# 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 (⊥);

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]

${\displaystyle A\leftrightarrow \neg \neg A.}$

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]

G-formulae are defined as follows:[4]

• Atomic formulae are G-formulae, including truth(⊤);

## References

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4. 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