Proof Normalisation in a Logic Identifying Isomorphic Propositions

We define a fragment of propositional logic where isomorphic propositions, such as $A\land B$ and $B\land A$, or $A\Rightarrow (B\land C)$ and $(A\Rightarrow B)\land(A\Rightarrow C)$ are identified. We define System I, a proof language for this logic, and prove its normalisation and consistency

Proof Normalisation in a Logic Identifying Isomorphic Propositions

We define a fragment of propositional logic where isomorphic propositions, such as A wedge B and B wedge A, or A ==> (B wedge C) and (A ==> B) wedge (A ==> C) are identified. We define System I, a proof language for this logic, and prove its normalisation and consistency

Involutive automorphisms of N∘∘N_\circ^\circ groups of finite Morley rank

We classify a large class of small groups of finite Morley rank: $N_\circ^\circ$-groups which are the infinite analogues of Thompson's $N$-groups. More precisely, we constrain the $2$-structure of groups of finite Morley rank containing a definable, normal, non-soluble, $N_\circ^\circ$-subgroup

Graph Abstraction and Abstract Graph Transformation

Many important systems like concurrent heap-manipulating programs, communication networks, or distributed algorithms are hard to verify due to their inherent dynamics and unboundedness. Graphs are an intuitive representation of states of these systems, where transitions can be conveniently described by graph transformation rules. We present a framework for the abstraction of graphs supporting abstract graph transformation. The abstraction method naturally generalises previous approaches to abstract graph transformation. The set of possible abstract graphs is finite. This has the pleasant consequence of generating a finite transition system for any start graph and any finite set of transformation rules. Moreover, abstraction preserves a simple logic for expressing properties on graph nodes. The precision of the abstraction can be adjusted according to properties expressed in this logic to be verified

Extensional proofs in a propositional logic modulo isomorphisms

System I is a proof language for a fragment of propositional logic where isomorphic propositions, such as $A\wedge B$ and $B\wedge A$, or $A\Rightarrow(B\wedge C)$ and $(A\Rightarrow B)\wedge(A\Rightarrow C)$ are made equal. System I enjoys the strong normalisation property. This is sufficient to prove the existence of empty types, but not to prove the introduction property (every closed term in normal form is an introduction). Moreover, a severe restriction had to be made on the types of the variables in order to obtain the existence of empty types. We show here that adding $\eta$-expansion rules to System I permits to drop this restriction, and yields a strongly normalising calculus with enjoying the full introduction property.Comment: 15 pages plus references and appendi

Deduction modulo theory

This paper is a survey on Deduction modulo theor

Quantum logic and decohering histories

An introduction is given to an algebraic formulation and generalisation of the consistent histories approach to quantum theory. The main technical tool in this theory is an orthoalgebra of history propositions that serves as a generalised temporal analogue of the lattice of propositions of standard quantum logic. Particular emphasis is placed on those cases in which the history propositions can be represented by projection operators in a Hilbert space, and on the associated concept of a `history group'.Comment: 14 pages LaTeX; Writeup of lecture given at conference ``Theories of fundamental interactions'', Maynooth Eire 24--26 May 1995

Synonymy and Identity of Proofs - A Philosophical Essay

The main objective of the dissertation is to investigate from a strictly philosophical perspective different approaches and results related to the problem of identity of proofs, which is a problem of general proof theory at the intersection of mathematics and philosophy. The author characterizes,compares and evaluates a range of formal criteria of proof-identity that have been proposed in the proof-theoretic literature. While these proposals come from mathematical logicians, the author’s background in both mathematical logic and philosophy allows him to present and discuss these proposals in a manner that is accessible to and fruitful for philosophers, especially those working in logic and philosophy of mathematics, as well as mathematical logicians. The dissertation is structured into a prologue and five sections. In the prologue, the author traces the development of the concept of a proof in ancient philosophy, culminating in the work of Aristotle. In Section I, the author turns to the roots of proof theory in modern philosophy, offering a detailed interpretation of Kant’s “Die falsche Spitzfindigkeit der vier syllogistischen Figuren”, which uncovers interesting links between Kant’s inferences of understanding and of reason and modern proof-theoretic semantics. In Section II, the author turns from historical to systematic considerations concerning different kinds of identity-criteria of proofs, ranging from overly liberal criteria that trivialize proof identity to overly strict, syntactical criteria. In Section III, the heart of the dissertation, the author offers a thorough philosophical discussion of the normalisation thesis. In Section IV, the author considers the difficulties encountered in his discussion of identity of proofs --- particularly of the normalisation thesis --- through the lens of a discussion of the notion of synonymy, and compares this thesis with other possible formal accounts of identity of proofs. In particular, by recourse to Carnap’s notion of synonymy, developed in “Meaning and Necessity”, the author proposes a notion of synonymy of proofs. In Section V, the final substantial section, the author compares the normalisation thesis to the Church-Turing thesis, thereby adducing another dimension of evaluation of the former