|
|
| Line 1: |
Line 1: |
| In [[intuitionistic logic]], the '''Harrop formulae''', named after [[Ronald Harrop]], are the class of formulae inductively defined as follows:<ref name=dummett>{{cite book | last = Dummett | first = Michael | title = Elements of Intuitionism | publisher = [[Oxford University Press]] | edition = 2nd | date = 2000 | pages = 227 | isbn = 0-19-850524-8 }}</ref><ref name="basic proof theory">{{cite book | authors=A. S. Troelstra, H. Schwichtenberg | title=Basic proof theory | publisher = [[Cambridge University Press]] | isbn = 0-521-77911-1 }}</ref><ref>{{cite journal | author = Ronald Harrop | year = 1956 | title = On disjunctions and existential statements in intuitionistic systems of logic | journal = Mathematische Annalen | volume = 132, Number 4 | doi=10.1007/BF01360048}}</ref>
| | My hobby is mainly Machining. <br>I also to learn Japanese in my spare time.<br><br>My website; [http://gnlhookup.com/activity/p/1564147/ перейти на сайт] |
| | |
| * Atomic formulae are Harrop, including falsity (⊥);
| |
| | |
| * <math>A \wedge B</math> is Harrop provided <math>A</math> and <math>B</math> are;
| |
| | |
| * <math>\neg F</math> is Harrop for any well-formed formula <math>F</math>;
| |
| | |
| * <math>F \rightarrow A</math> is Harrop provided <math>A</math> is, and <math>F</math> is any well-formed formula;
| |
| | |
| * <math>\forall x. A</math> is Harrop provided <math>A</math> is.
| |
| | |
| By excluding disjunction and existential quantification (except in the [[antecedent (logic)|antecedent]] of implication), [[Constructivism (mathematics)|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:<ref name="dummett" />
| |
| | |
| : <math>A \leftrightarrow \neg \neg A.</math>
| |
| | |
| Harrop formulae were introduced around 1956 by Ronald Harrop and independently by [[Helena Rasiowa]].<ref name="basic proof theory"/> Variations of the fundamental concept are used in different branches of [[Constructivism (mathematics)|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 clause]]s, 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:<ref name=handbook>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</ref>
| |
| | |
| * Rigid atomic formulae, i.e. constants <math>r</math> or formulae <math>r(t_1,...,t_n)</math>, are hereditary Harrop;
| |
| | |
| * <math>A \wedge B</math> is hereditary Harrop provided <math>A</math> and <math>B</math> are;
| |
| | |
| * <math>\forall x. A</math> is hereditary Harrop provided <math>A</math> is;
| |
| | |
| * <math>G \rightarrow A</math> is hereditary Harrop provided <math>A</math> is rigidly atomic, and <math>G</math> is a ''G''-formula.
| |
| | |
| ''G''-formulae are defined as follows:<ref name=handbook />
| |
| | |
| * Atomic formulae are ''G''-formulae, including truth(⊤);
| |
| | |
| * <math>A \wedge B</math> is a ''G''-formula provided <math>A</math> and <math>B</math> are;
| |
| | |
| * <math>A \vee B</math> is a ''G''-formula provided <math>A</math> and <math>B</math> are;
| |
| | |
| * <math>\forall x. A</math> is a ''G''-formula provided <math>A</math> is;
| |
| | |
| * <math>\exists x. A</math> is a ''G''-formula provided <math>A</math> is;
| |
| | |
| * <math>H \rightarrow A</math> is a ''G''-formula provided <math>A</math> is, and <math>H</math> is hereditary Harrop.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Constructivism (mathematics)]]
| |
| [[Category:Intuitionism]]
| |
My hobby is mainly Machining.
I also to learn Japanese in my spare time.
My website; перейти на сайт