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[MA375] 6.1-7.2: Strings, FSMs, ... , Exam 1 Material: FORMAL Definitions to Know
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Terms in this set (11)
Kleene closure
For a given language A over ∑,
A* = A⁺ ∪ {λ} is the ___ of A.
Positive closure
Concatenation
For an alphabet ∑ and languages A and B over ∑, this operation is applied:
AB = {ab|a∈A, b∈B}
Finite State Machine
A 5-tuple M={S,I,O,v,w}, where S is the set of internal machine states, I is the input alphabet, O is the output alphabet,
v is a *function* which takes in current state and the input to tell (spit out) the next state of the machine (i.e. v : S × I → S),
and w is a *function* which takes in the current state and the input to tell (spit out) the output of the machine (i.e. w : S × I → O).
Regular language
A language that can be written using Kleene closures, unions, and concatenations. These are the ONLY languages that can be recognized by FSMs.
Composite relation
If A,B,and C are sets and we have relations R₁⊆ A × B and R₂⊆B × C, then this is a relation from A to C defined by:
R₁∘R₂ = {(x,z)|∃y ((x,y)∈R₁ and (y,z)∈R₂)}.
Aⁿ
For a given language A over ∑,
___ = {w₁,w₂,...,wₙ|wᵢ∈A (i∈ℤ⁺)}
Reflexive
a relation R on A is ____ if (a,a) ∈ R for all a ∈ A
Symmetric
∀x,y∈A if (x,y)∈R then (y,x)∈R
Transitive
∀x,y,z∈A if (x,y)∈R and (y,z)∈R then (x,z)∈R
Antisymmetric
∀x,y∈A if (x,y)∈R and (y,x)∈R, then x=y.
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