Lucas Dixon

Constructing Induction Rules for Deductive Synthesis Proofs

We describe novel computational techniques for constructing induction rules for deductive synthesis proofs. Deductive synthesis holds out the promise of automated construction of correct computer programs from specifications of their desired behaviour. Synthesis of programs with iteration or recursion requires inductive proof, but standard techniques for the construction of appropriate induction rules are restricted to recycling the recursive structure of the specifications. What is needed is induction rule construction techniques that can introduce novel recursive structures. We show that a combination of rippling and the use of meta-variables as a least-commitment device can provide such novelty.

IsaPlanner: A Prototype Proof Planner in Isabelle

IsaPlanner is a generic framework for proof planning in the interactive theorem prover Isabelle. It facilitates the encoding of reasoning techniques, which can be used to conjecture and prove theorems automatically. This paper introduces our approach to proof planning, gives and overview of IsaPlanner, and presents one simple yet effective reasoning technique.

Conjecture Synthesis for Inductive Theories

We have developed a program for inductive theory formation, called IsaCoSy, which synthesises conjectures ‘bottom-up’ from the available constants and free variables. The synthesis process is made tractable by only generating irreducible terms, which are then filtered through counter-example checking and passed to the automatic inductive prover IsaPlanner. The main technical contribution is the presentation of a constraint mechanism for synthesis. As theorems are discovered, this generates additional constraints on the synthesis process. We evaluate IsaCoSy as a tool for automatically generating the background theories one would expect in a mature proof assistant, such as the Isabelle system. The results show that IsaCoSy produces most, and sometimes all, of the theorems in the Isabelle libraries. The number of additional un-interesting theorems are small enough to be easily pruned by hand.

Open-graphs and monoidal theories

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph . While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor , and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits ‘enough adhesivity’ from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are ‘type safe’ in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discrete and finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parameterised by graphical signatures, which are similar to the monoidal signatures of Joyal and Street, and define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs and more general monoidal theories. Thus, open-graphs give us a tool for mechanised reasoning in monoidal categories.

Higher Order Rippling in IsaPlanner

We present an account of rippling with proof critics suitable for use in higher order logic in Isabelle/ISAPLANNER. We treat issues not previously examined, in particular regarding the existence of multiple annotations during rippling. This results in an efficient mechanism for rippling that can conjecture and prove needed lemmas automatically as well as present the resulting proof plans as Isar style proof scripts.

Graphical reasoning in compact closed categories for quantum computation

Compact closed categories provide a foundational formalism for a variety of important domains, including quantum computation. These categories have a natural visualisation as a form of graphs. We present a formalism for equational reasoning about such graphs and develop this into a generic proof system with a fixed logical kernel for reasoning about compact closed categories. A salient feature of our system is that it provides a formal and declarative account of derived results that can include ‘ellipses’-style notation. We illustrate the framework by instantiating it for a graphical language of quantum computation and show how this can be used to perform symbolic computation.

Towards formal proof script refactoring

We propose proof script refactorings as a robust tool for constructing, restructuring, and maintaining formal proof developments. We argue that a formal approach is vital, and illustrate by defining and proving correct a number of valuable refactorings in a simplified proof script and declarative proof language of our own design.

Case-Analysis for rippling and inductive proof

Rippling is a heuristic used to guide rewriting and is typically used for inductive theorem proving. We introduce a method to support case-analysis within rippling. Like earlier work, this allows goals containing if-statements to be proved automatically. The new contribution is that our method also supports case-analysis on datatypes. By locating the case-analysis as a step within rippling we also maintain the termination. The work has been implemented in IsaPlanner and used to extend the existing inductive proof method. We evaluate this extended prover on a large set of examples from Isabelles theory library and from the inductive theorem proving literature. We find that this leads to a significant improvement in the coverage of inductive theorem proving. The main limitations of the extended prover are identified, highlight the need for advances in the treatment of assumptions during rippling and when conjecturing lemmas.

Best-First rippling

Rippling is a form of rewriting that guides search by only performing steps that reduce the differences between formulae. Termination is normally ensured by a defined measure that is required to decrease with each step. Because of these restrictions, rippling will fail to prove theorems about, for example, mutual recursion where steps that temporarily increase the differences are necessary. Best-first rippling is an extension to rippling where the restrictions have been recast as heuristic scores for use in best-first search. If nothing better is available, previously illegal steps can be considered, making best-first rippling more flexible than ordinary rippling. We have implemented best-first rippling in the IsaPlanner system together with a mechanism for caching proof-states that helps remove symmetries in the search space, and machinery to ensure termination based on term embeddings. Our experiments show that the implementation of best-first rippling is faster on average than IsaPlanners version of traditional depth-first rippling, and solves a range of problems where ordinary rippling fails.

Open Graphs and Computational Reasoning

We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of half-edges (edges which are drawn with an unconnected end) and enjoy rich compositional principles by connecting graphs along these half-edges. In particular, this allows equations and rewrite rules to be specified between graphs. Particular computational models can then be encoded as an axiomatic set of such rules. Further rules can be derived graphically and rewriting can be used to simulate the dynamics of a computational system, e.g. evaluating a program on an input. Examples of models which can be formalised in this way include traditional electronic circuits as well as recent categorical accounts of quantum information.We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of half-edges (edges which are drawn with an unconnected end) and enjoy rich compositional principles by connecting graphs along these half-edges. In particular, this allows equations and rewrite rules to be specified between graphs. Particular computational models can then be encoded as an axiomatic set of such rules. Further rules can be derived graphically and rewriting can be used to simulate the dynamics of a computational system, e.g. evaluating a program on an input. Examples of models which can be formalised in this way include traditional electronic circuits as well as recent categorical accounts of quantum information.