Automated deduction with built-in theories: completeness results and constraint solving techniques

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Strategic Issues, Problems and Challenges in Inductive Theorem Proving

Abstract(Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving. Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems. Nowadays, the field of TP has reached a certain degree of maturity and powerful TP systems are widely available and used. The situation with ITP is strikingly different, in the sense that proving inductive theorems in an essentially automatic way still is a very challenging task, even for the most advanced existing ITP systems. Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings. In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging. More generally, we will systematically isolate, investigate and classify the main problems and challenges in ITP w.r.t. automation, on different levels and from different points of views. Finally, based on this analysis we will present some theses about the state of the art in the field, possible criteria for what could be considered as substantial progress, and promising lines of research for the future, towards (more) automated ITP

Introduction to the Literature On Programming Language Design

This is an introduction to the literature on programming language design and related topics. It is intended to cite the most important work, and to provide a place for students to start a literature search

Modal Logics for Nominal Transition Systems

We define a general notion of transition system where states and action labels can be from arbitrary nominal sets, actions may bind names, and state predicates from an arbitrary logic define properties of states. A Hennessy-Milner logic for these systems is introduced, and proved adequate and expressively complete for bisimulation equivalence. A main technical novelty is the use of finitely supported infinite conjunctions. We show how to treat different bisimulation variants such as early, late, open and weak in a systematic way, explore the folklore theorem that state predicates can be replaced by actions, and make substantial comparisons with related work. The main definitions and theorems have been formalised in Nominal Isabelle

Survey of local algorithms

A local algorithm is a distributed algorithm that runs in constant time, independently of the size of the network. Being highly scalable and fault-tolerant, such algorithms are ideal in the operation of large-scale distributed systems. Furthermore, even though the model of local algorithms is very limited, in recent years we have seen many positive results for non-trivial problems. This work surveys the state-of-the-art in the field, covering impossibility results, deterministic local algorithms, randomised local algorithms, and local algorithms for geometric graphs.Peer reviewe

Techniques for Modelling Structured Operational and Denotational Semantics Definitions with Term Rewriting Systems

A fundamental requirement for the application of automatic proof support for program verification is that the semantics of programs be appropriately formalized using the object language underlying the proof tool. This means that the semantics definition must not only be stated as syntactically correct input for the proof tool to be used, but also in such a way that the desired proofs can be performed without too many artificial complications. And it must be clear, of course, that the translation from mathematical metalanguage into the object language is correct. The objective of this work is to present methods for the formalization of structured operational and denotational semantics definitions that meet these requirements. It combines techniques known from implementation of the $\lambda$-calculus with a new way to control term rewriting on object level, thus reaching a conceptually simple representation based on unconditional rewriting. This deduction formalism is available within many of the existent proof tools, and therefore application of the representation methods is not restricted to a particular tool. Correctness of the representations is achieved by proving that the non-trivial formalizations yield results that are equivalent to the meta-level definitions in a strong sense. Since the representation algorithms have been implemented in form of executable programs, there is no need to carry out tedious coding schemes by hand. Semantics definitions can be stated in a format very close to the usual meta language format, and they can be transformed automatically into an object-level representation that is accessible to proof tools. The formalizations of the two semantics definition styles are designed in a consistent way, both making use of the same modelling of the underlying mathematical basis. Therefore, they can be used simultaneously in proofs. This is demonstrated in a larger example, where an operational and a denotational semantics definition for a programming language are proved to be equivalent using the Larch Prover. This proof has been carried out by hand before, and so the characteristics of the automated proof can be made quite clear