Subformula Linking as an Interaction Method

International audienceCurrent techniques for building formal proofs interactively involve one or several proof languages for instructing an interpreter of the languages to build or check the proof being described. These linguistic approaches have a drawback: the languages are not generally portable, even though the nature of logical reasoning is universal. We propose a somewhat speculative alternative method that lets the user directly manipulate the text of the theorem, using non-linguistic metaphors. It uses a proof formalism based on linking subformulas, which is a variant of deep inference (inference rules are allowed to apply in any formula context) where the relevant formulas in a rule are allowed to be arbitrarily distant. We substantiate the design with a prototype implementation of a linking-based interactive prover for first-order classical linear logic

Integrating a Global Induction Mechanism into a Sequent Calculus

Most interesting proofs in mathematics contain an inductive argument which requires an extension of the LK-calculus to formalize. The most commonly used calculi for induction contain a separate rule or axiom which reduces the valid proof theoretic properties of the calculus. To the best of our knowledge, there are no such calculi which allow cut-elimination to a normal form with the subformula property, i.e. every formula occurring in the proof is a subformula of the end sequent. Proof schemata are a variant of LK-proofs able to simulate induction by linking proofs together. There exists a schematic normal form which has comparable proof theoretic behaviour to normal forms with the subformula property. However, a calculus for the construction of proof schemata does not exist. In this paper, we introduce a calculus for proof schemata and prove soundness and completeness with respect to a fragment of the inductive arguments formalizable in Peano arithmetic.Comment: 16 page

Constructing Fully Complete Models of Multiplicative Linear Logic

The multiplicative fragment of Linear Logic is the formal system in this family with the best understood proof theory, and the categorical models which best capture this theory are the fully complete ones. We demonstrate how the Hyland-Tan double glueing construction produces such categories, either with or without units, when applied to any of a large family of degenerate models. This process explains as special cases a number of such models from the literature. In order to achieve this result, we develop a tensor calculus for compact closed categories with finite biproducts. We show how the combinatorial properties required for a fully complete model are obtained by this glueing construction adding to the structure already available from the original category.Comment: 72 pages. An extended abstract of this work appeared in the proceedings of LICS 201

Managing the consistency of distributed documents

Many businesses produce documents as part of their daily activities: software engineers produce requirements specifications, design models, source code, build scripts and more; business analysts produce glossaries, use cases, organisation charts, and domain ontology models; service providers and retailers produce catalogues, customer data, purchase orders, invoices and web pages. What these examples have in common is that the content of documents is often semantically related: source code should be consistent with the design model, a domain ontology may refer to employees in an organisation chart, and invoices to customers should be consistent with stored customer data and purchase orders. As businesses grow and documents are added, it becomes difficult to manually track and check the increasingly complex relationships between documents. The problem is compounded by current trends towards distributed working, either over the Internet or over a global corporate network in large organisations. This adds complexity as related information is not only scattered over a number of documents, but the documents themselves are distributed across multiple physical locations. This thesis addresses the problem of managing the consistency of distributed and possibly heterogeneous documents. “Documents” is used here as an abstract term, and does not necessarily refer to a human readable textual representation. We use the word to stand for a file or data source holding structured information, like a database table, or some source of semi-structured information, like a file of comma-separated values or a document represented in a hypertext markup language like XML [Bray et al., 2000]. Document heterogeneity comes into play when data with similar semantics is represented in different ways: for example, a design model may store a class as a rectangle in a diagram whereas a source code file will embed it as a textual string; and an invoice may contain an invoice identifier that is composed of a customer name and date, both of which may be recorded and managed separately. Consistency management in this setting encompasses a number of steps. Firstly, checks must be executed in order to determine the consistency status of documents. Documents are inconsistent if their internal elements hold values that do not meet the properties expected in the application domain or if there are conflicts between the values of elements in multiple documents. The results of a consistency check have to be accumulated and reported back to the user. And finally, the user may choose to change the documents to bring them into a consistent state. The current generation of tools and techniques is not always sufficiently equipped to deal with this problem. Consistency checking is mostly tightly integrated or hardcoded into tools, leading to problems with extensibility with respect to new types of documents. Many tools do not support checks of distributed data, insisting instead on accumulating everything in a centralized repository. This may not always be possible, due to organisational or time constraints, and can represent excessive overhead if the only purpose of integration is to improve data consistency rather than deriving any additional benefit. This thesis investigates the theoretical background and practical support necessary to support consistency management of distributed documents. It makes a number of contributions to the state of the art, and the overall approach is validated in significant case studies that provide evidence of its practicality and usefulness

Quantified CTL: Expressiveness and Complexity

While it was defined long ago, the extension of CTL with quantification over atomic propositions has never been studied extensively. Considering two different semantics (depending whether propositional quantification refers to the Kripke structure or to its unwinding tree), we study its expressiveness (showing in particular that QCTL coincides with Monadic Second-Order Logic for both semantics) and characterise the complexity of its model-checking and satisfiability problems, depending on the number of nested propositional quantifiers (showing that the structure semantics populates the polynomial hierarchy while the tree semantics populates the exponential hierarchy)

Refinement Modal Logic

In this paper we present {\em refinement modal logic}. A refinement is like a bisimulation, except that from the three relational requirements only `atoms' and `back' need to be satisfied. Our logic contains a new operator 'all' in addition to the standard modalities 'box' for each agent. The operator 'all' acts as a quantifier over the set of all refinements of a given model. As a variation on a bisimulation quantifier, this refinement operator or refinement quantifier 'all' can be seen as quantifying over a variable not occurring in the formula bound by it. The logic combines the simplicity of multi-agent modal logic with some powers of monadic second-order quantification. We present a sound and complete axiomatization of multi-agent refinement modal logic. We also present an extension of the logic to the modal mu-calculus, and an axiomatization for the single-agent version of this logic. Examples and applications are also discussed: to software verification and design (the set of agents can also be seen as a set of actions), and to dynamic epistemic logic. We further give detailed results on the complexity of satisfiability, and on succinctness

Grafting Hypersequents onto Nested Sequents

We introduce a new Gentzen-style framework of grafted hypersequents that combines the formalism of nested sequents with that of hypersequents. To illustrate the potential of the framework, we present novel calculi for the modal logics $\mathsf{K5}$ and $\mathsf{KD5}$, as well as for extensions of the modal logics $\mathsf{K}$ and $\mathsf{KD}$ with the axiom for shift reflexivity. The latter of these extensions is also known as $\mathsf{SDL}^+$ in the context of deontic logic. All our calculi enjoy syntactic cut elimination and can be used in backwards proof search procedures of optimal complexity. The tableaufication of the calculi for $\mathsf{K5}$ and $\mathsf{KD5}$ yields simplified prefixed tableau calculi for these logic reminiscent of the simplified tableau system for $\mathsf{S5}$, which might be of independent interest