Naive Problem Solving and Naive Mathematics

AI problem solvers have almost always been given a complete and correct axiomatization of their problem domain and of the operators available to change it. Here I discuss a paradigm for problem solving in which the problem solver initially is given only a list of available operators, with no indication as to the structure of the world or the behavior of the operators. Thus, to begin it is "blind" and can only stagger about in the world tripping over things until it begins to understand what is going on. Eventually it will learn enough to solve problems in the world as well as if it the world had been explained to it initially. I call this paradigm naive problem solving. The difficulty of adequately formalizing all but the most constrained domains makes naive problem solving desirable. I have implemented a naive problem solver that learns to stack blocks and to use an elevator. It learns by finding instances of "naive mathematical cliches" which are common mental models that are likely to be useful in any domain.MIT Artificial Intelligence Laborator

Who Cares about Axiomatization? Representation, Invariance, and Formal Ontologies

The philosophy of science of Patrick Suppes is centered on two important notions that are part of the title of his recent book (Suppes 2002): Representation and Invariance. Representation is important because when we embrace a theory we implicitly choose a way to represent the phenomenon we are studying. Invariance is important because, since invariants are the only things that are constant in a theory, in a way they give the “objective” meaning of that theory. Every scientific theory gives a representation of a class of structures and studies the invariant properties holding in that class of structures. In Suppes’ view, the best way to define this class of structures is via axiomatization. This is because a class of structures is given by a definition, and this same definition establishes which are the properties that a single structure must possess in order to belong to the class. These properties correspond to the axioms of a logical theory. In Suppes’ view, the best way to characterize a scientific structure is by giving a representation theorem for its models and singling out the invariants in the structure. Thus, we can say that the philosophy of science of Patrick Suppes consists in the application of the axiomatic method to scientific disciplines. What I want to argue in this paper is that this application of the axiomatic method is also at the basis of a new approach that is being increasingly applied to the study of computer science and information systems, namely the approach of formal ontologies. The main task of an ontology is that of making explicit the conceptual structure underlying a certain domain. By “making explicit the conceptual structure” we mean singling out the most basic entities populating the domain and writing axioms expressing the main properties of these primitives and the relations holding among them. So, in both cases, the axiomatization is the main tool used to characterize the object of inquiry, being this object scientific theories (in Suppes’ approach), or information systems (for formal ontologies). In the following section I will present the view of Patrick Suppes on the philosophy of science and the axiomatic method, in section 3 I will survey the theoretical issues underlying the work that is being done in formal ontologies and in section 4 I will draw a comparison of these two approaches and explore similarities and differences between them

The Axiomatic Structure of Empirical Content

In this paper, we provide a formal framework for studying the empirical content of a given theory. We define the falsifiable closure of a theory to be the least weakening of the theory that makes only falsifiable claims. The falsifiable closure is our notion of empirical content. We prove that the empirical content of a theory can be exactly captured by a certain kind of axiomatization, one that uses axioms which are universal negations of conjunctions of atomic formulas. The falsifiable closure operator has the structure of a topological closure, which has implications, for example, for the behavior of joint vis a vis single hypotheses. The ideas here are useful for understanding theories whose empirical content is well-understood (for example, we apply our framework to revealed preference theory, and Afriat's theorem), but they can also be applied to theories with no known axiomatization. We present an application to the theory of multiple selves, with a fixed finite set of selves and where selves are aggregated according to a neutral rule satisfying independence of irrelevant alternatives. We show that multiple selves theories are fully falsifiable, in the sense that they are equivalent to their empirical content

Dynamic reasoning in a knowledge-based system

Any space based system, whether it is a robot arm assembling parts in space or an onboard system monitoring the space station, has to react to changes which cannot be foreseen. As a result, apart from having domain-specific knowledge as in current expert systems, a space based AI system should also have general principles of change. This paper presents a modal logic which can not only represent change but also reason with it. Three primitive operations, expansion, contraction and revision are introduced and axioms which specify how the knowledge base should change when the external world changes are also specified. Accordingly the notion of dynamic reasoning is introduced, which unlike the existing forms of reasoning, provide general principles of change. Dynamic reasoning is based on two main principles, namely minimize change and maximize coherence. A possible-world semantics which incorporates the above two principles is also discussed. The paper concludes by discussing how the dynamic reasoning system can be used to specify actions and hence form an integral part of an autonomous reasoning and planning system

Vienna Circle and Logical Analysis of Relativity Theory

In this paper we present some of our school's results in the area of building up relativity theory (RT) as a hierarchy of theories in the sense of logic. We use plain first-order logic (FOL) as in the foundation of mathematics (FOM) and we build on experience gained in FOM. The main aims of our school are the following: We want to base the theory on simple, unambiguous axioms with clear meanings. It should be absolutely understandable for any reader what the axioms say and the reader can decide about each axiom whether he likes it. The theory should be built up from these axioms in a straightforward, logical manner. We want to provide an analysis of the logical structure of the theory. We investigate which axioms are needed for which predictions of RT. We want to make RT more transparent logically, easier to understand, easier to change, modular, and easier to teach. We want to obtain deeper understanding of RT. Our work can be considered as a case-study showing that the Vienna Circle's (VC) approach to doing science is workable and fruitful when performed with using the insights and tools of mathematical logic acquired since its formation years at the very time of the VC activity. We think that logical positivism was based on the insight and anticipation of what mathematical logic is capable when elaborated to some depth. Logical positivism, in great part represented by VC, influenced and took part in the birth of modern mathematical logic. The members of VC were brave forerunners and pioneers.Comment: 25 pages, 1 firgure