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For example, the statement "d2 is the weekday following d1" can be seen as a truth function associating to each tuple (d2, d1) the value true or false. The extension of this truth function is, by convention, the set of all such tuples associated with the value true, i.e.
In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be ...
The input to Aleph is background knowledge, specified as a logic program, a language bias in the form of mode declarations, as well as positive and negative examples specified as ground facts. [2] As output it returns a logic program which, together with the background knowledge, entails all of the positive examples and none of the negative ...
(That set might be empty, currently.) For example, the extension of a function is a set of ordered pairs that pair up the arguments and values of the function; in other words, the function's graph. The extension of an object in abstract algebra, such as a group, is the underlying set of the object. The extension of a set is the set itself.
Each such rule can be read as an implication: … meaning "If each is true, then is true". Logic programs compute the set of facts that are implied by their rules. Many implementations of Datalog, Prolog, and related languages add procedural features such as Prolog's cut operator or extra-logical features such as a foreign function interface.
A definite clause grammar (DCG) is a way of expressing grammar, either for natural or formal languages, in a logic programming language such as Prolog. It is closely related to the concept of attribute grammars / affix grammars. DCGs are usually associated with Prolog, but similar languages such as Mercury also include DCGs.
Depending on the chosen foundation, some extensionality principles may imply another. For example it is well known that in univalent foundations, the univalence axiom implies both propositional and functional extensionality. Extensionality principles are usually assumed as axioms, especially in type theories where computational content must be ...
Given a goal clause, represented as the negation of a problem to be solved : with selected literal , and an input definite clause: . whose positive literal (atom) unifies with the atom of the selected literal , SLD resolution derives another goal clause, in which the selected literal is replaced by the negative literals of the input clause and the unifying substitution is applied: