253 research outputs found
Tight Size-Degree Bounds for Sums-of-Squares Proofs
We exhibit families of -CNF formulas over variables that have
sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank)
but require SOS proofs of size for values of from
constant all the way up to for some universal constant.
This shows that the running time obtained by using the Lasserre
semidefinite programming relaxations to find degree- SOS proofs is optimal
up to constant factors in the exponent. We establish this result by combining
-reductions expressible as low-degree SOS derivations with the
idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and
Riis'03], and then applying a restriction argument as in [Atserias, M\"uller,
and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a
generic method of amplifying SOS degree lower bounds to size lower bounds, and
also generalizes the approach in [ALN14] to obtain size lower bounds for the
proof systems resolution, polynomial calculus, and Sherali-Adams from lower
bounds on width, degree, and rank, respectively
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
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 Paris-harrington and off-diagonal ramsey tautologies
We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and
of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound
in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a
(very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2). We
show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial
in the number of propositional variables. The proof technique for the lower bound extends the
idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle
beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal
Ramsey principle is established. This is obtained by adapting some constructions due to Erdos and Mills. Ëť
We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles
Semantic Versus Syntactic Cutting Planes
In this paper, we compare the strength of the semantic and syntactic version of the cutting planes proof system. First, we show that the lower bound technique of [22] applies also to semantic cutting planes: the proof system has feasible interpolation via monotone real circuits, which gives an exponential lower bound on lengths of semantic cutting planes refutations. Second, we show that semantic refutations are stronger than syntactic ones. In particular, we give a formula for which any refutation in syntactic cutting planes requires exponential length, while there is a polynomial length refutation in semantic cutting planes. In other words, syntactic cutting planes does not p-simulate semantic cutting planes. We also give two incompatible integer inequalities which require exponential length refutation in syntactic cutting planes. Finally, we pose the following problem, which arises in connection with semantic inference of arity larger than two: can every multivariate non-decreasing real function be expressed as a composition of non-decreasing real functions in two variables
Monte Carlo evaluation of the impact of subsequent strokes on backflashover rate
The paper deals with the impact of subsequent strokes on the backflashover rate (BFR) of HV overhead transmission lines (OHLs), assessed by means of an ATP-EMTP Monte Carlo procedure. The application to a typical 150 kV Italian OHL is discussed, simulating several tower grounding system arrangements. Subsequent strokes parameters are added to the statistical simulation variables: Peak current, front time, time-to-half value, lightning polarity, line insulation withstand, lightning location and phase angle of the power frequency voltage. The input data are fed to an ATP-EMTP complete circuit model of the OHL, including line insulation, lightning representation and tower grounding system, the latter simulated by a pi-circuit model able to simulate the effects due to propagation and soil ionization, at modest computational costs. Numerical results evidence a non-negligible BFR increase (in relative terms) due to subsequent strokes: for spatially concentrated grounding systems the BFR increase approximatively vary in inverse proportion with the low frequency grounding resistance, whereas for spatially extended grounding systems the BFR increase depends on the grounding system behavior at high frequencies
An equivalent circuit for the evaluation of cross-country fault currents in medium voltage (MV) distribution networks
A Cross-Country Fault (CCF) is the simultaneous occurrence of a couple of Line-to-Ground Faults (LGFs), affecting different phases of same feeder or of two distinct ones, at different fault locations. CCFs are not uncommon in medium voltage (MV) public distribution networks operated with ungrounded or high-impedance neutral: despite the relatively small value of LGF current that is typical of such networks, CCF currents can be comparable to those that are found in Phase-To-Phase Faults, if the affected feeder(s) consists of cables. This occurs because the faulted cables' sheaths/screens provide a continuous, relatively low-impedance metallic return path to the fault currents. An accurate evaluation is in order, since the resulting current magnitudes can overheat sheaths/screens, endangering cable joints and other plastic sheaths. Such evaluation, however, requires the modeling of the whole MV network in the phase domain, simulating cable screens and their connections to the primary and secondary substation earth electrodes by suitable computer programs, such as ATP (which is the acronym for alternative transient program) or EMTP (the acronym for electromagnetic transient program), with substantial input data being involved. This paper presents a simplified yet accurate circuit model of the faulted MV network, taking into account the CCF currents' return path (cable sheaths/screens, ground conductors, and earthing resistances of secondary substations). The proposed CCF model can be implemented in a general-purpose simulation program, and it yields accurate fault currents estimates: for a 20 kV network case study, the comparison with accurate ATP simulations evidences mismatches mostly smaller than 2%, and never exceeding 5%
Graph Colouring is Hard for Algorithms Based on Hilbert's Nullstellensatz and Gr\"{o}bner Bases
We consider the graph -colouring problem encoded as a set of polynomial
equations in the standard way over -valued variables. We prove that there
are bounded-degree graphs that do not have legal -colourings but for which
the polynomial calculus proof system defined in [Clegg et al '96, Alekhnovich
et al '02] requires linear degree, and hence exponential size, to establish
this fact. This implies a linear degree lower bound for any algorithms based on
Gr\"{o}bner bases solving graph -colouring using this encoding. The same
bound applies also for the algorithm studied in a sequence of papers [De Loera
et al '08,'09,'11,'15] based on Hilbert's Nullstellensatz proofs for a slightly
different encoding, thus resolving an open problem mentioned in [De Loera et al
'08,'09,'11] and [Li '16]. We obtain our results by combining the polynomial
calculus degree lower bound for functional pigeonhole principle (FPHP) formulas
over bounded-degree bipartite graphs in [Mik\v{s}a and Nordstr\"{o}m '15] with
a reduction from FPHP to -colouring derivable by polynomial calculus in
constant degree
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