We define the notion of the uniform reduct of a propositional proof system as the set of those bounded formulas in the language of Peano Arithmetic which have polynomial size proofs under the Paris-Wilkietranslation. With respect to the arithmetic complexity of uniform reducts, we show that uniform reducts are Π1-hard and obviously in Σ2. We also show under certain regularity conditions that ea...

The aim of this work is to obtain an interactive proof environment based on Isabelle/HOL for reasoning formally about cryptographic protocols, expressed as processes of the spi calculus (a π-calculus with cryptographic primitives). To this end, we formalise syntax, semantics, and hedged bisimulation, an environment-sensitive bisimulation which can be used for proving security properties of prot...

In this note, we provide a formal proof of the fact that any protocol that is a zero-knowledge proof of knowledge for a relation R is also a secure protocol for the zero-knowledge proof of knowledge functionality, where the latter is defined according to the standard framework of stand-alone secure computation. Although this is a well-known fact, to the best of our knowledge, no full proof of t...

Interactive formal proof and automated veriication based on decision graphs are two contrasting formal hardware veriication techniques. In this paper, we compare these two approaches. In particular we consider HOL and MDG. The former is an interactive theorem proving system based on higher-order logic, while the latter is an automatic system based on Multiway Decision Graphs. As the basis for o...

We describe an implementation of the Display Logic calculus for relation algebra as an Isabelle theory. Our implementation is the rst mechanisation of any display calculus, but also provides a useful interactive proof assistant for relation algebra. The inference rules of Display Logic are coded directly as Isabelle theorems, thereby guaranteeing the correctness of all derivations. We describe ...

Sledgehammer is a highly successful subsystem of Isabelle/HOL that calls automatic theorem provers to assist with interactive proof construction. It requires no user configuration: it can be invoked with a single mouse gesture at any point in a proof. It automatically finds relevant lemmas from all those currently available. An unusual aspect of its architecture is its use of unsound translatio...

For a finite group $G$ let $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$. We give a short proof of a theorem of Brandl, which classifies finite groups with $nu(G)=1$.

We give a proof of the generalisation of Mendes-France and Van der Poorten's recent result over an arbitrary field of positive characteristic and then by extending a result of Carlitz, we shall introduce a class of algebraically independent series.