Algorithmic theory of natural numbers: Różnice pomiędzy wersjami

Wersja z 13:51, 11 lis 2024

The theory [math]\mathcal{ATN} [/math] of one constant 0, one one-argument functor [math]s[/math] and predicate of equality =.
Axioms
[math]\begin{array}{|l|}\hline \begin{align*} & \tag{I} \forall_n s(n) \neq 0 \\ & \tag{M} \forall_n \forall_m s(n)=s(m) \implies n=m \\ & \tag{S} \forall_n \{ \begin{array}{l}m:=0; \\ \mathbf{while}\ n\neq m\ \mathbf{do}\\ \quad \\ m:=s(m)\\ \mathbf{od} \end{array}\}(n=m) \\ \hline \end{align*} \end{array} [/math]
This set of formulas gives a complete specification of the structure [math]\mathfrak{N} [/math] of natural numbers with the successor operation. It means that any implementation whether hardware or software (e.g. by means of a class) that satisfies three axioms listed above is correct. It means also that if an algorithmic formula is valid in the standard model of these axioms then it has a proof with the use of program calculus.
One may extend this set adding five axioms that define operation of addition, subtraction predecessor and predicate < and operation of multiplication. [math]\begin{align*} \tag{A}\label{add} & x+y \stackrel{df}{=} \{t:=0; w:=x; \textbf{while }t\neq y\textbf{ do }t:=s(t); w:=s(w) \textbf{ od}\}w & \\ \tag{L}\label{less} & x \lt y \stackrel{df}{\equiv} \{ w:=0; \textbf{while }w\neq y\land w\neq x \textbf{ do } w:=s(w) \textbf{ od}\}(w=x \land w\neq y) & \\ \tag{P}\label{predec} & P(x)\stackrel{df}{=} \{\begin{array}{l}w:=0;\ \textbf{if }x \neq 0 \textbf{ then } \ \textbf{while } s(w)\neq x\textbf{ do } w:=s(w) \textbf{ od} \ \textbf{fi}\end{array} \}w & \\ \tag{O}\label{odejm} & x\stackrel{.}{\_\_}y \stackrel{df}{=} \{w:=x; t:=0; \mathbf{while }\ t\neq y\ \mathbf{ do }\ t:=s(t); w:=P(w)\ \mathbf{ od} \}w & \\ \tag{M}\label{mult} & x\cdot y \stackrel{df}{=} \{t:=0; w:=0; \textbf{while }t\neq y\textbf{ do }t:=s(t); w:=w+x \textbf{ od}\}w & \\ \end{align*} [/math]

Among theorems of this theory one can find theorem on correctness of Euclid's algoritm
[math] \tag{Eucl} \underbrace{\forall_{n\neq 0}\,\forall_{m\neq 0}\left\{\begin{array}{l}\textbf{while }n \neq m \textbf{ do }\\ \quad \textbf{if }n\geq m \\ \quad \textbf{ then } \\ \qquad n:=n-m \\ \quad \textbf{ else }\\ \qquad m:=m-n \\ \quad\textbf{ fi}\\ \textbf{ od} \end{array}\right\}(n=m)}_{halting\ formula\ of\ Euclid's\ algorithm} [/math] and the theorem stating that the structure of natural numbers enjoys the Archimedean property [math]\mathfrak{N} \models \tag{Archi} \underbrace{\forall_{n\neq 0}\,\forall_{m\neq 0}\left\{\begin{array}{l} a:=n; \\ \textbf{while }\ a \leq m\ \textbf{ do }\\ \quad a :=a+x \\ \textbf{ od} \end{array}\right\}(a \gt x)}_{ \ axiom\ of\ Archimedes\ } [/math]