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Orthogonality as a property of term rewriting systems (TRSs) describes where the reduction rules of the system are all left-linear, that is each variable occurs only once on the left hand side of each reduction rule, and there is no overlap between them, i.e. the TRS has no critical pairs.
Term rewriting systems can be employed to compute arithmetic operations on natural numbers.To this end, each such number has to be encoded as a term.The simplest encoding is the one used in the Peano axioms, based on the constant 0 (zero) and the successor function S.
Orthogonal term rewriting systems are confluent. In certain cases, the word normal is used to mean orthogonal , particularly in the geometric sense as in the normal to a surface . For example, the y -axis is normal to the curve y = x 2 {\displaystyle y=x^{2}} at the origin.
Parallel outermost and Gross-Knuth reduction are hypernormalizing for all almost-orthogonal term rewriting systems, meaning that these strategies will eventually reach a normal form if it exists, even when performing (finitely many) arbitrary reductions between successive applications of the strategy. [8]
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In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace.Formally, starting with a linearly independent set of vectors {v 1, ... , v k} in an inner product space (most commonly the Euclidean space R n), orthogonalization results in a set of orthogonal vectors {u 1, ... , u k} that generate the same subspace as the vectors v 1 ...
Given a set E of equations between terms, the following inference rules can be used to transform it into an equivalent convergent term rewrite system (if possible): [4] [5] They are based on a user-given reduction ordering (>) on the set of all terms; it is lifted to a well-founded ordering ( ) on the set of rewrite rules by defining (s → t) (l → r) if
A rewriting system has the unique normal form property (UN) if for all normal forms a, b ∈ S, a can be reached from b by a series of rewrites and inverse rewrites only if a is equal to b. A rewriting system has the unique normal form property with respect to reduction (UN →) if for every term reducing to normal forms a and b, a is equal to ...