Practical Automated Reasoningwith the Polarized Inverse Method

Sean McLaughlin

Thesis ProposalCarnegie Mellon University

School of Computer Science

Thesis Committee:

Jeremy Avigad CMU PhilosophyRobert Harper SCS

Dale Miller Ecole PolytechniqueFrank Pfenning SCSAndre Platzer SCS

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Outline

Proposed Contribution

A proof-producing meta-theorem prover for Twelf.

1. This is an important problem!I Twelf is the strongest proof assistant in its domainI Many applications, in theory and practiceI Requires more work than it should

2. We can (hopefully) solve it!I The inverse methodI FocusingI Polarization

3. We can solve other interesting problems along the way!I First-order logicI ConstraintsI LF

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Twelf

I A proof assistant for deductive systemsI Programming languagesI Logics

I ExamplesI Type safety for Standard ML (Lee, Crary, Harper)I Trusted computing/typed assembly language (Crary, Sarkar)I Certified compilers (Murphy, Crary)I Proof carrying code (Necula, et al.)I Security

I Proof-carrying authentication (Appel, Felten)I Proof-carrying file system (Garg, Pfenning)

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Example

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Simply Typed λ-Calculus

Language

Terms e ::= λx . e | e1 e2

Types τ ::= α | τ1 ⇒ τ2

Typing rules

Γ, x : τ ` x : τ

Γ, x : τ1 ` e : τ2

Γ ` λx . e : τ1 ⇒ τ2

Γ ` e1 : τ2 ⇒ τ Γ ` e2 : τ2

Γ ` e1 e2 : τ

Evaluation rules

x ↪→ x λx . e ↪→ λx . ee1 ↪→ λx . e3 [e2/x ]e3 ↪→ e

e1 e2 ↪→ e

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Theorem (Type Preservation)

If Γ ` e : τ and e ↪→ e ′ then Γ ` e ′ : τ

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Theorem (Type Preservation)

If Γ ` e : τ and e ↪→ e ′ then Γ ` e ′ : τ

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Twelf

X Support for variable binding

X Dependent types

X Many interesting applications

X Lack of automation (i.e. proof search)

Proofs account for %80 to %95 of code in the SML typepreservation theorem. (22 KLOC total)

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Proposed Work

Theory

abc

Implementation

I A proof-producing meta-theorem prover for Twelf.I New theorem provers for

I First-order propositional logicI Intuitionistic first-order logicI Intuitionistic first-order logic with constraintsI LF

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Outline

IntroductionTwelf

The Polarized Inverse MethodExample: Intutionistic LogicThe Inverse MethodFocusing and PolarizationImogen

ApplicationsConstraintsInductionLFM+

2

ConclusionRelated Work

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Outline

IntroductionTwelf

The Polarized Inverse MethodExample: Intutionistic LogicThe Inverse MethodFocusing and PolarizationImogen

ApplicationsConstraintsInductionLFM+

2

ConclusionRelated Work

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The Inverse Method

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The Inverse Method

Forward Search

I Why??I Local quantificationI SaturationI Redundancy elimination

Example

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The Inverse Method

Forward Search

I Why??I Local quantificationI SaturationI Redundancy elimination

Example

....

Γ −→ A

....

Γ −→ B...

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The Inverse Method

Forward Search

I Why??I Local quantificationI SaturationI Redundancy elimination

Example

abc

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The Inverse Method

Forward Search

I Why??I Local quantificationI SaturationI Redundancy elimination

Example

Γ,A −→ B Γ −→ B

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abc

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Focusing and Polarization

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Focusing and Polarization

Focused proofs

Inversion proofs

Polarized atoms,

conjunction

All proofs

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Imogen

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abc

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Outline

IntroductionTwelf

The Polarized Inverse MethodExample: Intutionistic LogicThe Inverse MethodFocusing and PolarizationImogen

ApplicationsConstraintsInductionLFM+

2

ConclusionRelated Work

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Outline

IntroductionTwelf

The Polarized Inverse MethodExample: Intutionistic LogicThe Inverse MethodFocusing and PolarizationImogen

ApplicationsConstraintsInductionLFM+

2

ConclusionRelated Work

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IntuitionisticPropositional Logic

IntuitionisticFirst-Order Logic

IntuitionisticLogic with Constraints

IntuitionisticLogic with Induction

LF

M+2

(Twelf)

Constraints

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abc

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Induction

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Inductive loop

Proof Stack

Filling

Splitting

Recursion

Q.E.D.

Unprovable

no

yes

no

no

yesyes

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LF

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abc

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M+2

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abc

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Outline

IntroductionTwelf

The Polarized Inverse MethodExample: Intutionistic LogicThe Inverse MethodFocusing and PolarizationImogen

ApplicationsConstraintsInductionLFM+

2

ConclusionRelated Work

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Related Work

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The End

Thanks for coming!

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