Fix some finite set of symbols \(\Sigma\). We define a string over \(\Sigma\) recursively as follows.

• Base case: The empty sequence, denoted \(\epsilon\), is a string.

• Recursive rule: If \(x\) is a string and \(a \in \Sigma\), then \(ax\) is a string.

We define a binary tree recursively as follows.

• Base case: A single node \(r\) is a binary tree with root \(r\).

• Recursive rule: If \(T'\) and \(T''\) are binary trees with roots \(r_1\) and \(r_2\), then \(T\), which has a node \(r\) adjacent to \(r_1\) and \(r_2\) is a binary tree with root \(r\).

  

An alphabet is a non-empty, finite set, and is usually denoted by \(\Sigma\). The elements of \(\Sigma\) are called symbols or characters.

Given an alphabet \(\Sigma\), a string (or word) over \(\Sigma\) is a (possibly infinite) sequence of symbols, written as \(a_1a_2a_3\ldots\), where each \(a_i \in \Sigma\). The string with no symbols is called the empty string and is denoted by \(\epsilon\).

The length of a string \(w\), denoted \(|w|\), is the number of symbols in \(w\). If \(w\) has an infinite number of symbols, then the length is undefined.

Let \(\Sigma\) be an alphabet. We denote by \(\Sigma^*\) the set of all strings over \(\Sigma\) consisting of finitely many symbols. Equivalently, using set notation, \[\Sigma^* = \{a_1a_2\ldots a_n : \text{ $n \in \mathbb N$, and $a_i \in \Sigma$ for all $i$}\}.\]

The reversal of a string \(w = a_1a_2\ldots a_n\), denoted \(w^R\), is the string \(w^R = a_na_{n-1}\ldots a_1\).

The concatenation of strings \(u\) and \(v\) in \(\Sigma^*\), denoted by \(uv\) or \(u \cdot v\), is the string obtained by joining together \(u\) and \(v\).

For \(n \in \mathbb N\), the \(n\)’th power of a string \(u\), denoted by \(u^n\), is the word obtained by concatenating \(u\) with itself \(n\) times.

We say that a string \(u\) is a substring of string \(w\) if \(w = xuy\) for some strings \(x\) and \(y\).

For strings \(x\) and \(y\), we say that \(y\) is a prefix of \(x\) if there exists a string \(z\) such that \(x = yz\). If \(z \neq \epsilon\), then \(y\) is a proper prefix of \(x\).

We say that \(y\) is a suffix of \(x\) if there exists a string \(z\) such that \(x = zy\). If \(z \neq \epsilon\), then \(y\) is a proper suffix of \(x\).

Let \(A\) be a set and let \(\Sigma\) be an alphabet. An encoding of the elements of \(A\), using \(\Sigma\), is an injective function \(\text{Enc}: A \to \Sigma^*\). We denote the encoding of \(a \in A\) by \(\left \langle a \right\rangle\).

If \(w \in \Sigma^*\) is such that there is some \(a \in A\) with \(w = \left \langle a \right\rangle\), then we say \(w\) is a valid encoding of an element in \(A\).

A set that can be encoded is called encodable.

Let \(\Sigma\) be an alphabet. Any function \(f: \Sigma^* \to \Sigma^*\) is called a function problem over the alphabet \(\Sigma\).

Let \(\Sigma\) be an alphabet. Any function \(f: \Sigma^* \to \{0,1\}\) is called a decision problem over the alphabet \(\Sigma\). The co-domain of the function is not important as long as it has two elements. One can think of the co-domain as \(\{\text{No}, \text{Yes}\},\) \(\{\text{False}, \text{True}\}\) or \(\{ \text{Reject}, \text{Accept}\}\).

Any (possibly infinite) subset \(L \subseteq \Sigma^*\) is called a language over the alphabet \(\Sigma\).

The reversal of a language \(L \subseteq \Sigma^*\), denoted \(L^R\), is the language \[L^R = \{w^R \in \Sigma^* : w \in L\}.\]

The concatenation of languages \(L_1, L_2 \subseteq \Sigma^*\), denoted \(L_1L_2\) or \(L_1 \cdot L_2\), is the language \[L_1L_2 = \{uv \in \Sigma^* : u \in L_1, v \in L_2\}.\]

For \(n \in \mathbb N\), the \(n\)’th power of a language \(L \subseteq \Sigma^*\), denoted \(L^n\), is the language obtained by concatenating \(L\) with itself \(n\) times, that is, \[L^n = \underbrace{L \cdot L \cdot L \cdots L}_{n \text{ times}}.\] Equivalently, \[L^n = \{u_1u_2\cdots u_n \in \Sigma^* : u_i \in L \text{ for all } i \in \{1,2,\ldots,n\}\}.\]

The star of a language \(L \subseteq \Sigma^*\), denoted \(L^*\), is the language \[L^* = \bigcup_{n \in \mathbb N} L^n.\] Equivalently, \[L^* = \{u_1u_2\cdots u_n \in \Sigma^* : n \in \mathbb N, u_i \in L \text{ for all } i \in \{1,2,\ldots,n\}\}.\] Note that we always have \(\epsilon\in L^*\). Sometimes we may prefer to not include \(\epsilon\) by default, so we define \[L^+ = \bigcup_{n \in \mathbb N^+} L^n.\]

We denote by \(\mathsf{ALL}\) the set of all languages (over the default alphabet \(\Sigma = \{{\texttt{0}}, {\texttt{1}}\}\)).

A deterministic finite automaton (DFA) \(M\) is a \(5\)-tuple \[M = (Q, \Sigma, \delta, q_0, F),\] where

• \(Q\) is a non-empty finite set (which we refer to as the set of states of the DFA);

• \(\Sigma\) is a non-empty finite set (which we refer to as the alphabet of the DFA);

• \(\delta\) is a function of the form \(\delta: Q \times \Sigma \to Q\) (which we refer to as the transition function of the DFA);

• \(q_0 \in Q\) is an element of \(Q\) (which we refer to as the start state of the DFA);

• \(F \subseteq Q\) is a subset of \(Q\) (which we refer to as the set of accepting states of the DFA).

Let \(M = (Q, \Sigma, \delta, q_0, F)\) be a DFA and let \(w = w_1w_2 \cdots w_n\) be a string over an alphabet \(\Sigma\) (so \(w_i \in \Sigma\) for each \(i \in \{1,2,\ldots,n\}\)). The computation path of \(M\) with respect to \(w\) is a specific sequence of states \(r_0, r_1, r_2, \ldots , r_n\) (where each \(r_i \in Q\)). We’ll often write this sequence as follows.

 
These states correspond to the states reached when the DFA reads the input \(w\) one character at a time. This means \(r_0 = q_0\), and \[\forall i \in \{1,2,\ldots,n\}, \quad \delta(r_{i-1},w_i) = r_i.\] An accepting computation path is such that \(r_n \in F\), and a rejecting computation path is such that \(r_n \not\in F\).

We say that \(M\) accepts a string \(w \in \Sigma^*\) (or “\(M(w)\) accepts”, or “\(M(w) = 1\)”) if the computation path of \(M\) with respect to \(w\) is an accepting computation path. Otherwise, we say that \(M\) rejects the string \(w\) (or “\(M(w)\) rejects”, or “\(M(w) = 0\)”).

Let \(f: \Sigma^* \to \{0,1\}\) be a decision problem and let \(M\) be a DFA over the same alphabet. We say that \(M\) solves (or decides, or computes) \(f\) if the input/output behavior of \(M\) matches \(f\) exactly, in the following sense: for all \(w \in \Sigma^*\), \(M(w) = f(w)\).

If \(L\) is the language corresponding to \(f\), the above definition is equivalent to saying that \(M\) solves (or decides, or computes) \(L\) if the following holds:

• if \(w \in L\), then \(M\) accepts \(w\) (i.e. \(M(w) = 1\));

• if \(w \not \in L\), then \(M\) rejects \(w\) (i.e. \(M(w) = 0\)).

A language \(L \subseteq \Sigma^*\) is called a regular language if there is a deterministic finite automaton \(M\) that solves \(L\).

We denote by \(\mathsf{REG}\) the set of all regular languages (over the default alphabet \(\Sigma = \{{\texttt{0}}, {\texttt{1}}\}\)).

Let \(M = (Q, \Sigma, \delta, q_0, F)\) be a DFA. We define the generalized transition function \(\delta_\wp: \wp(Q) \times \Sigma \to \wp(Q)\) as follows. For \(S \subseteq Q\) and \(\sigma \in \Sigma\), \[\delta_{\wp}(S, \sigma) = \{\delta(q, \sigma) : q \in S\}.\]

A Turing machine (TM) \(M\) is a 7-tuple \[M = (Q, \Sigma, \Gamma, \delta, q_0, q_\text{acc}, q_\text{rej}),\] where

• \(Q\) is a non-empty finite set
  (which we refer to as the set of states of the TM);

• \(\Sigma\) is a non-empty finite set that does not contain the blank symbol \(\sqcup\)
  (which we refer to as the input alphabet of the TM);

• \(\Gamma\) is a finite set such that \(\sqcup\in \Gamma\) and \(\Sigma \subset \Gamma\)
  (which we refer to as the tape alphabet of the TM);

• \(\delta\) is a function of the form \(\delta: Q \times \Gamma \to Q \times \Gamma \times \{\text{L}, \text{R}\}\)
  (which we refer to as the transition function of the TM);

• \(q_0 \in Q\) is an element of \(Q\)
  (which we refer to as the initial state of the TM or the starting state of the TM);

• \(q_\text{acc}\in Q\) is an element of \(Q\)
  (which we refer to as the accepting state of the TM);

• \(q_\text{rej}\in Q\) is an element of \(Q\) such that \(q_\text{rej}\neq q_\text{acc}\)
  (which we refer to as the rejecting state of the TM).

Let \(M\) be a Turing machine where \(Q\) is the set of states, \(\sqcup\) is the blank symbol, and \(\Gamma\) is the tape alphabet. To understand how \(M\)’s computation proceeds we generally need to keep track of three things: (i) the state \(M\) is in; (ii) the contents of the tape; (iii) where the tape head is. These three things are collectively known as the “configuration” of the TM. More formally: a configuration for \(M\) is defined to be a string \(uqv \in (\Gamma \cup Q)^*\), where \(u, v \in \Gamma^*\) and \(q \in Q\). This represents that the tape has contents \(\cdots \sqcup\sqcup\sqcup uv \sqcup\sqcup\sqcup\cdots\), the head is pointing at the leftmost symbol of \(v\), and the state is \(q\). A configuration is an accepting configuration if \(q\) is \(M\)’s accept state and it is a rejecting configuration if \(q\) is \(M\)’s reject state.

Suppose that \(M\) reaches a certain configuration \(\alpha\) (which is not accepting or rejecting). Knowing just this configuration and \(M\)’s transition function \(\delta\), one can determine the configuration \(\beta\) that \(M\) will reach at the next step of the computation. (As an exercise, make this statement precise.) We write \[\alpha \vdash_M \beta\] and say that “\(\alpha\) yields \(\beta\) (in \(M\))”. If it is obvious what \(M\) we’re talking about, we drop the subscript \(M\) and just write \(\alpha \vdash \beta\).

Given an input \(x \in \Sigma^*\) we say that \(M(x)\) halts if there exists a sequence of configurations (called the computation path) \(\alpha_0, \alpha_1, \dots, \alpha_{T}\) such that:

1. \(\alpha_0 = q_0x\), where \(q_0\) is \(M\)’s initial state;

2. \(\alpha_t \vdash_M \alpha_{t+1}\) for all \(t = 0, 1, 2, \dots, T-1\);

3. \(\alpha_T\) is either an accepting configuration (in which case we say \(M(x)\) accepts) or a rejecting configuration (in which case we say \(M(x)\) rejects).

Otherwise, we say \(M(x)\) loops.

Let \(f: \Sigma^* \to \{0,1\}\) be a decision problem and let \(M\) be a TM with input alphabet \(\Sigma\). We say that \(M\) solves (or decides, or computes) \(f\) if the input/output behavior of \(M\) matches \(f\) exactly, in the following sense: for all \(w \in \Sigma^*\), \(M(w) = f(w)\).

If \(L\) is the language corresponding to \(f\), the above definition is equivalent to saying that \(M\) solves (or decides, or computes) \(L\) if the following holds:

• if \(w \in L\), then \(M\) accepts \(w\) (i.e. \(M(w) = 1\));

• if \(w \not \in L\), then \(M\) rejects \(w\) (i.e. \(M(w) = 0\)).

If \(M\) solves a decision problem (language), \(M\) must halt on all inputs. Such a TM is called a decider.

A language \(L\) is called decidable (or computable) if there exists a TM that decides (i.e. solves) \(L\).

We denote by \(\mathsf{R}\) the set of all decidable languages (over the default alphabet \(\Sigma = \{{\texttt{0}}, {\texttt{1}}\}\)).

Let \(M\) be a TM with input alphabet \(\Sigma\). We say that on input \(x\), \(M\) outputs the string \(y\) if the following hold:

• \(M(x)\) halts with the halting configuration being \(uqv\) (where \(q \in \{q_\text{acc}, q_\text{rej}\}\)),

• the string \(uv\) equals \(y\).

In this case we write \(M(x) = y\).

We say \(M\) solves (or computes) a function problem \(f : \Sigma^* \to \Sigma^*\) if for all \(x \in \Sigma^*\), \(M(x) = f(x)\).

Let \(\Sigma\) be some finite alphabet. A universal Turing machine \(U\) is a Turing machine that takes \(\left \langle M, x \right\rangle\) as input, where \(M\) is a TM and \(x\) is a word in \(\Sigma^*\), and has the following high-level description:

\[\begin{aligned} \hline\hline\\[-12pt] &\textbf{def}\;\; U(\left \langle \text{TM } M, \; \text{string } x \right\rangle):\\ &{\scriptstyle 1.}~~~~\texttt{Simulate $M$ on input $x$ (i.e. run $M(x)$)}\\ &{\scriptstyle 2.}~~~~\texttt{If it accepts, accept.}\\ &{\scriptstyle 3.}~~~~\texttt{If it rejects, reject.}\\ \hline\hline\end{aligned}\]

Note that if \(M(x)\) loops forever, then \(U\) loops forever as well. To make sure \(M\) always halts, we can add a third input, an integer \(k\), and have the universal machine simulate the input TM for at most \(k\) steps.

Fix some alphabet \(\Sigma\).

• We call a DFA \(D\) satisfiable if there is some input string that \(D\) accepts. In other words, \(D\) is satisfiable if \(L(D) \neq \varnothing\).

• We say that a DFA \(D\) self-accepts if \(D\) accepts the string \(\left \langle D \right\rangle\). In other words, \(D\) self-accepts if \(\left \langle D \right\rangle \in L(D)\).

We define the following languages: \[\begin{aligned} \text{ACCEPTS}_{\text{DFA}}& = \{\left \langle D,x \right\rangle : \text{$D$ is a DFA that accepts the string $x$}\}, \\ \text{SA}_{\text{DFA}}= \text{SELF-ACCEPTS}_{\text{DFA}}& = \{\left \langle D \right\rangle : \text{$D$ is a DFA that self-accepts}\}, \\ \text{SAT}_{\text{DFA}}& = \{\left \langle D \right\rangle : \text{$D$ is a satisfiable DFA} \}, \\ \text{NEQ}_{\text{DFA}}& = \{ \left \langle D_1, D_2 \right\rangle : \text{$D_1$ and $D_2$ are DFAs with $L(D_1) \neq L(D_2)$} \}.\end{aligned}\]

We can relax the notion of a TM \(M\) solving a language \(L\) in the following way. We say that \(M\) semi-decides \(L\) if for all \(w \in \Sigma^*\), \[w \in L \quad \Longleftrightarrow \quad M(w) \text{ accepts}.\] Equivalently,

• if \(w \in L\), then \(M\) accepts \(w\) (i.e. \(M(w) = 1\));

• if \(w \not \in L\), then \(M\) does not accept \(w\) (i.e. \(M(w) \in \{0,\infty\}\)).

(Note that if \(w \not \in L\), \(M(w)\) may loop forever.)

We call \(M\) a semi-decider for \(L\).

A language \(L\) is called semi-decidable if there exists a TM that semi-decides \(L\).

We denote by \(\mathsf{RE}\) the set of all semi-decidable languages (over the default alphabet \(\Sigma = \{{\texttt{0}}, {\texttt{1}}\}\)).

Let \(X\) and \(Y\) be two (possibly infinite) sets.

• A function \(f: X \to Y\) is called injective if for any \(x,x' \in X\) such that \(x \neq x'\), we have \(f(x) \neq f(x')\). We write \(X \hookrightarrow Y\) if there exists an injective function from \(X\) to \(Y\).

• A function \(f: X \to Y\) is called surjective if for all \(y \in Y\), there exists an \(x \in X\) such that \(f(x) = y\). We write \(X \twoheadrightarrow Y\) if there exists a surjective function from \(X\) to \(Y\).

• A function \(f: X \to Y\) is called bijective (or one-to-one correspondence) if it is both injective and surjective. We write \(X \leftrightarrow Y\) if there exists a bijective function from \(X\) to \(Y\).

Let \(X\) and \(Y\) be two (possibly infinite) sets.

• We write \(|X| = |Y|\) if \(X \leftrightarrow Y\).

• We write \(|X| \leq |Y|\) (or \(|Y| \geq |X|\)) if \(X \hookrightarrow Y\), or equivalently, if \(Y \twoheadrightarrow X\).

• We write \(|X| < |Y|\) (or \(|Y| > |X|\)) if it is not the case that \(|X| \geq |Y|\).

• A set \(S\) is called countable if \(|S| \leq |\mathbb N|\).

• A set \(S\) is called countably infinite if it is countable and infinite.

• A set \(S\) is called uncountable if it is not countable, i.e. \(|S| > |\mathbb N|\).

Let \(\Sigma\) be some finite alphabet. We denote by \(\Sigma^\infty\) the set of all infinite length words over the alphabet \(\Sigma\).

We say that a TM \(M\) self-accepts if \(M(\left \langle M \right\rangle) = 1\). The self-accepts problem is defined as the decision problem corresponding to the language \[\text{SA}_{\text{TM}}= \text{SELF-ACCEPTS}_{\text{TM}}= \{\left \langle M \right\rangle : \text{$M$ is a TM which self-accepts} \}.\] The complement problem corresponds to the language \[\overline{\text{SA}_{\text{TM}}}= \overline{\text{SELF-ACCEPTS}_{\text{TM}}}= \{\left \langle M \right\rangle : \text{$M$ is a TM which does not self-accept}\}.\] Note that if a TM \(M\) does not self-accept, then \(M(\left \langle M \right\rangle) \in \{0, \infty\}\).

We define the following languages: \[\begin{aligned} \text{ACCEPTS}_{\text{TM}}& = \{\left \langle M,x \right\rangle : \text{$M$ is a TM which accepts input $x$}\}, \\ \text{HALTS}_{\text{TM}}& = \{\left \langle M,x \right\rangle : \text{$M$ is a TM which halts on input $x$} \}, \\ \end{aligned}\]

A TM \(M\) is satisfiable if there is some input string that \(M\) accepts. In other words, \(M\) is satisfiable if \(L(M) \neq \varnothing\). We now define the following languages: \[\begin{aligned} \text{SAT}_{\text{TM}}& = \{\left \langle M \right\rangle : \text{$M$ is a satisfiable TM}\}, \\ \text{NEQ}_{\text{TM}}& = \{\left \langle M_1,M_2 \right\rangle : \text{$M_1$ and $M_2$ are TMs with $L(M_1) \neq L(M_2)$}\}, \\ \text{HALTS-EMPTY}_{\text{TM}}& = \{\left \langle M \right\rangle : \text{$M$ is a TM such that $M(\epsilon)$ halts}\}, \\ \text{FINITE}_{\text{TM}}& = \{\left \langle M \right\rangle : \text{$M$ is a TM such that $L(M)$ is finite}\}.\end{aligned}\]

Given some decision problem \(g : \Sigma^* \to \{0,1\}\), we define a \(g\)-oracle Turing machine (or simply \(g\)-OTM, or just OTM if \(g\) is understood from the context) as a Turing machine with an additional oracle instruction which can be used in the TM’s high-level description. The additional instruction has the following form. For any string \(x\) and any variable name \(y\) of our choosing, we are allowed to use \[y = g(x).\] After this instruction, the variable \(y\) holds the value \(g(x)\) and can be used in the rest of the Turing machine’s description. In this context, the function \(g\) is called an oracle.

We’ll use a superscript of \(g\) (e.g. \(M^g\)) to indicate that a machine is a \(g\)-OTM.

Similar to TMs, we write \(M^g(x) = 1\) if the oracle TM \(M^g\), on input \(x\), accepts. We write \(M^g(x) = 0\) if it rejects. And we write \(M^g(x) = \infty\) if it loops forever.

If \(f : \Sigma^* \to \{0,1\}\) is a decision problem such that for all \(x \in \Sigma^*\), \(M^g(x) = f(x)\), then we say \(M^g\) describes \(f\) (and the language corresponding to \(f\)).

Let \(L\) and \(K\) be two languages, and let \(k\) be the decision problem corresponding to \(K\). We say that \(L\) Turing-reduces to \(K\), written \(L \leq_T K\), if there is an oracle Turing machine \(M^k\) describing \(L\).

Let \(\text{PS}\) be a proof system and let \(S\) be a statement in \(\text{PS}\).

• Statement \(S\) is provable means there is a proof of \(S\) in \(\text{PS}\).

  • If \(S\) is provable, we say \(\text{PS}\) proves \(S\).

  • We use \(\text{Provable}(\cdot)\) to denote the function such that \(\text{Provable}(\left \langle S \right\rangle)\) returns True if and only if \(S\) is provable. We can view \(\text{Provable}(\cdot)\) as a decision problem.

• Statement \(S\) is refutable means \(\text{PS}\) proves the negation of \(S\), (i.e. \({\neg S}\)).

• Statement \(S\) is resolvable means that \(S\) is either provable or refutable.

• If statement \(S\) is not resolvable, we say \(S\) is independent of \(\text{PS}\).

Let \(\text{PS}\) be a proof system.

• \(\text{PS}\) is consistent means for all \(S\), if \(S\) is provable, then \({\neg S}\) is not provable. Or equivalently, for all \(S\), at most one of \(S\) and \({\neg S}\) is provable.

• \(\text{PS}\) is sound means for all \(S\), if \(S\) is provable, then \(S\) is true (i.e. \(\text{Provable}(\left \langle S \right\rangle) \to S\)). This is equivalent to saying \(\text{PS}\) does not prove false statements.

• \(\text{PS}\) is complete means every statement in \(\text{PS}\) is resolvable. Or equivalently, for all \(S\), at least one of \(S\) and \({\neg S}\) is provable.

• \(\text{PS}\) is decidable means that the \(\text{Provable}\) decision problem is decidable.

We say proving \(S\) reduces to proving \(T\) (or simply \(S\) reduces to \(T\)) if \(T \to S\) is provable.

Let \(V_\text{ZFC}\) be the verifier TM in the ZFC proof system. A language \(L\) is ZFC-verifiable (or ZFC-provable) if for all \(x \in \Sigma^*\) we have:

• (completeness) if \(x \in L\), there exists \(y \in \Sigma^*\) such that \(V_\text{ZFC}(\left \langle \text{``}x \in L\text{''}, y \right\rangle)\) accepts;

• (soundness) if \(x \not \in L\), for all \(y \in \Sigma^*\), \(V_\text{ZFC}(\left \langle \text{``}x \in L\text{''}, y \right\rangle)\) rejects.

Let \(V\) be a decider TM and let \(L\) be a language over \(V\)’s input alphabet \(\Sigma\). We say that \(V\) verifies \(L\) if for all \(x \in \Sigma^*\) the following holds:

• (completeness) if \(x \in L\), there exists \(y \in \Sigma^*\) such that \(V(\left \langle x, y \right\rangle)\) accepts;

• (soundness) if \(x \not \in L\), for all \(y \in \Sigma^*\), \(V(\left \langle x, y \right\rangle)\) rejects.

We call \(V\) a verifier for \(L\). We also call (and think of) \(V\) as computational proof system for \(L\).

A language \(L\) is called verifiable if there exists a TM that verifies \(L\).

For \(f: \mathbb R^+ \to \mathbb R^+\) and \(g: \mathbb R^+ \to \mathbb R^+\), we write \(f(n) = O(g(n))\) if there exist constants \(C > 0\) and \(n_0 > 0\) such that for all \(n \geq n_0\), \[f(n) \leq Cg(n).\] In this case, we say that \(f(n)\) is big-O of \(g(n)\).

For \(f: \mathbb R^+ \to \mathbb R^+\) and \(g: \mathbb R^+ \to \mathbb R^+\), we write \(f(n) = \Omega(g(n))\) if there exist constants \(c > 0\) and \(n_0 > 0\) such that for all \(n \geq n_0\), \[f(n) \geq cg(n).\] In this case, we say that \(f(n)\) is big-Omega of \(g(n)\).

For \(f: \mathbb R^+ \to \mathbb R^+\) and \(g: \mathbb R^+ \to \mathbb R^+\), we write \(f(n) = \Theta(g(n))\) if \[f(n) = O(g(n)) \quad \quad \text{and} \quad \quad f(n) = \Omega(g(n)).\] This is equivalent to saying that there exists constants \(c,C,n_0 > 0\) such that for all \(n \geq n_0\), \[cg(n) \leq f(n) \leq Cg(n).\] In this case, we say that \(f(n)\) is Theta of \(g(n)\).

Suppose we are using some computational model in which what constitutes a step in an algorithm is understood. Suppose also that for any input \(x\), we have an explicit definition of its length. The worst-case running time of an algorithm \(A\) is a function \(T_A : \mathbb N\to \mathbb N\) defined by \[T_A(n) = \max_{\substack{\text{instances/inputs $x$} \\ \text{of length $n$}}} \text{ number of steps $A$ takes on input $x$}.\] We drop the subscript \(A\) and just write \(T(n)\) when \(A\) is clear from the context.

\[\begin{aligned} \text{Constant time:} & \quad T(n) = O(1). \\ \text{Logarithmic time:} & \quad T(n) = O(\log n). \\ \text{Linear time:} & \quad T(n) = O(n). \\ \text{Quadratic time:} & \quad T(n) = O(n^2). \\ \text{Polynomial time:} & \quad T(n) = O(n^k) \text{ for some constant $k > 0$}. \\ \text{Exponential time:} & \quad T(n) = O(2^{n^k}) \text{ for some constant $k > 0$}.\end{aligned}\]

We denote by \(\mathsf{P}\) the set of all languages that can be decided in polynomial-time, i.e., in time \(O(n^k)\) for some constant \(k>0\).

In the integer addition problem, we are given two \(n\)-bit numbers \(x\) and \(y\), and the output is their sum \(x+y\). In the integer multiplication problem, we are given two \(n\)-bit numbers \(x\) and \(y\), and the output is their product \(xy\).

An undirected graph \(G\) is a pair \((V,E)\), where

• \(V\) is a finite non-empty set called the set of vertices (or nodes),

• \(E\) is a set called the set of edges, and every element of \(E\) is of the form \(\{u,v\}\) for distinct \(u, v \in V\).

Let \(G=(V,E)\) be a graph, and \(e = \{u,v\} \in E\) be an edge in the graph. In this case, we say that \(u\) and \(v\) are neighbors or adjacent. We also say that \(u\) and \(v\) are incident to \(e\). For \(v \in V\), we define the neighborhood of \(v\), denoted \(N(v)\), as the set of all neighbors of \(v\), i.e. \(N(v) = \{u : \{v,u\} \in E\}\). The size of the neighborhood, \(|N(v)|\), is called the degree of \(v\), and is denoted by \(\deg(v)\).

A graph \(G = (V,E)\) is called \(d\)-regular if every vertex \(v \in V\) satisfies \(\deg(v) = d\).

Let \(G=(V,E)\) be a graph. A path of length \(k\) in \(G\) is a sequence of distinct vertices \[v_0,v_1,\ldots,v_k\] such that \(\{v_{i-1}, v_i\} \in E\) for all \(i \in \{1,2,\ldots,k\}\). In this case, we say that the path is from vertex \(v_0\) to vertex \(v_k\) (or that the path is between \(v_0\) and \(v_k\)).

A cycle of length \(k\) (also known as a \(k\)-cycle) in \(G\) is a sequence of vertices \[v_0, v_1, \ldots, v_{k-1}, v_0\] such that \(v_0, v_1, \ldots, v_{k-1}\) is a path, and \(\{v_0,v_{k-1}\} \in E\). In other words, a cycle is just a “closed” path. The starting vertex in the cycle is not important. So for example, \[v_1, v_2, \ldots, v_{k-1},v_0,v_1\] would be considered the same cycle. Also, if we list the vertices in reverse order, we consider it to be the same cycle. For example, \[v_0, v_{k-1}, v_{k-2} \ldots, v_1, v_0\] represents the same cycle as before.

A graph that contains no cycles is called acyclic.

Let \(G = (V,E)\) be a graph. We say that two vertices in \(G\) are connected if there is a path between those two vertices. A connected graph \(G\) is such that every pair of vertices in \(G\) is connected.

A subset \(S \subseteq V\) is called a connected component of \(G\) if \(G\) restricted to \(S\), i.e. the graph \(G' = (S, E' = \{\{u,v\} \in E : u,v \in S\})\), is a connected graph, and \(S\) is disconnected from the rest of the graph (i.e. \(\{u,v\} \not \in E\) when \(u \in S\) and \(v \not \in S\)). Note that a connected graph is a graph with only one connected component.

Let \(G=(V,E)\) be a graph. The distance between vertices \(u, v \in V\), denoted \(\text{dist}(u,v)\), is defined to be the length of the shortest path between \(u\) and \(v\). If \(u = v\), then \(\text{dist}(u,v) = 0\), and if \(u\) and \(v\) are not connected, then \(\text{dist}(u,v)\) is defined to be \(\infty\).

A graph satisfying two of the following three properties is called a tree:

1. connected,

2. \(m = n-1\),

3. acyclic.

A vertex of degree 1 in a tree is called a leaf. And a vertex of degree more than 1 is called an internal node.

A directed graph \(G\) is a pair \((V,A)\), where

• \(V\) is a non-empty finite set called the set of vertices (or nodes),

• \(A\) is a finite set called the set of directed edges (or arcs), and every element of \(A\) is a tuple \((u,v)\) for \(u, v \in V\). If \((u,v) \in A\), we say that there is a directed edge from \(u\) to \(v\). Note that \((u,v) \neq (v,u)\) unless \(u = v\).

Let \(G = (V,A)\) be a directed graph. For \(u \in V\), we define the neighborhood of \(u\), \(N(u)\), as the set \(\{v \in V : (u,v) \in A\}\). The vertices in \(N(u)\) are called the neighbors of \(u\). The out-degree of \(u\), denoted \(\deg_\text{out}(u)\), is \(|N(u)|\). The in-degree of \(u\), denoted \(\deg_\text{in}(u)\), is the size of the set \(\{v \in V : (v,u) \in A\}\). A vertex with out-degree 0 is called a sink. A vertex with in-degree 0 is called a source.

The arbitrary-first search algorithm, denoted AFS, is the following generic algorithm for searching a given graph. Below, “bag” refers to an arbitrary data structure that allows us to add and retrieve objects. The algorithm, given a graph \(G\) and some vertex \(s\) in \(G\), traverses all the vertices in the connected componenet of \(G\) containing \(s\).

\[\begin{aligned} \hline\hline\\[-12pt] &\textbf{def}\;\; \text{AFS}(\langle \text{graph } G = (V,E), \; \text{vertex } s \in V \rangle):\\ &{\scriptstyle 1.}~~~~\texttt{Put $s$ into bag.}\\ &{\scriptstyle 2.}~~~~\texttt{While bag is non-empty:}\\ &{\scriptstyle 3.}~~~~~~~~\texttt{Pick an arbitrary vertex $v$ from bag.}\\ &{\scriptstyle 4.}~~~~~~~~\texttt{If $v$ is unmarked:}\\ &{\scriptstyle 5.}~~~~~~~~~~~~\texttt{Mark $v$.}\\ &{\scriptstyle 6.}~~~~~~~~~~~~\texttt{For each neighbor $w$ of $v$:}\\ &{\scriptstyle 7.}~~~~~~~~~~~~~~~~\texttt{Put $w$ into bag.}\\ \hline\hline\end{aligned}\]

Note that when a vertex \(w\) is added to the bag, it gets there because it is the neighbor of a vertex \(v\) that has been just marked by the algorithm. In this case, we’ll say that \(v\) is the parent of \(w\) (and \(w\) is the child of \(v\)). Explicitly keeping track of this parent-child relationship is convenient, so we modify the above algorithm to keep track of this information. Below, a tuple of vertices \((v,w)\) has the meaning that vertex \(v\) is the parent of \(w\). The initial vertex \(s\) has no parent, so we denote this situation by \((\perp, s)\).

\[\begin{aligned} \hline\hline\\[-12pt] &\textbf{def}\;\; \text{AFS}(\langle \text{graph } G = (V,E), \; \text{vertex } s \in V \rangle):\\ &{\scriptstyle 1.}~~~~\texttt{Put $(\perp, s)$ into bag.}\\ &{\scriptstyle 2.}~~~~\texttt{While bag is non-empty:}\\ &{\scriptstyle 3.}~~~~~~~~\texttt{Pick an arbitrary tuple $(p, v)$ from bag.}\\ &{\scriptstyle 4.}~~~~~~~~\texttt{If $v$ is unmarked:}\\ &{\scriptstyle 5.}~~~~~~~~~~~~\texttt{Mark $v$.}\\ &{\scriptstyle 6.}~~~~~~~~~~~~\text{parent}(v) = p.\\ &{\scriptstyle 7.}~~~~~~~~~~~~\texttt{For each neighbor $w$ of $v$:}\\ &{\scriptstyle 8.}~~~~~~~~~~~~~~~~\texttt{Put $(v,w)$ into bag.}\\ \hline\hline\end{aligned}\]

The breadth-first search algorithm, denoted BFS, is AFS where the bag is chosen to be a queue data structure.

The depth-first search algorithm, denoted DFS, is AFS where the bag is chosen to be a stack data structure.

In the minimum spanning tree problem, the input is a connected undirected graph \(G = (V,E)\) together with a cost function \(c : E \to \mathbb R^+\). The output is a subset of the edges of minimum total cost such that, in the graph restricted to these edges, all the vertices of \(G\) are connected. For convenience, we’ll assume that the edges have unique edge costs, i.e. \(e \neq e' \implies c(e) \neq c(e')\).

A matching in a graph \(G=(V,E)\) is a subset of the edges that do not share an endpoint. A maximum matching in \(G\) is a matching with the maximum number of edges among all possible matchings. A maximal matching is a matching with the property that if we add any other edge to the matching, it is no longer a matching. A perfect matching is a matching that covers all the vertices of the graph.

In the maximum matching problem the input is an undirected graph \(G=(V,E)\) and the output is a maximum matching in \(G\).

Let \(G = (V,E)\) be a graph and let \(M \subseteq E\) be a matching in \(G\). An augmenting path in \(G\) with respect to \(M\) is a path such that

1. the path is an alternating path, which means that the edges in the path alternate between being in \(M\) and not in \(M\),

2. the first and last vertices in the path are not a part of the matching \(M\).

A graph \(G = (V,E)\) is called bipartite if there is a partition of \(V\) into sets \(X\) and \(Y\) such that all the edges in \(E\) have one endpoint in \(X\) and the other in \(Y\). Sometimes the bipartition is given explicitly and the graph is denoted by \(G = (X, Y, E)\).

Let \(G = (V,E)\) be a graph. Let \(k \in \mathbb{N}^+\). A \(k\)-coloring of \(V\) is just a map \(\chi : V \to C\) where \(C\) is a set of cardinality \(k\). (Usually the elements of \(C\) are called colors. If \(k = 3\) then \(C = \{\text{red},\text{green},\text{blue}\}\) is a popular choice. If \(k\) is large, we often just call the “colors” \(1,2, \dots, k\).) A \(k\)-coloring is said to be legal for \(G\) if every edge in \(E\) is bichromatic, meaning that its two endpoints have different colors. (I.e., for all \(\{u,v\} \in E\) it is required that \(\chi(u)\neq\chi(v)\).) Finally, we say that \(G\) is \(k\)-colorable if it has a legal \(k\)-coloring.

An instance of the stable matching problem is a tuple of sets \((X, Y)\) with \(|X| = |Y|\), and a preference list for each element of \(X\) and \(Y\). A preference list for an element in \(X\) is an ordering of the elements in \(Y\), and a preference list for an element in \(Y\) is an ordering of the elements of \(X\). Below is an example of an instance of the stable matching problem:

The output of the stable matching problem is a stable matching, which is a subset \(S\) of \(\{(x, y) : x \in X, y \in Y\}\) with the following properties:

• The matching is a perfect matching, which means every \(x \in X\) and every \(y \in Y\) appear exactly once in \(S\). If \((x,y) \in S\), we say \(x\) and \(y\) are matched.

• There are no unstable pairs. A pair \((x,y)\) where \(x \in X\) and \(y \in Y\) is called unstable if \((x,y) \not \in S\), but they both prefer each other to the elements they are matched to.

Consider an instance of the stable matching problem. We say that \(x \in X\) is a valid partner of \(y \in Y\) (or \(y\) is a valid partner of \(x\)) if there is some stable matching in which \(x\) and \(y\) are matched. For \(z \in X \cup Y\), we define the best valid partner of \(z\), denoted \(\text{best}(z)\), to be the highest ranked valid partner of \(z\). Similarly, we define the worst valid partner of \(z\), denoted \(\text{worst}(z)\), to be the lowest ranked valid partner of \(z\).

In the \(k\)-coloring problem, the input is an undirected graph \(G=(V,E)\), and the output is True if and only if the graph is \(k\)-colorable (see Definition (\(k\)-colorable graphs)). We denote this problem by \(k\text{COL}\). The corresponding language is \[\{\langle G \rangle: \text{$G$ is a $k$-colorable graph}\}.\]

Let \(G = (V,E)\) be an undirected graph. A subset \(S\) of the vertices is called a clique if for all \(u, v \in S\) with \(u \neq v\), \(\{u,v\} \in E\). We say that \(G\) contains a \(k\)-clique if there is a subset of the vertices of size \(k\) that forms a clique.

In the clique problem, the input is an undirected graph \(G=(V,E)\) and a number \(k \in \mathbb N^+\), and the output is True if and only if the graph contains a \(k\)-clique. We denote this problem by \(\text{CLIQUE}\). The corresponding language is \[\{\langle G, k \rangle: \text{$G$ is a graph, $k \in \mathbb N^+$, $G$ contains a $k$-clique}\}.\]

Let \(G = (V,E)\) be an undirected graph. A subset of the vertices is called an independent set if there is no edge between any two vertices in the subset. We say that \(G\) contains an independent set of size \(k\) if there is a subset of the vertices of size \(k\) that forms an independent set.

In the independent set problem, the input is an undirected graph \(G=(V,E)\) and a number \(k \in \mathbb N^+\), and the output is True if and only if the graph contains an independent set of size \(k\). We denote this problem by \(\text{IS}\). The corresponding language is \[\{\langle G, k \rangle: \text{$G$ is a graph, $k \in \mathbb N^+$, $G$ contains an independent set of size $k$}\}.\]