Rules of Inductive Inference

Logical formulations of inductive inference rules were developed by [Genesereth and Nilsson 88] and by [van Lamsweerde 91]. This chapter is based on these papers, as well as on [Angluin and Smith 83]. 

Definition 3-2 Given background knowledge (B and a set of examples E, a hypothesis // is an inductive conclusion of and iff the following four conditions hold  

Note that inductive inference is not necessarily sound u =H does not hold 

Since there are many possible inductive conclusions for any background knowledge and example set, it is necessary to prune the generalization search space so as to get useful inductive conclusions. Interesting approaches are based on induction biases  

We here only focus on instance-to-class generalization. Other kinds of generalization are class-to-class generalization and part-to-whole generalization. 

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In Section 3.2.1, we suggest a terminology for the components of an empirical learning system. Then, in Section 3.2.2, we define rules of inductive inference, and survey their usage in empirical learning from examples in Section 3.2.3. Finally, in... 

Learning can be abstracted as the search through a state space, where states correspond to hypotheses, and operators correspond to rules of inductive inference. 

Rules of inductive inference mostly selective necessarily constructive... 

The INDUCE systems (see [Michalski 84]) were among the first to address the issues of using constructive rules of inductive inference. The learning mechanism is a bottom-up approximation-driven one, though heavily based on heuristics. 

Note that deductive inference is always sound. Typical rules of deductive inference are modusponens, universal instantiation, resolution, mathematical induction, and so on. The branch of artificial intelligence research that is bent on automating deductive inference is called automated theorem proving. 

Observe that we have in this procedure worked out some of the steps previously left to the THEOREM PROVER, The previous procedure involves having the progranmer select a set of inductive assertions and critical points, and then feed this into the computer parts a VERIFICATION CONDITION GENERATOR and a THEOREM PROVER. In this alternative construction we still need inductive assertions as the nature of the Rule of Iteration for WHILE statements shows. Now the inductive assertions are fed directly into the THEOREM PROVER which las been augmented by the special axioms and rules D0,D1,D2,D3 and D4 in addition to all of the usual arithmetic axioms, rules of inference, rules for handling identities and special axioms for the primitives in question (such as the factorial axioms in our example). In effect the THEOREM PROVER works backwards from the output condition and the various inductive assertions using DO - D3 to find what amounts to path verification conditions -... 

Inductive Inference Module Performs generation of rules on the basis of structure and activity data based on algorithms of the logical structural approach [20] and provides tools for automated selection of biophores (pharmacophores) and interactive building of 3D QSAR models. The module performs statistical evaluation of the predictive and discriminating power of selected biophores and models. 

We say that the grounds support the claim on the basis of the existence of a warrant that explains the connection between the grounds and the claim. It is easy to relate the structure of these basic elements with the process of inference, whether inductive or deductive, in classical logic. The warrants are the set of rules of inference, and the grounds and claim are the set of well-defined propositions or statements. It will be only the sequence and procedures, as used to formulate the three basic elements and their structure in a logical fashion, that will determine the type of inference that is used. 

A more sophisticated extension would be the handling of proofs-by-induction. This requires additional rules of inference, such as those of [Kanamori and Fujita 86] or [Fribourg 93]. Such proofs are sometimes necessary, as shown in [Flener 93]. 

Generalize into properties, if possible, the examples where X is of a size equal to or less than some integer n, where n is the largest size where this leads to properties without recursion and without redundancy of information. Set m to A2 + d, where d is the decomposition decrement. The most useful generalization technique is the maximally repeated application of the replacing-a-constant-by-a-variable inductive inference rule to an example this often requires a subsequent specialization by introduction of a body to the resulting unit clause. 

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