Languages of Dot-depth One over Infinite Words

Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order, successor, minimum, and maximum predicates. Knast (1983) has shown that it is decidable whether a language has dot-depth one. We extend Knast's result to infinite words. In particular, we describe the class of languages definable in alternation-free first-order logic over infinite words, and we give an effective characterization of this fragment. This characterization has two components. The first component is identical to Knast's algebraic property for finite words and the second component is a topological property, namely being a Boolean combination of Cantor sets. As an intermediate step we consider finite and infinite words simultaneously. We then obtain the results for infinite words as well as for finite words as special cases. In particular, we give a new proof of Knast's Theorem on languages of dot-depth one over finite words.Comment: Presented at LICS 201

On -maximality

AbstractThis paper investigates a connection between the semantic notion provided by the ordering ◁∗ among theories in model theory and the syntactic (N)SOPn hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP2 and SOP1. It is shown here that SOP3 implies SOP2 implies SOP1. In Shelah's article (Ann. Pure Appl. Logic 80 (1996) 229) it was shown that SOP3 implies ◁∗-maximality and we prove here that ◁∗-maximality in a model of GCH implies a property called SOP2″. It has been subsequently shown by Shelah and Usvyatsov that SOP2″ and SOP2 are equivalent, so obtaining an implication between ◁∗-maximality and SOP2. It is not known if SOP2 and SOP3 are equivalent.Together with the known results about the connection between the (N)SOPn hierarchy and the existence of universal models in the absence of GCH, the paper provides a step toward the classification of unstable theories without the strict order property

Discovering qualitative empirical laws

In this paper we describe GLAUBER, an AI system that models the scientific discovery of qualitative empirical laws. We have tested the system on data from the history of early chemistry, and it has rediscovered such concepts as acids, alkalis, and salts, as well as laws relating these concepts. After discussing GLAUBER we examine the program's relation to other discovery systems, particularly methods for conceptual clustering and language acquisition