Bundy, Alan

English

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      Automating inductive proofCitation for published version:Bundy, A 2014, 'Automating inductive proof'. in M Baaz & S Hetzl (eds), Perspectives on Induction: Specialsession of the Logic Colloquium at the Vienna Summer of Logic. Vienna Summer of Logic, Vienna, Austria,pp. 19.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Author final version (often known as postprint)Published In:Perspectives on InductionGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 20. Feb. 2015I ALAN BUNDY, Automating inductive proof.School of Informatics, University of Edinburgh, 10 Crichton St, Edinburgh, UK.E-mail: A.Bundy@ed.ac.uk.URL Address: http://homepages.inf.ed.ac.uk/bundy/.The automation of inductive proof plays a pivotal role in the formal developmentof ict systems: both software and hardware. It is required to reason about all formsof repetition, which arises in: recursive and iterative programs; parameterised hard-ware; traces of program runs; program invariants; etc. Since formal proof is a highlyskilled and time-consuming activity, industry requires as much automation as possibleto enable formal methods to be used cost effectively.Unfortunately, inductive reasoning is much harder to automate than, for instance,first-order reasoning. Negative results from mathematical logic underpin these difficul-ties. These results include incompleteness, the undecidability of termination and theabsence of cut elimination. Of these, the absence of cut elimination creates the mostpractical problems. The proofs of even some very simple and obviously true conjec-tures require the injection of cut formulae. These formulae typically take the form ofintermediate lemmas, generalisations of the conjecture or non-standard induction rules.Cut rule steps are generally assumed to require human intervention with an interactiveprover to provide an appropriate cut formula.We have developed a proof technique called rippling [Bundy et al, 2005] that guidesthe manipulation of the induction conclusion until the induction hypothesis can beused in its proof. In fact, rippling can be used in any situation where a given embedsin a goal. It rewrites the goal while preserving and re-grouping the embedding until aninstance of the given appears as a sub-expression of the goal.The main contribution of rippling, however, is not its guidance of the step case, butthe way it informs the application of the cut rule. It provides a strong expectation ofthe direction of the proof, but is not always successful. When it fails, an analysis ofthe failure suggests an appropriate application of cut: the form of a missing lemma, ageneralisation or a non-standard induction rule [Ireland & Bundy, 1996]. This increasesthe scope of inductive-proof automation, which has economic implications for the useof formal methods in the ict industry.[Bundy et al, 2005]Bundy, A., Basin, D., Hutter, D. and Ireland, A. (2005). Rippling:Meta-level Guidance for Mathematical Reasoning, volume 56 of Cambridge Tracts inTheoretical Computer Science. Cambridge University Press.[Ireland & Bundy, 1996]Ireland, A. and Bundy, A. (1996). Productive use of failurein inductive proof. Journal of Automated Reasoning, 16(1–2):79–111.

Automating inductive proof

Abstract

Similar works

Full text

Available Versions

Edinburgh Research Explorer