Equational deduction is generalised within a category-based abstract model theory framework, and proved complete under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and valuations as model morphisms rather than functions. Applications include many- and order-sorted (conditional) equational logics, Horn clause logic, equational deduction modulo a theory, constraint logics, and more, as well as any possible combination among them. In the cases of equational deduction modulo a theory and of constraint logic the completeness result is new. One important consequence is an abstract version of Herbrand's Theorem, which provides an abstract model theoretic foundation for equational and constraint logic programming.

The Hybrid system (Ambler et al. 2002b), implemented within Isabelle/HOL, allows object logics to be represented using higher order abstract syntax (HOAS), and reasoned about using tactical theorem proving in general, and principles of (co)induction in particular. The form of HOAS provided by Hybrid is essentially a lambda calculus with constants.

Of fundamental interest is the form of the lambda abstractions provided by Hybrid. The user has the convenience of writing lambda abstractions using names for the binding variables. However, each abstraction is actually a definition of a de Bruijn expression, and Hybrid can unwind the user's abstractions (written with names) to machine friendly de Bruijn expressions (without names). In this sense the formal system contains a hybrid of named and nameless bound variable notation.

In this paper, we present a formal theory in a logical framework, which can be viewed as a model of core Hybrid, and state and prove that the model is representationally adequate for HOAS. In particular, it is the canonical translation function from λ-expressions to Hybrid that witnesses adequacy. We also prove two results that characterise how Hybrid represents certain classes of λ-expression.

We provide the first detailed proof to be published that proper locally nameless de Bruijn expressions and α-equivalence classes of λ-expressions are in bijective correspondence. This result is presented as a form of de Bruijn representational adequacy, and is a key component of the proof of Hybrid adequacy.

The Hybrid system contains a number of different syntactic classes of expression, and associated abstraction mechanisms. Hence, this paper also aims to provide a self-contained theoretical introduction to both the syntax and key ideas of the system. Although this paper will be of considerable interest to those who wish to work with Hybrid in Isabelle/HOL, a background in automated theorem proving is not essential.

This paper initiates the study of a process algebra based on atomic actions that are assigned resources, and that supports true concurrency. By true concurrency we mean that the parallel composition of concurrent processes does not rely on an interleaving of concurrent actions for its definition. Our process algebra includes a number of interesting operators that can be defined using resources of atomic actions to control their behaviour: of particular note is a (weak) sequential composition operator that exploits the truly concurrent nature of the semantics; this operator extends significantly the operation of prefixing by atomic actions that is supported in most truly concurrent semantics. Our language also includes a parallel composition operator that allows local events to execute asynchronously, while requiring synchronising events to execute simultaneously. In addition, the language supports a restriction operator and includes (unguarded) recursion.

We present both a denotational semantics and a companion operational semantics for our language. The denotational semantics supports true concurrency, so that parallel composition is defined without non-determinism or interleaving. This semantics also is novel for its treatment of recursion. The meaning of a recursive process is defined using a least fixed point on a subdomain that is determined by the body of the recursion, and that varies from one process to another. Nonetheless, the recursion operators in the language have continuous interpretations in the denotational model. In fact, our denotational model is based on a domain-theoretic generalisation of Mazurkiewicz traces in which the concatenation operator, as well as the other operators from our language, can be given continuous interpretations.

The operational model is presented in a natural SOS style. We prove a congruence theorem relating the two semantics, which implies the operational model itself is compositional. The congruence theorem also implies the denotational model is adequate with respect to the operational semantics, and we characterise the relatively mild conditions under which the denotational semantics is fully abstract with respect to the operational semantics.

Interaction nets have proved to be a useful tool for the study of computational aspects of various formalisms (e.g. λ-calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semi-simple interaction nets as a programming language, and present a type assignment system using intersection types. First we show that interactions preserve types (i.e., the system enjoys subject reduction), and we compare this type assignment system with the intersection systems for λ-calculus and term rewriting systems. Then we define a recursion scheme that ensures termination of all interaction sequences. By relaxing the scheme and using the type assignment system, we derive another sufficient condition for termination of interaction nets. Finally, we show that although the type system based on general intersection types is not decidable, its restriction to rank 2 types is, and we give an algorithm that computes principal types for nets.

We give a constructive proof showing that every finitely generated polynomial ideal has a Gröbner basis, provided the ring of coefficients is Noetherian in the sense of Richman and Seidenberg. That is, we give a constructive termination proof for a variant of the well-known algorithm for computing the Gröbner basis. In combination with a purely order-theoretic result we have proved in a separate paper, this yields a unified constructive proof of the Hilbert basis theorem for all Noether classes: if a ring belongs to a Noether class, then so does the polynomial ring. Our proof can be seen as a constructive reworking of one of the classical proofs, in the spirit of the partial realisation of Hilbert's programme in algebra put forward by Coquand and Lombardi. The rings under consideration need not be commutative, but are assumed to be coherent and strongly discrete: that is, they admit a membership test for every finitely generated ideal. As a complement to the proof, we provide a prime decomposition for commutative rings possessing the finite-depth property.

Computers are used extensively to simulate continuous dynamical systems. However, different conceptual and mathematical structures underlie discrete machines and continuous dynamics, so the question arises as to the ability of the computer to simulate or, more generally, to check the properties of a continuous system.

We discuss and compare two notions of stability for a continuous dynamical system, viz. shadowing and robustness, and relate them to both the practical and theoretical computability of the system. We first discuss what we can learn from the stability of a system, using a finite-precision machine. We then show, following the work in Collins (2005), that shadowing fails but robustness succeeds in ensuring the checkability of a reachability property.

The theory of algebraic module specifications and modular systems was developed initially mainly on the basis of equational algebraic specifications. We show that it is in fact almost independent of what kind of underlying specification framework is chosen. More specifically, we present a formulation where this framework appears as an indexed category or, equivalently, specification frame. The ensuing theory is called the theory of abstract module specifications. We are able to prove main results concerning the correctness and compositionality of abstract module specifications in a purely categorical way, assuming the existence of pushouts of morphisms between abstract specifications that allow model amalgamation, functor extension and/or suitable free constructions. Then, by instantiating the theory of abstract module specifications to the behaviour specification frame in the sense of Nivela and Orejas, we obtain a theory of behaviour module specifications.

Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.

Mixins are modules that may contain deferred components, that is, components not defined in the module itself; moreover, in contrast to parameterised modules (like ML functors), they can be mutually dependent and allow their definitions to be overridden. In a preceding paper we defined a syntax and denotational semantics of a kernel language of mixin modules. Here, we take instead an axiomatic approach, giving a set of algebraic laws expressing the expected properties of a small set of primitive operators on mixins. Interpreting axioms as rewriting rules, we get a reduction semantics for the language and prove the existence of normal forms. Moreover, we show that the model defined in the earlier paper satisfies the given axiomatisation.

We present the dual to Birkhoff's variety theorem in terms of predicates over the carrier of a cofree coalgebra (that is, in terms of ‘coequations’). We then discuss the dual to Birkhoff's completeness theorem, showing how closure under deductive rules dualises to yield two modal operators acting on coequations. We discuss the properties of these operators and show that they commute. We prove as our main result the invariance theorem, which is the formal dual of Birkhoff's completeness theorem.

In this paper, aimed at dependently typed programmers, we present a novel connection between automated and interactive theorem proving paradigms. The novelty is that the connection offers a better trade-off between usability, efficiency and soundness when compared to existing techniques. This technique allows for a powerful interactive proof framework that facilitates efficient verification of finite domain theorems and guided construction of the proof of infinite domain theorems. Such situations typically occur with industrial verification. As a case study, an embedding of SAT and CTL model checking is presented, both of which have been implemented for the dependently typed proof assistant Agda.

Finally, an example of a real world railway control system is presented, and shown using our proof framework to be safe with respect to an abstract model of trains not colliding or derailing. We demonstrate how to formulate safety directly and show using interactive theorem proving that signalling principles imply safety. Therefore, a proof by an automated theorem prover that the signalling principles hold for a concrete system implies the overall safety. Therefore, instead of the need for domain experts to validate that the signalling principles imply safety they only need to make sure that the safety is formulated correctly. Therefore, some of the validation is replaced by verification using interactive theorem proving.

In this paper, we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorially as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO rewriting without interfaces, confluence for terminating rewriting systems is, in general, undecidable. Nevertheless, we show here that confluence for DPOI, and hence string diagram rewriting, is decidable. We apply this result to give effective procedures for deciding local confluence of symmetric monoidal theories with and without Frobenius structure by critical pair analysis. For the latter, we introduce the new notion of path joinability for critical pairs, which enables finitely many joins of a critical pair to be lifted to an arbitrary context in spite of the strong non-local constraints placed on rewriting in a generic symmetric monoidal theory.

In this paper we define a logical framework, called &;lambda;TT, that is well suited for semantic analysis. We introduce the notion of a fibration [Lscr]1 : [Fscr]1 [xrarr ] [Cscr]1 being internally definableThe definability as used in this paper should not be confused with Bénabou's ‘definability’ (Bénabou 1985). in a fibration [Lscr]2 : [Fscr]2 [xrarr ] [Cscr]2. This notion amounts to distinguishing an internal category Lin [Lscr]2 and relating [Lscr]1 to the externalization of L through a pullback. When both [Lscr]1 and [Lscr]2 are term models of typed calculi [Lscr]1 and [Lscr]2, respectively, we say that [Lscr]1 is an internal typed calculus definable in the frame language [Lscr]2. We will show by examples that if an object language is adequately represented in λTT, then it is an internal typed calculus definable in the frame language λTT. These examples also show a general phenomenon: if the term model of an object language has categorical structure S, then an adequate encoding of the language in λTT imposes an explicit internal categorical structure S in the term model of λTT and the two structures are related via internal definability. Our categorical investigation of logical frameworks indicates a sensible model theory of encodings.

We investigate the use of conjunctive non-standard type inference for the elimination of useless code in higher-order typed functional programs. In particular, we present a non-standard type assignment system for detecting useless code together with a mapping that simplifies a program by removing the useless code detected using the system.

Iterative monads were introduced by Calvin Elgot in the 1970's and are those ideal monads in which every guarded system of recursive equations has a unique solution. We prove that every ideal monad has an iterative reflection, that is, an embedding into an iterative monad with the expected universal property. We also introduce the concept of iterativity for algebras for the monad , following in the footsteps of Evelyn Nelson and Jerzy Tiuryn, and prove that is iterative if and only if all free algebras for are iterative algebras.

We show how coalgebras can be presented by operations and equations. This is a special case of Linton's approach to algebras over a general base category ${\cal X}$, namely where ${\cal X}$ is taken as the dual of sets. Since the resulting equations generalise coalgebraic coequations to situations without cofree coalgebras, we call them coequations. We prove a general co-Birkhoff theorem describing covarieties of coalgebras by means of coequations. We argue that the resulting coequational logic generalises modal logic. This relies on the fact that coalgebraic operations respect an appropriate notion of bisimulation and can be considered as modal operators.

Divisibility monoids are a natural algebraic generalisation of Mazurkiewicz trace monoids, namely monoids in which the distributivity of the underlying lattices is kept as a hypothesis, but the relations between the generators are not necessarily supposed to be commutations. Garside monoids are a natural algebraic generalisation of spherical Artin monoids, namely monoids in which the existence of common multiples is kept as a hypothesis, but the relations between the generators are not necessarily supposed to be of Coxeter type. Here, we show that the quasi-centre of these monoids can be studied and described in a similar way, and then we exhibit the intersection between the two classes of monoids.

We propose a variant of public announcement logic for asynchronous systems. To capture asynchrony, we introduce two different modal operators for sending and receiving messages. The natural approach to defining the semantics leads to a circular definition, but we describe two restricted cases in which we solve this problem. The first case requires the Kripke model representing the initial epistemic situation to be a finite tree, and the second one only allows announcements from the existential fragment. After establishing some validities, we study the model checking problem and the satisfiability problem in cases where the semantics is well-defined, and we provide several complexity results.

We give a short and elementary proof of the Schröder–Simpson Theorem, which states that the space of all continuous maps from a given space X to the non-negative reals with their Scott topology is the cone-theoretic dual of the probabilistic powerdomain on X.

We present a type system based on session types that works on a conventional pi calculus. Types are equipped with a constructor that describes the two ends of a single communication channel, this being the only type available for describing the behaviour of channels. Session types, in turn, describe the behaviour of each individual channel end, as usual. A novel notion of typing context split allows for typing processes not typable with extant type systems. We show that our system guarantees that typed processes do not engage in races for linear resources. We assess the expressiveness of the type system by providing three distinct encodings – from the pi calculus with polarized variables, from the pi calculus with accept and request primitives, and from the linear pi calculus – into our system. For each language we present operational and typing correspondences, showing that our system effectively subsumes foregoing works on linear and session types. In the case of the linear pi calculus we also provide a completeness result.