Intership presentation

Transcript

1. On the Automatability of Tree-like Res(k) Gaia CARENINI (Department of Computer Science, ENS Paris)
2. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Collaboration This presentation is based on a joint work with my supervisor: Susanna F. de Rezende Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 2/18
3. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Overview 1 Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question 2 Motivation Scientific Interest of the Problem P vs NP Problem 3 Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof 4 Conclusion, Further Results of the Internship & Open Questions Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 3/18
4. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions SAT Problem Given a CNF formula : (¬y ∨ ¬z)∧(¬x ∨ ¬z)∧( literal x ∨y)∧(x ∨ ¬y ∨ z)∧(¬x ∨ z) clauses Is there a truth-value assignment to the variables (x,y,z) that satisfies all clauses in ? YES → ∃ an assignment to variables that satisfies NO → ∃ a proof of unsatisfability of F, i.e. a refutation of Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 4/18
5. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions A Proof System: Tree-like Res(k) Definition of Res(k)[Kra01] Res(k) proof system is completely defined by: ∧-introduction A ∨ 1 B ∨ ( 2 ∧ ··· ∧ s ) A ∨ B ∨ ( 1 ∧ ... s ) Cut A ∨ ( 1 ∨ ··· ∨ s ) B ∨ ( 1 ∧ ··· ∧ s ) A ∨ B Refutations obtained are tree-like if every clause appears at most once. Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 5/18
6. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Automatability Definition of Automatability [BPR97] A proof system is automatable in time when there exists an algorithm that takes as input an unsatisfiable CNF formula F and outputs an -refutation of F in time in the size of the shortest -refutation of F (plus the size of F). input unsat F → Automating algorithm for runs in (size of F + size of smallest refutation of F) → output -Refutation of F Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 6/18
7. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions A Hardness Assumption: rETH Definition of rETH [DHM+14] The randomized exponential time hypothesis states that there exists > 0 and c > 1 such that 3-SAT upon n variables and c ·n clauses can’t be decided in randomized time 2 ·n. Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 7/18
8. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Our Research Question Tree-like Resolution, i.e. Tree-like Res(1), is NOT automatable in less then subexponential time under rETH [dR21]. Problem Instance What about tree-like Res(k)? Is it automatable in less then subexponential time unless rETH is false? Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 8/18
9. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Scientific Interest of the Problem The main motivations behind automatability studies lie in: Automated Theorem Proving; State-of-the-art SAT Solving (DPLL algorithms & conflict driven clause learning (CDCL)); P vs NP Problem. (a) A. Church (b) A. Turing (c) H. Putnam Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 9/18
10. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions P vs NP Problem The P vs NP problem can be decoupled into two sub-problems: Is there a polynomial size proof of unsatisfability for every unsatisfiable formula? If an unsatisfiable formula F admits polynomial size proofs of unsatisfiability, can we obtain one in polynomial time? YES ∧ YES ⇐⇒ P = NP P versus NP – a gift to mathematics from computer science (Steve Smale) Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 10/18
11. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Non Automatability of Tree-like Res(k) under rETH Main Result If tree-like Res(k) is automatable in time no( k log(k) ·log(n)) then rETH is false. Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 11/18
12. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Overview of the Proof Main Theorem Let be an n-variate formula, be tree-like Res(k). → subexponential time ShortcutRef( ) → ShortcutRef( ) has -refutation of size at most 2 n/k·log(k) · poly(n) if is SAT ShortcutRef( ) has -refutation of size at least 2Ω(n) if is UNSAT Main Theorem Main Result By contradiction! Solving 3-SAT in 2 ·n randomized time thanks to ShortcutRef( ). Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 12/18
13. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Building bricks of ShortcutRef( ) Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 13/18
14. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Expander Graph Figure: Image retrieved from [DC10] Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 14/18
15. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Architecture of ShortcutRef( ) Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 15/18
16. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Conclusions & Further Results of the Internship Non automatability of tree-like Res(k) in no((k/log(k))·log(n)); The proof of this work gives non automatability results for several other systems; Improvement of the reduction to pigeonhole principle of the Ref(F) defined in [dRGN+21]; Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 16/18
17. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Open Questions Futher Non-Automatability Results What can we say about automatability of Res(⊕) or tree-like cutting planes? Are they NP-hard to automate? Search for Automating Algorithms What other proof systems, perhaps tree-like ones, are automatable in subexponential time? Constructions of Expanders This result relies heavily on the construction of an unbalanced expander of constant degree, can we find better construction for them? Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 17/18
18. Introduction SAT Problem A Proof System: Tree-like Res(k) Automatability A Hardness Assumption: rETH Our Research Question Motivation Scientific Interest of the Problem P vs NP Problem Main Result & Overview of the Proof Non Automatability of Tree-like Res(k) under rETH Overview of the Proof Conclusion, Further Results of the Internship & Open Questions Contacts Interested in the topic? Hoping for further explanations? Please, feel free to reach out, contacting me or my advisor : Gaia Carenini at gaia.carenini@ens.psl.eu Susanna F. de Rezende at susanna.rezende@cs.lth.se Thank you! Gaia Carenini Automatability of Tree-like Res(k) 5th September 2022 18/18