Type-alpha DPLs

This paper introduces Denotational Proof Languages (DPLs). DPLs are languages for presenting, discovering, and checking formal proofs. In particular, in this paper we discus type-alpha DPLs---a simple class of DPLs for which termination is guaranteed and proof checking can be performed in time linear in the size of the proof. Type-alpha DPLs allow for lucid proof presentation and for efficient proof checking, but not for proof search. Type-omega DPLs allow for search as well as simple presentation and checking, but termination is no longer guaranteed and proof checking may diverge. We do not study type-omega DPLs here. We start by listing some common characteristics of DPLs. We then illustrate with a particularly simple example: a toy type-alpha DPL called PAR, for deducing parities. We present the abstract syntax of PAR, followed by two different kinds of formal semantics: evaluation and denotational. We then relate the two semantics and show how proof checking becomes tantamount to evaluation. We proceed to develop the proof theory of PAR, formulating and studying certain key notions such as observational equivalence that pervade all DPLs. We then present NDL, a type-alpha DPL for classical zero-order natural deduction. Our presentation of NDL mirrors that of PAR, showing how every basic concept that was introduced in PAR resurfaces in NDL. We present sample proofs of several well-known tautologies of propositional logic that demonstrate our thesis that DPL proofs are readable, writable, and concise. Next we contrast DPLs to typed logics based on the Curry-Howard isomorphism, and discuss the distinction between pure and augmented DPLs. Finally we consider the issue of implementing DPLs, presenting an implementation of PAR in SML and one in Athena, and end with some concluding remarks.

Learning by Failing to Explain

Explanation-based Generalization requires that the learner obtain an explanation of why a precedent exemplifies a concept. It is, therefore, useless if the system fails to find this explanation. However, it is not necessary to give up and resort to purely empirical generalization methods. In fact, the system may already know almost everything it needs to explain the precedent. Learning by Failing to Explain is a method which is able to exploit current knowledge to prune complex precedents, isolating the mysterious parts of the precedent. The idea has two parts: the notion of partially analyzing a precedent to get rid of the parts which are already explainable, and the notion of re-analyzing old rules in terms of new ones, so that more general rules are obtained.