A Note on Finding One-Variable Patterns Consistent with Examples and Counterexamples

We consider the problem of finding one-variable patterns consistent with given positive examples and negative examples. We try to give some evidence that the pattern finding problem is computationally difficult by finding an NP-complete graph problem (called MCP) such that the pattern finding problem is a subproblem of MCP. We also give sufficient conditions such that the pattern finding problem is polynomial-time computable and show that some of the conditions are related with solving word-equations in one variable.