133 research outputs found
Representations of Monotone Boolean Functions by Linear Programs
We introduce the notion of monotone linear-programming circuits (MLP circuits), a model of computation for partial Boolean functions. Using this model, we prove the following results.
1. MLP circuits are superpolynomially stronger than monotone Boolean circuits.
2. MLP circuits are exponentially stronger than monotone span programs.
3. MLP circuits can be used to provide monotone feasibility interpolation theorems for Lovasz-Schrijver proof systems, and for mixed Lovasz-Schrijver proof systems.
4. The Lovasz-Schrijver proof system cannot be polynomially simulated by the cutting planes proof system. This is the first result showing a separation between these two proof systems.
Finally, we discuss connections between the problem of proving lower bounds on the size of MLPs and the problem of proving lower bounds on extended formulations of polytopes
On the Relative Strength of Pebbling and Resolution
The last decade has seen a revival of interest in pebble games in the context
of proof complexity. Pebbling has proven a useful tool for studying
resolution-based proof systems when comparing the strength of different
subsystems, showing bounds on proof space, and establishing size-space
trade-offs. The typical approach has been to encode the pebble game played on a
graph as a CNF formula and then argue that proofs of this formula must inherit
(various aspects of) the pebbling properties of the underlying graph.
Unfortunately, the reductions used here are not tight. To simulate resolution
proofs by pebblings, the full strength of nondeterministic black-white pebbling
is needed, whereas resolution is only known to be able to simulate
deterministic black pebbling. To obtain strong results, one therefore needs to
find specific graph families which either have essentially the same properties
for black and black-white pebbling (not at all true in general) or which admit
simulations of black-white pebblings in resolution. This paper contributes to
both these approaches. First, we design a restricted form of black-white
pebbling that can be simulated in resolution and show that there are graph
families for which such restricted pebblings can be asymptotically better than
black pebblings. This proves that, perhaps somewhat unexpectedly, resolution
can strictly beat black-only pebbling, and in particular that the space lower
bounds on pebbling formulas in [Ben-Sasson and Nordstrom 2008] are tight.
Second, we present a versatile parametrized graph family with essentially the
same properties for black and black-white pebbling, which gives sharp
simultaneous trade-offs for black and black-white pebbling for various
parameter settings. Both of our contributions have been instrumental in
obtaining the time-space trade-off results for resolution-based proof systems
in [Ben-Sasson and Nordstrom 2009].Comment: Full-length version of paper to appear in Proceedings of the 25th
Annual IEEE Conference on Computational Complexity (CCC '10), June 201
Understanding space in resolution: optimal lower bounds and exponential trade-offs
We continue the study of tradeoffs between space and length of
resolution proofs and focus on two new results:
begin{enumerate}
item
We show that length and space in resolution are uncorrelated. This
is proved by exhibiting families of CNF formulas of size that
have proofs of length but require space . Our
separation is the strongest possible since any proof of length
can always be transformed into a proof in space , and
improves previous work reported in [Nordstr"{o}m 2006, Nordstr"{o}m and
H{aa}stad 2008].
item We prove a number of trade-off results for space in the range
from constant to , most of them superpolynomial or even
exponential. This is a dramatic improvement over previous results in
[Ben-Sasson 2002, Hertel and Pitassi 2007, Nordstr"{o}m 2007].
end{enumerate}
The key to our results is the following, somewhat surprising, theorem:
Any CNF formula can be transformed by simple substitution
transformation into a new formula such that if has the right
properties, can be proven in resolution in essentially the same
length as but the minimal space needed for is lower-bounded
by the number of variables that have to be mentioned simultaneously in
any proof for . Applying this theorem to so-called pebbling
formulas defined in terms of pebble games over directed acyclic graphs
and analyzing black-white pebbling on these graphs yields our results
Narrow proofs may be maximally long
We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n(Omega(w)). This shows that the simple counting argument that any formula refutable in width w must have a proof in size n(O(w)) is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.Peer ReviewedPostprint (author's final draft
On the proof complexity of logics of bounded branching
We investigate the proof complexity of extended Frege (EF) systems for basic
transitive modal logics (K4, S4, GL, ...) augmented with the bounded branching
axioms . First, we study feasibility of the disjunction property
and more general extension rules in EF systems for these logics: we show that
the corresponding decision problems reduce to total coNP search problems (or
equivalently, disjoint NP pairs, in the binary case); more precisely, the
decision problem for extension rules is equivalent to a certain special case of
interpolation for the classical EF system. Next, we use this characterization
to prove superpolynomial (or even exponential, with stronger hypotheses)
separations between EF and substitution Frege (SF) systems for all transitive
logics contained in or under some
assumptions weaker than . We also prove analogous
results for superintuitionistic logics: we characterize the decision complexity
of multi-conclusion Visser's rules in EF systems for Gabbay--de Jongh logics
, and we show conditional separations between EF and SF for all
intermediate logics contained in .Comment: 58 page
Going Meta on the Minimum Circuit Size Problem: How Hard Is It to Show How Hard Showing Hardness Is?
The Minimum Circuit Size Problem (MCSP) is a problem with a long history in computational complexity theory which has recently experienced a resurgence in attention. MCSP takes as input the description of a Boolean function f as a truth table as well as a size parameter s, and outputs whether there is a circuit that computes f of size ≤ s. It is of great interest whether MCSP is NP-complete, but there have been shown to be many technical obstacles to proving that it is. Most of these results come in the following form: If MCSP is NP-complete under a certain type of reduction, then we get a breakthrough in complexity theory that seems well beyond current techniques. These results indicate that it is unlikely we will be able to show MCSP is NP-complete under these kinds of reductions anytime soon.
I seek to add to this line of work, in particular focusing on an approximation version of MCSP which is central to some of its connections to other areas of complexity theory, as well as some other variants on the problem. Let f indicate an n-ary Boolean function that thus has a truth table of size 2n. I have used the approach of Saks and Santhanam (2020) to prove that if on input f approximating MCSP within a factor superpolynomial in n is NP-complete under general polynomial-time Turing reductions, then E ⊈ P/poly (a dramatic circuit lower bound). This provides a barrier to Hirahara (2018)\u27s suggested program of using the NP-completeness of a 2(1-)n-approximation version of MCSP to show that if NP is hard in the worst case (P ≠NP), it is also hard on average (i.e., to rule out Heuristica). However, using randomized reductions to do so remains potentially tractable.
I also extend the results of Saks and Santhanam (2020) to what I define as Σk-MCSP and Q-MCSP, getting stronger circuit lower bounds, namely E ⊈ ΣkP/poly and E ⊈ PH/poly, just from their NP-hardness. Since Σk-MCSP and Q-MCSP seem to be harder problems than MCSP, at first glance one might think it would be easier to show that Σk-MCSP or Q-MCSP is NP-hard, but my results demonstrate that the opposite is true
Narrow Proofs May Be Maximally Long
We prove that there are 3-CNF formulas over n variables that can be refuted
in resolution in width w but require resolution proofs of size n^Omega(w). This
shows that the simple counting argument that any formula refutable in width w
must have a proof in size n^O(w) is essentially tight. Moreover, our lower
bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams,
implying that the corresponding size upper bounds in terms of degree and rank
are tight as well. Our results do not extend all the way to Lasserre, however,
where the formulas we study have proofs of constant rank and size polynomial in
both n and w
On lengths of proofs in non-classical logics
AbstractWe give proofs of the effective monotone interpolation property for the system of modal logic K, and others, and the system IL of intuitionistic propositional logic. Hence we obtain exponential lower bounds on the number of proof-lines in those systems. The main results have been given in [P. Hrubeš, Lower bounds for modal logics, Journal of Symbolic Logic 72 (3) (2007) 941–958; P. Hrubeš, A lower bound for intuitionistic logic, Annals of Pure and Applied Logic 146 (2007) 72–90]; here, we give considerably simplified proofs, as well as some generalisations
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