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In this thesis we investigate the applicability of proof planning to automate the verification of hardware systems. Proof planning is a meta-level reasoning technique which captures patterns of proof common to a family of theorems. It contributes to the automation of proof by incorporating and extending heuristics found in the Nqthm theorem prover and using them to guide a tactic-based theorem prover in the search for a proof. We have addressed the automation of proof for hardware verification from a proof planning perspective, and have applied the strategies and search control mechanisms of proof planning to generate automatically customised tactics which prove conjectures about the correctness of many types of circuits. The contributions of this research can be summarised as follows: (1) we show by experimentation the applicability of the proof planning ideas to verify automatically hardware designs;(2)we develop and use a methodology based on the concept of proof engineering using proof planning to verify various combinational and sequential circuits which include: arithmetic circuits (adders, subtracters, multipliers, dividers, factorials), data-path components arithmetic logic units shifters, processing units) and a simple microprocessor system; and (3) we contribute to the profiling of the Clam proof planning system by improving its robustness and efficiency in handling large terms and proofs. In verifying hardware, the user formalises a problem by writing the specification, the implementation and the conjecture, using a logic language, and asks Clam to compose a tactic to prove the conjecture. This tactic is then executed by the Oyster prover. To compose a tactic, Clam uses a set of methods which implement the heuristics that specify general-purpose tactics, and AI planning mechanisms. Search is controlled by a type of annotated rewriting called rippling, which controls the selective application of rewrite scaled wave rules. We have extended some of the Clam's methods to verify circuits. The size of the proofs were orders of magnitude larger than the proofs that had been attempted before with proof planning, and are comparable with similar verification proofs obtained by other systems but using fewer lemmas and less interaction. Proof engineering refers to the application of formal proof for system design and verification. We propose a proof engineering methodology which consists of partitioning the automation of formal proof into three different kind of tasks: user, proof and systems tasks. User tasks have to do with formalising a particular verification problem and using a formal tool to obtain a proof. Proof tasks refer to the tuning of proof techniques (e.g. methods and tactics)to help obtain a proof. Systems tasks have to do with the modification of a formal tool system. By making this distinction explicit, proof development is more manageable. We conjecture that our approach is widely applicable and can be integrated into formal verification environments to improve automation facilities, and be utilised to verify commercial and safety-critical hardware systems in industrial settings.

Proof planning is a paradigm for the automation of proof that focuses on encoding intelligence to guide the proof process. The idea is to capture common patterns of reasoning which can be used to derive abstract descriptions of proofs known as proof plans. These can then be executed to provide fully formal proofs. This thesis concerns the development and analysis of a novel approach to proof planning that focuses on an explicit representation of choices during search. We embody our approach as a proof planner for the generic proof assistant Isabelle and use the Isar language, which is human-readable and machine-checkable, to represent proof plans. Within this framework we develop an inductive theorem prover as a case study of our approach to proof planning. Our prover uses the difference reduction heuristic known as rippling to automate the step cases of the inductive proofs. The development of a flexible approach to rippling that supports its various modifications and extensions is the second major focus of this thesis. Here, our inductive theorem prover provides a context in which to evaluate rippling experimentally. This work results in an efficient and powerful inductive theorem prover for Isabelle as well as proposals for further improving the efficiency of rippling. We also draw observations in order to direct further work on proof planning. Overall, we aim to make it easier for mathematical techniques, and those specific to mechanical theorem proving, to be encoded and applied to problems.

A key problem in automating proof by mathematical induction is choosing an induction rule suitable for a given conjecture. Since Boyer & Moore’s NQTHM system the standard approach has been based on recursion analysis, which uses a combination of induction rules based on the relevant recursive function definitions. However, there are practical examples on which such techniques are known to fail. Recent research has tried to improve automation by delaying the choice of inductive rule until later in the proof, but these techniques suffer from two serious problems. Firstly, a lack of search control: specifically, in controlling the application of ‘speculative’ proof steps that partially commit to a choice of induction rule. Secondly, a lack of generality: they place significant restrictions on the form of induction rule that can be chosen. In this thesis we describe a new delayed commitment strategy for inductive proof that addresses these problems. The strategy dynamically creates an appropriate induction rule by proving schematic proof goals, where unknown rule structure is represented by meta-variables which become instantiated during the proof. This is accompanied by a proof that the generated rule is valid. The strategy achieves improved control over speculative proof steps via a novel speculation critic. It also generates a wider range of useful induction rules than other delayed commitment techniques, partly because it removes unnecessary restrictions on the individual proof cases, and partly because of a new technique for generating the rule’s overall case structure. The basic version of the strategy has been implemented using the lamdaClam proof planner. The system was extended with a novel proof critics architecture for this purpose. An evaluation shows the strategy is a useful and practical technique, and demonstrates its advantages.

The discovery of unknown lemmas, case-splits and other so called eureka steps are challenging problems for automated theorem proving and have generally been assumed to require user intervention. This thesis is mainly concerned with the automated discovery of inductive lemmas. We have explored two approaches based on failure recovery and theory formation, with the aim of improving automation of firstand higher-order inductive proofs in the IsaPlanner system. We have implemented a lemma speculation critic which attempts to find a missing lemma using information from a failed proof-attempt. However, we found few proofs for which this critic was applicable and successful. We have also developed a program for inductive theory formation, which we call IsaCoSy. IsaCoSy was evaluated on different inductive theories about natural numbers, lists and binary trees, and found to successfully produce many relevant theorems and lemmas. Using a background theory produced by IsaCoSy, it was possible for IsaPlanner to automatically prove more new theorems than with lemma speculation. In addition to the lemma discovery techniques, we also implemented an automated technique for case-analysis. This allows IsaPlanner to deal with proofs involving conditionals, expressed as if- or case-statements.

This thesis presents an investigation into the structure of proof in non-standard analysis using proof-planning. The theory of non-standard analysis, developed by Robinson in the 1960s, offers a more algebraic way of looking at proof in analysis. Proof-planning is a technique for reasoning about proof at the meta-level. In this thesis, we use it to encapsulate the patterns of reasoning that occur in non-standard analysis proofs. We first introduce in detail the mathematical theory and the proof-planning architecture. We then present our research methodology, describe the formal framework, which includes an axiomatisation, and develop suitable evaluation criteria. We then present our development of proof-plans for theorems involving limits, continuity and differentiation. We then explain how proof-planning applies to theorems which combine induction and non-standard analysis. Finally we give a detailed evaluation of the results obtained by combining the two attractive approaches of proof-planning and non-standard analysis, and draw conclusions from the work.