Concepts inSafety of recursive Horn clauses with infinite relations
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an (ordered) -tuple is a sequence (or ordered list) of elements, where is a positive integer. There is also one 0-tuple, an empty sequence. An -tuple is defined inductively using the construction of an ordered pair. Tuples are usually written by listing the elements within parentheses "" and separated by commas; for example, denotes a 5-tuple.
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Finite set
In mathematics, a finite set is a set that has a finite number of elements. For example, is a finite set with five elements. The number of elements of a finite set is a natural number, and is called the cardinality of the set. A set that is not finite is called infinite. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting.
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Undecidable problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is impossible to construct a single algorithm that always leads to a correct yes-or-no answer. A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns yes.
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Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers. In common usage, it refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers.
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Term (logic)
In mathematical logic, universal algebra, and rewriting systems, terms are expressions which can be obtained from constant symbols, variables and function symbols. Constant symbols are the 0-ary functions, so no special syntactic class is needed for them. Terms that do not contain variables are known as ground terms; they are used to form ground expressions. Terms in first-order logic are essentially defined this way.
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Horn clause
In mathematical logic, a Horn clause is a clause with at most one positive literal. They are named after the logician Alfred Horn, who first pointed out the significance of such clauses in 1951. Horn clauses play a basic role in logic programming and are important for constructive logic. A Horn clause with exactly one positive literal is a definite clause; in universal algebra definite clauses appear as quasiidentities.
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Infinity
Infinity refers to something without any limit, and is a concept relevant in a number of fields, predominantly mathematics and physics. Having a recognizable history in these disciplines reaching back into the time of ancient Greek civilization, the term in the English language derives from Latin infinitas, which is translated as "unboundedness". In mathematics, "infinity" is often treated as if it were a number but it is not the same sort of number as the real numbers.
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Necessity and sufficiency
In logic, necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.
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