A shorter model theory
On syntactic congruences for &ohgr;-languages
Theoretical Computer Science - Special issue: formal language theory
Mixed real-integer linear quantifier elimination
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Efficient minimization of deterministic weak &ohgr;-automata
Information Processing Letters
Representing Arithmetic Constraints with Finite Automata: An Overview
ICLP '02 Proceedings of the 18th International Conference on Logic Programming
On the Automata Size for Presburger Arithmetic
LICS '04 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science
An effective decision procedure for linear arithmetic over the integers and reals
ACM Transactions on Computational Logic (TOCL)
Theory of Computation (Texts in Computer Science)
Theory of Computation (Texts in Computer Science)
LIRA: handling constraints of linear arithmetics over the integers and the reals
CAV'07 Proceedings of the 19th international conference on Computer aided verification
Don't care words with an application to the automata-based approach for real addition
CAV'06 Proceedings of the 18th international conference on Computer Aided Verification
A BDD-Like implementation of an automata package
CIAA'04 Proceedings of the 9th international conference on Implementation and Application of Automata
Ehrenfeucht--Fraïssé goes automatic for real addition
Information and Computation
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Automata-based decision procedures have proved to be a particularly useful tool for infinite-state model checking, where automata are used to represent sets of real and integer values. However, not all theoretical aspects of these decision procedures are completely understood. We establish triple exponential upper bounds on the automata size for FO$({\mathbb Z} ,+,and FO$({\mathbb R},Z,+,. While a similar bound for Presburger Arithmetic, i.e., FO$({\mathbb Z},+,was obtained earlier using a quantifier elimination based approach, the bound for FO$({\mathbb R},Z,+,is novel. We define two graded back-and-forth systems, and use them to derive bounds on the automata size by establishing a connection between those systems and languages that can be described by formulas in the respective logics. With these upper bounds that match the known lower bounds, the theoretical background for automata-based decision procedures for linear arithmetics becomes more complete.