Elimination is by case analysis: given \(\phi \lor \psi\), if both \(\phi \vdash \theta\) and \(\psi \vdash \theta\), conclude \(\theta\).

\(\rightarrow i\) discharges the assumption \(\phi\). \(\rightarrow e\) is modus ponens.

\(\neg i\) discharges \(\phi\). \(\bot e\) is ex falso quodlibet: from falsehood, anything follows.

A proof of a formula \(\phi\) from assumptions \(\Gamma\) is a sequence of formulas following the inference rules such that all assumptions used in the proof belong to \(\Gamma\). We write \(\Gamma \vdash \phi\).

Computation rules: \(\pi_1(s, t) \longrightarrow s\) and \(\pi_2(s, t) \longrightarrow t\).

Computation rule (beta reduction): \((\lambda x : A . t)\, u \longrightarrow t[x := u]\).

Substitution requires first renaming bound variables in \(t\) so they are distinct from free variables in \(u\).

Elimination by case analysis:

Computation rules:

Extends propositional logic with predicates. \[ \phi ::= p(t_1, \dots, t_n) \mid \phi \land \phi \mid \phi \lor \phi \mid \phi \rightarrow \phi \mid \bot \mid \forall x . \phi \mid \exists x . \phi \] We add predicates expressing relations and constants and function symbols expressing elements. Every function symbol has an arity \(\geq 1\). Likewise we interpret formulas as having an arity \(\geq 0\).

Terms express elements: \[ t ::= x \mid c \mid f(t_1, \dots, t_n) \] where \(c\) represents constants, \(f\) a function symbol of arity \(n\).

To integrate predicate logic and type theory, introduce a special type \(\Prop\) — the type of propositions, called the propositional universe.

Often we make the context explicit, e.g.: \[ p : \Prop,\; q : \Prop \vdash p \land q \rightarrow q \land p : \Prop \]

An alternative is to include connectives and quantifiers as typed constants: \[ \rightarrow,\; \land,\; \lor : \Prop \rightarrow \Prop \rightarrow \Prop \] \[ \exists,\; \forall : (A \rightarrow \Prop) \rightarrow \Prop \]

• In \(\forall i\), \(z\) must not appear free outside the box (i.e. not free in the context).
• In \(\forall e\), every free occurrence of \(x\) in \(\phi\) is replaced by \(t\), with renaming of bound variables if necessary.

In \(\exists e\): \(\psi : \Prop\) must be provable outside the box, and \(\psi\) cannot contain \(x\) free.

The type \(\operatorname{nat}\) is given by

Alternatively, we can think of this type as given by

1. A constant \(\operatorname{nil} : \operatorname{List} A\)
2. A map \(\operatorname{cons} : A \rightarrow \operatorname{List} A \rightarrow \operatorname{List} A\)

Alternatively, we can think of it as given by

1. A constant \(\operatorname{nil} : \operatorname{Vec} 0 \, A\)
2. A map \(\operatorname{cons} : A \rightarrow \operatorname{Vec} A \, n \rightarrow \operatorname{Vec} A \, (\operatorname{succ} n)\)

The map including the dependent \(n\) means we really have a family of maps, which we give as a dependent product: \[ \operatorname{cons} : \prod(n : \operatorname{nat}) .\, A \rightarrow \operatorname{Vec} A \, n \rightarrow \operatorname{Vec} A \, (\operatorname{succ} n) \] Hence why \(\operatorname{cons}\) takes the additional argument \(n\). Vectors form a dependent type.

We also have the computation rule \[ (\lambda x : A . t) u \longrightarrow t[u/x]. \]

Note that these rules match to the rules for \(\rightarrow\). In type theory, once you have dependent product, we don't need the \(\rightarrow\) type as a primitive anymore, because it is defineable. In particular, reduces to the ordinary function type \(A \rightarrow B\) when \(B\) does not depend on \(x\), \(\Pi(x : A) . B\).

We also define the computation rules

Intuitively, \(\Sigma(x : A). B\) is the type where elements are tagged elements \[ \operatorname{in}_a(b) \] where \(a : A\) and \(b : B[a/x]\). Equivalently, we can give the tagging with pairs \((a, b)\) for \(a : A\) and \(b : B[a/x]\).

\[ \operatorname{Ind} : P : \operatorname{nat} \rightarrow\Prop .\, \left( P0 \rightarrow (\forall x : \operatorname{nat} .\, Px \rightarrow P (\operatorname{succ} x)) \rightarrow \forall x : \operatorname{nat} .\, Px \right) \]

\[ R_{\operatorname{nat}} : \prod P : \operatorname{nat} \rightarrow \Type .\, \left( P0 \rightarrow \left( \prod x : \operatorname{nat} .\, Px \rightarrow P(\operatorname{succ} x) \right) \right) \rightarrow \prod x : \operatorname{nat}.\, Px \]

Say we have \(P : \operatorname{nat} \rightarrow \Type\), \(b : P_0\), \(f : \prod (x : \operatorname{nat} .\,Px \rightarrow P(\operatorname{succ} x))\). Then we have a term \[ R_{\operatorname{nat}}P\, b\, f\, : \prod(x : \operatorname{nat}) .\, Px \] which we can apply as

An encoding is a function \[ e : \text{ inputs} \rightarrow \left\{ 0, 1 \right\}^{ * }. \]

(We could use some other alphabet, but we use binary encodings on the computer.)

We assume a fixed explicit encodings in this course. This means that we always have an explicit input, e.g. we could give a number recursively in terms of its digits, but we still always assume it to be given explicitly.

A computational problem with boolean output.

The fiber of true becomes interesting as an object of study. When we view it in an encoding, it is \[ L = \left\{ w \in \left\{ 0, 1 \right\}^{ * } \mid P(w) \right\} , \] which is a language. So decision problems are recognising a language. (For now?) we only care about decidable languages.

An additional detail to sweep under the rug is that we assume \(P(w)\) also checks the validity of \(w\), i.e. we never think about encoding problems as such.

\[ \begin{aligned} f &= O(g(n)) \iff \exists C \in \R .\exists n_0 \in \N . \forall n \geq n_0 . f(n) \leq C \cdot g(n) \\ f &= \Omega(g) \iff g = O(f) \\ f &= \Theta(g) \iff f = O(g) \text{ in } g = O(f) \end{aligned} \] We also use the little \(o\) notation, which indicates a weak upper bound: \[ f = o(g) \iff \exists n_0 \in \N . \forall C > 0 . \forall n \geq n_0 . f(n) \leq C g(n) \] i.e. all since we have enough additional complexity, it dominates any constant. We get \[ f = o(g) \implies \lim_{n \rightarrow \infty} \frac{f}{g} = 0 \]

Likewise we define \[ f = \omega(g) \iff g = o(f). \]

We define it as \((\Gamma, \sqcup, Q, q_s, q_h, \delta)\) where \(\delta : \Gamma \times Q \rightarrow \Gamma \times Q \times \left\{ -1, 0, 1 \right\}\) is the transition function.

The definition of the TM doesn't itself speak of the tape state. We also define a configuration of the TM \(M\) as \((q, t, i)\) where \(q\) is the state of the control unit, \(t : \Z \rightarrow \Gamma\) the state of the tape and \(i \in \Z\) the position of the head. The initial configuration is then \((q_s, in, 0)\). The input is always right from 0.

A TM \(M\) recognizes a language \(L\) if

1. For every input, \(M\) halts
2. \(\forall \omega \in L .\,\) applying \(M\) to \(\omega\) finishes with \(q_{\text{accept}}\)
3. \(\forall \omega \not\in L .\,\) applying \(M\) to \(\omega\) finishes with \(q_{\text{reject}}\)

Let \(M\) be a TM. We define

It computes a function \[ f : \Sigma_1^{ * } \rightarrow \Sigma_2^{ * } \] if

• \(M\) always finishes for \(\omega \in \Sigma_1^{ * }\)
• \(M\) has a halting state \(q_{h}\)
• At the end the tape contains \(f(\omega)\) and all blanks around.

In particular this means we need \(\Gamma \supseteq \Sigma_1 \cup \Sigma_2 \) .

This is more general than decideability, which reduces to the case \(f : \Sigma^{ * } \rightarrow \left\{ 0, 1 \right\}\).

A \(k\)-tape Turing machine has \(k\) tapes. It's equivalent to consider transition functions which move just one or all tape-heads at once. Sometimes one of the tapes is considered to be read-only and considered as an input tape, likewise sometimes we only have an output tape, etc.

We say that \(\operatorname{DTIME}(f(n))\) is the set of languages \(L\) such that there exists a TM \(M\) that recognizes \(L\) and has \[ \operatorname{time}_M(n) = O(f(n)). \]

We usually take the \(TM\) to be a \(k\)-tape machine for some constant \(k\).

We define \[ \operatorname{P} = \bigcup\limits_{d \geq 1} \operatorname{DTIME}(n^d). \]

We define \[ \operatorname{EXP} = \bigcup\limits_{d \geq 1} \operatorname{DTIME} \left( 2^{n^{d}} \right) \]

An Universal Turing machine is a TM \(\mathcal{U}\) that takes as input \(\left< M, x \right>\), where \(M\) is (a description of) a Turing machine, \(x\) is an input word to \(M\), and returns \(M(x)\), i.e. what the TM \(M\) would give for input \(x\).

Furthermore, \(\mathcal{U}\) simulates the computation of \(M(x)\).

A nondeterministic Turing machine with 1 tape has transition function \[ \delta : Q \times \Gamma \rightarrow \pot(Q\times \Gamma \times \left\{ 0, -1, 1 \right\}) \] which maps a state to a set of states, corresponding to the options of the next state. Note that we have no explicit choice over the branching.

A Nondeterministic Turing machine accepts a language \(L\) if

• It always finishes its computation
• If \(\omega \in L\), there must exist a computation path that finishes with \(q_{\text{accept}}\).
• If \(\omega \not\in L\), all possible branches must finish with \(q_{\text{reject}}\).

Let \(M\) be a NTM. Then

Our definition takes the maximum over all inputs and choices. One can also take the definition where we take the minimum over the choices, but we must be more careful in this instance. For time complexity, both definitions are equivalent - we can always add a counter and keep track of the steps and memory.

\[ \operatorname{NP} = \bigcup\limits_{d \geq 1} \operatorname{NTIME}(n^d). \] This is the decision problems for which there exists a NTM \(M\) that solves the problem in \(O(n^d)\) for some constant \(d \leq 1\).

Note that \(\operatorname{P} \subseteq \operatorname{NP}\), since a deterministic TM is also a nondeterministic TM.

\(\N\) je definirana kot najmanjša množica, kjer je

• \(0 \in \N\)
• \(n \in \N \implies n^{ + } \in \N\)

To zapišemo tudi s pravili izpeljave:

Aritmetične izraze lahko podamo kot \[ e ::= \underline{n} \mid -e \mid e_1 + e_2 \mid e_1 * e_2, \] ali s pravili izpeljave:

Z drevesom izpeljave lahko ekonomično pišemo dokaze vsebovanosti.

Induktivna definicija sodosti naravnih števil:

Induktivna definicija relacije manjšega ali enakosti:

Alternativna (ekvivalentna) definicija:

Obe definiciji podata ekvivalentno relacijo.

Induktivno relacijo \(R \subseteq X\) lahko predstavimo z družino induktivnih množic \((D_x)_{x \in X}\), kjer \(D_x\) predstavlja množico dokazov, da je \(x \in R\).

Obratno, iz vsake družine \((D_x)_{x \in X}\) lahko definiramo relacijo \(R \subseteq X\) kot \[ R = \left\{ x \in X \mid D_x \neq \emptyset \right\}. \]

Algebra za funktor \(F\) je

• Množica \(X\)
• Preslikava \(\alpha : FX \rightarrow X\)

Homomorfizem algebr je homomorfizm algebr.

Za množico lokacij \(l \in \mathbb{L}\) definiramo sinkakso jezika

Every infinite subset of \(\R\) is either in bijection with \(\N\) or in bijection with \(\R\).

This is independent from ZFC.

\[ 2^{\aleph_0} = \aleph_1 \] as an equation in cardinal arithmetic. Note that this version implies the regular continuum hypothesis.

We give the axioms in a way that mirrors the way mathematicians actually use them.

Our axioms will tell us which classes can be sets:

• Assume a universe \(\mathcal{U}\) of all mathematical entities that we may use as elements of sets.
• A collection of elements of the universe is called a class.
• If \(P(x)\) is a property of elements of \(\mathcal{U}\), then \(\{x \mid P(x)\}\) satisfies \[ y \in \{x \mid P(x)\} \iff P(y). \]
• A set is a class that is itself an element of \(\mathcal{U}\).

A product of classes \(\underline{X}\) and \(\underline{Y}\) is a class \(\underline{X} \times \underline{Y}\) together with class functions \(\pi_1 : \underline{X} \times \underline{Y} \rightarrow \underline{X}\) and \(\pi_2 : \underline{X} \times \underline{Y} \rightarrow \underline{Y}\) satisfying \[ \forall x \in \underline{X}.\, \forall y \in \underline{Y}.\, \exists! z \in \underline{X} \times \underline{Y}.\, \pi_1(z) = x \land \pi_2(z) = y. \]

Suppose \(F : \underline{X} \rightarrow \underline{Y}\) is a class function. Its graph is \[ \Gamma(F) := \{(x, y) \in \underline{X} \times \underline{Y} \mid y = F(x)\} \subseteq \underline{X} \times \underline{Y}. \] If \(X\) is a set and \(F : X \rightarrow Y\) is a class function, then \(\Gamma(F)\) is a set by Replacement.

The exponential class is defined as \[ \underline{Y}^{X} := \left\{ \Gamma(F) \mid F : X \rightarrow \underline{Y} \text{ is a function} \right\}. \]

A natural number structure is a triple \((\underline{N}, 0, S)\) where \(\underline{N}\) is a class, \(0 \in \underline{N}\), and \(S : \underline{N} \rightarrow \underline{N}\) satisfying:

1. \(S\) is injective: \(\forall x, y \in \underline{N}.\, S(x) = S(y) \implies x = y\).
2. Acyclicity: \(\forall x \in \underline{N}.\, S(x) \neq 0\).
3. Induction: if \(\underline{X}\) satisfies \(0 \in \underline{X}\) and \(\forall x \in \underline{N}.\, x \in \underline{X} \implies S(x) \in \underline{X}\), then \(\underline{N} \subseteq \underline{X}\).

Apply the PRT with \(\mathcal{A} = \underline{N}\), \(\mathcal{Z} = \underline{N}\), \(B(z) = z\), and \(F(z, x) = S(x)\). The unique resulting \[ R : \underline{N} \times \underline{N} \rightarrow \underline{N} \] satisfies

This \(R\) is addition on \(\underline{N}\).

For a set \(n\) define \(n^{+} := n \cup \{n\}\). A set \(X\) is a vN-number if:

1. For every subset \(Y \subseteq X\) satisfying \[ \emptyset \in X \implies \emptyset \in Y \quad \text{and} \quad \forall z \in Y.\, z^{ + } \in X \implies z^{ + } \in Y, \] it holds that \(Y = X\).
2. \(X = \emptyset\) or \(\exists z \in X\) such that \(X = z^{+}\).

The class of all vN-numbers is \[ \mathrm{vN} := \{x \mid x \text{ is a set and a vN-number}\}. \]

Two sets \(A, B\) have the same cardinality (bijective correspondene, isomorphic, equipotent), if there exists a bijective function \[ f : A \rightarrow B. \]

Our notation will be \[ \abs{A} = \abs{B}. \] We don't yet have an independent meaning for \(\abs{A}\).

A set \(A\) has cardinality less than or equal to a set \(B\) if there exists an injective function \[ i : A\rightarrowtail B. \]

We write \[ \abs{A} \leq \abs{B}. \]

A set \(A\) has cardinality strictly less than a set \(B\) if \[ \abs{A} < \abs{B} :\iff \abs{A} \leq \abs{B} \land \abs{A} \neq \abs{B}. \]

\(\abs{A} \ll \abs{B} :\iff \abs{A} \leq \abs{B}\) and there is no surjection from \(A\) to \(B\).

If \(\abs{A} \ll \abs{B}\), then \(\abs{A} < \abs{B}\), and we again have the sandwiching property \[ \abs{A} \leq \abs{B} \ll \abs{C} \leq \abs{D} \implies \abs{A} \ll \abs{D}. \]

The standard finite sets are, for \(n \in \N\), \[ [n] = \{m \in \N \mid m < n\}. \]

Herein we define \[ [\cdot] : \N \rightarrow \pot \N \] using the recursion theorem.

A set \(A\) is finite if there exists \(n \in \N\) such that \(\abs{A} = \abs{[n]}\). In this case we define \(\abs{A} = n\).

Note that this is well defined, since there is a unique such \(n\).

A set \(A\) is infinite if it is not finite.

We define \(\aleph_0 = \abs{\N}\).

• A set \(X\) is countable if \(\abs{X} \leq \aleph_0\).
• If \(X\) is countable and infinite, then we call it countably infinite.

Examples of countable sets are \(\N, \Z, \Q, \N \times \N\).

Countable sets are closed under finite product, disjoint finite union, countable union, countable product, \(\dots\) Also if \(A\) is countable and \(A \twoheadrightarrow B\), then \(B\) is countable. Likewise if \(B\) is countable and \(A \rightarrowtail B\), then \(A\) is countable.

A set \(X\) is uncountable if it is not countable.

The covariant powerclass operation is functorial. That is, if \(f : X \rightarrow Y\) is a function, then we have a function \[ \mathcal{P}f : \mathcal{P}X \rightarrow \mathcal{P}Y \] defined by \(\mathcal{P}f(A) = \{f(a) \mid a \in A\}\). Note that \(\mathcal{P}\) preserves identities and composition.

We've already defined \(n\) as the cardinality \(\abs{[n]}\) and \(\aleph_0 = \abs{\N}\). We also define

We say that \(\abs{X}\) is idempotent (or its own square) if \(\abs{X} = \abs{X} \cdot \abs{X}\).

For instance, \(\aleph_0\) is idempotent.