regular language - How does "δ:Q×Σ→Q" read in the definition of a DFA (deterministic finite automaton)? -


how δ: q × Σ → q in english? describing × , → mean help.

δ a mathematical function called transition function . like. 

z = f(x, y)

a function in mathematical defines mapping of elements in 1 set set. in function set of input arguments called domain of function , output rage.

[answer]   

in expression "δ:q×Σ → q",

× means cartesian product (that set), , → mapping.
"δ:q×Σ → q" says δ transition function defined mapping q×Σ q. where, domain of δ q × Σ , range q.

note: cartesian product mathematical possible order pair (mapping) between 2 sets.

you can say:

δ transition function defined mapping between(or associates) cartesian product of set of statesq , language symbolsΣ set of stateq. abbreviated δ: q×Σ → q

here, q finite set of states , Σ finite set of language symbols.

additionally in automated can represent transition function in tree ways.
1. transition table
2. transition graph or state diagram.
3. transition function: finite set of mapping rules. e.g. {δ(q0, a) → q1, δ(q1, a) → q2}
all same purpose define maping

in dfa. δ:q×Σ → q can written δ(q,Σ) → q it's similar function. in δ function 2 input arguments state q , language symbol Σ , returned value q.

what meaning of δ(q,Σ) → q

suppose in set of transition function δ have element δ(q0, a) → q1 means. if present state q0 consuming a symbol can shift state q1. , state-diagram δ(q0, a) → q1: 

(q0)---a---►(q1)

and transition table is: 

+----+----+ |q\Σ |  | +----+----+ | q0 | q1 | +----+----+

and defines mapping (q0, a) (q1).

some authors write δ ⊆ q×Σ → q in formal dfa definition means δ partial function (not defined on full domain q×Σ). can defined δ on full domain required sometime example find complement dfa. here(complement dfa), wrote 2 dfas same language 1 partial dfa other complement dfa.


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