Turing complete is a term used in computability theory to describe abstract machines. These are usually called automata. An automaton is Turing complete if it can be used to emulate a Turing machine. It is also called computationally universal.
Most modern programming languages are Turing-complete.
The standard regular expressions, which most programming languages use, are not Turing complete. This is because regular expression engines have been adapted to include back-references, and a finite automaton cannot handle back references.
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