Short Proofs are Hard to Find Ian Mertz University of Toronto - PowerPoint PPT Presentation

Short Proofs are Hard to Find Ian Mertz University of Toronto Joint work w/ Toni Pitassi, Hao Wei IAS, December 5, 2017 Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 1 / 35

Introduction Proof complexity overview Proof complexity Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 2 / 35

Introduction Proof complexity overview Proof complexity How long is the shortest P -proof of τ ? Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 2 / 35

Introduction Proof complexity overview Proof complexity How long is the shortest P -proof of τ ? Can we find short P -proofs of τ ? Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 2 / 35

Introduction Proof complexity overview Proof complexity How long is the shortest P -proof of τ ? Can we find short P -proofs of τ ? Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 2 / 35

Introduction Proof complexity overview Proof systems Propositional proof system [Cook-Reckhow] A propositional proof system is an onto map from proofs to tautologies checkable in polynomial time. Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 3 / 35

Introduction Proof complexity overview Proof systems Propositional proof system [Cook-Reckhow] A propositional proof system is an onto map from refutations to unsatisfiable formulas checkable in polynomial time. Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 3 / 35

Introduction Proof complexity overview Proof systems Propositional proof system [Cook-Reckhow] A propositional proof system is an onto map from refutations to unsatisfiable formulas checkable in polynomial time. Polynomially-bounded PPS [Cook-Reckhow] A PPS P is polynomially bounded if for every unsatisfiable k -CNF τ with n variables and poly( n ) clauses ( k = O (log n )), there exists a P -proof π such that | π | ≤ poly( n ). Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 3 / 35

Introduction Proof complexity overview Proof systems Propositional proof system [Cook-Reckhow] A propositional proof system is an onto map from refutations to unsatisfiable formulas checkable in polynomial time. Polynomially-bounded PPS [Cook-Reckhow] A PPS P is polynomially bounded if for every unsatisfiable k -CNF τ with n variables and poly( n ) clauses ( k = O (log n )), there exists a P -proof π such that | π | ≤ poly( n ). Theorem (Cook-Reckhow) NP = coNP iff there exists a polynomially-bounded PPS. Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 3 / 35

Introduction Proof complexity overview Resolution Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 4 / 35

Introduction Proof complexity overview Relations between proof systems Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 5 / 35

Introduction Automatizability Automatizability Automatizability [Bonet-Pitassi-Raz] A proof system P is automatizable if there exists an algorithm A : UNSAT → P that takes as input τ and returns a P -refutation of τ in time poly( n , S ), where S := S P ( τ ). Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 6 / 35

Introduction Automatizability Automatizability Automatizability [Bonet-Pitassi-Raz] A proof system P is f -automatizable if there exists an algorithm A : UNSAT → P that takes as input τ and returns a P -refutation of τ in time f ( n , S ), where S := S P ( τ ). Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 6 / 35

Introduction Automatizability Automatizability Automatizability [Bonet-Pitassi-Raz] A proof system P is f -automatizable if there exists an algorithm A : UNSAT → P that takes as input τ and returns a P -refutation of τ in time f ( n , S ), where S := S P ( τ ). Automatizability is connnected to many problems in computer science... theorem proving and SAT solvers ([Davis-Putnam-Logemann-Loveland], [Pipatsrisawat-Darwiche]) algorithms for PAC learning ([Kothari-Livni], [Alekhnovich-Braverman-Feldman-Klivans-Pitassi]) algorithms for unsupervised learning ([Bhattiprolu-Guruswami-Lee]) approximation algorithms (many works...) Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 6 / 35

Introduction Automatizability Known automatizability results any polynomially bounded PPS is not automatizable if NP �⊆ P/poly ([Ajtai]; [Impagliazzo],[BPR]) Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 7 / 35

Introduction Automatizability Known automatizability results any polynomially bounded PPS is not automatizable if NP �⊆ P/poly ([Ajtai]; [Impagliazzo],[BPR]) approximating S P ( τ ) to within 2 log 1 − o (1) n is NP-hard ([Alekhnovich-Buss-Moran-Pitassi]) Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 7 / 35

Introduction Automatizability Known automatizability results any polynomially bounded PPS is not automatizable if NP �⊆ P/poly ([Ajtai]; [Impagliazzo],[BPR]) approximating S P ( τ ) to within 2 log 1 − o (1) n is NP-hard ([Alekhnovich-Buss-Moran-Pitassi]) lower bounds against strong (Frege/Extended Frege) systems under cryptographic assumptions ([Bonet-Domingo-Gavald` a-Maciel-Pitassi],[BPR],[Kraj´ ı˘ cek-Pudl´ ak]) Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 7 / 35

Introduction Automatizability Known automatizability results first lower bounds against automatizability for Res , TreeRes by [Alekhnovich-Razborov] Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 8 / 35

Introduction Automatizability Known automatizability results first lower bounds against automatizability for Res , TreeRes by [Alekhnovich-Razborov] extended to Nullsatz , PC by [Galesi-Lauria] Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 8 / 35

Introduction Automatizability Known automatizability results first lower bounds against automatizability for Res , TreeRes by [Alekhnovich-Razborov] extended to Nullsatz , PC by [Galesi-Lauria] Rest of this talk: a new version of [AR] + [GL] simplified stronger lower bounds (near quasipolynomial) works for more systems (Res, TreeRes, Nullsatz, PC, Res(k)) Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 8 / 35

Introduction Automatizability Our results Theorem (Main Theorem for GapETH) Assuming GapETH , P is not n ˜ o (log log S ) -automatizable for P = Res , TreeRes , Nullsatz , PC . Theorem (Main Theorem for ETH) o (log 1 / 7 − o (1) log S ) -automatizable for P = Res , Assuming ETH , P is not n ˜ TreeRes , Nullsatz , PC . Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 9 / 35

Introduction Automatizability Our results Theorem (Main Theorem for GapETH) Assuming GapETH , P is not n ˜ o (log log S ) -automatizable for P = Res , TreeRes , Nullsatz , PC . Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 9 / 35

Introduction Automatizability Known automatizability results System Assumption Result Ref 2 log 1 − o (1) n Any PPS NP-hard [ABMP] Any poly PPS NP �⊆ P/poly superpoly( n , S ) [A]; [I],[BPR] AC 0 -Frege Diffie-Hellman requires superpoly( n , S ) [BDGMP] circuits of size 2 n ǫ Frege Factoring Blum integers superpoly( n , S ) [BPR] requires circuits of size n ω (1) E. Frege Discrete log is not in P/poly superpoly( n , S ) [KP] W[P] � = FPT Res, TreeRes superpoly( n , S ) [AR] Nullsatz, PC W[P] � = FPT superpoly( n , S ) [GL] n ˜ Ω(log log S ) Res, TreeRes, GapETH this work Ω(log 1 / 7 − o (1) log S ) n ˜ Nullsatz, PC ETH Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 10 / 35

Introduction Automatizability A note on width automatizability Theorem (Observation) If τ has a width d TreeRes or Res refutation, it can be found in time n O ( d ) . Proof: brute force (repeatedly resolve all pairs of available clauses) Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 11 / 35

Introduction Automatizability A note on width automatizability Theorem (Clegg-Edmonds-Impagliazzo) If τ has a degree d Nullsatz or PC refutation, it can be found in time n O ( d ) . Proof: Groebner basis algorithm Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 11 / 35

Introduction Automatizability A note on width automatizability Theorem (Sherali-Adams; Shor, Parrilo-Lasserre) If τ has a degree d SA or SoS refutation, it can be found in time n O ( d ) . Proof: linear/semidefinite programming Ian Mertz (U. of Toronto) Short Proofs are Hard to Find IAS, December 5, 2017 11 / 35