# Harrop formula

In intuitionistic logic, the Harrop formulae, named after Ronald Harrop, are the class of formulae inductively defined as follows:[1][2][3]

• Atomic formulae are Harrop, including falsity (⊥);
• $A \wedge B$ is Harrop provided $A$ and $B$ are;
• $\neg F$ is Harrop for any well-formed formula $F$;
• $F \rightarrow A$ is Harrop provided $A$ is, and $F$ is any well-formed formula;
• $\forall x. A$ is Harrop provided $A$ is.

By excluding disjunction and existential quantification (except in the antecedent of implication), non-constructive predicates are avoided, which has benefits for computer implementation. From a constructivist point of view, Harrop formulae are "well-behaved." For example, in Heyting arithmetic, Harrop formulae satisfy a classical equivalence not usually satisfied in constructive logic:[1]

$A \leftrightarrow \neg \neg A.$

Harrop formulae were introduced around 1956 by Ronald Harrop and independently by Helena Rasiowa.[2] Variations of the fundamental concept are used in different branches of constructive mathematics and logic programming.

## Hereditary Harrop formulae and logic programming

A more complex definition of hereditary Harrop formulae is used in logic programming as a generalisation of horn clauses, and forms the basis for the language λProlog. Hereditary Harrop formulae are defined in terms of two (sometimes three) recursive sets of formulae. In one formulation:[4]

• Rigid atomic formulae, i.e. constants $r$ or formulae $r(t_1,...,t_n)$, are hereditary Harrop;
• $A \wedge B$ is hereditary Harrop provided $A$ and $B$ are;
• $\forall x. A$ is hereditary Harrop provided $A$ is;
• $G \rightarrow A$ is hereditary Harrop provided $A$ is rigidly atomic, and $G$ is a G-formula.

G-formulae are defined as follows:[4]

• Atomic formulae are G-formulae, including truth(⊤);
• $A \wedge B$ is a G-formula provided $A$ and $B$ are;
• $A \vee B$ is a G-formula provided $A$ and $B$ are;
• $\forall x. A$ is a G-formula provided $A$ is;
• $\exists x. A$ is a G-formula provided $A$ is;
• $H \rightarrow A$ is a G-formula provided $A$ is, and $H$ is hereditary Harrop.

## References

1. ^ a b Dummett, Michael (2000). Elements of Intuitionism (2nd ed.). Oxford University Press. p. 227. ISBN 0-19-850524-8.
2. ^ a b A. S. Troelstra, H. Schwichtenberg. Basic proof theory. Cambridge University Press. ISBN 0-521-77911-1.
3. ^ Ronald Harrop (1956). "On disjunctions and existential statements in intuitionistic systems of logic". Mathematische Annalen. 132, Number 4. doi:10.1007/BF01360048.
4. ^ a b Dov M. Gabbay, Christopher John Hogger, John Alan Robinson, Handbook of Logic in Artificial Intelligence and Logic Programming: Logic programming, Oxford University Press, 1998, p 575, ISBN 0-19-853792-1