The Theorems project aims at extending current computer algebra systems by facilities jor supporting mathematical proving. The present early-prototype version of the Theorems software system is implemented in Mathetnatica 3.0. The system consists of a general higher-order predicate logic prover and a collection of special provers that call each other depending on the particular proof situations. The individual provers imitate the proof style of human mathematicians and aim at producing human-readable proofs in natuml language presented in nested cells that facilitate studying the computer-generated proofs at various levels of detail. The special provers are intimately connected with the junctors that build up the various mathematical domains.1 The Objectives of the Theorems ProjectThe Tlaeorema project aims at providing a uniform (logic and software) frame for computing, solving, and proving. In a simplified view, given a "knowledge base" K of formulae (and a logical / computational derivation mechanism L), q q q a " comput e~' for K (in an abstract sense) enables the user to provide an expression (term, formula, program) T with a free variable z and a value v (from an appropriate domain) and "evaluates" Tr+. (T with rJsubstituted for x) w.r.t. K (within the calculus L), a " solve~' for K enables the user to provide an expression T with a free variable z and produces (all) values v for which Tz+" holds (in L) w.r. t. K, and a "prover" for K enables the user to provide an expression T with a free variable z and decides whether, for all values v, Tz+v holds (in L) w.r.t. K. qThis paper was preparedduring a stay of the first author as a visiting researchfellow at the Universityof Tsukuba, Japan, Chair of ProfessorTetsuo Ida, and was supportedby TARA (Tsukuba Advanced Research Alliance), IPA-AITP (Advanced Information Technology Program of the Information-TechnologyPromotion Agency,

No abstract

No abstract

Origami is the centuries-old art of folding paper, and recently, it is investigated as computer science: Given an origami with creases, the problem to determine if it can be flat after folding all creases is NP-hard. Another hundreds-old art of folding paper is a pop-up book. A model for the pop-up book design problem is given, and its computational complexity is investigated. We show that both of the opening book problem and the closing book problem are NP-hard.

No abstract