Space Complexity of Polynomial Calculus Massimo Lauria Sapienza - PowerPoint PPT Presentation

Space Complexity of Polynomial Calculus Massimo Lauria Sapienza – Universit` a di Roma L OGICAL A PPROACHES TO B ARRIERS IN C OMPLEXITY II C AMBRIDGE 2012 (joint work with Y. Filmus, J. Nordstr¨ om, N.Thapen and N. Zewi)

“SAT has been solved” —the daring practitioner— “SAT is clearly hard” —the savvy theorist—

“SAT solving can get better. . . ” —the relentless coder—

Next step is to study memory requirements of algebraic sat-solvers/theorem provers.

Modern SAT solvers are based on Resolution but algebraic reasoning can be beneficial.

shorter proofs

shorter proofs simple proof structure (e.g. Polynomial Calculus)

shorter proofs simple proof structure (e.g. Polynomial Calculus) proof search (e.g. Buchberger)

shorter proofs simple proof structure (e.g. Polynomial Calculus) proof search (e.g. Buchberger) implementations (e.g P OLY B ORI )

. . . NOT ENOUGH TO SWITCH

. . . NOT ENOUGH TO SWITCH modern solvers are very optimized

. . . NOT ENOUGH TO SWITCH modern solvers are very optimized to change paradigm is bad for SAT races

. . . NOT ENOUGH TO SWITCH modern solvers are very optimized to change paradigm is bad for SAT races their main issue is space , not proof length

S PACE IN MODERN SAT - SOLVERS during a running, solvers learn clauses memory fills very quickly retaining the right clauses if fundamental

D OES ALGEBRA HELP ? memory requirements relation between space and time

O UR RESULTS width variables space n 2 Ω( n ) n PCR [Alekhnovich et al. 02] n 2 Θ( n ) 3 PC pigeonhole principle O ( 1 ) O ( n ) n PC any formula Ω( n ) 2 log n n log n PCR bit-pigeonhole principle n 2 Ω( n ) 4 PCR xor-pigeonhole principle

O UR RESULTS width variables space n 2 Ω( n ) n PCR [Alekhnovich et al. 02] n 2 Θ( n ) 3 PC pigeonhole principle O ( 1 ) O ( n ) n PC any formula Ω( n ) 2 log n n log n PCR bit-pigeonhole principle n 2 Ω( n ) 4 PCR xor-pigeonhole principle I N THIS TALK WE SKETCH THIS PROOF !

O UTLINE Algebraic proof system PCR 1 Model space complexity in PCR refutations 2 We sketch the proof of a space lower bound for PCR 3

A LGEBRAIC PROOF SYSTEM

P ROOF S YSTEMS Deterministic polynomial time P ( · , · ) if F ∈ U NSAT then P ( F , π ) = 1 for some π ∈ { 0 , 1 } ∗ if F �∈ U NSAT then P ( F , π ) = 0 for all π ∈ { 0 , 1 } ∗

P ROOF SYSTEMS IN COMPLEXITY THEORY There is P where any unsat formula has a “short” refutation in P ⇐ ⇒ NP=coNP Cook-Reckhow program (1979) Prove proof length lower bound for stronger and stronger system in order to prove NP � = CO NP

P ROOF SYSTEMS AND SAT SOLVERS The trace of “SAT-solver( F )= unsat ” is a refutation for F . − → DLL tree-like resolution Clause Learning − → regular WRTL [BHJ ’08] − → CL + Restarts resolution C RYPTO M INI S AT − → fragments of PCR on GF ( 2 ) − → P OLYBORI PC on GF ( 2 )

P OLYNOMIAL C ALCULUS (P CR ) CNF formula − → set of polynomials SAT assignments − → common roots − → true 0 false − → 1 − → x , ¯ variable x x x 2 − x x ∈ { true , false } − → x − 1 x + ¯ x ∨ ¬ y ∨ ¬ z ∨ s ∨ t − → x · ¯ y · ¯ z · s · t

P OLYNOMIAL C ALCULUS (P CR ) L INEAR COMBINATION M ULTIPLICATION p p q xp α p + β q

F ⊢ 1 iff F ∈ U NSAT (S OUNDNESS ) I NFERENCE PRESERVES COMMON ROOTS (C OMPLETENESS ) S IMULATES DECISION TREES

PC defined in [CEI96] and PCR in [ABRW02]; PCR strictly better than resolution in proof length; Size-Degree Trade-off [IPS99,GL10a]; Exponential lower bounds on length are known [Raz98,AR03, BGIP01, BI10, IPS99, Raz98]; Proof search is hard [GL10b] based on [AR08].

S PACE COMPLEXITY OF PCR

  xz + yz · · · → xz − 1  

    xz + yz xz + yz  → · · · → xz − 1 xz − 1    1 − yz inference step from polynomials in memory

      xz + yz xz + yz xz + yz  →  → · · · → xz − 1 xz − 1 —     1 − yz 1 − yz inference step from polynomials in memory erasure of a polynomial

        xz + yz xz + yz xz + yz xz + yz x 2 − x  →  →  →  · · · · · · → xz − 1 xz − 1     1 − yz 1 − yz 1 − yz inference step from polynomials in memory erasure of a polynomial logical axiom/initial polynomial download

Space measure: #monomials in a configuration

Space measure: #monomials in a configuration   xz + yz   xz − 1     1 − yz (this configuration counts as space six)

Space measure: #monomials in a configuration   xz + yz   xz − 1     1 − yz (this configuration counts as space six) Roads not taken O ( 1 ) polynomials are always sufficient (#polynomials) too expensive compared to implementations (#symbols)

L ITTLE IS KNOWN FOR PCR SPACE Lower bounds for wide CNFs [Alekhnovich et al. 2002] [Huynh, Nordstr¨ Length-Space trade-offs om, 2012] Lower bounds for narrow CNFs [FLNTZ 2012]

L OWER BOUND FOR NARROW CNF S

B IT -P IGEONHOLE PRINCIPLE Fix n = 2 m : there is no injective function F : [ n + 1 ] → { 0 , 1 } m .

B IT -P IGEONHOLE PRINCIPLE Fix n = 2 m : there is no injective function F : [ n + 1 ] → { 0 , 1 } m . For each: two pigeons a and b hole s ∈ { 0 , 1 } m � ( f a , 1 � = s 1 ) ∨ · · · ∨ ( f a , m � = s m ) ( f b , 1 � = s 1 ) ∨ · · · ∨ ( f b , m � = s m ) � �� � � �� � F ( a ) � = s F ( b ) � = s

E XAMPLE F ( 1 ) � = � 1101 � or F ( 3 ) � = � 1101 � translates to � ( ¬ f 1 , 1 ∨ ¬ f 1 , 2 ∨ f 1 , 3 ∨ ¬ f 1 , 4 ) ( ¬ f 3 , 1 ∨ ¬ f 3 , 2 ∨ f 3 , 3 ∨ ¬ f 3 , 4 )

T HEOREM Any PCR refutation of the “Bit” pigeohole principle has a configuration with at least n / 8 monomials.

Proof Sketch         1 = 0 . . . . . . .  .   .   .   .  . . . .        

Proof Sketch         . . . .  .   .   .   .  . . . .                 1 = 0 . . . . . . .  .   .   .   .  . . . .         1 a parallel sequence of ( . . . ) such that ( . . . ) [ . . . ] ;

Proof Sketch         . . . .  .   .   .   .  . . . .                 1 = 0 . . . . . . .  .   .   .   .  . . . .         1 a parallel sequence of ( . . . ) such that ( . . . ) [ . . . ] ; 2 size of ( . . . ) if at most twice the size of [ . . . ] ; (assuming monomial space ≤ n / 8 : )

Proof Sketch         . . . .  .   .   .   .  . . . .                 1 = 0 . . . . . . .  .   .   .   .  . . . .         1 a parallel sequence of ( . . . ) such that ( . . . ) [ . . . ] ; 2 size of ( . . . ) if at most twice the size of [ . . . ] ; (assuming monomial space ≤ n / 8 : ) 3 ( . . . ) of size ≤ n / 4 are all satisfiable;

Proof Sketch         . . . .  .   .   .   .  . . . .                 1 = 0 . . . . . . .  .   .   .   .  . . . .         1 a parallel sequence of ( . . . ) such that ( . . . ) [ . . . ] ; 2 size of ( . . . ) if at most twice the size of [ . . . ] ; (assuming monomial space ≤ n / 8 : ) 3 ( . . . ) of size ≤ n / 4 are all satisfiable; 4 contradiction since ( . . . ) [ 1 = 0 ]

S PECIAL CONFIGURATIONS (2-CNFs where no pigeon is mentioned twice)   ¬ f 1 , 3 ∨ f 4 , 2 ∨  f 5 , 1 f 7 , 3    .  .  .   ¬ f 6 , 5 ∨ ¬ f 2 , 3 A satisfying assignment : satisfies the 2-CNFs no collision on the occurring pigeons

Observation Any special configuration with at most n / 4 clauses is satisfiable. Proof. at most n / 2 pigeons;   1 ¬ f 1 , 3 ∨ f 4 , 2 at least n / 2 + 1 free holes per pigeon; ∨ 2  f 5 , 1 f 7 , 3    .   . at least one free hole per satisfied 3 .   literal. ¬ f 6 , 5 ∨ ¬ f 2 , 3

Input: ( M 0 , M 1 , . . . , M l ) with | M i | ≤ n / 8 Output: ( S 0 , S 1 , . . . , S l ) such that | S i | ≤ 2 | M i | and S i implies M i

Input: ( M 0 , M 1 , . . . , M l ) with | M i | ≤ n / 8 Output: ( S 0 , S 1 , . . . , S l ) such that | S i | ≤ 2 | M i | and S i implies M i S 0 := ∅ [Initial configuration] S i + 1 := S i [Inference] S i + 1 := S i [Logical axioms]