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 (⊥);
  • is Harrop provided and are;
  • is Harrop for any well-formed formula ;
  • is Harrop provided is, and is any well-formed formula;
  • is Harrop provided 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]

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 or formulae , are hereditary Harrop;
  • is hereditary Harrop provided and are;
  • is hereditary Harrop provided is;
  • is hereditary Harrop provided is rigidly atomic, and is a G-formula.

G-formulae are defined as follows:[4]

  • Atomic formulae are G-formulae, including truth(⊤);
  • is a G-formula provided and are;
  • is a G-formula provided and are;
  • is a G-formula provided is;
  • is a G-formula provided is;
  • is a G-formula provided is, and is hereditary Harrop.

See also

References

  1. Dummett, Michael (2000). Elements of Intuitionism (2nd ed.). Oxford University Press. p. 227. ISBN 0-19-850524-8.
  2. A. S. Troelstra, H. Schwichtenberg. Basic proof theory. Cambridge University Press. ISBN 0-521-77911-1.CS1 maint: uses authors parameter (link)
  3. Ronald Harrop (1956). "On disjunctions and existential statements in intuitionistic systems of logic". Mathematische Annalen. 132 (4): 347. doi:10.1007/BF01360048.
  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
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