Week 10 Flashcards
(18 cards)
What is Finite State Automata (FSA)?
What is Finite State Automata (FSA)?
Definition:
A model of computation that changes states based on inputs until reaching a final or acceptor state
Key Points:
Represents systems that react to sequences of inputs
Can be deterministic (DFA) or nondeterministic (NFA)
What are the uses of Finite Automata
Hardware: Circuit design, pattern matching.
Software: Lexical analyzers, language parsing, text search engines.
What are the core concepts in Automata Theory?
Alphabet (Σ):
A finite, nonempty set of symbols.
Example: Σ = {a, b, c, …, z}.
Strings:
Finite sequences of symbols from Σ.
Example: “bccd” is a string over Σ = {a, b, …, z}.
Languages:
A subset of all possible strings over Σ (denoted as Σ*).
Example: L = {abc, def} where Σ = {a, …, z}.
What are the components of a Deterministic Finite Automaton (DFA)?
Finite set of states, Q
Alphabet, Σ
Transition function, δ(p,a)=q.
Start state, q0
Accepting states, F⊂Q
What is a transition function in DFA?
Maps the current state p and input symbol a to a new state q
Notation:
δ(p,a)=q
Example:
If Q = {p,q,r} and Σ = {0,1}
δ(p,0)=q
δ(q,1)=r
What is Nondeterministic Finite Automata (NFA)?
NFAs can transition to multiple states for the same input.
NFAs can transition without any input (epsilon transitions).
Easier to construct but harder to implement directly.
Question: Are NFAs and DFAs equivalent?
Answer: Yes, every NFA has an equivalent DFA.
What is a transition table?
How is a DFA represented in a transition table?
Example table:
State Input: 0 Input: 1
→p q p
q r q
r* r r
Legend:
→: Start state.
∗: Accepting state.
Can you describe an example DFA?
Question: How does a DFA recognize a string of interleaved ‘0’s and ‘1’s starting and ending with ‘1’?
States:
Q = {p,q,r}
q0 = p(start state)
F = {r} (accepting state)
Transition Function:
δ(p,1)=q
δ(q,0)=r
Can you create a DFA for a specific language?
How would you create a DFA for strings where every ‘a’ is followed by a ‘b’ over Σ = {a,b}?
Answer:
Task for practice: Draw a transition table or diagram for this language.
What is a Regular Expression?
A regular expression is an algebraic and declarative description of a deterministic or nondeterministic finite state automaton (DFA or NFA)
It is used to specify patterns in strings, defining languages
What are the main operations in Regular Expressions?
Closure(*)::
Denotes L*, the set of strings formed by concatenating zero or more strings from language L
Example: 0* means any number of 0’s (including none)
Concatenation:
Denoted LM, where L and M are languages
Forms strings by appending any string from L with any string from M
Example: if L = {a} and M = {b}, LM = {ab}
Union(+):
Denotes L + M, where a string can belong to L, M or both
Example: 0 + 1 matches either 0 or 1
What is the precedence of operators in Regular Expressions?
Closure (*) has the highest precedence and applies to the smallest sequence of symbols to its left
Concatenation has the next highest precedence; expressions juxtaposed are grouped together
Union (+) has the lowest precedence; expressions are grouped by union in any order
What does the Closure Operator do?
Denoted L*
Represents all possible strings formed by concatenating zero or more strings from language L
Example: If L = {a} then L* = {ϵ,a,aa,aaa,…}
(ϵ is the empty string)
What does the Concatenation Operator do?
Denoted LM
Forms strings by appending any string from language L, with any string from language M
Example:
if L = {0} and M = {1}, then LM = {01}
What does the Union Operator do?
Denoted L + M
Represents the set of strings that are in L, M or both
Example:
If L = {a} and M = {b}, L + M = {a,b}
Can you provide examples of Regular Expressions?
0* + 1: Matches any number of 0’s or a single 1
101*: Matches 10 followed by any number of 1’s
10 + 01: Matches 10 or 01
0(10)*: Matches 0 followed by any number of 10’s
How do Regular Expressions relate to Automata?
Regular expressions are an algebraic representation of patterns recognized by finite state automata (DFA or NFA)
Each regular expression corresponds to an automaton that accepts the language it defines
What regular expression describes a DFA that recognizes strings starting and ending with 1?
Regular Expression 1(0 + 1)*1
Starts and ends with 1
Contains any combination of 0’s and 1’s in between
