form w  T* i.e. one with no non-terminals. The language defined by G is the set of all sentences of G, denoted L(G). aaaSbbb => aaaaSbbbb. S =>* aaaabbbb. aaaSbbb is a sentential form of G aaaabbbb is a sentence of G. L(G) = {, ab, aabb, aaabbb,...}, which is {a n b n : n ≥ 0}

Definitions and Notation (cont.) Notation: we normally order the set of productions, and assign them numbers. If x => y by using rule number i, then we write x => i y  ->  1 |  2 |  3... |  n is shorthand for  ->  1  ->  2 :  ->  n In general, non-terminals will be uppercase, while terminals will be lowercase. A context-free grammar (CFG) is one in which all productions are of the form  -> , where   N - i.e. the left-hand side is a single non-terminal. A context-free language (CFL) is one that can be defined by a context-free grammar.

Context-Free Grammars A CFG is called context-free because the left-hand side of all productions contain only single symbols, and so a production can be applied to a symbol without needing to consider the symbol's context. We only consider context-free grammars in this course. Some languages are not context-free. Example: {a n b n c n : n ≥ 0} Some languages cannot be defined by any grammar. It is believed that these are the same languages that cannot be defined by any algorithm or effective procedure.

Example CFG G = ({S}, {a, +, *, (, )}, S, { S -> S+S | S*S | (S) | a} )

Example CFG G = ({S}, {a, +, *, (, )}, S, { S -> S+S | S*S | (S) | a} ) This is a grammar of algebraic expressions. The productions are: 1) S -> S + S 2) S -> S * S 3) S -> (S) 4) S -> a. Example derivation: S => S * S => a * S => a * (S) => a * (S + S) => a * (a + S) => a * (a + a). Note that there are many other ways of deriving the same string.

Why Grammar? In English, the grammar is the set of conventions defining the structure of sentences - e.g. a sentence must have a subject and an object verbs must agree with nouns e.g. "John walks" & "John and Mary walk" adjectives come before nouns e.g. "the red car" and not "the car red" We have shown a formalisation of this notion. We now can write explicit clear statements of what sentences are in a language. Grammars can be used in the processing of natural language by computer (4th year option), in formalising design, in pattern recognition, and many other areas.

A grammar for a small part of English S -> NP VP NP -> Det NP1 | PN NP1 -> Adj NP1| N Det -> a | the PN -> peter | paul | mary Adj -> large | black N -> dog | cat | horse VP -> V NP V -> is | likes | hates Can you derive: peter is a large black cat

A grammar for a small part of English S -> NP VP NP -> Det NP1 | PN NP1 -> Adj NP1| N Det -> a | the PN -> peter | paul | mary Adj -> large | black N -> dog | cat | horse VP -> V NP V -> is | likes | hates Example derivations: S => NP VP => PN VP => mary VP => mary V NP => mary hates NP => mary hates Det NP1 => mary hates the NP1 => mary hates the N => mary hates the dog S => NP VP => NP V NP => NP V Det NP1 => NP V a NP1 => NP V a Adj NP1 => NP is a Adj NP1 => NP is a Adj Adj NP1 => NP is a large Adj NP1 => NP is a large Adj N => NP is a large black N => NP is a large black cat => PN is a large black cat => peter is a large black cat

Regular Grammars A grammar is regular if each production is of the form: (i) A -> t or (ii) A -> tB (iii) A -> where A, B  N, t  T. Example: S -> aA | bB A -> aS | a B -> bS | b Is this s sentence of the language? aaaabb

Regular Grammars A grammar is regular if each production is of the form: (i) A -> t or (ii) A -> tB (iii) A -> where A, B  N, t  T. Example: S -> aA | bB A -> aS | a B -> bS | b S => aA => aaS => aaaA => aaaaS => aaaabB => aaaabb

Regular Grammars A grammar is regular if each production is of the form: (i) A -> t or (ii) A -> tB (iii) A -> where A, B  N, t  T. Example: S -> aA | bB A -> aS | a B -> bS | b S => aA => aaS => aaaA => aaaaS => aaaabB => aaaabb The language generated by this grammar is the language denoted by …..

Regular Grammars A grammar is regular if each production is of the form: (i) A -> t or (ii) A -> tB (iii) A -> where A, B  N, t  T. Example: S -> aA | bB A -> aS | a B -> bS | b S => aA => aaS => aaaA => aaaaS => aaaabB => aaaabb The language generated by this grammar is the language denoted by (aa + bb) +

Regular Grammars and Regular Languages Thus we now have three different definitions of the one class of languages:  regular expressions  finite state automata  regular grammars Theorem: (stated here without proof) A language is regular iff it can be defined by a regular grammar. All three are useful in Computing Science

Example CFG (2) 1) S -> XaaX 2) X -> aX 3) X -> bX 4) X -> S => XaaX => bXaaX => baXaaX => babXaaX => babaaX => babaaaX => babaaabX => babaaab This grammar defines the language: ………

Example CFG (2) 1) S -> XaaX 2) X -> aX 3) X -> bX 4) X -> S => XaaX => bXaaX => baXaaX => babXaaX => babaaX => babaaaX => babaaabX => babaaab This grammar defines the language (a + b)*aa(a + b)*

...as a Regular Grammar 1) S -> aS 2) S -> bS 3) S -> aM 4) M -> aB 5) B -> aB 6) B -> bB 7) B -> S => bS => baS => babS => babaM => babaaB => babaaaB => babaaabB => babaaab S => bS => baM => baaB => baa

Backus-Naur Form A notation devised for defining the language Algol 60. PASCAL syntax rules are often presented in this form. Example: ::= ::= real | integer | boolean ::= identifier | identifier This formalism is equivalent to CFG's, where names enclosed in are non-terminals, names in bold are terminals, and ::= is the same as the -> notation.

Constructing Grammars Suppose we wanted to construct a grammar for the language of all strings of the form accc...cb or abab...abcc....cabab...ab n times We need to find rules to create: (i) sequences of strings - ccc....c (ii) bracketed strigs - accc...cb, and (iii) nested strings - abab...ab abab...ab Sequencing A -> aA | or A -> Aa |  e.g. A => aA => aaA =>... => aaaaaA => aaaaa Bracketing A -> aBb or A -> Bb B ->  xB  B -> ax | Bx e.g. A => aBb => axBb => axxBb =>... => axxxxxb

S -> abSab | abBab B -> cB | c What language does this generate? (Say it precisely) Constructing Grammars (cont.) Nesting A -> aAb | B B -> xB | e.g. A => aAb => aaAbb => aaaAbbb =>... => aaaaaAbbbbb => aaaaaBbbbbb =>... => aaaaaxxxBbbbbb => aaaaaxxxbbbbb Example:

S -> abSab | abBab B -> cB | c What language does this generate? The language (ab) n +c m +(ab) n (where n>0 and m>0) Constructing Grammars (cont.) Nesting A -> aAb | B B -> xB | e.g. A => aAb => aaAbb => aaaAbbb =>... => aaaaaAbbbbb => aaaaaBbbbbb =>... => aaaaaxxxBbbbbb => aaaaaxxxbbbbb Example:

S -> abSab | abBab B -> cB | c Example derivations: S => abBab => abcBab =>... abccccab S => abSab => ababSabab =>abababSababab => abababBababab => abababcBababab =>... => abababccccababab Constructing Grammars (cont.) Nesting A -> aAb | B B -> xB | e.g. A => aAb => aaAbb => aaaAbbb =>... => aaaaaAbbbbb => aaaaaBbbbbb =>... => aaaaaxxxBbbbbb => aaaaaxxxbbbbb Example: