Automata and complexity

Slides

Intro

Some problems are undecidable: program termination, post correspondence problem

Some are NP-complete - travelling salesman, satisfiability in prop. logic

Word: finite sequence of symbols from finite alphabet Σ

conventions:

• symbols: a, b, c
• words: u, v, w, x, y, z
• empty word: λ

computer program is a word, takes input word, produces output word

can concatenate words, |v| means length of word

power v^k is k concatenations of v’s

reverse (a₁ … a_n)^R = a_n … a₁

Formal languages

Formal language: set of words.

Σ^* is set of all words over Σ, formal language L is subset of Σ^*

Since it’s a set, the usual set operations have meaning (complement, union, intersection, etc.)

nth power of language is Lⁿ⁺¹ = Lⁿ L

Kleene star: $L^{* = \bigcup_{i=0}}{\infty} L^{i$, $L}+ = \bigcup_{i=1}^{{\infty} L}i$

$\overline{L}^{R$ == $\overline{L}R}$

But sets are not the best for language description, so let’s use something else.

Deterministic Finite Automata (DFA)

Consists of:

• Q: finite set of states
• Σ: finite input alphabet
• δ : Q × Σ → Q: transition function
• q₀ ∈ Q: starting state
• set F ⊂ Q of final states

If automaton in state q reads symbol a, resulting state is δ(q, a).

$(q, aw) \vdash (q’, w) \quad if \quad \delta(q, a) = q’$

You can write transition function δ in form of table:

DFA transition graphs

DFA can be visualised as transition graph:

• states are nodes of graph
• arows with labels from Σ
• starting state: extra incoming arrow
• final states: double circle
• arrow q → q' with label a iff δ(q, a) = q'

if $(q, w) \vdash^* (q’, λ)$, can write $q \xrightarrow{w} q’$

A DFA defines a regular language based on what it accepts (or rejects).

To get a complement, you can just invert final states.

Determinism

Deterministic: for every state and symbol, any state q has only one outgoing arrow with label a

For every input word, there's exactly one path from starting state through transition graph

Theorems for regular languages

A language is regular if there is a DFA M = (Q, Σ, δ, q₀, F) with L(M) = L

If L regular, $\overline{L}$ also regular.

If L₁ and L₂ regular, then L₁ ∪ L₂ regular.

If L regular, L^R regular.

Every finite language is regular.