Resolution with Counting: Dag-Like Lower Bounds and Different Moduli
Resolution over linear equations is a natural extension of the popular resolution refutation system, augmented with the ability to carry out basic counting. Denoted Res ( lin R ) , this refutation system operates with disjunctions of linear equations with Boolean variables over a ring R , to refute...
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| Published in: | Computational complexity Vol. 30; no. 1 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
01.06.2021
Springer Nature B.V |
| Subjects: | |
| ISSN: | 1016-3328, 1420-8954 |
| Online Access: | Get full text |
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| Summary: | Resolution over linear equations
is a natural extension of the popular resolution refutation system, augmented with the ability to carry out basic counting. Denoted
Res
(
lin
R
)
, this refutation system operates with disjunctions of linear equations with Boolean variables over a ring
R
, to refute unsatisfiable sets of such disjunctions. Beginning in the work of Raz & Tzameret (2008), through the work of Itsykson & Sokolov (2020) which focused on tree-like lower bounds, this refutation system was shown to be fairly strong. Subsequent work (cf. Garlik & Kołodziejczyk 2018; Itsykson & Sokolov 2020; Krajícek 2017; Krajícek & Oliveira 2018) made it evident that establishing lower bounds against general
Res
(
lin
R
)
refutations is a challenging and interesting task since the system captures a ``minimal'' extension of resolution with counting gates for which no super-polynomial lower bounds are known to date.
We provide the first super-polynomial size lower bounds against general (dag-like) resolution over linear equations refutations in the large characteristic regime. In particular, we prove that the subset-sum principle
1
+
∑
i
=
1
n
2
i
x
i
=
0
requires refutations of exponential size over
Q
. We use a novel lower bound technique: We show that under certain conditions every refutation of a subset-sum instance
f
=
0
must pass through a fat clause consisting of the equation
f
=
α
for every
α
in the image of
f
under Boolean assignments, or can be efficiently reduced to a proof containing such a clause. We then modify this approach to prove exponential lower bounds against tree-like refutations of any subset-sum instance that depends on
n
variables, hence also separating tree-like from dag-like refutations over the rationals.
We then turn to the finite fields regime, showing that the work of Itsykson & Sokolov (2020), where tree-like lower bounds over
F
2
were obtained, can be carried over and extended to every finite field. We establish new lower bounds and separations as follows: (
i
) For every pair of distinct primes
p
,
q
, there exist CNF formulas with short tree-like refutations in
Res
(
lin
F
p
)
that require exponential-size tree-like
Res
(
lin
F
q
)
refutations; (
ii
) random
k
-CNF formulas require exponential-size tree-like
Res
(
lin
F
p
)
refutations, for every prime
p
and constant
k
; and (
iii
) exponential-size lower bounds for tree-like
Res
(
lin
F
)
refutations of the pigeonhole principle, for
every
field
F
. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1016-3328 1420-8954 |
| DOI: | 10.1007/s00037-020-00202-x |