Arrows for Knowledge-Based Circuits

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1 Arrows for Knowledge-Based Circuits Peter Gammie A thesis submitted for the degree of Doctor of Philosophy of The Australian National University. February 2013 COLLEGE OF ENGINEERING AND COMPUTER SCIENCE

2 Declaration The work in this thesis is my own except where otherwise stated. Peter Gammie This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit

3 Acknowledgements I thank my parents, Liz and Richard, for their imperturbable support, and Clem Baker-Finch and John Lloyd for their encouragement and perspective. I am grateful to Kai Engelhardt for patiently rubbishing all sorts of half baked stuff over the years, and also to Pete Rickwood, Andrew Taylor and David Wahlstedt for many valuable discussions. I am indebted to Gerwin Klein for teaching me what I always wanted to know about a proof assistant, and have benefited from the expertise of Florian Haftmann, Brian Huffman, Andreas Lochbihler and others on the Isabelle mailing list. I have greatly enjoyed discussing synchronous programming languages with Tim Bourke and Grégoire Hamon over many years. The debt this project owes to Ron van der Meyden is obvious. He introduced me to formally reasoning about knowledge while we worked on the MCK system; I build on that experience here. John Hughes taught me about Arrows at the Advanced Functional Programming Summer School in Estonia in I thank Antti Valmari for his expert advice on automata minimisation. I value the insightful comments of Andreas Abel, Tim Bourke, Raj Goré, David Greenaway, Oleg Kiselyov, Ben Lippmeier, Adam Megacz, Toby Murray, Bernie Pope, Colin Runciman and Josef Svenningsson on various parts of this work. Some of this work was carried out while I was an Australian Youth Ambassador for Development in T.P. Hồ Chí Minh, Việt Nam, funded by the Australian Government via AusAID. I thank Bạch Việt, Bích, Cử, Daz, Dung, Mai, Mike, Nhu, anh Thien, Tigôn and especially Loan for a year of welcome distractions. Cảm ơn nhiều! I am also grateful to June Andronick, Gerwin Klein, Toby Murray, Thomas Sewell and Simon Winwood of the L4.verified team at NICTA 1 for a pleasant six months of proofs and tea in the afternoon. I thank Andrew Taylor for sending paid work my way, and the Australian National University for the funding to complete this project. Our last conclusion is to recall a principle that has been so often fruitful in Computer Science and that is central to Scott s theory of computation: a good concept is one that is closed 1. under arbitrary composition 2. under recursion. Gilles Kahn (1974) 1 NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.

4 Abstract Knowledge-based programs (KBPs) are a formalism for directly relating agents knowledge and behaviour in a way that has proven useful for specifying distributed systems. Here we present a scheme for compiling KBPs to executable automata in finite environments with a proof of correctness in Isabelle/HOL. We use Arrows, a functional programming abstraction, to structure a prototype domain-specific synchronous language embedded in Haskell. By adapting our compilation scheme to use symbolic representations we can apply it to several examples of reasonable size.

5 Contents 1 Introduction Knowledge in system design Reactive systems and synchronous digital circuits Synopsis How to read this thesis Reasoning about knowledge Modal logics of knowledge Dynamic epistemic logic Knowledge-based programs Model-checking knowledge Explicit-state model checking Model checking using Boolean decision diagrams Model checking using SAT Verifying KBP implementations by model checking Concluding remarks A theory of knowledge-based programs in Isabelle/HOL Proof overview A modal logic of knowledge Satisfaction Generated models Simulations Knowledge-based programs Environments and views Canonical structures Automata construction Incremental views Automata and the notion of implementation Automata using equivalence classes Automata using simulations Generic DFS Finite map operations An algorithm for automata construction Concrete views The clock view The synchronous perfect-recall view Perfect recall for a single agent Perfect recall in deterministic broadcast environments v

6 vi CONTENTS Perfect recall in non-deterministic broadcast environments Examples The autonomous robot The Muddy Children Concluding remarks Synchronous digital circuits as functional programs Circuit Semantics Circuits and Functional Programming µfp Hardware synthesis from first-order recursion equations Hydra Lava Lava Other Lavas Hawk Cryptol Jazz High-level Hardware Synthesis Concluding remarks Related Work Synchronous Languages Algebraic Techniques Relational models Other models of boxes and wires On formal functional models for synchronous digital circuits Concluding remarks Arrows for synchronous digital circuits What are Arrows? Command combinators A pattern of Arrows for reinterpretation Circuit Arrows The ArrowComb class The ArrowMux class The ArrowDelay class The ArrowCombLoop class Meta-circuits Two examples Datatypes and the need for generics Sized saturated natural numbers Concluding Remarks Interpretations of Circuit Descriptions Netlists Simulation Constructivity Analysis Kesterel: Esterel as an Arrow Transformer The Esterel Language Implementation as an Arrow Transformer

7 CONTENTS vii 5.6 Concluding remarks Knowledge-based circuits and applications Arrows for knowledge-based circuits Symbolic algorithms The Clock case The Single-Agent Perfect Recall case The Multi-Agent Broadcast Perfect Recall cases Automata Minimisation The Robot redux Logic puzzles The Muddy Children Mr. S and Mr. P Concluding remarks Model checking the Dining Cryptographers Cache coherency protocols Kesterel model Verification Concluding remarks Concluding remarks Conclusions and future work Arrows for Knowledge-based Circuits The finally-tagless approach to open syntax Staging in EDSLs Sharing in EDSLs Capturing information Concluding remarks Representations and implementation techniques The KBP formalism A Model Checking Knowledge and Linear Time: PSPACE Cases 168 A.1 Introduction A.2 Basic definitions A.3 Main results A.4 An algorithm scheme A.5 Model checking with respect to perfect recall A.5.1 Formulas of L {2,U,K } A.5.2 Multi-agent broadcast and L {2,U,K 1,...,K n,c } with perfect recall A.6 Formulas of L {2,U,K 1,...,K n,c } for the clock and observational views A.7 Conclusion B The Worker/Wrapper Transformation 189 B.1 Fixed-point theorems for program transformation B.2 The transformation according to Gill and Hutton B.2.1 Worker/wrapper fusion is partially correct B.2.2 A non-strict unwrap may go awry B.3 A totally-correct fusion rule B.4 Backtracking using lazy lists and continuations

8 viii CONTENTS Bibliography 222

Application of Definitive Scripts to Computer Aided Conceptual Design

University of Warwick Department of Engineering Application of Definitive Scripts to Computer Aided Conceptual Design Alan John Cartwright MSc CEng MIMechE A thesis submitted in compliance with the regulations