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22
Fresh Logic
 Journal of Applied Logic
, 2007
"... Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with resp ..."
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Cited by 186 (22 self)
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Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with respect to metavariables. We present oneandahalfthorder logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of oneandahalfthorder logic derivability, show them equivalent, show that the derivations satisfy cutelimination, and prove correctness of an interpretation of firstorder logic within it. We discuss the technicalities in a wider context as a casestudy for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation
Merging HOL with Set Theory  preliminary experiments
, 1994
"... Set theory is the standard foundation for mathematics, but the majority of general purpose mechanised proof assistants support versions of type theory (higher order logic). Examples include Alf, Automath, Coq, EHDM, HOL, IMPS, LAMBDA, LEGO, Nuprl, PVS and Veritas. For many applications type theory w ..."
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Cited by 11 (1 self)
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Set theory is the standard foundation for mathematics, but the majority of general purpose mechanised proof assistants support versions of type theory (higher order logic). Examples include Alf, Automath, Coq, EHDM, HOL, IMPS, LAMBDA, LEGO, Nuprl, PVS and Veritas. For many applications type theory works well and provides, for specification, the benefits of typechecking that are wellknown in programming. However, there are areas where types get in the way or seem unmotivated. Furthermore, most people with a scientific or engineering background already know set theory, whereas type theory may appear inaccessable and so be an obstacle to the uptake of proof assistants based on it. This paper describes some experiments (using HOL) in combining set theory and type theory; the aim is to get the best of both worlds in a single system. Three approaches have been tried, all based on an axiomatically specified type V of ZFlike sets: (i) HOL is used without any additions besides V; (ii) an emb...
Reasoning with Constraint Diagrams
 School of Computing, Mathematical and Information Sciences
, 2004
"... Constraint diagrams are designed for the formal specification of software systems. However, their applications are broader than this since constraint diagrams are a logic that can be used in any formal setting. This document summarizes the main results presented in my PhD thesis, the focus of which ..."
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Cited by 11 (3 self)
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Constraint diagrams are designed for the formal specification of software systems. However, their applications are broader than this since constraint diagrams are a logic that can be used in any formal setting. This document summarizes the main results presented in my PhD thesis, the focus of which is on a fragment of the constraint diagram language, called spider diagrams, and constraint diagrams themselves. In the thesis, sound and complete systems of spider diagrams and constraint diagrams are presented and the expressiveness of the spider diagram language is established. 1
A Comparison of Additivity Axioms in Timed Transition Systems
, 1993
"... This paper discusses some axioms from the literature which have been used to define properties of timed transition systems. The additivity axiom proposed by (amongst others) Wang, and Nicollin and Sifakis is compared with the trajectory axiom of Lynch and Vaandrager. Some conditions for an additive ..."
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Cited by 9 (2 self)
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This paper discusses some axioms from the literature which have been used to define properties of timed transition systems. The additivity axiom proposed by (amongst others) Wang, and Nicollin and Sifakis is compared with the trajectory axiom of Lynch and Vaandrager. Some conditions for an additive transition system to be trajectoried are discussed. These are proved sufficient by using some simple terminology from category theory to show how this problem about timed transition systems can be turned into an equivalent problem about monotone functions on partially ordered sets. We also discuss trajectory (bi)simulation, which is a variant of HoStuart's path bisimulation, and use similar techniques to discuss when (bi)simulation is equivalent to trajectory (bi)simulation.
Integrated Formal Methods with Richer Methodological Profiles for the Development of MultiPerspective Systems
, 1996
"... The thesis investigates some of the traditional problems with the established formal methods, such as requirements elicitation, the validation problem, divergence from current industrial practice, adverse effects on early problem solving and the incompleteness of perspective. Recent approaches to so ..."
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Cited by 8 (2 self)
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The thesis investigates some of the traditional problems with the established formal methods, such as requirements elicitation, the validation problem, divergence from current industrial practice, adverse effects on early problem solving and the incompleteness of perspective. Recent approaches to solving some of these problems are reviewed, including structured and formal methods integration, hybrid formal methods and multiparadigmed approaches. The definition of a method first used by Kronlof is adopted and two reasons for integrating methods are hypothesised: ffl The integration of methods which result in a richer methodological profile, such as methods which address different stages of the lifecycle, and; ffl The integration of methods which result in a wider overall perspective, and are thus effective over a wider number of prospective problems, such as methods which consider different orthogonal aspects of requirements. Two pieces of work are then presented, one for each hypot...
A General Mathematics of Names
 Information and Computation
, 2007
"... We introduce FMG (FraenkelMostowski Generalised) set theory, a generalisation of FM set theory which allows binding of infinitely many names instead of just finitely many names. We apply this generalisation to show how three presentations of syntax — de Bruijn indices, FM sets, and namecarrying sy ..."
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Cited by 8 (5 self)
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We introduce FMG (FraenkelMostowski Generalised) set theory, a generalisation of FM set theory which allows binding of infinitely many names instead of just finitely many names. We apply this generalisation to show how three presentations of syntax — de Bruijn indices, FM sets, and namecarrying syntax — have a relation generalising to all sets and not only sets of syntax trees. We also give syntaxfree accounts of Barendregt representatives, scope extrusion, and other phenomena associated to αequivalence. Our presentation uses a novel presentation based not on a theory but on a concrete model U.
Proofs Without Syntax
 Annals of Mathematics
"... [M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatori ..."
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Cited by 6 (0 self)
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[M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graphtheoretic), rather than syntactic. It defines a combinatorial proof of a proposition φ as a graph homomorphism h: C → G(φ), where G(φ) is a graph associated with φ and C is a coloured graph. The main theorem is soundness and completeness: φ is true if and only if there exists a combinatorial proof h: C → G(φ). 1.
A study of substitution, using nominal techniques and FraenkelMostowki sets
"... FraenkelMostowski (FM) set theory delivers a model of names and alphaequivalence. This model, now generally called the ‘nominal ’ model, delivers inductive datatypes of syntax with alphaequivalence — rather than inductive datatypes of syntax, quotiented by alphaequivalence. The treatment of name ..."
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Cited by 4 (1 self)
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FraenkelMostowski (FM) set theory delivers a model of names and alphaequivalence. This model, now generally called the ‘nominal ’ model, delivers inductive datatypes of syntax with alphaequivalence — rather than inductive datatypes of syntax, quotiented by alphaequivalence. The treatment of names and alphaequivalence extends to the entire sets universe. This has proven useful for developing ‘nominal ’ theories of reasoning and programming on syntax with alphaequivalence, because a sets universe includes elements representing functions, predicates, and behaviour. Often, we want names and alphaequivalence to model captureavoiding substitution. In this paper we show that FM set theory models captureavoiding subsitution for names in much the same way as it models alphaequivalence; as an operation valid for the entire sets universe which coincides with the usual (inductively defined) operation on inductive datatypes. In fact, more than one substitution action is possible (they all agree on sets representing
Model Theory of Finite Difference Fields and Simple Groups
, 2007
"... The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without prop ..."
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Cited by 3 (0 self)
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The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. 2 Asymptotic classes are classes of finite structures which have uniformly definable estimates for the cardinalities of their firstorder definable sets akin to those in finite fields given by the LangWeil estimates. The goal of the thesis is to prove that the finite simple groups of a fixed Lie type and Lie rank form asymptotic classes. This requires the following: 1. The introduction describes the background. 2. Chapter 4 shows a general method of generating one asymptotic class of structures from another through the notion of biinterpretability. Specifically, the notions
Mechanical verification of mutually recursive procedures for parsing expressions generated by a LL(1) grammar using separation logic
 In Preparation. 2006
, 2006
"... This paper adds support for mutually recursive procedures on top of a predicate transformer semantics of imperative programs with pointers implemented in PVS theorem prover. We define and prove correct a collection of mutually recursive procedures which constructs the parsing tree of an expression g ..."
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Cited by 2 (2 self)
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This paper adds support for mutually recursive procedures on top of a predicate transformer semantics of imperative programs with pointers implemented in PVS theorem prover. We define and prove correct a collection of mutually recursive procedures which constructs the parsing tree of an expression generated by a context free grammar. We use separation logic to specify and verify these procedures; the parsing tree is represented in memory using pointers and the specification predicates are defined using separation logic.