1. Practical Automated Reasoning
with the Polarized Inverse Method
Sean McLaughlin
Thesis Proposal
Carnegie Mellon University
School of Computer Science
Thesis Committee:
Jeremy Avigad CMU Philosophy
Robert Harper SCS
Dale Miller ´
Ecole Polytechnique
Frank Pfenning SCS
Andr´ Platzer
e SCS
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2. Outline
Proposed Contribution
A proof-producing meta-theorem prover for Twelf.
1. This is an important problem!
Twelf is the strongest proof assistant in its domain
Many applications, in theory and practice
Requires more work than it should
2. We can (hopefully) solve it!
The inverse method
Focusing
Polarization
3. We can solve other interesting problems along the way!
First-order logic
Constraints
LF
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3. 3 / 32

4. Twelf
A proof assistant for deductive systems
Programming languages
Logics
Examples
Type safety for Standard ML (Lee, Crary, Harper)
Trusted computing/typed assembly language (Crary, Sarkar)
Certified compilers (Murphy, Crary)
Proof carrying code (Necula, et al.)
Security
Proof-carrying authentication (Appel, Felten)
Proof-carrying file system (Garg, Pfenning)
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5. Example
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6. Simply Typed λ-Calculus
Language
Terms e ::= λx. e | e1 e2
Types τ ::= α | τ1 ⇒ τ2
Typing rules
Γ, x : τ1 e : τ2 Γ e1 : τ2 ⇒ τ Γ e2 : τ2
Γ, x : τ x :τ Γ λx. e : τ1 ⇒ τ2 Γ e1 e2 : τ
Evaluation rules
e1 → λx. e3 [e2 /x]e3 → e
x →x λx. e → λx. e e1 e2 → e
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7. Theorem (Type Preservation)
If Γ e : τ and e → e then Γ e :τ
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8. Theorem (Type Preservation)
If Γ e : τ and e → e then Γ e :τ
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9. Twelf
Support for variable binding
Dependent types
Many interesting applications
Lack of automation (i.e. proof search)
Proofs account for %80 to %95 of code in the SML type
preservation theorem. (22 KLOC total)
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10. Proposed Work
Theory
abc
Implementation
A proof-producing meta-theorem prover for Twelf.
New theorem provers for
First-order propositional logic
Intuitionistic first-order logic
Intuitionistic first-order logic with constraints
LF
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11. Outline
Introduction
Twelf
The Polarized Inverse Method
Example: Intutionistic Logic
The Inverse Method
Focusing and Polarization
Imogen
Applications
Constraints
Induction
LF
+
M2
Conclusion
Related Work
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12. Outline
Introduction
Twelf
The Polarized Inverse Method
Example: Intutionistic Logic
The Inverse Method
Focusing and Polarization
Imogen
Applications
Constraints
Induction
LF
+
M2
Conclusion
Related Work
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13. 12 / 32

14. The Inverse Method
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15. The Inverse Method
Forward Search
Why??
Local quantification
Saturation
Redundancy elimination
Example
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16. The Inverse Method
Forward Search
Why??
Local quantification
Saturation
Redundancy elimination
Example
.
. .
.
.
. .
.
Γ −→ A Γ −→ B
.
.
.
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17. The Inverse Method
Forward Search
Why??
Local quantification
Saturation
Redundancy elimination
Example
abc
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18. The Inverse Method
Forward Search
Why??
Local quantification
Saturation
Redundancy elimination
Example
Γ, A −→ B Γ −→ B
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19. abc
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20. Focusing and Polarization
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21. Focusing and Polarization
Polarized
atoms,
conjunction
Focused
proofs
Inversion
proofs
All proofs
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22. Imogen
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23. abc
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24. Outline
Introduction
Twelf
The Polarized Inverse Method
Example: Intutionistic Logic
The Inverse Method
Focusing and Polarization
Imogen
Applications
Constraints
Induction
LF
+
M2
Conclusion
Related Work
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25. Outline
Introduction Intuitionistic
Propositional Logic
Twelf
The Polarized Inverse Method
Intuitionistic
Example: Intutionistic Logic First-Order Logic
The Inverse Method
Focusing and Polarization
Imogen Intuitionistic
Logic with Constraints
Applications Intuitionistic
Logic with Induction
Constraints
LF
Induction
LF
+
M2 +
M2
(Twelf)
Conclusion
Related Work
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26. Constraints
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27. abc
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28. Induction
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29. Inductive loop
no
Proof Stack Q.E.D.
yes yes
Filling
no
no
Splitting Unprovable
yes
Recursion
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30. LF
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31. abc
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32. +
M2
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33. abc
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34. Outline
Introduction
Twelf
The Polarized Inverse Method
Example: Intutionistic Logic
The Inverse Method
Focusing and Polarization
Imogen
Applications
Constraints
Induction
LF
+
M2
Conclusion
Related Work
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35. Related Work
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36. The End
Thanks for coming!
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