If {A(V)} is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V 1 and V 2 are spacelike separated spacetime regions, then the system (A(V 1), A(V 2), f) is said to satisfy the Weak Reichenbach’s Common Cause Principle iff for every pair of projections A ¥ A(V 1), B ¥ A(V 2) correlated in the normal state f there exists a projection C belonging to a von Neumann algebra associated with a spacetime region V contained in the union of the backward light cones of V 1 and V 2 and disjoint from both V 1 and V 2, a projection having the properties of a Reichenbachian common cause of the correlation between A and B. It is shown that if the net has the local primitive causality property then every local system (A(V 1), A(V 2), f) with a locally normal and locally faithful state f and suitable bounded V 1 and V 2 satisfies the

Introduction: Characterizing the Questions of causal reasoning This chapter describes the mechanism approach to the study of causal reasoning. We will first offer a characterization of the central issues in human causal reasoning, and will discuss how the mechanism approach addresses these issues. In the course of this presentation, we will frequently compare the mechanism approach with alternative accounts based on analyses of covariation, or what is often termed the regularity view. The aims of this chapter are the following: to explain why covariation and mechanism are different, to discuss why such a distinction is actually a useful tool for our understanding of causal reasoning, and to explicate the complementary nature of the two views. Before presenting these two approaches, it is necessary first to offer a description of the domain or problem itself : namely, what are these alternative approaches to? Although there are a number of different ways of characterizing the study of

Intuitive notions about the advantages of modularity for evolvability run into the problem of how we parse the organism into traits. In order to resolve the “question of multiplicity”, there needs to be a way to get the human observer out of the way, and define modularity in terms of physical processes. I will offer two candidate ideas towards this resolution: • the dimensionality of phenotypic variation, and • the causal screening off of phenotypic variables by other phenotypic variables. With this framework, the evolutionary advantages that have been attributed to modularity do not derive from modularity per se. Rather, they require that there be an “alignment ” between the spaces of phenotypic variation, and the selection gradients that are available to the organism. Modularity may facilitate such alignment, but it is not sufficient; the appropriate phenotype-fitness map in conjunction with the genotypephenotype map is also necessary for evolvability. 1 The Question of Multiplicity A good deal of work in recent years has shown that the structure of the genotype-phenotype map is of fundamental importance to the process of evolution. The variational properties

developed a theory of statistical causal inference. In his presentation at the Notre Dame

Evolutionary theory is awash with probabilities. For example, natural selection is said to occur when there is variation in fitness, and fitness is standardly decomposed into two components, viability and fertility, each of which is understood probabilistically. With respect to viability, a fertilized egg is said to have a certain chance of surviving to reproductive age; with respect to fertility, an adult is said to have an expected number of offspring. There is more to evolutionary theory than the theory of natural selection, and here too one finds probabilistic concepts aplenty. When there is no selection, the theory of neutral evolution says that a gene’s chance of eventually reaching fixation is 1/(2N), where N is the number of organisms in the generation of the diploid population to which the gene belongs. The evolutionary consequences of mutation are likewise conceptualized in terms of the probability per unit time a gene has of changing from one state to another. The examples just mentioned are all “forwarddirected” probabilities; they describe the probability of later events, conditional on earlier events. However, evolutionary theory also uses “backwards probabilities ” that describe the probability of a cause conditional on its effects; for example, coalescence theory allows one to calculate the expected number of generations in the past that the genes in the present generation find their most recent common

It is increasingly common to encounter time-varying random fields on networks (metabolic networks, sensor arrays, distributed computing, etc.). This paper considers the problem of optimal, nonlinear prediction of these fields, showing from an information-theoretic perspective that it is formally identical to the problem of finding minimal local sufficient statistics. I derive general properties of these statistics, show that they can be composed into global predictors, and explore their recursive estimation properties. For the special case of discrete-valued fields, I describe a convergent algorithm to identify the local predictors from empirical data, with minimal prior information about the

This paper proposes a model for commonsense causal reasoning, based on the basic idea of neural networks. After an analysis of the advantages and limitations of existing accounts of causality, a fuzzy logic based formalism FEL is proposed that takes into account the inexactness and the cumulative evidentiality of commonsense causal reasoning, overcoming the limitations of existing accounts. Analyses concerning how FEL handles various aspects of commonsense causal reasoning are performed, in an abstract way. FEL can be implemented (naturally) in a neural (connectionist) network. This work also serves to link rule-based reasoning with neural network models, in that a rule-encoding scheme (FEL) is equated directly to a neural network model. 1 1 I wish to thank Dave Waltz, James Pustejovsky, and Tim Hickey for discussions, comments, and suggestions, during the early stage of this and related work. Thanks also go to the anonymous reviewers for their helpful suggestions. 1 Introduction ...

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This report is based on lecture notes written for CS 262C, Spring 1998, and is organized as follows. Following a review of the SL framework (Section 2) Section 3 provides a comparison to other approaches to causation and suggests an explanation of why the notion of actual cause has encountered difficulties in those approaches. Section 3 defines "actual cause" and illustrates, through examples, how the "probability that event X = x actually caused event

A major issue in Complexity Science is the formal description of emer- gent properties and behaviours in terms of lower level properties and behaviours. As a consequence, there are few techniques for empirically investigating specific emer- gent properties. In this paper, we introduce a general compositional approach to specifying such properties, using constraints to define representative sets of com- positions. More specifically, we propose a calculus of complex events, which are compositions of events generated from component-level rule executions. Complex event types can be assembled hierarchically, giving a formal means of relating be- haviours at different levels of abstraction. In being able to specify and then identify complex events of different types in systems and simulations, we have a method for empirically discovering relationships between behaviours defined at different levels. The formalism offers two important practical advantages. Firstly, higher level prop- erties can be defined with different degrees of specificity so they can be defined with limited knowledge; we can then further sub-classify properties after they have been detected to discover differences in their constituent properties. Secondly, the formal- ism is related directly to the rules driving component behaviour so that all higher level behaviours can ultimately be decomposed into rule executions; this is partic- ularly important for desirable and dysfunctional properties, and in circumstances where intervention at the component rule level is possible.