Symbolic computation or algebraic computation or computer algebra relates to algorithms and software for manipulating mathematical expressions and equations in symbolic form, as opposed to manipulating the approximations of specific numerical quantities represented by those symbols. Software applications that perform symbolic calculations are called computer algebra systems.
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Boolean algebra
Boolean algebra, as developed in 1854 by George Boole in his book An Investigation of the Laws of Thought, is a variant of ordinary elementary algebra differing in its values, operations, and laws. Instead of the usual algebra of numbers, Boolean algebra is the algebra of truth values 0 and 1, or equivalently of subsets of a given set. The operations are usually taken to be conjunction ¿, disjunction ¿, and negation ¬, with constants 0 and 1.
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Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra.
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Computer algebra system
A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.
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Reduce (computer algebra system)
Reduce is a general-purpose computer algebra system geared towards applications in physics. The development of the Reduce computer algebra system was started in the 1960s by Anthony C. Hearn. Since then, many scientists from all over the world have contributed to its development under his direction. Reduce is written entirely in its own LISP dialect called Standard LISP, expressed in an Algol-like syntax called RLISP. The latter is used as a basis for Reduce's user-level language.
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Theory (mathematical logic)
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. Usually a deductive system is understood from context. An element of a theory is then called an axiom of the theory, and any sentence that follows from the axioms is called a theorem of the theory. Every axiom is also a theorem. A first-order theory is a set of first-order sentences.
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First-order logic
First-order logic is a formal system used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic (a less precise term). First-order logic is distinguished from propositional logic by its use of quantified variables.
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Formal language
In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols. The alphabet of a formal language is the set of symbols, letters, or tokens from which the strings of the language may be formed; frequently it is required to be finite. The strings formed from this alphabet are called words, and the words that belong to a particular formal language are sometimes called well-formed words or well-formed formulas.
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