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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 ...
(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.
An early example of answer set programming was the planning method proposed in 1997 by Dimopoulos, Nebel and Köhler. [3] [4] Their approach is based on the relationship between plans and stable models. [5] In 1998 Soininen and Niemelä [6] applied what is now known as answer set programming to the problem of product configuration. [4]
If an atom A belongs to a stable model of a logic program P then A is the head of one of the rules of P. Minimality Any stable model of a logic program P is minimal among the models of P relative to set inclusion. The antichain property If I and J are stable models of the same logic program then I is not a proper subset of J.
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 ...
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.
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 ...