Abstract. Whilst it is generally accepted as a positive criterion, affordance only gives the weakest of hints for interactive systems designers. This paper shows how useful it is to consider affordance as generated by a correspondence between program symmetries and user interface symmetries. Symmetries in state spaces (for instance, as might be visualised in statecharts) can be carried through to user interfaces and into user manuals, with beneficial results. Exploiting affordances, understood in this way, in addition to their well known user interface benefits, makes programs simpler and more reliable, and makes user manuals shorter. 1

Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs. © 2001 Academic Press Key Words: formal proofs; logical flow graphs; cut elimination; bridge groups; Baumslag–Solitar groups; Gersten groups.

The logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be non-elementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an acyclic proof with O(k n+1 ) lines. In particular, there is a quadratic time algorithm which eliminates a single cycle from a proof. These results are motivated by the search for general methods on proving lower bounds on proof size and by the design of more efficient heuristic algorithms for proof search.

Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. In our approach, "logic" is often forgotten and combinatorial properties of graphs are taken into account to explain logical phenomena.

What is mathematics, exactly? This is a somewhat complicated question, with no simple answer. In any event, mathematics is like a large place, with many regions and villages, and many different ways of doing things. One can also try to make up new ways of doing things, in connection with whatever might be of interest. Let us look here at a few points which can come up naturally in ordinary life, as well as involve some substantial mathematics (in some of their forms). Differences between answers of “yes ” and “no” Imagine the following kind of question: “Does so-and-so have a pencil in his or her office?” If someone finds a pencil in the office, then that provides a way in which an answer of “yes ” can be clearly established. The pencil can simply be shown. An answer of “no ” is quite different, and apparently more complicated. How can one establish an answer of “no”, without just going through all of the contents of the office?

topics pertaining to algebras of linear operators