C A RIB OO An Induction Based Proof Tool for Termination with - - PowerPoint PPT Presentation

CARIBOO

An Induction Based Proof Tool for Termination with Strategies

• O. Fissore, I. Gnaedig, H. Kirchner

PROTHEO Project

Nancy-Saarbruecken Workshop on Logic, Proofs and Programs June 17–18, 2004

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Overview

• 1. Introduction: Motivations and goals
• 2. Theory: An induction-based method for proving termination
• 3. CARIBOO: An induction-based tool for proving termination
• 4. Conclusion: Going further

2

Proof and validation tools for rule-based languages

Goal: to provide programmers with adapted and accessible tools for confluence, sufficient completeness, termination, . . . For this: finer proof techniques and algorithms than those existing:

• specific to the working domain (ground term algebras)
• specific to the evaluation strategies: innermost, outermost, lazy evaluation, local

strategies, . . . And: to make them easily accessible to non specialists:

• “expertise-encapsulated” tools avoiding the user to know the underlined theory
• avoiding user interaction (as often as possible)

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Termination for rule based languages

The universal termination problem: a lot of techniques

• Syntactic and semantic methods (applying directly to the TRS)

Path orderings, polynomial interpretations...

• Transformational approaches (transforming one TRS into another)

Semantic labelling, dependency pairs... Rule based languages: one needs more specific tools

• termination specific to the ground term algebra
• termination with strategies when the standard rewriting relation diverges

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Overview

• 1. Introduction: Motivations and goals
• 2. Theory: An induction-based method for proving termination
• 3. CARIBOO: An induction-based tool for proving termination
• 4. Conclusion: Going further

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An original approach: explicit induction on the termination property

• t
• t terminates ⇐

⇒ there is no infinite rewrite chain starting from t

• t ∈ T (F)
• >: noetherian ordering on T (F)
• (∀t′ < t, t′ terminates =

⇒ t terminates)

• > is incrementally constructed along the inductive proof

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Simulating the derivation trees

f(0, 1, x) → f(x, x, x) g(x, y) → x g(x, y) → y f(t, u, v)

• f(t↓, u↓, v↓)

t↓=0 u↓=1

• f(v↓, v↓, v↓)

g(t, u)

• g(t↓, u↓)
• t↓

u↓

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Explicit induction on the termination property: a double mechanism (1/2)

abstraction (simulating normalization) applying the induction hypothesis on the direct subterms g(t1, . . . , tm)

• g(X1, . . . , Xm)

if tref ≻ t1, . . . , tm

• rdering constraints

(tref > ti) stored in C abstraction constraints (ti↓= Xi) stored in A

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Explicit induction on the termination property: a double mechanism (2/2)

narrowing (simulating reduction) top narrowing in all possible ways in one step ({g(X1, . . . , Xm)}, A, C)

• ({t′

1}, σ1A, C)

. . . ({t′

l}, σlA, C)

if g(X1, . . . , Xm) ǫ

σi t′ i and σi compatible with A

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Simulation of the derivation trees by induction

t

• detection of

a normal form

StopN

• . . .

u = f(u1, . . . , un)

t≻u1,...,un

• Abstract
• v = g(v1, . . . , vm)

t≻v

• f(u1↓, . . . , un↓)
• Narrow
• ×

branch cut by induction

Stop . . . . . .

• 10

Overview

• 1. Introduction: Motivations and goals
• 2. Theory: An induction-based method for proving termination
• 3. CARIBOO: An induction-based tool for proving termination
• 4. Conclusion: Going further

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CARIBOO: From theory to practice

• CARIBOO implements our techniques for the:

– innermost strategy – outermost strategy – local strategies

• CARIBOO is written in ELAN for ELAN
• CARIBOO is a clickable tool, accessible to non specialists
• CARIBOO is a proof assistant, accessible to specialists
• CARIBOO connects to other proof tools

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CARIBOO: A user-friendly tool

specification file signature variables TRS

• ELAN proof procedure

inference rules + strategies

• queries

delegated

features user’s answers

• proof tree (internal file)
• Java GUI
• proof tree

(display) proof options rewriting strategy interaction level

• edition tool

for the specification file

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CARIBOO: ELAN proof procedure

Strategies

Abstract Narrow StopA StopN Stop

Integrated features Inference rules

embedding ordering usable rules narrowing unsatisfiability of A

• rdering constraints solving

termination property of a term satisfiability of A

Delegated features

ELAN strategy operators

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Related work

• The innermost case: [Gnaedig, Kirchner & Fissore 2001]

Related work: the dependency pair method [Arts & Giesl 1996, 2000]: – no other strategies

• Local strategies on operators: [Fissore, Gnaedig & Kirchner 2001]

Related work: context-sensitive rewriting [Lucas 1996, 2001, 2002, Zantema 1997, Giesl & Middeldorp 1999]

• Lazy evaluation: Related work:

– also studied for functional languages [Panitz & Schmidt-Schauss 1997] – on-demand strategy annotations [Alpuente, Escobar, Gramlich & Lucas 2002] – lazy rewriting inspired by graph rewriting with labels [Lucas 2002]

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Overview

• 1. Introduction: Motivations and goals
• 2. Theory: An induction-based method for proving termination
• 3. CARIBOO: An induction-based tool for proving termination
• 4. Conclusion: Going further

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Termination of rewriting with strategies – future work –

• Extensions of the inductive proof method:

– To other rewriting strategies (lazy, needed reductions, . . . ) – To equational theories (AC) – To conditional rewriting – To the caracterisation of terminating subsets – To a proof of non-termination

• Evolutions of CARIBOO:

– Connection with other proof tools – Automatic criterion for (un)satisfiability of constraints – Deduction of good strategies

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