1. Resolution over Linear Equations: (Partial) Survey & Open Problems
Iddo Tzameret
Royal Holloway,
University of London
(Based mainly on Raz-T. 2008, Itsykson-Sokolov 2014,
and ongoing joint work with Fedor Part)

2. Resolution over Linear Equations R(linℜ)
Proof-lines are disjunction of linear equations over ring ℜ:
𝐿 1 = 𝑎 1 ∨…∨ 𝐿 𝑚 = 𝑎 𝑚
Rules
Resolution: 𝐷∨𝐿=𝑎 𝐸∨𝐿′=𝑏
𝐷∨𝐸∨(𝐿− 𝐿 ′ =𝑎−𝑏)
Simplification:
𝐷∨ 𝑏=𝑎 𝐷
An R(linℜ) refutation of a collection of disjunctions of linear equations K is a proof of the empty disjunction from K.
Introduced by Raz, T (over ℤ; in unary representation).
See also, R(CP*) in Krajicek 1998
Weakening: 𝐷
𝐷 ∨𝐸
if b≠a
Boolean Axiom: ( 𝑥 𝑖 =0)∨ (𝑥 𝑖 =1)

3. Example: R(linℤ) Refuting CNFs: Replace positive literals by 𝑥 𝑖 =1
and negative literals by 𝑥 𝑖 =0
Size is number of symbols with integers in unary
Example: R(linℤ)
Refute: ( 𝑥 1 + 𝑥 2 =3)
𝑥 1 =0 ∨ 𝑥 1 = 𝑥 2 =0 ∨ 𝑥 2 =1
𝑥 2 =0 ∨ 𝑥 2 =1
𝑥 1 +𝑥 2 =0 ∨ 𝑥 1 =1 ∨ 𝑥 2 =1
𝑥 1 +𝑥 2 =0 ∨ 𝑥 1 +𝑥 2 =1 ∨ 𝑥 2 =1
If x2=0
𝑥 1 =0 ∨ 𝑥 1 = 𝑥 2 =0 ∨ 𝑥 2 =1
𝑥 2 =0 ∨ 𝑥 2 =1
𝑥 1 +𝑥 2 =1 ∨ 𝑥 1 =1 ∨ 𝑥 2 =0
𝑥 1 +𝑥 2 =1 ∨ 𝑥 1 +𝑥 2 =2 ∨ 𝑥 2 =0
If x2=1
𝑥 1 +𝑥 2 =0 ∨ 𝑥 1 +𝑥 2 =1 ∨ 𝑥 1 +𝑥 2 =2
𝑥 1 +𝑥 2 =0 ∨ 𝑥 1 +𝑥 2 =1 ∨ 𝑥 1 +𝑥 2 =2 ∨ 0=1
( 𝑥 1 + 𝑥 2 =3) ⎕

4. Natural (“minimal”) extension of resolution that can “count”.
Motivation
Natural (“minimal”) extension of resolution that can “count”.
First step towards Frege+Counting Gates lower bounds:
R(lin 𝔽 2 ): “weakest” subsystem of AC0[2]-Frege for which we don’t know lower bounds.

5. Some Upper Bounds R(linℤ) ⊢* PHP (in CNF) (Raz-T. 2008)
R(linℤ) ⊢* Tseitin (mod q) (in CNF) (Raz-T )
Simulations
R(linℤ) simulates CP with small coefficients (Raz-T. 2008)
R(linℤ) simulates R(lin 𝔽 2 ) (Itsykson-Sokolov 2014)

6. Lower Bounds R0(linℜ): restrict R(linℜ) to operate with
Constant many distinct linear forms in a clause (excluding single variables, that can occur freely);
Coefficients of variables are constants.
Exponential lower bounds on R0(linℜ) via monotone interpolation: clique/coloring tautologies (Raz-T. 2008)

7. R(lin 𝔽 2 ) Most results from Itsykson-Sokolov 2014
Focused on tree-like refutations
Over 𝔽2, so don’t need Boolean axioms

8. Some Upper Bounds
Tree-like R(lin 𝔽 2 ) ⊢* unsatisfiable 𝐴 𝒙 = 𝒃 (Itsykson-Sokolov 2014)
Note: no disjunctions in initial clauses
Tree-like R(lin 𝔽 2 ) ⊢* Graph Matching Principle
“no perfect matching in graphs with odd number of nodes”

9. Linear Decision Trees
A linear decision tree for unsat CNF C is a tree with:
Linear forms f on nodes;
f=0 go to left child; f=1 go to right child;
Clause 𝐶 𝑖 on leaf, if system of equations on its path imply ¬ 𝐶 𝑖 .
𝑥 1 +𝑥 3 +𝑥 5 +𝑥 6 =0
𝑥 1 +𝑥 3 +𝑥 5 +𝑥 6 =1
𝑥 1 +𝑥 2 =1
𝑥 1 +𝑥 3 =1
𝑥 2 +𝑥 4 =1
𝑥 2 +𝑥 4 =0
𝒕𝒉𝒆 𝒑𝒂𝒕𝒉⊨¬𝐶 1
𝒕𝒉𝒆 𝒑𝒂𝒕𝒉⊨¬𝐶 3

10. Linear Decision Trees
Linear decision tree for unsat CNF C ≈ Tree-like R(lin 𝔽 2 ).
𝑥 1 +𝑥 3 +𝑥 5 +𝑥 6 =0
𝑥 1 +𝑥 3 +𝑥 5 +𝑥 6 =1
𝑥 1 +𝑥 2 =1
𝑥 1 +𝑥 3 =1
𝑥 2 +𝑥 4 =1
𝑥 2 +𝑥 4 =0
𝒕𝒉𝒆 𝒑𝒂𝒕𝒉⊨¬𝐶 1
𝒕𝒉𝒆 𝒑𝒂𝒕𝒉⊨¬𝐶 3

11. Tree-like R(lin 𝔽 2 ) ⊢* unsat 𝐴 𝒙 = 𝒃
𝐴 𝒙 = 𝒃 is written as a CNF (assume number of rows=4)
Just build the linear decision tree
𝐴 1 𝒙 =0
𝐴 1 𝒙 = 𝑏 1
𝐴 2 𝒙 = 𝑏 2
Full decision tree for variables
in 𝐴 1 𝒙 is proportional to CNF
refpresentation of 𝐴 1 𝒙
𝐴 3 𝒙 = 𝑏 3
𝐴 4 𝒙 = 𝑏 4
𝐴 4 𝒙 =1
Full decision tree for variables
in 𝐴 4 𝒙 is proportional to CNF
refpresentation of 𝐴 4 𝒙
this node is not reached, by assumption on unsatisfiability of the system.
so we put the dt on the parent node

12. Lower Bounds
Exponential lower bounds on tree-like R(lin 𝔽 2 ) for the n+1 to n PHP (IS 2014)
More tree-like R(lin 𝔽 2 ) lower bounds

13. Tree-like R(lin 𝔽 2 ) PHP Lower Bounds
Use Impagliazzo-Pudlak game technique:
Given an unsatisfiable CNF, Prover and Delayer play in turns:
Prover: Asks Delayer the value of some linear form
Delayer:
Answers 0/1.
Answers “you choose!”, earning 1 point.
Game ends when Prover exposes a contradiction between equations accumulated to an initial clause (or if equations accumulated are unsatisfiable).
Thm (IS14): If Delayer has a strategy to always earn k points, then linear decision tree is ≥ 2k.
Delayer can earn at least (n-1)/2 points. And so the lower bound on decision trees is 2(n-1)/2.

14. Open Problems

15. Open Problems
R(lin 𝔽 2 ) lower bounds
Good candidate: PHP

16. Depth-3 IPS over 𝔽2 simulates tree-like R(lin 𝔽 2 )
Algebraic Approach
Depth-3 IPS over 𝔽2 simulates tree-like R(lin 𝔽 2 )
Probably: Depth-4 IPS over 𝔽2 simulates dag-like R(lin 𝔽 2 )
Use Grigoriev-Razborov 2000 lower bound (cf. Kumar-Sapsharishi 2017)?
Feasible Monotone Interpolation
Krajicek 2017, Krajicek-Oliviera 2017
R(lin 𝔽 2 ) lower bound is reduced to a monotone circuit lower bound with oracle access

17. Communication Complexity Approaches Sokolov 2017
Communication complexity protocol (tree) generalized into a DAG; lower bounds attempt on this
Further Related Results
Garlik and Kolodziejczyky, 2017: “Some subsystems of constant-depth Frege with parity”
If R(linℤ) is (weakly) automatizable then random 3CNFs with O(n1.4) clauses can be refuted in polynomial-deterministic time (T. 2014)
‘’Simulation’’ of Feige-Kim-Ofek (2006) witnesses.

18. Other Open Problems Random 3CNF lower bounds for tree-like R(lin 𝔽 2 )
Weak automatizability of R(lin 𝔽 2 ) implies PTIME refutation algorithm for random 3CNFs with O(n1.4) clauses? (Known for R(linℤ))

19. Thanks for Listening!

20. Appendix

21. Some Upper Bounds Proof of m to n PHP (for any m) [DAG like proof]:
Pigeon Axioms:
Hole Axioms:
For every pigeon i:
For every hole j:
Summing by pigeons (1), all variables sum up to a value from m,m+1,…,nm
Summing by holes (2), all variables sum up to a value from 0,1,…,n

22. Tree-like R(lin 𝔽 2 ) PHP Lower Bounds
Lemma: Let 𝐴 𝒙 = 𝒃 (over all variables 𝑥𝑖𝑗).
Assume number of equations ≤ (𝑛−1)/2.
For each pigeon 𝑖: if there is a proper solution, then there is a proper solution that satisfies pigeon 𝑖 axiom.
Proof. If we have a proper solution to 𝐴 𝒙 = 𝒃 , then since number of linear equations is ≤ (𝑛−1)/2, we have enough “slack” to sufficiently modify the assignment while forcing pigeon 𝑖 to some hole.

23. Tree-like R(lin 𝔽 2 ) PHP Lower Bounds
Lemma: Let 𝐴 𝒙 = 𝒃 (over all variables 𝑥𝑖𝑗).
Assume number of equations ≤ (𝑛−1)/2.
For each pigeon 𝑖: if there is a proper solution, then there is a proper solution that satisfies pigeon 𝑖 axiom.
Concluding the lower bound: Prover strategy:
Delayer asks value of f:
If accumulated equations T properly imply f=a, answer a;
If T has proper solution then T ⋃ {f=a} has proper solution
Otherwise, answer “You choose!”.

24. Tree-like R(lin 𝔽 2 ) PHP Lower Bounds
Concluding the lower bound: Prover strategy:
Delayer asks value of f:
If accumulated equations T properly imply f=a, answer a;
If T has proper solution then T ⋃ {f=a} has proper solution
Otherwise, answer “You choose!”.
Delayer earns > (𝑛−1)/2 points:
While Delayer points ≤ (𝑛−1)/2, we have:
Consider only chosen accumulated equations T (other equations are propery implied by it).
Since T has ≤ (𝑛−1)/2 equations, for every pigeon i there’s a proper solution that maps it somewhere, so no pigeon axiom is falsified.
Since T has proper solution, hole axioms are not falsified.