Lower bounds for a proof system with an exponential speed-up over constant-depth Frege systems and over polynomial calculus

1997 ◽  
pp. 85-90 ◽  
Author(s):  
Jan Krajiček
Keyword(s):  
Lower Bounds ◽  
Proof System ◽  
Constant Depth ◽  
Frege Systems ◽  
Speed Up ◽  
Polynomial Calculus

Lower bounds for modal logics

2007 ◽  
Vol 72 (3) ◽  
pp. 941-958 ◽  
Author(s):  
Pavel Hrubeš
Keyword(s):  
Lower Bound ◽  
Modal Logic ◽  
Lower Bounds ◽  
Modal Logics ◽  
Proof System ◽  
Speed Up ◽  
Exponential Lower Bound

AbstractWe give an exponential lower bound on number of proof-lines in the proof system K of modal logic, i.e., we give an example of K-tautologies ψ1, ψ2, … s.t. every K-proof of ψi must have a number of proof-lines exponential in terms of the size of ψi. The result extends, for the same sequence of K-tautologies, to the systems K4, Gödel–Löb's logic, S andS4. We also determine some speed-up relations between different systems of modal logic on formulas of modal-depth one.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

Lower bounds for the polynomial calculus

1998 ◽  
Vol 7 (4) ◽  
pp. 291-324 ◽  
Author(s):  
A.A. Razborov
Keyword(s):  
Lower Bounds ◽  
Polynomial Calculus

scholarly journals Randomized feasible interpolation and monotone circuits with a local oracle

2018 ◽  
Vol 18 (02) ◽  
pp. 1850012 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Interpolation Theorem ◽  
Proof System ◽  
Symbolic Logic ◽  
Approximate Counting ◽  
Small Width ◽  
Randomized Protocols ◽  
Monotone Circuits

The feasible interpolation theorem for semantic derivations from [J. Krajíček, Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, J. Symbolic Logic 62(2) (1997) 457–486] allows to derive from some short semantic derivations (e.g. in resolution) of the disjointness of two [Formula: see text] sets [Formula: see text] and [Formula: see text] a small communication protocol (a general dag-like protocol in the sense of Krajíček (1997) computing the Karchmer–Wigderson multi-function [Formula: see text] associated with the sets, and such a protocol further yields a small circuit separating [Formula: see text] from [Formula: see text]. When [Formula: see text] is closed upwards, the protocol computes the monotone Karchmer–Wigderson multi-function [Formula: see text] and the resulting circuit is monotone. Krajíček [Interpolation by a game, Math. Logic Quart. 44(4) (1998) 450–458] extended the feasible interpolation theorem to a larger class of semantic derivations using the notion of a real communication complexity (e.g. to the cutting planes proof system CP). In this paper, we generalize the method to a still larger class of semantic derivations by allowing randomized protocols. We also introduce an extension of the monotone circuit model, monotone circuits with a local oracle (CLOs), that does correspond to communication protocols for [Formula: see text] making errors. The new randomized feasible interpolation thus shows that a short semantic derivation (from a certain class of derivations larger than in the original method) of the disjointness of [Formula: see text], [Formula: see text] closed upwards, yields a small randomized protocol for [Formula: see text] and hence a small monotone CLO separating the two sets. This research is motivated by the open problem to establish a lower bound for proof system [Formula: see text] operating with clauses formed by linear Boolean functions over [Formula: see text]. The new randomized feasible interpolation applies to this proof system and also to (the semantic versions of) cutting planes CP, to small width resolution over CP of Krajíček [Discretely ordered modules as a first-order extension of the cutting planes proof system, J. Symbolic Logic 63(4) (1998) 1582–1596] (system R(CP)) and to random resolution RR of Buss, Kolodziejczyk and Thapen [Fragments of approximate counting, J. Symbolic Logic 79(2) (2014) 496–525]. The method does not yield yet lengths-of-proofs lower bounds; for this it is necessary to establish lower bounds for randomized protocols or for monotone CLOs.

Lower bounds for cutting planes proofs with small coefficients

1997 ◽  
Vol 62 (3) ◽  
pp. 708-728 ◽  
Author(s):  
Maria Bonet ◽  
Toniann Pitassi ◽  
Ran Raz
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Proof System ◽  
Small Weight ◽  
Deduction Rule ◽  
Special Case ◽  
Exponential Lower Bound

AbstractWe consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution.We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other.Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

scholarly journals Lower Bounds and Separations for Constant Depth Multilinear Circuits

2009 ◽  
Vol 18 (2) ◽  
pp. 171-207 ◽  
Author(s):  
Ran Raz ◽  
Amir Yehudayoff
Keyword(s):  
Lower Bounds ◽  
Constant Depth

scholarly journals Uniform derandomization from pathetic lower bounds

2012 ◽  
Vol 370 (1971) ◽  
pp. 3512-3535 ◽  
Author(s):  
Eric Allender ◽  
V. Arvind ◽  
Rahul Santhanam ◽  
Fengming Wang
Keyword(s):  
Word Problem ◽  
Lower Bounds ◽  
Black Box ◽  
Arithmetic Circuits ◽  
List Type ◽  
Probabilistic Algorithms ◽  
Constant Depth ◽  
Subexponential Time ◽  
Polynomial Size ◽  
Threshold Circuits

The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where ‘pathetic’ lower bounds of the form n 1+ ϵ would suffice to derandomize interesting classes of probabilistic algorithms. We show the following: — If the word problem over S 5 requires constant-depth threshold circuits of size n 1+ ϵ for some ϵ >0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size). — If there are no constant-depth arithmetic circuits of size n 1+ ϵ for the problem of multiplying a sequence of n  3×3 matrices, then, for every constant d , black-box identity testing for depth- d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC 0 circuits of subexponential size).

scholarly journals Augmenting the Power of (Partial) MaxSat Resolution with Extension

2020 ◽  
Vol 34 (02) ◽  
pp. 1561-1568 ◽  
Author(s):  
Javier Larrosa ◽  
Emma Rollon
Keyword(s):  
Lower Bound ◽  
Objective Function ◽  
Lower Bounds ◽  
Polynomial Time ◽  
Proof System ◽  
Extension Rule ◽  
Resolution Proof

The refutation power of SAT and MaxSAT resolution is challenged by problems like the soft and hard Pigeon Hole Problem PHP for which short refutations do not exist. In this paper we augment the MaxSAT resolution proof system with an extension rule. The new proof system MaxResE is sound and complete, and more powerful than plain MaxSAT resolution, since it can refute the soft and hard PHP in polynomial time. We show that MaxResE refutations actually subtract lower bounds from the objective function encoded by the formulas. The resulting formula is the residual after the lower bound extraction. We experimentally show that the residual of the soft PHP (once its necessary cost of 1 has been efficiently subtracted with MaxResE) is a concise, easy to solve, satisfiable problem.

scholarly journals Tight rank lower bounds for the Sherali–Adams proof system

2009 ◽  
Vol 410 (21-23) ◽  
pp. 2054-2063 ◽  
Author(s):  
Stefan Dantchev ◽  
Barnaby Martin ◽  
Mark Rhodes
Keyword(s):  
Lower Bounds ◽  
Proof System
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