Short Proofs are Hard to Find Ian Mertz University of Toronto Joint work w/ Toniann Pitassi, Hao Wei ICALP, July 10, 2019 Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 1 / 20
Introduction Proof complexity overview Proof propositional complexity Whitehead, A. N., & Russell, B. (1925). Principia mathematica . Cambridge [England]: The University Press. pp.379 Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 2 / 20
Introduction Proof complexity overview Proof propositional complexity Whitehead, A. N., & Russell, B. (1925). Principia mathematica . Cambridge [England]: The University Press. pp.379 How long is the shortest P -proof of τ ? Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 2 / 20
Introduction Proof complexity overview Proof propositional complexity Whitehead, A. N., & Russell, B. (1925). Principia mathematica . Cambridge [England]: The University Press. pp.379 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 ICALP, July 10, 2019 2 / 20
Introduction Proof complexity overview Resolution One of the simplest and most important proof systems Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 3 / 20
Introduction Proof complexity overview Resolution One of the simplest and most important proof systems SAT solvers ([Davis-Putnam-Logemann-Loveland], [Pipatsrisawat-Darwiche]) automated theorem proving model checking planning/inference Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 3 / 20
Introduction Proof complexity overview Resolution Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 4 / 20
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 P ( τ )), where S P ( τ ) is the size of the shortest P -refutation of τ . Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 5 / 20
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 P ( τ )), where S P ( τ ) is the size of the shortest P -refutation of τ . Automatizability is connnected to many problems in computer science... theorem proving and SAT solvers 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 ICALP, July 10, 2019 5 / 20
Introduction Automatizability Known automatizability lower bounds General results and results for strong systems approximating S P ( τ ) to within 2 log 1 − o (1) n is NP-hard for all “reasonable” P ([Alekhnovich-Buss-Moran-Pitassi]) Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 6 / 20
Introduction Automatizability Known automatizability lower bounds General results and results for strong systems approximating S P ( τ ) to within 2 log 1 − o (1) n is NP-hard for all “reasonable” P ([Alekhnovich-Buss-Moran-Pitassi]) lower bounds against different 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 ICALP, July 10, 2019 6 / 20
Introduction Automatizability Known automatizability lower bounds Results for weak systems first lower bounds against automatizability for Res , TreeRes by [Alekhnovich-Razborov] Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 7 / 20
Introduction Automatizability Known automatizability lower bounds Results for weak systems 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 ICALP, July 10, 2019 7 / 20
Introduction Automatizability Known automatizability lower bounds Results for weak systems 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 construction and proofs stronger lower bounds via ETH assumption results also hold for Res(r) Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 7 / 20
Introduction Automatizability Our results Theorem (Main Theorem) Assuming ETH , P is not n ˜ o (log log S P ( τ )) -automatizable for P = Res , TreeRes , Nullsatz , PC . Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 8 / 20
Introduction Automatizability Our results Theorem (Main Theorem) Assuming ETH , P is not n ˜ o (log log S P ( τ )) -automatizable for P = Res , TreeRes , Nullsatz , PC . Theorem (Main Theorem for Res(r)) o (log log S P ( τ ) / exp ( r 2 )) -automatizable for Assuming ETH , Res(r) is not n ˜ r ≤ ˜ O (log log log n ) . Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 8 / 20
Introduction Automatizability Our results Theorem (Main Theorem) Assuming ETH , P is not n ˜ o (log log S P ( τ )) -automatizable for P = Res , TreeRes , Nullsatz , PC . Theorem (Atserias-Muller’19) Assuming P � = NP , Res is not automatizable. Assuming ETH , Res is not automatizable in subexponential time. Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 8 / 20
Introduction Automatizability Our results Theorem (Main Theorem) Assuming ETH , P is not n ˜ o (log log S P ( τ )) -automatizable for P = Res , TreeRes , Nullsatz , PC . Theorem (Atserias-Muller’19) Assuming P � = NP , Res is not automatizable. Assuming ETH , Res is not automatizable in subexponential time. Theorem (Bonet-Pitassi; Ben-Sasson-Wigderson) TreeRes is n O (log S P ( τ )) -automatizable. Res is n O ( √ n log S P ( τ )) -automatizable. Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 8 / 20
Our results Overview Getting an automatizability lower bound Recipe: (1) Hard gap problem G (2) Turn an instance of G into a tautology τ such that “yes” instances have small proofs “no” instances have no small proofs (3) Run automatizing algorithm Aut on τ and see how long the output is Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 9 / 20
Our results Overview Gap hitting set S = { S 1 . . . S n } over [ n ] Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 10 / 20
Our results Overview Gap hitting set S = { S 1 . . . S n } over [ n ] hitting set : H ⊆ [ n ] s.t. H ∩ S i � = ∅ for all i ∈ [ n ] Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 10 / 20
Our results Overview Gap hitting set S = { S 1 . . . S n } over [ n ] hitting set : H ⊆ [ n ] s.t. H ∩ S i � = ∅ for all i ∈ [ n ] γ ( S ) is the size of the smallest H Gap hitting set : given S , distinguish whether γ ( S ) ≤ k or γ ( S ) > k 2 Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 10 / 20
Our results Overview Gap hitting set S = { S 1 . . . S n } over [ n ] hitting set : H ⊆ [ n ] s.t. H ∩ S i � = ∅ for all i ∈ [ n ] γ ( S ) is the size of the smallest H Gap hitting set : given S , distinguish whether γ ( S ) ≤ k or γ ( S ) > k 2 Theorem (Chen-Lin) Assuming ETH the gap hitting set problem cannot be solved in time n o ( k ) for k = ˜ O (log log n ) Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 10 / 20
Our results Overview From gap hitting set to automatizability Theorem (Main Technical Lemma) For k = ˜ O (log log n ) , there exists a polytime algorithm mapping S to τ S s.t. if γ ( S ) ≤ k then S P ( τ S ) ≤ n O (1) if γ ( S ) > k 2 then S P ( τ S ) ≥ n Ω( k ) where P ∈ { TreeRes , Res , Nullsatz , PC } . Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 11 / 20
Our results Overview Proof sketch of main theorem Theorem (Main Theorem) Assuming ETH , P is not n ˜ o (log log S P ( τ )) -automatizable. Proof: Let Aut be the automatizing algorithm for P running in time o (log log S ) , and let k = ˜ f ( n , S ) = n ˜ Θ(log log n ). Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 12 / 20
Our results Overview Proof sketch of main theorem Theorem (Main Theorem) Assuming ETH , P is not n ˜ o (log log S P ( τ )) -automatizable. Proof: Let Aut be the automatizing algorithm for P running in time o (log log S ) , and let k = ˜ f ( n , S ) = n ˜ Θ(log log n ). Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 12 / 20
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