Lecture 1¶
Introduction¶
1.Automata theory¶
- finite automata
- pushdown automata
- Turing machines
2. Compuatability Theory¶
3. Complexity theory: hardness and complexity class.¶
Problem¶
Definition
Optimization Problem
e.g. Given \(G: (V, E, W)\), what is the minimum spanning tree?
Search Problem
e.g. Given \(G\) and an integer \(k\), find a spanning tree whose weight is at most \(k\) or tells such a tree not exist.
Decision Problem
e.g. Given \(G\) and \(k\), is there a spanning tree with weight at most \(k\)?
Counting Problem
e.g. Given \(G\) and \(k\), what is the number of spanning trees with weight at most \(k\)?
Since decision problem is the easiest among these four, and it's relatively easy to answer (just yes or no), the problem we discuss is limited to decision problem. If decision problem is hard, so as others.
If we specify \(G\) and \(k\) of the decision problem above, we have an instance of the problem. If the answer of the instance is yes, then we call it yes-instance, otherwise no-instance.
For computer, we need to encode the problem to make computer understand. So the decision problem above is encoded to
Given a string \(W\), is \(w \in \{\text{encode}<G, k>: <G, k> \text{is a yes-instance}\}\)?
So we abstract the problem to a set, which we call a language.
Definition | Alphabet and String
An alphabet \(\Sigma\) is a finite set of symbols. e.g. \(\Sigma = \{0 , 1\}\), \(\Sigma = \emptyset\).
A string is a finite sequence of symbols from some \(\Sigma\).
- The length of a string \(|w|\) is the number of symbols in \(w\).
- Specially, we use \(e\) to denote an empty string with \(|e| = 0\).
\(\Sigma^i\) is the set of all strings of length \(i\) over \(\Sigma\).
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e.g. for \(\Sigma = \{0, 1\}\), \(\Sigma^0 = \{e\}\), \(\Sigma^1 = \{0, 1\}\), \(\Sigma^2 = \{00, 01, 10, 11\}\).
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Specially, \(\Sigma^* = \bigcup\limits_{i \ge 0} \Sigma^i\), \(\Sigma^+ = \bigcup\limits_{i \ge 1} \Sigma^i\).
Definition | Operation of String
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concatnation
- \(u = a_1\dots a_n, v = b_1\dots b_m\), \(w = uv = a_1\dots a_nb_1\dots b_m\).
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exponentiation
- \(w^i = \underbrace{w \cdots w}_{i\ \text{times}}\).
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reversal
- \(w = a_1\dots a_n\), \(w^R = a_n\dots a_1\).
Definition | Language
A language over \(\Sigma\) is a subset \(L \subseteq \Sigma^*\).
Thesis
Decision problems is equivalent to languages.
Finite Automata¶
The circle with an arrow pointing to it is initial state and the double circle is final state. Initial state is unique while final state can be several.
Definition
A deterministic finite automata (DFA) is \(M = (K, \Sigma, \delta, s, F)\), where
- \(K\) is a finite set of states.
- \(\Sigma\) is the input alphabet.
- \(s \in k\) is initial state.
- \(F \subseteq k\) is the set of final states.
- \(\delta\) is a transition function. It describes in the current state, when read a symbol, what will the next state be.
Definition
A configuration is an element of \(K \times \Sigma^*\) (current state and unread input). It pairs the current state the machine is in with the remaining input that it still has to process.
Yields represents the transition from one configuration to another.
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Yields in one step is a transition to the next state over 1 symbol of input. This is indicated by the yields or right tack symbol: \(\vdash\). For a specific automata \(M\), we denote it as \(\vdash_M\).
- \((q, w) \vdash_M (q', w')\) if \(w = aw'\) for some \(a \in \Sigma\) and \(\delta(q, a) = q'\).
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For abbreviating the transition across multiple steps, we use \(\vdash_M^*\) (Yields).
- \((q, w) \vdash_M^* (q', w')\) if \((q, w) = (q', w')\) or \((q, w) \vdash_M \cdots \vdash_M (q', w')\).
Example
For the automata in upper part of the state diagram above, if the configuration is \((q_0, 1001)\). We can describe the yields as
Definition
\(M\) accepts \(w \in \Sigma^*\) if \((s, w) \vdash_M^* (q, e)\) for some \(q \in F\).
For \(L(M) = \{ w \in \Sigma^*: M \text{ accepts } w \}\), we say \(M\) accepts \(L(M)\).
Definition
A language is regular（正则的） if it is accepted by some FA (finite automata).
Definition
Regular Operations
If \(A\) and \(B\) are languages,
- Union \(A \cup B = \{w | w \in A \text{ or } w \in B\}\).
- Concatnation \(A \cdot B = \{ab | a \in A \text{ and } b \in B\}\) (Also denoted by \(A \circ B\)).
- Kleene Star \(A^* = \{w_1w_2\cdots w_k| w_i \in A \text{ and } k \ge 0\}\).
Closure Property¶
Theorem
If \(A\) and \(B\) are regular, so is \(A \cup B\).
Proof
\(\exists M_A = (K_A, \Sigma, \delta_A, s_A, F_A)\) accepts \(A\), and \(\exists M_B = (K_B, \Sigma, \delta_B, s_B, F_B)\) accepts \(A\).
We construct \(\exists M_\cup = (K_\cup, \Sigma, \delta_\cup, s_\cup, F_\cup)\) where
- \(K_\cup = K_A \times K_B\).
- \(s_\cup = (s_A, s_B)\).
- \(F_\cup = \{(q_A, q_B) \in K_A \times K_B | q_A \in F_A \text{ or } q_B \in F_B\}\).
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\(\delta_\cup\): for any \(q_A \in K_A\), \(q_B \in K_B\), \(a \in \Sigma\),
\[ \delta((q_A, q_B), a) = (\delta_A(q_A, a), \delta_B(q_B, a)) \]
Then \(M_\cup\) accepts \(A \cup B\).
Theorem
If \(A\) and \(B\) are regular, so is \(A \cap B\). If \(A\) is regular, so is \(\overline{A}\).
创建日期: 2023.09.23 14:49:51 CST