Nonmonotonic Logics and Their Algebraic Foundations

Abstract

Abstract. The goal of this note is to provide a background and references for the invited lecture presented at Computer Science Logic 2006. We briefly discuss motivations that led to the emergence of nonmonotonic logics and introduce two major nonmonotonic formalisms, default and autoepistemic logics. We then point out to algebraic principles behind the two logics and present an abstract algebraic theory that unifies them and provides an effective framework to study properties of nonmonotonic reasoning. We conclude with comments on other major research directions in nonmonotonic logics. Why nonmonotonic logicsIn the late 1970s, research on languages for knowledge representation, and considerations of basic patterns of commonsense reasoning brought attention to rules of inference that admit exceptions and are used only under the assumption of normality of the world in which one functions or to put it differently, when things are as expected.For instance, a knowledge base concerning a university should support an inference that, given no information that might indicate otherwise, if Dr. Jones is a professor at that university, then Dr. Jones teaches. Such conclusion might be sanctioned by an inference rule stating that normally university professors teach. In commonsense reasoning rules with exceptions are ubiquitous. Planning our day and knowing we are to have lunch with a friend, we might use the following rule: normally, lunches end by 1:00pm. If nothing we know indicates that the situation we are in is not normal, we use this rule and conclude that our lunch will be over by 1:00pm.The problem with such rules is that they do not lend themselves in any direct way to formalizations in terms of first-order logic, unless all exceptions are known and explicitly represented -an unrealistic expectation in practice. The reason is that standard logical inference is monotone: whenever a sentence α is a consequence of a set T of sentences then α is also a consequence of any set of sentences T ′ such that T ⊆ T ′ . On the other hand, it is clear that reasoning with normality rules when complete information is unavailable, is not monotone. In our lunch scenario, we may conclude that the lunch will be over by 1:00pm. However, if we learn that our friend will be delayed, the normality assumption is no longer valid our earlier inference is unsupported; we have to withdraw it.Such reasoning, where additional information may invalidate conclusions, is called nonmonotonic. As we briefly noted above, it is common. It has been a focus of extensive

La herencia de propiedades como mecanismo inferencial en el contexto del razonamiento no-monotono

“…Por ejemplo, podríamos necesitar hacer inferencias a partir de una base de datos sobre los diferentes vuelos de un aeropuerto, en cuyo caso hemos de suponer que contamos con completitud de la información (Reiter, 1980). En otro contexto, podríamos querer sacar conclusiones sobre los atributos de una persona que recién conocemos, en cuyo caso contamos con información escasa e incierta (Truszczyński, 2006). El rasgo común de estos ejemplos consiste en la naturaleza retractable de las conclusiones a las que llegamos.…”