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The latter corresponds to a hierarchical structure ("syntax tree") which is unique for the given expression. The compiler generates machine code from the tree in such a way that operations originating at the lowest hierarchy level are executed first. Examples: !A + !B is interpreted as (!A) + (!B) ++A + !B is interpreted as (++A) + (!B)
The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]
Algebra includes the study of algebraic structures, which are sets and operations defined on these sets satisfying certain axioms. The field of algebra is further divided according to which structure is studied; for instance, group theory concerns an algebraic structure called group. Outline of algebra; Glossary of field theory; Glossary of ...
An illustration of how the levels of the hierarchy interact and where some basic set categories lie within it. In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them.
In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set , though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets .
Leading-order terms; NURBS order, a number one greater than the degree of the polynomial representation of a non-uniform rational B-spline; Order of convergence, a measurement of convergence; Order of derivation; Order of an entire function; Order of a power series, the lowest degree of its terms; Ordered list, a sequence or tuple
Word problem from the Līlāvatī (12th century), with its English translation and solution. In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation.
The word problem was one of the first examples of an unsolvable problem to be found not in mathematical logic or the theory of algorithms, but in one of the central branches of classical mathematics, algebra. As a result of its unsolvability, several other problems in combinatorial group theory have been shown to be unsolvable as well.