The key result underlying both NP upper bounds is that given two states of a finite-state automaton encoding the set of solutions of a system of linear Diophantine equations, one can decide whether one state reaches the other in NP in the size of the encoding of the system (and without explicitly constructing the automaton). The NP upper bounds are the non-trivial part in those results, since, unlike, for instance, in existential Presburger arithmetic, the encoding of smallest solutions can grow super-polynomially. The authorвЂ™s opinion drastically changed when appealing to automata-based approaches recently allowed for settling long-standing open problems about the complexity of the existential fragments of BГјchi arithmetic and linear arithmetic over p-adic fields, which were both shown NP-completeВ . Given this history, it is not surprising that, until recently, the author was of the opinion that automata should better be avoided when attempting to prove complexity upper bounds for arithmetic theories. It took, for instance, 50 years to show that BГјchiвЂ™s seminal approach for deciding Presburger arithmetic using finite-state automata runs in triply-exponential time and thus matches the upper bound of quantifier-elimination algorithmsВ . However, understanding the algorithmic properties of automata-based decision procedures turned out to be surprisingly difficult and tedious, see e.g.В . Automata-based decision procedures for arithmetic theories have also been of remarkable practical use and have been implemented inВ tools such as LASHВ or TaPASВ . Finite-state automata over finite and infinite words provide an elegant method for deciding linear arithmetic theories such as Presburger arithmetic or linear real arithmetic.

Finite state automata key