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We prove several decidability and undecidability results for thesatisfiability and validity problems for languages that can express solutionsto word equations with length constraints. The atomic formulas over thislanguage are equality over string terms (word equations), linear inequalityover the length function (length constraints), and membership in regular sets.These questions are important in logic, program analysis, and formalverification. Variants of these questions have been studied for many decades bymathematicians. More recently, practical satisfiability procedures (aka SMTsolvers) for these formulas have become increasingly important in the contextof security analysis for string-manipulating programs such as web applications.  We prove three main theorems. First, we give a new proof of undecidabilityfor the validity problem for the set of sentences written as a forall-existsquantifier alternation applied to positive word equations. A corollary of thisundecidability result is that this set is undecidable even with sentences withat most two occurrences of a string variable. Second, we consider Booleancombinations of quantifier-free formulas constructed out of word equations andlength constraints. We show that if word equations can be converted to a solvedform, a form relevant in practice, then the satisfiability problem for Booleancombinations of word equations and length constraints is decidable. Third, weshow that the satisfiability problem for quantifier-free formulas over wordequations in regular solved form, length constraints, and the membershippredicate over regular expressions is also decidable.

(Un)Decidability Results for Word Equations with Length and Regular Expression Constraints