Abstract Natural-language proof assistants such as Naproche and our own system Natty can translate a mathematical text in controlled natural language into a series of logical formulas to be verified. Natty also contains an automatic prover that is designed to quickly verify formulas representing proof steps. Natty’s prover is based on superposition, but uses a variety of techniques that are unusual for superposition-based provers, including commutative unification and a form of rewriting that sometimes preserves the rewritten clause. To evaluate this prover and others, we have produced a test suite called TextbookMath containing over 150 theorems and their proofs, which we transcribed from a classic number systems textbook by Mendelson into N, the controlled natural language of Natty. Natty can read this text and generate a set of over 900 conjectures of higher-order logic, each corresponding to a single proof step in the original Mendelson text. We find that established high-order provers such as E and Vampire can prove only about 85% of these conjectures using a single strategy with a 5-second timeout. Surprisingly, some of the conjectures they fail to prove look relatively easy and should be provable with only a few superposition steps. Natty’s automatic prover can prove about 91% of these conjectures under a similar time restriction. If we expand the Mendelson text with various intermediate proof steps and lemmas, Natty can completely verify a textbook development of the natural numbers, integers and rationals including 5 of Wiedijk’s well-known list of 100 theorems.

This paper proposes an new method for unsupervised segmentation of images which does not rely on parametric modelling of the observed images. Furthermore, the problem of finding the number of image classes is carried out as an integral part of the segmentation process, rather than by resorting to goodness-of-fit cluster validation measures, such as AIC or MDL. A brief overview of the algorithm is given, as well as examples of its application to both synthetic and real images.

Most computer algebra systems provide little assistance in working with expressions involving functions with complex branch cuts. Worse, by their ignorance of the existence of branch cuts, algebra systems sometimes simplify complex expressions incorrectly. We propose a computer representation for branch cuts; we show how a complex expression's branch cuts may be mechanically computed, and how an expression with branch cuts may sometimes be algebraically simplified within each of its branches.