Modal Logics with Counting

Abstract. We present a modal language that includes explicit operators to count the number of elements that a model might include in the extension of a formula, and we discuss how this logic has been previously investigated under different guises. We show that the language is related to graded modalities and to hybrid logics. We illustrate a possible application of the language to the treatment of plural objects and queries in natural language. We investigate the expressive power of this logic via bisimulations, discuss the complexity of its satisfiability problem, define a new reasoning task that retrieves the cardinality bound of the extension of a given input formula, and provide an algorithm to solve it. 1 Counting, Modally Suppose there are at least two apples (say, on the table, but we don’t care at the moment where the apples are). First-order logic (FOL) with equality has no problem expressing this fact1 : ∃x.∃y.(x = y ∧ Apple(x) ∧ Apple(y)). We can actually dispen...

• Type: Conference
• Year: 2010
• Where: WOLLIC
• Authors: Carlos Areces, Guillaume Hoffmann, Alexandre Denis
• Keywords: Artificial Intelligence, Counting Quantifier, Logic, Modal Languages, WOLLIC 2010