34 research outputs found
A feasible interpolation for random resolution
Random resolution, defined by Buss, Kolodziejczyk and Thapen (JSL, 2014), is
a sound propositional proof system that extends the resolution proof system by
the possibility to augment any set of initial clauses by a set of randomly
chosen clauses (modulo a technical condition). We show how to apply the general
feasible interpolation theorem for semantic derivations of Krajicek (JSL, 1997)
to random resolution. As a consequence we get a lower bound for random
resolution refutations of the clique-coloring formulas.Comment: Preprint April 2016, revised September and October 201
Consistency of circuit lower bounds with bounded theories
Proving that there are problems in that require
boolean circuits of super-linear size is a major frontier in complexity theory.
While such lower bounds are known for larger complexity classes, existing
results only show that the corresponding problems are hard on infinitely many
input lengths. For instance, proving almost-everywhere circuit lower bounds is
open even for problems in . Giving the notorious difficulty of
proving lower bounds that hold for all large input lengths, we ask the
following question: Can we show that a large set of techniques cannot prove
that is easy infinitely often? Motivated by this and related
questions about the interaction between mathematical proofs and computations,
we investigate circuit complexity from the perspective of logic.
Among other results, we prove that for any parameter it is
consistent with theory that computational class , where is one of
the pairs: and , and , and
. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory already formalizes
sophisticated arguments, such as a proof of the PCP Theorem. These consistency
statements are unconditional and improve on earlier theorems of [KO17] and
[BM18] on the consistency of lower bounds with
The Cook-Reckhow definition
The Cook-Reckhow 1979 paper defined the area of research we now call Proof
Complexity. There were earlier papers which contributed to the subject as we
understand it today, the most significant being Tseitin's 1968 paper, but none
of them introduced general notions that would allow to make an explicit and
universal link between lengths-of-proofs problems and computational complexity
theory. In this note we shall highlight three particular definitions from the
paper: of proof systems, p-simulations and the pigeonhole principle formula,
and discuss their role in defining the field. We will also mention some related
developments and open problems
On monotone circuits with local oracles and clique lower bounds
We investigate monotone circuits with local oracles [K., 2016], i.e.,
circuits containing additional inputs that can perform
unstructured computations on the input string . Let be
the locality of the circuit, a parameter that bounds the combined strength of
the oracle functions , and
be the set of -cliques and the set of complete -partite graphs,
respectively (similarly to [Razborov, 1985]). Our results can be informally
stated as follows.
1. For an appropriate extension of depth- monotone circuits with local
oracles, we show that the size of the smallest circuits separating
(triangles) and (complete bipartite graphs) undergoes two phase
transitions according to .
2. For , arbitrary depth, and , we
prove that the monotone circuit size complexity of separating the sets
and is , under a certain restrictive
assumption on the local oracle gates.
The second result, which concerns monotone circuits with restricted oracles,
extends and provides a matching upper bound for the exponential lower bounds on
the monotone circuit size complexity of -clique obtained by Alon and Boppana
(1987).Comment: Updated acknowledgements and funding informatio
Information in propositional proofs and algorithmic proof search
We study from the proof complexity perspective the (informal) proof search
problem:
Is there an optimal way to search for propositional proofs?
We note that for any fixed proof system there exists a time-optimal proof
search algorithm. Using classical proof complexity results about reflection
principles we prove that a time-optimal proof search algorithm exists w.r.t.
all proof systems iff a p-optimal proof system exists.
To characterize precisely the time proof search algorithms need for
individual formulas we introduce a new proof complexity measure based on
algorithmic information concepts. In particular, to a proof system we
attach {\bf information-efficiency function} assigning to a
tautology a natural number, and we show that:
- characterizes time any -proof search algorithm has to use on
and that for a fixed there is such an information-optimal algorithm,
- a proof system is information-efficiency optimal iff it is p-optimal,
- for non-automatizable systems there are formulas with short
proofs but having large information measure .
We isolate and motivate the problem to establish {\em unconditional}
super-logarithmic lower bounds for where no super-polynomial size
lower bounds are known. We also point out connections of the new measure with
some topics in proof complexity other than proof search.Comment: Preliminary version February 202
On the existence of strong proof complexity generators
Cook and Reckhow 1979 pointed out that NP is not closed under complementation
iff there is no propositional proof system that admits polynomial size proofs
of all tautologies. Theory of proof complexity generators aims at constructing
sets of tautologies hard for strong and possibly for all proof systems. We
focus at a conjecture from K.2004 in foundations of the theory that there is a
proof complexity generator hard for all proof systems. This can be equivalently
formulated (for p-time generators) without a reference to proof complexity
notions as follows:
* There exist a p-time function stretching each input by one bit such
that its range intersects all infinite NP sets.
We consider several facets of this conjecture, including its links to bounded
arithmetic (witnessing and independence results), to time-bounded Kolmogorov
complexity, to feasible disjunction property of propositional proof systems and
to complexity of proof search. We argue that a specific gadget generator from
K.2009 is a good candidate for . We define a new hardness property of
generators, the -hardness, and shows that one specific gadget
generator is the -hardest (w.r.t. any sufficiently strong proof
system). We define the class of feasibly infinite NP sets and show, assuming a
hypothesis from circuit complexity, that the conjecture holds for all feasibly
infinite NP sets.Comment: preliminary version August 2022, revised July 202