Lecture 8
Deterministic Finite Automata
CS 241: Foundations of Sequential Programs
Spring 2017
Troy Vasiga et al. (where Dan Holtby ∈ al)
University of Waterloo
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Review
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formal languages give a theoretical basis for communication and
organizing processes
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terminology (alphabet, word, language)
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specification vs. recognition
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studying language levels that increase in power/complexity
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regular languages are composed of union, concatenation and
repetition
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Recognizers: Finite Automata
Regular languages can be recognized by finite automata.
We begin with deterministic finite automata, also called DFAs.
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alphabet (Σ)
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finite set of states (or it wouldn’t be a finite automata)
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start state
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accepting (final) states(s)
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transition function (or table, or diagram)
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Finite Automata Example 1
Example: selected opcodes from MIPS assembly language, where
alphabet is the ASCII characters.
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Observations About Finite Automata
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ability to trace
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transitions out of a state are unique
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errors
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size of this language
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DFA M and language L(M)
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Finite Automata Example 2
Example: MIPS labels, where the alphabet is ASCII characters.
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More Finite Automata Examples
Let Σ = {a, b, c}.
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strings with one a, one b and one c
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strings with at least one a
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string with an even number of a’s
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More Finite Automata Examples
Let Σ = {a, b, c}.
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strings with an even number a’s and odd number of b’s
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strings with an even number a’s or odd number of b’s
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More Finite Automata Examples
Let Σ = {a − z}.
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All strings that contain ”dan”
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All strings that contain ”adam”
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DFA summary
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a.k.a. finite state machines
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start state
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final/accepting states
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implicit error state
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accepted and rejected words
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L(M) – the language recognized by DFA M
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Notice that L(M) = L(M 0 ) even though M 6= M 0
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Formal Definition
A DFA is a 5-tuple (Σ, Q, q0 , A, δ) where
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finite alphabet Σ
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finite set of states Q
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start state q0
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set of final/accepting states A ⊆ Q
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transition function: δ : Q × Σ → Q
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DFA Interpreter Algorithm
Input: A word w = w1 w2 ...wn where each wi ∈ Σ
Output: true if accepted, false if rejected
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Implementing DFAs
Need to implement the transition function somehow
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Where are DFAs used?
Everywhere!
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A microwave is a DFA (probably).
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A Turing machine is a DFA plus infinite memory (magnetic tape)
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A CPU is a DFA plus finite memory (registers)
A computer is a CPU (DFA) plus a lot of finite memory (RAM)
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4GB of RAM has 234359738368 states
Not a practical way to model a computer
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