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

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

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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

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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

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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

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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

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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

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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

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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

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Introduction Proof complexity overview

Resolution

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 4 / 20

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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, SP(τ)), where SP(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

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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, SP(τ)), where SP(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

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Introduction Automatizability

Known automatizability lower bounds

General results and results for strong systems approximating SP(τ) to within 2log1−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

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Introduction Automatizability

Known automatizability lower bounds

General results and results for strong systems approximating SP(τ) to within 2log1−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

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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

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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

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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

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Introduction Automatizability

Our results

Theorem (Main Theorem) Assuming ETH, P is not n˜

  • (log log SP(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

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Introduction Automatizability

Our results

Theorem (Main Theorem) Assuming ETH, P is not n˜

  • (log log SP(τ))-automatizable for P = Res,

TreeRes, Nullsatz, PC. Theorem (Main Theorem for Res(r)) Assuming ETH, Res(r) is not n˜

  • (log log SP(τ)/exp(r2))-automatizable for

r ≤ ˜ O(log log log n).

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 8 / 20

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Introduction Automatizability

Our results

Theorem (Main Theorem) Assuming ETH, P is not n˜

  • (log log SP(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

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Introduction Automatizability

Our results

Theorem (Main Theorem) Assuming ETH, P is not n˜

  • (log log SP(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 nO(log SP(τ))-automatizable. Res is nO(√

n log SP(τ))-automatizable.

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 8 / 20

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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

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Our results Overview

Gap hitting set

S = {S1 . . . Sn} over [n]

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 10 / 20

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Our results Overview

Gap hitting set

S = {S1 . . . Sn} over [n] hitting set: H ⊆ [n] s.t. H ∩ Si = ∅ for all i ∈ [n]

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 10 / 20

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Our results Overview

Gap hitting set

S = {S1 . . . Sn} over [n] hitting set: H ⊆ [n] s.t. H ∩ Si = ∅ 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) > k2

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 10 / 20

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Our results Overview

Gap hitting set

S = {S1 . . . Sn} over [n] hitting set: H ⊆ [n] s.t. H ∩ Si = ∅ 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) > k2 Theorem (Chen-Lin) Assuming ETH the gap hitting set problem cannot be solved in time no(k) for k = ˜ O(log log n)

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 10 / 20

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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 τS s.t. if γ(S) ≤ k then SP(τS) ≤ nO(1) if γ(S) > k2 then SP(τS) ≥ nΩ(k) where P ∈ {TreeRes, Res, Nullsatz, PC}.

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 11 / 20

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Our results Overview

Proof sketch of main theorem

Theorem (Main Theorem) Assuming ETH, P is not n˜

  • (log log SP(τ))-automatizable.

Proof: Let Aut be the automatizing algorithm for P running in time f (n, S) = n˜

  • (log log S), and let k = ˜

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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Our results Overview

Proof sketch of main theorem

Theorem (Main Theorem) Assuming ETH, P is not n˜

  • (log log SP(τ))-automatizable.

Proof: Let Aut be the automatizing algorithm for P running in time f (n, S) = n˜

  • (log log S), and let k = ˜

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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Our results Overview

Proof sketch of main theorem

Theorem (Main Theorem) Assuming ETH, P is not n˜

  • (log log SP(τ))-automatizable.

Proof: Let Aut be the automatizing algorithm for P running in time f (n, S) = n˜

  • (log log S), and let k = ˜

Θ(log log n). Theorem (Main Technical Lemma) if γ(S) ≤ k then SP(τ) ≤ nO(1) if γ(S) > k2 then SP(τ) ≥ nΩ(k)

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 12 / 20

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Our results Overview

Proof sketch of main theorem

Theorem (Main Theorem) Assuming ETH, P is not n˜

  • (log log SP(τ))-automatizable.

Proof: Let Aut be the automatizing algorithm for P running in time f (n, S) = n˜

  • (log log n) = no(k), and let k = ˜

Θ(log log n). Theorem (Main Technical Lemma) if γ(S) ≤ k then SP(τ) ≤ nO(1) if γ(S) > k2 then SP(τ) ≥ nΩ(k)

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 12 / 20

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Our results Overview

Proof sketch of main theorem

Theorem (Main Theorem) Assuming ETH, P is not n˜

  • (log log SP(τ))-automatizable.

Proof: Let Aut be the automatizing algorithm for P running in time f (n, S) = n˜

  • (log log S), and let k = ˜

Θ(log log n). Theorem (Main Technical Lemma) if γ(S) ≤ k then SP(τ) ≤ nO(1) if γ(S) > k2 then SP(τ) ≥ nΩ(k)

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 12 / 20

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Our results Main Technical Lemma I: Defining τS

Detour: universal sets

Am×m is (m, q)-universal if for all I ⊆ [m], |I| ≤ q, all 2|I| possible column vectors appear in A restricted to the rows I

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 13 / 20

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Our results Main Technical Lemma I: Defining τS

Detour: universal sets

Am×m is (m, q)-universal if for all I ⊆ [m], |I| ≤ q, all 2|I| possible column vectors appear in A restricted to the rows I Am×m is (m, q)-dual universal if for all J ⊆ [m], |J| ≤ q, all 2|J| possible row vectors appear in A restricted to the columns J

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 13 / 20

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Our results Main Technical Lemma I: Defining τS

Detour: universal sets

Am×m is (m, q)-universal if for all I ⊆ [m], |I| ≤ q, all 2|I| possible column vectors appear in A restricted to the rows I Am×m is (m, q)-dual universal if for all J ⊆ [m], |J| ≤ q, all 2|J| possible row vectors appear in A restricted to the columns J constructions like the Paley graph work for q = log m

4

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 13 / 20

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Our results Main Technical Lemma I: Defining τS

Defining τS

Variables of τS will implicitly define two matrices using A and S

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 14 / 20

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Our results Main Technical Lemma I: Defining τS

Defining τS

Variables of τS will implicitly define two matrices using A and S

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 14 / 20

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Our results Main Technical Lemma I: Defining τS

Defining τS

Variables of τS will implicitly define two matrices using A and S

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 14 / 20

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Our results Main Technical Lemma I: Defining τS

Defining τS

τS will state that there exist α, β such that there is no i, j where Q[i, j] = R[i, j] = 1

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 15 / 20

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Our results Main Technical Lemma I: Defining τS

Defining τS

τS will state that there exist α, β such that there is no i, j where Q[i, j] = R[i, j] = 1

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 15 / 20

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Our results Main Technical Lemma II: Upper bound

Upper bound on SP(τS)

Lemma (Upper bound on SP(τS)) If γ(S) ≤ k ≤ log m

4 , then τS is unsatisfiable and S(τS) ≤ mkn for TreeRes.

High-level idea: the universal property of A guarantees some column of Q will be a hitting set.

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 16 / 20

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Our results Main Technical Lemma II: Upper bound

Upper bound on SP(τS)

Lemma (Upper bound on SP(τS)) If γ(S) ≤ k ≤ log m

4 , then τS is unsatisfiable and S(τS) ≤ mkn for TreeRes.

High-level idea: the universal property of A guarantees some column of Q will be a hitting set.

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 16 / 20

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Our results Main Technical Lemma II: Upper bound

Upper bound on SP(τS)

Lemma (Upper bound on SP(τS)) If γ(S) ≤ k ≤ log m

4 , then τS is unsatisfiable and S(τS) ≤ mkn for TreeRes.

High-level idea: the universal property of A guarantees some column of Q will be a hitting set.

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 16 / 20

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Our results Main Technical Lemma II: Upper bound

Upper bound on SP(τS)

Lemma (Upper bound on SP(τS)) If γ(S) ≤ k ≤ log m

4 , then τS is unsatisfiable and S(τS) ≤ mkn for TreeRes.

High-level idea: the universal property of A guarantees some column of Q will be a hitting set.

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 16 / 20

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Our results Main Technical Lemma II: Upper bound

Upper bound on SP(τS)

Lemma (Upper bound on SP(τS)) If γ(S) ≤ k ≤ log m

4 , then τS is unsatisfiable and S(τS) ≤ mkn for TreeRes.

High-level idea: the universal property of A guarantees some column of Q will be a hitting set. Size of the proof: mkn

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 16 / 20

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Our results Main Technical Lemma II: Upper bound

Upper bound on SP(τS)

Lemma (Upper bound on SP(τS)) If γ(S) ≤ k ≤ log m

4 , then τS is unsatisfiable and S(τS) ≤ n2 for TreeRes.

High-level idea: the universal property of A guarantees some column of Q will be a hitting set. Size of the proof: mkn = n2 for m = n1/k

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 16 / 20

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Our results Main Technical Lemma III: Lower bound

Lower bound on SP(τS)

High-level idea 1: any proof π must query all rows in some hitting set

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 17 / 20

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Our results Main Technical Lemma III: Lower bound

Lower bound on SP(τS)

High-level idea 1: any proof π must query all rows in some hitting set Res/TreeRes - prover-delayer game [Pudl´ ak, Atserias-Lauria-Nordstr¨

  • m]

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 17 / 20

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Our results Main Technical Lemma III: Lower bound

Lower bound on SP(τS)

High-level idea 1: any proof π must query all rows in some hitting set Res/TreeRes - prover-delayer game [Pudl´ ak, Atserias-Lauria-Nordstr¨

  • m]

Nullsatz/PC - linear operator [Galesi-Lauria]

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 17 / 20

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Our results Main Technical Lemma III: Lower bound

Lower bound on SP(τS)

High-level idea 1: any proof π must query all rows in some hitting set Res/TreeRes - prover-delayer game [Pudl´ ak, Atserias-Lauria-Nordstr¨

  • m]

Nullsatz/PC - linear operator [Galesi-Lauria] Res(k) - switching lemma [Buss-Impagliazzo-Segerlend]

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 17 / 20

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Our results Main Technical Lemma III: Lower bound

Lower bound on SP(τS)

High-level idea 1: any proof π must query all rows in some hitting set Res/TreeRes - prover-delayer game [Pudl´ ak, Atserias-Lauria-Nordstr¨

  • m]

Nullsatz/PC - linear operator [Galesi-Lauria] Res(k) - switching lemma [Buss-Impagliazzo-Segerlend] TreeCP - lifting [upcoming work]

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 17 / 20

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Our results Main Technical Lemma III: Lower bound

Lower bound on SP(τS)

High-level idea 2: π knows nothing about a row or column without setting lots of variables

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 18 / 20

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Our results Main Technical Lemma III: Lower bound

Lower bound on SP(τS)

High-level idea 2: π knows nothing about a row or column without setting lots of variables Error-correcting codes xi ∈ {0, 1}6 log m, yj ∈ {0, 1}6 log n fx : {0, 1}6 log m → [m], fy : {0, 1}6 log n → [n]

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 18 / 20

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Conclusion

Open problems

Better hard k in gap hitting set → better non-automatizability result

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 19 / 20

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Conclusion

Open problems

Better hard k in gap hitting set → better non-automatizability result Theorem (Chen-Lin) Assuming ETH the gap hitting set problem cannot be solved in time no(k) for k = O(log1/7−o(1) log n) Theorem (Main Technical Lemma) For k = O(√log n), there exists a polytime algorithm mapping S to τS . . .

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 19 / 20

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Conclusion

Thank you!

O:’t6m@taIz@’bIlIti O:t6’mætaIz@’bIlIti

Ian Mertz (U. of Toronto) Short Proofs are Hard to Find ICALP, July 10, 2019 20 / 20