Research design and methods

Basic concepts 

Grammars 

Research design and methods 

Ivona Kučerová 

COGSCIL 750 

September 21, 2009 

Ivona Kučerová: Research design and methods COGSCIL 750


Grammars 

◮ what does it mean when we say that a sentence is 

grammatical? 



Grammars 


grammatical? 

◮ in technical sense, a sentence is grammatical only with respect 

to a grammar 



Grammars 


grammatical? 

◮ in technical sense, a sentence is grammatical only with respect 

to a grammar 

◮ a sentence is grammatical if it can be generated by the 

grammar 



Grammars 

◮ a natural language as a set of strings 



Grammars 


◮ the task: an explicit device that can decide whether a string 

belongs to the set or not 



Grammars 


◮ the task: an explicit device that can decide whether a string 

belongs to the set or not 

◮ we will call such a device grammar 



Grammars 

Strings 

◮ Given a finite set A, a string on (or over) A is a finite 

sequence of occurrences from A 



Grammars 

Strings 

◮ A = {a,b,c} 

◮ examples of strings on A: 

◮ acbaab 

◮ aacbbac 

◮ bbaccaab 

◮ . . . 



Grammars 

Strings 

◮ strings are finite 

◮ A is assumed to be finite 



Grammars 

Strings 

◮ the set from which strings are formed, A, is called vocabulary 

or alphabet 



Grammars 

Strings 

◮ the length of a string is the number of occurrences of symbols 

in it (number of tokens!) 

◮ for example, the length of aabbcab is 7 



Grammars 

Strings 

◮ careful: a string is a linearly ordered sequence 

◮ but not a linearly ordered set 



Grammars 

Strings 

◮ we can define a string of length n over the alphabet A to be a 

function mapping the first n positive integers into A 



Grammars 

Strings 

◮ An example 

◮ acbaab can be defined as the function 

{< 1,a >,< 2,c >,< 3,b >,< 4,a >,< 5,a >,< 6,b >} 



Grammars 

Strings 

◮ An example 

◮ acbaab can be defined as the function 

{< 1,a >,< 2,c >,< 3,b >,< 4,a >,< 5,a >,< 6,b >} 

◮ How do you define bbcabca? 



Grammars 

Strings 

◮ a special string is the unique string of length 0 

◮ empty string 

◮ e 



Grammars 

Operations on strings 

◮ concatenation is associative: 

◮ for any string alpha, β, γ, (α ⌢ β) ⌢ γ = α ⌢ (β ⌢ γ) 



Grammars 


◮ concatenation is not commutative, since in general 

α ⌢ β ≠ β ⌢ α 



Grammars 


◮ A ∗ = def the set of all strings over A 



Grammars 


◮ reversal 

◮ x R def the reversal of a string x 

◮ (acbab) R = babca 

◮ e R = e 



Grammars 


(1) Given an alphabet A 

a. If x is a string of length 0, then x R = x (i.e., e R = e) 

b. Kf x is a string of length k + 1, then it is of the form 

wa, where α ∈ A and w ∈ A ∗ , then 

x R = (wa) R = aw R . 



Grammars 


◮ the relation between concatenation and reversal: 

(2) For all strings x and y, (x ⌢ y) R = y R ⌢ x R 

(3) (bca ⌢ ca) R = (ca) R ⌢ (bca) R = ac ⌢ acb = acacb 



Grammars 


◮ given a string x, a substring of x is any string formed from 

contiguous occurrences of symbols in x taken in the same 

order in which they occur in x 

◮ formally, y is a substring of x iff there exists strings z and w 

such that x = z ⌢ y ⌢ w 

◮ notice that both z and w may be empty 



Grammars 


◮ prefix def : an initial substring 

◮ suffix def : a final substring 



Grammars 

Language 

◮ Language (over a vocabulary A) def : any subset of A ∗ 



Grammars 

Language 

◮ A ∗ is a denumerably infinite set 

◮ the cardinality of A ∗ (|A ∗ |) is ℵ 0 

◮ its power set, i.e., the set of all languages over A, has 

cardinality 2 ℵ 0 

◮ the set of all languages over A is nondenumerably infinite 



Grammars 

Language 

◮ learnability? 



Grammars 

Language 

◮ not every language can be characterized by a finite device (a 

grammar) 



Grammars 

Language 

◮ when we study formal languages, we study a scale of possible 

complex strings that can be distinguished by finite resources 

◮ we will talk about hierarchy of grammars and languages 



Grammars 

Grammars 

◮ a formal grammar is a deductive system of axioms and rules of 

inference, which generates the sentences of a language as its 

theorems 

◮ by convention, a grammar contains: 

◮ one axiom – the initial symbol (S) 

◮ finite number of rules of the form φ → ω, where φ and ω are 

strings (R) 



Grammars 

Grammars 

◮ grammars have two alphabets: 

◮ a terminal alphabet (VT ) 

◮ a non-terminal alphabet (VN ) 

◮ the strings we are interested in are strings over the terminal 

alphabet 



Grammars 

An example 

V T (the terminal alphabet) = {a,b} 

V N (the nonterminal alphabet) = {S,A,B} 

S (the initial symbol → a member of V N ) 

R(the set of rules) = 

{ 

S → ABS 

S → e 

AB → BA 

BA → AB 

A → a 

B → b 

} 



Grammars 

Grammars 

(4) Let Σ = V T ∪ V N . A formal grammar G is a quadruple 

< V T ,V N ,S,R >, where V T and V N are finite disjoint 

sets, S is a distinguished member of V N , and R is a finite 

set of ordered pairs in Σ ∗ V N Σ × Σ ∗ . 



Grammars 

Grammars 

(5) Given a grammar G =< V T ,V N ,S,R >, a derivation is a 

sequence of strings x 1 ,x 2 ,...,x n (n ≥ 1) such that x 1 = S 

and for each x i (2 ≤ i ≤ n), x i is obtained from x i−1 by one 

application of some rule in R. 



Grammars 

Grammars 

(6) A grammar G generates a string x ∈ VT ∗ if there is a 

derivation x 1 ,....,x n by G such that x n = x. 



Grammars 

Grammars 

(7) The language generated by a grammar G, denoted L(G), is 

the set of all strings generated by G.

Research design and methods

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?