Provability of weak circuit lower bounds J an Pich Kurt G odel - PowerPoint PPT Presentation

Proof complexity, Dagstuhl 1st February 2018 Provability of weak circuit lower bounds J´ an Pich Kurt G¨ odel Research Center University of Vienna based on a joint work with Moritz M¨ uller 1 / 11

Constructive proofs of circuit lower bounds Known circuit lower bounds for f given explicitly: AC 0 , AC 0 [ p ], etc. very constructive: p-time algorithm often recognizing when f is hard a.k.a natural proofs 2 / 11

Constructive proofs of circuit lower bounds Known circuit lower bounds for f given explicitly: AC 0 , AC 0 [ p ], etc. very constructive: p-time algorithm often recognizing when f is hard a.k.a natural proofs Razborov-Rudich: Cryptography works → no natural proof of P � =NP. 2 / 11

Constructive proofs of circuit lower bounds Known circuit lower bounds for f given explicitly: AC 0 , AC 0 [ p ], etc. very constructive: p-time algorithm often recognizing when f is hard a.k.a natural proofs Razborov-Rudich: Cryptography works → no natural proof of P � =NP. Mathematical logic: ◦ upper bounds: Prove all known circuit lower bounds in a constructive mathematical theory, e.g. PV 1 (p-time reasoning). - exhibit a structure of algorithms recognizing hard functions? 2 / 11

Constructive proofs of circuit lower bounds Known circuit lower bounds for f given explicitly: AC 0 , AC 0 [ p ], etc. very constructive: p-time algorithm often recognizing when f is hard a.k.a natural proofs Razborov-Rudich: Cryptography works → no natural proof of P � =NP. Mathematical logic: ◦ upper bounds: Prove all known circuit lower bounds in a constructive mathematical theory, e.g. PV 1 (p-time reasoning). - exhibit a structure of algorithms recognizing hard functions? ◦ lower bounds: PV 1 �⊢ strong circuit lower bounds? - stronger ’natural proofs’ barrier: P=NP consistent with PV 1 ? - circuit lower bounds as hard tautologies witnessing NP � =coNP? 2 / 11

Bounded arithmetic and propositional logic PV 1 : first-order theory formalizing p-time reasoning (Cook ’75) APC 1 : formalizes probabilistic p-time reasoning (Jeˇ r´ abek ’07) ∈ SIZE(2 ǫ n )” APC 1 := PV 1 + “ ∃ f / 3 / 11

Bounded arithmetic and propositional logic PV 1 : first-order theory formalizing p-time reasoning (Cook ’75) APC 1 : formalizes probabilistic p-time reasoning (Jeˇ r´ abek ’07) ∈ SIZE(2 ǫ n )” APC 1 := PV 1 + “ ∃ f / If PV 1 ⊢ ∀ xA ( x ) for a p-time predicate A , then tautologies expressing ∀ xA ( x ) have p-size Extended Frege EF proofs If APC 1 ⊢ ∀ xA ( x ) for a p-time predicate A , then tautologies expressing ∀ xA ( x ) have p-size WF proofs ∈ SIZE (2 ǫ n )” (Jeˇ WF : EF + “ ∃ f / r´ abek ’04) 3 / 11

How to express circuit lower bounds formally First-order formulation : ’every circuit of size n k fails to compute function f ’ LB ( f , n k ): 4 / 11

How to express circuit lower bounds formally First-order formulation : ’every circuit of size n k fails to compute function f ’ LB ( f , n k ): ∀ n > n 0 ∀ circuit C of size ≤ n k ∃ input y , | y | = n ; C ( y ) � = f ( y ) where n 0 , k are fixed constants ◦ If f ∈ NP , then LB ( f , n k ) is Π p 3 (i.e. ∀∃∀ statement) 4 / 11

How to express circuit lower bounds formally First-order formulation : ’every circuit of size n k fails to compute function f ’ LB ( f , n k ): ∀ n > n 0 ∀ circuit C of size ≤ n k ∃ input y , | y | = n ; C ( y ) � = f ( y ) where n 0 , k are fixed constants ◦ If f ∈ NP , then LB ( f , n k ) is Π p 3 (i.e. ∀∃∀ statement) different scaling : LB tt ( f , n k ): ∀ m , n > n 0 , | m | = 2 n ∀ circuit C of size ≤ n k ∃ y , | y | = n ; C ( y ) � = f ( y ) ◦ If f = SAT, then LB tt ( f , n k ) is coNP 4 / 11

How to express circuit lower bounds formally First-order formulation : ’every circuit of size n k fails to compute function f ’ LB ( f , n k ): ∀ n > n 0 ∀ circuit C of size ≤ n k ∃ input y , | y | = n ; C ( y ) � = f ( y ) where n 0 , k are fixed constants ◦ If f ∈ NP , then LB ( f , n k ) is Π p 3 (i.e. ∀∃∀ statement) different scaling : LB tt ( f , n k ): ∀ m , n > n 0 , | m | = 2 n ∀ circuit C of size ≤ n k ∃ y , | y | = n ; C ( y ) � = f ( y ) ◦ If f = SAT, then LB tt ( f , n k ) is coNP Easier to reason about LB tt ( f , n k ) than about LB ( f , n k ). 4 / 11

Propositional formulation : � (expresses LB tt ( f , n k )) tt ( f , n k ): f ( y ) � = C ( y ) y ∈{ 0 , 1 } n 2 n bits f ( y ), poly ( n ) variables for circuit C of size n k , total size: 2 O ( n ) 5 / 11

Propositional formulation : � (expresses LB tt ( f , n k )) tt ( f , n k ): f ( y ) � = C ( y ) y ∈{ 0 , 1 } n 2 n bits f ( y ), poly ( n ) variables for circuit C of size n k , total size: 2 O ( n ) ∈ SIZE ( n O (1) ) ⇒ ∃ A ⊆ { 0 , 1 } n of size poly ( n ) s.t. Lipton-Young: f / � y ∈ A f ( y ) � = C ( y ) 5 / 11

Propositional formulation : � (expresses LB tt ( f , n k )) tt ( f , n k ): f ( y ) � = C ( y ) y ∈{ 0 , 1 } n 2 n bits f ( y ), poly ( n ) variables for circuit C of size n k , total size: 2 O ( n ) ∈ SIZE ( n O (1) ) ⇒ ∃ A ⊆ { 0 , 1 } n of size poly ( n ) s.t. Lipton-Young: f / � y ∈ A f ( y ) � = C ( y ) lb A ( f , n k ): � y ∈ A f ( y ) � = C ( y ) - same meaning as tt ( f , n k ) but poly ( n ) size 5 / 11

Propositional formulation : � (expresses LB tt ( f , n k )) tt ( f , n k ): f ( y ) � = C ( y ) y ∈{ 0 , 1 } n 2 n bits f ( y ), poly ( n ) variables for circuit C of size n k , total size: 2 O ( n ) ∈ SIZE ( n O (1) ) ⇒ ∃ A ⊆ { 0 , 1 } n of size poly ( n ) s.t. Lipton-Young: f / � y ∈ A f ( y ) � = C ( y ) lb A ( f , n k ): � y ∈ A f ( y ) � = C ( y ) - same meaning as tt ( f , n k ) but poly ( n ) size lb w ( f , n k ): poly ( n ) size formula expressing LB ( f , n k ) with existential quantifiers witnessed feasibly by w 5 / 11

Propositional formulation : � (expresses LB tt ( f , n k )) tt ( f , n k ): f ( y ) � = C ( y ) y ∈{ 0 , 1 } n 2 n bits f ( y ), poly ( n ) variables for circuit C of size n k , total size: 2 O ( n ) ∈ SIZE ( n O (1) ) ⇒ ∃ A ⊆ { 0 , 1 } n of size poly ( n ) s.t. Lipton-Young: f / � y ∈ A f ( y ) � = C ( y ) lb A ( f , n k ): � y ∈ A f ( y ) � = C ( y ) - same meaning as tt ( f , n k ) but poly ( n ) size lb w ( f , n k ): poly ( n ) size formula expressing LB ( f , n k ) with existential quantifiers witnessed feasibly by w Possible witnessing w of LB ( f , n k ): a p-time algorithm with input : circuit C of size n k output : y s.t. C ( y ) � = f ( y ) 5 / 11

Propositional formulation : � (expresses LB tt ( f , n k )) tt ( f , n k ): f ( y ) � = C ( y ) y ∈{ 0 , 1 } n 2 n bits f ( y ), poly ( n ) variables for circuit C of size n k , total size: 2 O ( n ) ∈ SIZE ( n O (1) ) ⇒ ∃ A ⊆ { 0 , 1 } n of size poly ( n ) s.t. Lipton-Young: f / � y ∈ A f ( y ) � = C ( y ) lb A ( f , n k ): � y ∈ A f ( y ) � = C ( y ) - same meaning as tt ( f , n k ) but poly ( n ) size lb w ( f , n k ): poly ( n ) size formula expressing LB ( f , n k ) with existential quantifiers witnessed feasibly by w - ∃ w follows e.g. from PV 1 ⊢ LB ( f , n k ) 5 / 11

Propositional formulation : � (expresses LB tt ( f , n k )) tt ( f , n k ): f ( y ) � = C ( y ) y ∈{ 0 , 1 } n 2 n bits f ( y ), poly ( n ) variables for circuit C of size n k , total size: 2 O ( n ) ∈ SIZE ( n O (1) ) ⇒ ∃ A ⊆ { 0 , 1 } n of size poly ( n ) s.t. Lipton-Young: f / � y ∈ A f ( y ) � = C ( y ) lb A ( f , n k ): � y ∈ A f ( y ) � = C ( y ) - same meaning as tt ( f , n k ) but poly ( n ) size lb w ( f , n k ): poly ( n ) size formula expressing LB ( f , n k ) with existential quantifiers witnessed feasibly by w - ∃ w follows e.g. from PV 1 ⊢ LB ( f , n k ) Fact: If tt ( f , n k ) has no poly-size constant-depth Frege proofs, then lb A ( f , n k ) has no poly-size (full) Frege proofs. 5 / 11

Previous results Lower bounds: Razborov: S 2 2 ( α ) �⊢ “ LB tt ( SAT , n k )” unless cryptography breaks P.: VNC 1 �⊢ LB ( SAT , n k ) unless SIZE ( n k ) ⊆ approx “subexp NC 1 ” cek-Oliveira: ∀ k ∃ f ∈ P s.t. PV 1 �⊢ f ∈ SIZE ( cn k ) Kraj´ ıˇ Buss: PV 1 �⊢ NP = co NP unless P=NP “folklore”: V 0 �⊢ SAT ∈ P / poly 6 / 11

Previous results Lower bounds: Razborov: S 2 2 ( α ) �⊢ “ LB tt ( SAT , n k )” unless cryptography breaks P.: VNC 1 �⊢ LB ( SAT , n k ) unless SIZE ( n k ) ⊆ approx “subexp NC 1 ” cek-Oliveira: ∀ k ∃ f ∈ P s.t. PV 1 �⊢ f ∈ SIZE ( cn k ) Kraj´ ıˇ Buss: PV 1 �⊢ NP = co NP unless P=NP “folklore”: V 0 �⊢ SAT ∈ P / poly Razborov-Kraj´ ıˇ cek: Propositional systems with feasible interpolation property have no p-size proofs of tt ( f , n k ) unless cryptography breaks. Raz: Resolution has no p-size proofs of tt ( f , n k ) (unconditionally). Razborov: Res ( ǫ log n ) does not have p-size proofs of tt ( f , n ω (1) ). 6 / 11