Lec 01. Finite-state automata Eunjung Kim F UNDAMENTAL QUESTIONS FOR - - PowerPoint PPT Presentation

COMPUTABILITY AND COMPLEXITY, 2020 FALL SEMESTER

Lec 01. Finite-state automata

Eunjung Kim

FUNDAMENTAL QUESTIONS FOR CS

What do we mean by "computation"? "Computation is to solve a problem by an effective manner." Vague.

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FUNDAMENTAL QUESTIONS FOR CS

What do we mean by "computation"? "Computation is to solve a problem by an effective manner." Vague. What is a computer? What can it do and cannot?

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FUNDAMENTAL QUESTIONS FOR CS

What do we mean by "computation"? "Computation is to solve a problem by an effective manner." Vague. What is a computer? What can it do and cannot? Can anyone on earth devise a fundamentally more powerful computer? (An alien? In another universe?)

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COMPUTATION: WHAT AND HOW

Any well-formulated information can be represented as a string of 0 and 1, or any finite alphabet Σ. The object for computation can be stated as a function.

COMPUTE WHAT

computational problem ⇔ compute a function f : {0, 1}∗ → {0, 1}∗.

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COMPUTATION: WHAT AND HOW

Compute HOW Let us agree: "computing a function f" means "there is an effective methodalgorithm which outputs f(x) for each input x". The concept of "algorithm" is still vague.

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COMPUTATION: WHAT AND HOW

What do we expect for an algorithm, intuitively? a finite number of finitely describable instructions. each instruction and what to do next are unambiguous. all the basic operation should be executable by the concerned executor. terminates at some point (i.e. in finite number of steps)

TOWARD A RIGOROUS NOTION OF ALGORITHM

A mathematically rigorous description of an executor (computing device/machine...) and instructions is needed.

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MODEL OF COMPUTATION

Exercutor(machine) constituents: an alphabet Σ it recognizes, a gadget to read an input x ∈ Σ∗, a finite set of states to recognize its status ("where am I?"), memory to write and read later. Basic operation: read one alphabet from input tape (or from memory), update its internal state, move the header (only in one fixed direction, or both direction, or neither) on input tape or memory, write/change on memory tape.

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SET-UP

Concatenation xy of x and y. Cartesian product A × B Notations: Σ, Σi, ǫ, Σ∗. Computing a function f : Σ∗ → Γ∗ means ... Special function f : Σ∗ → {0, 1} language. Language: a subset A of Σ∗, indicator function fA. Computing fA ⇔ membership test for A

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FINITE (STATE) AUTOMATA

Example: vending machine Model of computation mimicking a simple computing device no/limited memory, basic operations: read one symbol from the input, update the state, and move to the next position in input.

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STATE DIAGRAM

Figure 1.4, Sipser 2012.

STRINGS ACCEPTED BY M

The set of all w ∈ {0, 1}∗ such that...

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FORMAL DEFINITION

A FINITE AUTOMATA IS A 5-TUPLE (Q, Σ,δ, q0, F)

Q a finite set called the states, Σ a finite set called the alphabet,

δ a function from Q × Σ to Q called the transition function,

q0 ∈ Q the start state, F ⊆ Q the set of accept states.

Figure 1.4, Sipser 2012.

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LANGUAGE RECOGNIZED BY FA

DEFINITION

Let M be a finite automata. A string w ∈ Σ∗ is accepted by a finite automata M if M ends in an accept state upon w. L(M) denotes the set of all strings accepted by M. A language A is said to be recognized by M if A = L(M).

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EXAMPLES OF FINITE AUTOMATA

Figure 1.7, 9, 12 from Sipser 2012.

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FORMAL DEFINITION OF COMPUTATION

Configuration of a finite automata M = (Q, Σ,δ, q0, F) is a pair (q, w) ∈ Q × Σ∗. We interpret a configuration (q, w) as... (q, w) M (q′, w′) if... A sequence of configuration is a computation history if... (q, w) ∗

M (q′, w′) if there is...

A sequence of configurations is an accepting computation history if...

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LANGUAGE RECOGNIZED BY FA

REVISITED

DEFINITION: LANGUAGE RECOGNIZED BY FA

Let M be a finite automata. A string w ∈ Σ∗ is accepted by a finite automata M if M ends in an accept state upon w: using the formal notation, if there is an accepting computation history starting from (q0, w) ending in (q, ǫ) for some q ∈ F. L(M) denotes the set of all strings accepted by M. A language A is said to be recognized by M if A = L(M).

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REGULAR OPERATIONS

FIVE OPERATIONS

Languages A, B over an alphabet Σ, we define the following opera- tions. Union: A ∪ B = {w : w ∈ A or w ∈ B}. Concatenation: A ◦ B = {xy : x ∈ A, y ∈ B}. Kleene star: A∗ =

k≥0{w1w2 · · · wk : wi ∈ A}.

Complementation: ¯ A = {w ∈ Σ∗ : w / ∈ A}. Intersection: A ∩ B = {w : w ∈ A and w ∈ B}.

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CLOSURE UNDER OPERATIONS

THEOREM

The class of all languages recognizable by FA is closed under the union operation (concatenation, Kleene star, complementation, inter- section).

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REFERENCE

Lots of interesting chronicle and discussion on the search for definition of algorithm/computation can be found online, for example, here, here.

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