The Word Problem in Combinatorial Calculus
The simplest example of an undecidable word problem occurs in combinatory logic: when are two strings of combinators equivalent? Because combinators encode all possible Turing machines, and the equivalence of two Turing machines is undecidable, it follows that the equivalence of two strings of combinators is undecidable.
Likewise, one has essentially the same problem in lambda calculus: given two distinct lambda expressions, there is no algorithm which can discern whether they are equivalent or not; equivalence is undecidable.
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