Correct Reasoning about Logic Programs

In this PhD project, we present an approach to the problem of determinacy inference in logic programs with cut, which treats cut uniformly and contextually. The overall aim is to develop a theoretical analysis, abstract it to a suitable domain and prove both the concrete analysis and the abstraction correct in a formal theorem prover (Coq). A crucial advantage of this approach, besides the guarantee of correctness, is the possibility of automatically extracting an implementation of the analysis

Nominal Logic Programming

Nominal logic is an extension of first-order logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, alpha-equivalence). This article investigates logic programming based on nominal logic. We describe some typical nominal logic programs, and develop the model-theoretic, proof-theoretic, and operational semantics of such programs. Besides being of interest for ensuring the correct behavior of implementations, these results provide a rigorous foundation for techniques for analysis and reasoning about nominal logic programs, as we illustrate via examples.Comment: 46 pages; 19 page appendix; 13 figures. Revised journal submission as of July 23, 200

Multi-agent Confidential Abductive Reasoning

In the context of multi-agent hypothetical reasoning, agents typically have partial knowledge about their environments, and the union of such knowledge is still incomplete to represent the whole world. Thus, given a global query they collaborate with each other to make correct inferences and hypothesis, whilst maintaining global constraints. Most collaborative reasoning systems operate on the assumption that agents can share or communicate any information they have. However, in application domains like multi-agent systems for healthcare or distributed software agents for security policies in coalition networks, confidentiality of knowledge is an additional primary concern. These agents are required to collaborately compute consistent answers for a query whilst preserving their own private information. This paper addresses this issue showing how this dichotomy between "open communication" in collaborative reasoning and protection of confidentiality can be accommodated. We present a general-purpose distributed abductive logic programming system for multi-agent hypothetical reasoning with confidentiality. Specifically, the system computes consistent conditional answers for a query over a set of distributed normal logic programs with possibly unbound domains and arithmetic constraints, preserving the private information within the logic programs. A case study on security policy analysis in distributed coalition networks is described, as an example of many applications of this system

Proofs as stateful programs: A first-order logic with abstract Hoare triples, and an interpretation into an imperative language

We introduce an extension of first-order logic that comes equipped with additional predicates for reasoning about an abstract state. Sequents in the logic comprise a main formula together with pre- and postconditions in the style of Hoare logic, and the axioms and rules of the logic ensure that the assertions about the state compose in the correct way. The main result of the paper is a realizability interpretation of our logic that extracts programs into a mixed functional/imperative language. All programs expressible in this language act on the state in a sequential manner, and we make this intuition precise by interpreting them in a semantic metatheory using the state monad. Our basic framework is very general, and our intention is that it can be instantiated and extended in a variety of different ways. We outline in detail one such extension: A monadic version of Heyting arithmetic with a wellfounded while rule, and conclude by outlining several other directions for future work.Comment: 29 page

An Implementation of Separation Logic in Coq

For certain applications, the correctness of software involved is crucial, particularly if human life is in danger. In order to achieve correctness, common practice is to gather evidence for program correctness by testing the system. Even though testing may find certain errors in the code, it cannot guarantee that the program is error-free. The program of formal verification is the act of proving or disproving the correctness of the system with respect to a formal specification. A logic for program verification is the so-called Hoare Logic. Hoare Logic can deal with programs that do not utilize pointers, i.e., it allows reasoning about programs that do not use shared mutable data structures. Separation Logic extends Hoare logic that allows pointers, including pointer arithmetic, in the programming language. It has four-pointer manipulating commands which perform the heap operations such as lookup, allocation, deallocation, and mutation. We introduce an implementation of separation logic in the interactive proof system Coq. Besides verifying that separation logic is correct, we will provide several examples of programs and their correctness proof

Fred:An Approach to Generating Real, Correct, Reusable Programs from Proofs

In this paper we describe our system for automatically extracting "correct" programs from proofs using a development of the Curry-Howard process. Although program extraction has been developed by many authors (see, for example, [HN88], [Con97] and [HKPM97]), our system has a number of novel features designed to make it very easy to use and as close as possible to ordinary mathematical terminology and practice. These features include 1. the use of Henkin's technique [Hen50] to reduce higher-order logic to many-sorted (first-order) logic; 2. the free use of new rules for induction subject to certain conditions; 3. the extensive use of previously programmed (total, recursive) functions; 4. the use of templates to make the reasoning much closer to normal mathematical proofs and 5. a conceptual distinction between the computational type theory (for representing programs) and the logical type theory (for reasoning about programs). As an example of our system we give a constructive proof of the well known theorem that every graph of even parity, which is non-trivial in the sense that it does not consist of isolated vertices, has a cycle. Given such a graph as input, the extracted program produces a cycle as promised