This paper describes the lessons learned from an application of formal specification techniques in the development of a security-critical system within a UK company. The authors advocate the gradual introduction of formal methods, beginning with an appreciation of existing development processes, and discuss the rôle played by non-software professionals, executable specifications, formal proof, ...

This paper presents an extension of Gentzen’s LK, called LPGK, which is suitable for expressing projective geometry and for deducing theorems of plane projective geometry. The properties of this calculus are investigated and the cut elimination theorem for LPGK is proven. A formulization of sketches is presented and the equivalence between sketches and formal proofs is demonstrated.

This article presents a formulation of the fan theorem in Martin-Löf’s type theory. Starting from one of the standard versions of the fan theorem we gradually introduce reformulations leading to a final version which is easy to interpret in type theory. Finally we describe a formal proof of that final version of the fan theorem.

A formal proof of convergence of a class of algorithms for reducing inconsistency of pairwise comparisons (PC) method is presented. The design of such algorithms is proposed. The convergence of the algorithms justifies making an inference that iterated modifications of the pc matrix made by human experts should also converge. This is instrumental for credibility of practical applications of the...

It is well known that cold dense quark matter is a color superconductor. Color superconducting phases are characterized by a non-vanishing gap function, which can be calculated self-consistently from the SchwingerDyson equation (gap equation). At asymptotic densities, calculations from first principles (QCD) should be possible. However, calculations of the gap function seem to give different re...

Cut-elimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cut-elimination method CERES (cut-elimination by resolution) works by extracting a set of clauses from a proof with cuts. Any resolution refutation of this set then serves as a skeleton of...

When one goes from a geometrical statement to an algebraic statement, the immediate translation is to replace every point by a pair of coordinates, if in the plane (or more as required). A statement with N points is then a statement with 2N (or 3N or more) variables, and the complexity of tools like cylindrical algebraic decomposition is doubly exponential in the number of variables. Hence one ...

ii Abstract (in Finnish) iiiin Finnish) iii

Ordinal embedding refers to the following problem: all we know about an unknown set of points x1, . . . , xn ∈ R are ordinal constraints of the form ‖xi−xj‖ < ‖xk−xl‖; the task is to construct a realization y1, . . . , yn ∈ R that preserves these ordinal constraints. It has been conjectured since the 1960ies that upon knowledge of all ordinal constraints a large but finite set of points can be ...

The paper re-opens the controversial case of the right frontier constraint. A formal proof of its applicability in models that use certain types of trees as modalities to represent the structure of discourse is given. Then its violation, reported to occur in discourses “under pressure”, is revisited and a proposal for discourse structure representation that deviates from classical trees is made.