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In its pure form, PROLOG-EBG a deductive, rather than inductive, learning prois cess That is, by calculating the weakest preimage of the explanation it produces a hypothesis h that follows deductively from the domain theory B, while covering the training data D To be more precise, PROLOG-EBG outputs a hypothesis h that satisfies the following two constraints:

SPECIALTY General and Family Practice Subspecialty: community health EDUCATION BROWN UNIVERSITY MD 1986 Program in Medicine Clerkships: -Acute care Roger Williams General Hospital -Pediatrics The Emma Pendleton Bradly Hospital DARTMOUTH MEDICAL SCHOOL 1984 The Brown-Dartmouth Program in Medical Education Providence, RI

where the training data D consists of a set of training examples in which xi is the ith training instance and f ( x i ) is its target value (f is the target function) Notice the first of these constraints is simply a formalization of the usual requirement in machine learning, that the hypothesis h correctly predict the target value f ( x i ) for each instance xi in the training datat Of course there will, in general, be many

This application is not complete, but if you look back at the original requirements, you have hit them all Let s review those priorities: I must record the client and the task that I am working on today (Done) I must be able to check who I worked for and what I did on any day (Done) I should know who owes me for work and how much (Partially done, but we have the information to work it out) I should be able to keep track of my clients and their contact information (Done) I could do with knowing how much to invoice any client (Not done, though we have the raw data) It could be useful to get a profile of my work across clients (Not done, but we have the raw data) I don t need to bill my time by the hour just by the day.

t ~ e r we include P R O L O G - Snegation-by-failure in our definition of entailment (F), so that exe ~Y~ amples are entailed to be negative examples if they cannot be proven to be positive

alternative hypotheses that satisfy this first constraint The second constraint describes the impact of the domain theory in PROLOG-EBL: output hypothesis is The further constrained so that it must follow from the domain theory and the data This second constraint reduces the ambiguity faced by the learner when it must choose a hypothesis Thus, the impact of the domain theory is to reduce the effective size of the hypothesis space and hence reduce the sample complexity of learning Using similar notation, we can state the type of knowledge that is required by PROLOG-EBG its domain theory In particular, PROLOG-EBG for assumes the domain theory B entails the classifications of the instances in the training data:

(Done) I don t want an accounting system (And we don t have one) So where is this application going from here Well, a few ideas come to mind, particularly looking at the pieces in this list that are not yet done First, it would be nice to be able to do some more analysis for example, get a breakdown of earnings by Consultant and client It might be a good idea to add some enhancements to the features, such as seeing whether you have invoiced a client for a particular piece of work It might be worth looking at aggregating journal entries into invoicable blocks of work However, that will have to wait for another time This chapter is not about a final polished product, but about getting you to see the development as an iterative process: conceptualize, design, build, repeat.

This constraint on the domain theory B assures that an explanation can be constructed for each positive example It is interesting to compare the PROLOG-EBG learning setting to the setting for inductive logic programming (ILP) discussed in 10 In that chapter, we discussed a generalization of the usual inductive learning task, in which background knowledge B' is provided to the learner We will use B' rather than B to denote the background knowledge used by ILP, because it does not typically satisfy the constraint given by Equation (1 13) ILP is an inductive learning system, whereas PROLOG-EBG deductive ILP uses its background knowledge B' to enis large the set of hypotheses to be considered, whereas PROLOG-EBG its domain uses theory B to reduce the set of acceptable hypotheses As stated in Equation (102), ILP systems output a hypothesis h that satisfies the following constraint: