Proofs Without Syntax [37] introduced polynomial-time checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. The paper lifts a simple, strongly normalising cut elimination from combinatori...

We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations. Normalising with regard to the Keplerian symmetry, we uncover resonances and conjecture that the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1:1 resonance corresponds to the ortho...

In both traditional and modern societies, twinship, as an unusual mode of reproduction, involves difficulties for social systems in maintaining consistent classification systems. It is proposed that the most prevalent response to twinship involves various 'strategies of normalisation' to defuse and contain the potential disruption. This proposition is illustrated and analysed in relation to eth...

We present a normalisation framework for linguistic representations and illustrate its use by normalising the Stanford Dependency graphs (SDs) produced by the Stanford parser into Labelled Stanford Dependency graphs (LSDs). The normalised representations are evaluated both on a testsuite of constructed examples and on free text. The resulting representations improve on standard Predicate/Argume...

We consider equational theories of binary relations, in a language expressing composition, converse, and lattice operations. We treat the equations valid in the standard model of sets and also define a hierarchy of equational axiomatisations stratifying the standard theory. By working directly with a presentation of relation-expressions as graphs we are able to define a notion of reduction whic...

Large-scale knowledge graphs (KGs) are widely used in industry and academia, and provide excellent use-cases for ontologies. We find, however, that popular ontology languages, such as OWL and Datalog, cannot express even the most basic relationships on the normalised data format of KGs. Existential rules are more powerful, but may make reasoning undecidable. Normalising them to suit KGs often a...

In this paper we introduce a cut-elimination procedure for classical logic, which is both strongly normalising and consisting of local proof transformations. Traditional cut-elimination procedures, including the one by Gentzen, are formulated so that they only rewrite neighbouring inference rules; that is they use local proof transformations. Unfortunately, such local proof transformation, if d...

We present a proof (of the main parts of which there is a formal version, checked with the Isabelle proof assistant) that, for a G3-style calculus covering all of intuitionistic zero-order logic, with an associated term calculus, and with a particular strongly normalising and confluent system of cut-reduction rules, every reduction step has, as its natural deduction translation, a sequence of z...

In this paper we extend the preliminary work developed elsewhere and investigate how to characterise many aspects of the compliance problem in business process modeling. We first define a formal and conceptually rich language able to represent, and reason about, chains of reparational obligations of various types. Second, we devise a mechanism for normalising a system of legal norms. Third, we ...

Many functions have to be written over and over again for different datatypes, either because datatypes change during the development of programs, or because functions with similar functionality are needed on diierent datatypes. Examples of such functions are pretty printers, debug-gers, equality functions, uniiers, pattern matchers, rewriting functions, etc. Such functions are called polytypic...