Approaching complexity through logic

1. Approaching complexity through logic University of Oxford Ján Pich Department of Theoretical Computer Science and Mathematical Logic Charles University in Prague 27 March 2019

2. Goal: understand the powerof logic and algorithms answer ‘‘universal’’ questions, e.g. Theorem Proving, Learning Algorithms

3. Plan of the Talk 1. History Fundamental problems: P vs NP vs coNP etc.(70s) Golden age of Complexity Lower Bounds (80s) Barriers (90s) 2. My work Barriers via Logic ⟾ Breaking Barriers 3. Future directions + connections toCryptography, SAT solving, etc.

4. 1. History

5. Fundamental questions [~70s] P vs NP: If a solution can be verified efficiently, can it be also found efficiently? Equivalent:∃ fast algorithm for Theorem Proving Problem? input: n bits & ZFC statement F output: decide if F has a ZFC proof of length n Cook Levin NP vs coNP: Do propositional tautologies admit short proofs? Can we learn efficiently? ... Gödel Valiant

6. Most direct approach: Complexity lower bounds for concrete computational models Boolean circuit: - It computes a function -This circuit has sizes = 8 (gates). After 50+ years of intense research: Every problem in NTIME[2n] might be computable by circuits with just 5n gates!

7. Golden Age of Circuit Complexity [80s] Circuits of polynomial-size and constant depth cannot compute PARITY Adding PARITY gates does not help much Circuits of polynomial-size without negation gates cannot solve problems in P etc. Ajtai (1983) Furst-Saxe-Sipser (1984) Razborov (1987) Smolensky (1987) Razborov (1985) P ≠ NP behind the corner?

8. Fall of the Circuit Lower Bounds Program [90s] Natural proofs of Razborov and Rudich (1994): All known circuit lower bounds on explicit Boolean functions are very constructive (imply efficient algorithms recognizing hard Boolean functions) Example. Given a as a string of all its values (truth-table) we can recognize in time if it is computable by a small circuit of constant-depth Barrier:Cryptography works ⟾ no natural proof of P ≠ NP proof: natural proof of P ≠ NP distinguishes random functions from pseudorandom functions (those with small circuits)

9. Logic Enters Natural proofs barrier is ad hoc Comprehensive understanding of barriers in the framework of Mathematical Logic: Razborov (1995). Cryptography works ⟾ P ≠ NP unprovable in theory S²₂(𝛼) • upper bounds: prove known lower bounds in a fragment of Peano Arithmetic • lower bounds:show that P ≠ NP is unprovable in such theories Unfortunately, S²₂(𝛼) very weak: not known to formalize any lower bound

10. NP vs coNP view tautologies encoding ∀x A(x) admit short proofs in a propositional proof system P T⊢∀x A(x) for a p-time predicate A First-order theories⟺ Propositional proof systems ⟾ Razborov (1995)[reformulated]. Cryptography works ⟾ P ≠ NP (expressed in tautologies) hard to prove in constructive propositional proof systems. Attacking from both sides: P ≠ NP hard to prove ⟾ P = NP consistent & approaching NP ≠ coNP Problem: Krajíček-Pudlák (1998). Strong proof systems not constructive unless RSA insecure.

11. Strong proof systems Razborov’s conjecture (2003): Standard hardness assumption ⟾ Nisan-Wigderson generator hard for Frege Frege - textbook system for propositional logic Circuit-Frege - Frege operating with circuits instead of formulas Proof complexity generators (Alekhnovich, Ben-Sasson, Razborov, Wigderson [2001] Krajíček [2001]) g:{0,1}ⁿ↦{0,1}ˢ is hard for system P iff ∀b∈{0,1}ˢ, formulas b∉ Rng(g) hard for P Hard to find even candidate hard tautologies for these systems. and P ≠ NP hard as well Pseudorandom generator

12. 2. My work

13. P. [2010] Razborov’s conjecture holds for constructive proof systems e.g.: Resolution (the underlying system of most existing SAT solvers) Cutting planes (captures linear programming) • Previously only special instances of Nisan-Wigderson generator were known to be hard for some weak systems. However, recall: Frege not constructive unless crypto breaks (Krajíček-Pudlák)

14. Approaching strong systems P. [2013] Theories weaker than PV cannot prove P ≠ NP unless hardness breaks. proof: Exploits witnessing. Based on Krajíček’s model-theoretic evidence for Razborov’s conjecture - another rich area Conceptual shift:Is P ≠ NP feasibly true? e.g. If P ≠ NP, can you also witness failures of all potential circuits for SAT? otherwise, some circuit might behave as if it was solving SAT Formally:PV⊢ P ≠ NP? PV: Cook’s theory (1975) formalizing p-time reasoning.

15. P.[2014] PV is powerful: it proves the PCP theorem PCP theorem:mathematical proofs can be verified whp by reading just 20 bits. • one of the highest achievements of Complexity theory. • sophisticated proof: a culmination of many innovative ideas (error-correcting codes, expanding graphs, interactive proofs, etc.) • but has small logical complexity let’s reconsider positive aspects of the provability of lower bounds P.[2014]PCP theorem can be expressed by tautologies having short Circuit-Frege proofs. Perhaps PV is too strong: formalizes most of Complexity Theory

16. Lower bounds imply learning Carmosino Impagliazzo Kabanets Kolokolova (2016). Natural proofs ⟾learning algorithms P.- Muller [2017]. Efficiently generating Circuit-Frege proofs of known circuit lower bounds more suitable for learning from random examples Learning model: access a function computable by a small circuit output a circuit computing the function whp

17. Emergence of Hardness Magnification weak system strong system P.- Muller [2017]. (proof complexity magnification) P ≠ NPslightly hard for constant-depth Frege⟾ P ≠ NPhard for Frege Hardness magnificationovercomes natural proofsbarrier! Oliveira Santhanam (2018). (circuit complexity magnification) “MCSP” hard for linear size circuits ⟾ P ≠ NP MCSP: given a function, decide if computable by a small circuit (ancient & fundamentalproblem)

18. Final Mystery Oliveira-P.-Santhanam [2018] MCSP hard for linear-size circuits ⟾ P ≠ NP MCSP hard for subquadratic-size parallelized circuits MCSP hard for linear-size almost-parallelized circuits ⟾ NP hard for parallelized polynomial-size circuits The assumption holds for PARITY but MCSP is much harder than PARITY!

19. 3. Future directions • Explain hardness magnification • Strengthen connections to learning • Implement in SAT solvers Prague: great place for collaborations on these topics! THANK YOU!