Presburger arithmetic

Presburger arithmetic is a weakened form of arithmetic which includes the structure $\\mathbb{N}$ , the constant $0$ , the unary function $S$ , the binary function $+$ , and the binary relation $<$ . Essentially, it is Peano arithmetic without multiplication.

The axioms are:

1.
$0\ot=Sx$
2.
$Sx=Sy\\rightarrow x=y$
3.
$x+0=x$
4.
$x+Sy=S(x+y)$
5.
For each first order formula $P(x)$ , $P(0)\\wedge\\forall x[P(x)\\rightarrow P(x+1)]\\rightarrow\\forall xP(x)$

Presburger arithmetic is decidable, but is consequently very limited in what it can express.

单词	PresburgerArithmetic
释义	Presburger arithmetic Presburger arithmetic is a weakened form of arithmetic which includes the structure $\\mathbb{N}$ , the constant $0$ , the unary function $S$ , the binary function $+$ , and the binary relation $<$ . Essentially, it is Peano arithmetic without multiplication. The axioms are: 1. $0\ot=Sx$ 2. $Sx=Sy\\rightarrow x=y$ 3. $x+0=x$ 4. $x+Sy=S(x+y)$ 5. For each first order formula $P(x)$ , $P(0)\\wedge\\forall x[P(x)\\rightarrow P(x+1)]\\rightarrow\\forall xP(x)$ Presburger arithmetic is decidable, but is consequently very limited in what it can express.
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