Learning First-Order Definitions of Functions

First-order learning involves finding a clause-form definition of a relation from examples of the relation and relevant background information. In this paper, a particular first-order learning system is modified to customize it for finding definitions of functional relations. This restriction leads to faster learning times and, in some cases, to definitions that have higher predictive accuracy. Other first-order learning systems might benefit from similar specialization.Comment: See http://www.jair.org/ for any accompanying file

Equational binary decision diagrams

We incorporate equations in binary decision diagrams (BDD). The resulting objects are called EQ-BDDs. A straightforward notion of ordered EQ-BDDs (EQ-OBDD) is defined, and it is proved that each EQ-BDD is logically equivalent to an EQ-OBDD. Moreover, on EQ-OBDDs satisfiability and tautology checking can be done in constant time. Several procedures to eliminate equality from BDDs have been reported in the literature. Typical for our approach is that we keep equalities, and as a consequence do not employ the finite domain property. Furthermore, our setting does not strictly require Ackermann's elimination of function symbols. This makes our setting much more amenable to combinations with other techniques in the realm of automatic theorem proving, such as term rewriting. We introduce an algorithm, which for any propositional formula with equations finds an EQ-OBDD that is equivalent to it. The algorithm is proved to be correct and terminating, by means of recursive path ordering. The algorithm has been implemented, and applied to benchmarks known from literature. The performance of a prototype implementation is comparable to existing proposals

A system of axiomatic set theory - Part VII

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now. Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I-III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of infinity. Thereby it becomes possible to set up the models on the basis of only I-III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II. On both these bases the Π0-system of Part VI, which satisfies the axioms I-V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did. Let us recall the main points of this procedure. For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be define

Metalevel and reflexive extension in mechanical theorem proving

In spite of many years of research into mechanical assistance for mathematics it is still much more difficult to construct a proof on a machine than on paper. Of course this is partly because, unlike a proof on paper, a machine checked proof must be formal in the strictest sense of that word, but it is also because usually the ways of going about building proofs on a machine are limited compared to what a mathematician is used to. This thesis looks at some possible extensions to the range of tools available on a machine that might lend a user more flexibility in proving theorems, complementing whatever is already available.In particular, it examines what is possible in a framework theorem prover. Such a system, if it is configured to prove theorems in a particular logic T, must have a formal description of the proof theory of T written in the framework theory F of the system. So it should be possible to use whatever facilities are available in F not only to prove theorems of T, but also theorems about T that can then be used in their turn to aid the user in building theorems of T.The thesis is divided into three parts. The first describes the theory FS₀, which has been suggested by Feferman as a candidate for a framework theory suitable for doing meta-theory. The second describes some experiments with FS₀, proving meta-theorems. The third describes an experiment in extending the theory PRA, declared in FS₀, with a reflection facility.More precisely, in the second section three theories are formalised: propositional logic, sorted predicate logic, and the lambda calculus (with a deBruijn style binding). For the first two the deduction theorem and the prenex normal form theorem are respectively proven. For the third, a relational definition of beta-reduction is replaced with an explicit function.In the third section, a method is proposed for avoiding the work involved in building a full Godel style proof predicate for a theory. It is suggested that the language be extended with quotation and substitution facilities directly, instead of providing them as definitional extensions. With this, it is possible to exploit an observation of Solovay's that the Lob derivability conditions are sufficient to capture the schematic behaviour of a proof predicate. Combining this with a reflection schema is enough to produce a non-conservative extension of PRA, and this is demonstrated by some experiments

All-Pairs Min-Cut in Sparse Networks

Algorithms are presented for the all-pairs min-cut problem in bounded treewidth, planar, and sparse networks. The approach used is to preprocess the input n-vertex network so that afterward, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute min-cuts subsequently. In particular, after an Onlog Ž n. preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for Ž 2 such networks the all-pairs min-cut problem can be solved in time On.. This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, �, of the input network. The parameter � varies between 1 and �Ž. n; the algorithms perform well when � � on. Ž. The value Ž 2 of a min-cut can be found in time On� � log �. and all-pairs min-cut can be Ž 2 4 solved in time On � � log �. for sparse networks. The corresponding runnin

Some topics in set theory

This thesis is divided into two parts. In the first of these we consider Ackermann-type set theories and many of our results concern natural models. We prove a number of results about the existence of natural models of Ackermann's set theory, A, and applications of this work are shown to answer several questions raised by Reinhardt in [56]. A+ (introduced in [56]) is another Ackermann-type set theory and we show that its set theoretic part is precisely ZF. Then we introduce the notion of natural models of A + and show how our results on natural models of A extend to these models. There are a number of results about other Ackermann-type set theories and some of the work which was already known for ZF is extended to A. This includes permutation models, which are shown to answer another of Reinhardt's questions. In the second part we consider the different approaches to set theory; dealing mainly with the more philosophical aspects. We reconsider Cantor's work, suggest that it has frequently been misunderstood and indicate how quasi-constructive set theories seem to use a definite part of Cantor's earlier ideas. Other approaches to set theory are also considered and criticised. The section on NF includes some more technical observations on ordered pairs. There is also an appendix, in which we outline some results on extended ordinal arithmetic.<p