Logic on Words

Quisani: Hello, I think we met before? Author: Yes, I visited Kevin Compton and Yuri Gurevich some time ago in Ann Arbor. Q: I remember it. You are not a logician, aren’t you? A: No, but I would like to talk with you about questions of logic related to the theory of finite automata. Q: I already read a column by Kevin Compton and Howard Straubing on a similar topic [13]. Are you too interested in circuits and complexity? A: I read Howard’s excellent book [42] on the subject, but it is not my main concern today. I would like to discuss more about the expressive power of fragments of Büchi’s sequential calculus and related decidability questions. Q: Would you remind me what this calculus is? A: You identify a word u = a0a1 . . . an−1 on the alphabet A with a relational structure Mu = (Dom(u), <, (Ra)a∈A), where Dom(u) = {0, . . . , n − 1}, < is the usual order on Dom(u) and Ra = {i ∈ Dom(u) | ai = a}. For instance, if u = abbaab, then Dom(u) = {0, 1, . . . , 5}, Ra = {0, 3, 4} and Rb = {1, 2, 5}. Q: I thought Büchi was interested in infinite words. A: You’re right. If u is an infinite word, you simply take Dom(u) = N. You may also consider biinfinite words, for which you would take Dom(u) = Z, or words over larger ordinals than ω, but we won’t consider these cases today. Now, with each sentence φ, is associated the set of words that satisfy φ: L(φ) = {u ∈ A | u satisfies φ} L(φ) = {u ∈ A | u satisfies φ}