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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 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]
Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept or object) represent the same object or not. For undecidability in axiomatic mathematics, see List of statements undecidable in ZFC.
The word problem is a well-known example of an undecidable problem. If A {\displaystyle A} is a finite set of generators for G {\displaystyle G} , then the word problem is the membership problem for the formal language of all words in A {\displaystyle A} and a formal set of inverses that map to the identity under the natural map from the free ...
Word problem (mathematics education), a type of textbook exercise or exam question to have students apply abstract mathematical concepts to real-world situations; Word problem (mathematics), a decision problem for algebraic identities in mathematics and computer science; Word problem for groups, the problem of recognizing the identity element ...
The Identity Correspondence Problem (ICP) asks whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. The problem is undecidable and equivalent to the following Group Problem: is the semigroup generated by a finite set of pairs of words (over a group alphabet) a group. [11]
In Microsoft Word, the feature is called "collapsible outlining". Many user interfaces provide disclosure widgets for code folding in a sidebar, indicated for example by a triangle that points sideways (if collapsed) or down (if expanded), or by a [-] box for collapsible (expanded) text, and a [+] box for expandable (collapsed) text.
The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. [3]: ND22, ND23