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User:Physis/Gödel-Herbrand-Kleene equational calculus

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Definition

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Defined in the follwing way.[1][2]

The calculus consists of a set (or system) of equations. An equation is of form <term> = <term>, where the left-hand side term may be required to contain a principial letter. Let the set of nonlogical symbols contain the unary f (besides the arithmetical ones 0 and s). This f is just an example of a particular signature, in generally, an arbitrary signature must be considered (which contains the signature of natural numbers at least as an 0 and s).

Syntax

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<equation> ::= <lhsterm> = <term>
<lhsterm> ::= 0
<lhsterm> ::= s <term>
<lhsterm> ::= f <term>
<lhsterm> ::= g <term> <term>
<term> ::= <variable>
<term> ::= <lhsterm>

Rules

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Expressing power

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Equivalent power with the theory of partial recursive functions.[3][2]

Notes

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  1. ^ Bezem & Klop & de Vrijer: 62
  2. ^ a b Monk 1976: 67
  3. ^ Bezem & Klop & de Vrijer: 63

References

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  • Bezem, Marc (2003). Term Rewriting Systems. Cambridge University Press. ISBN 0521391156. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  • Monk, J. Donald (1976). Mathematical Logic. Graduate Texts in Mathematics. New York • Heidelberg • Berlin: Springer-Verlag.
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