Greibach's theorem
In theoretical computer science, in particular in formal language theory, Greibach's theorem states that certain properties of formal language classes are undecidable. It is named after the computer scientist Sheila Greibach, who first proved it in 1963.
Given a set Σ, often called "alphabet", the (infinite) set of all strings built from members of Σ is denoted by Σ*. A formal language is a subset of Σ*. If L1 and L2 are formal languages, their product L1L2 is defined as the set { w1w2 : w1 ∈ L1, w2 ∈ L2 } of all concatenations of a string w1 from L1 with a string w2 from L2. If L is a formal language and a is a symbol from Σ, their quotient L/a is defined as the set { w : wa ∈ L } of all strings that can be made members of L by appending an a. Various approaches are known from formal language theory to denote a formal language by a finite description, such as a formal grammar or a finite-state machine.
For example, using an alphabet Σ = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }, the set Σ* consists of all (decimal representations of) natural numbers, with leading zeroes allowed, and the empty string, denoted as ε. The set Ldiv3 of all naturals divisible by 3 is an infinite formal language over Σ; it can be finitely described by the following regular grammar with start symbol S0:
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