Early efforts in runtime verification show that parametric regular and temporal logic specifications can be monitored efficiently. These approaches, however, have limited expressiveness: their specifications always reduce to monitors with finite state. More recent developments showed that parametric context-free properties can be efficiently monitored with overheads generally lower than 12-15%. While context-free grammars are more expressive than finite-state languages, they still do not allow every computable safety property. This paper presents a monitor synthesis algorithm for string rewriting systems (SRS). SRSs are well known to be Turing complete, allowing for the formal specification of any computable safety property. Earlier attempts at Turing complete monitoring have been relatively inefficient. This paper demonstrates that monitoring parametric SRSs is practical. The presented algorithm uses a modified version of Aho-Corasick string searching for quick pattern matching with an incremental rewriting approach that avoids reexamining parts of the string known to contain no redexes.
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