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Exponential Lower Bounds and Separation for Query Rewriting
"... We establish connections between the size of circuits and formulas computing monotone Boolean functions and the size of firstorder and nonrecursive Datalog rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower bounds and separation results from circuit complexity to prove ..."
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We establish connections between the size of circuits and formulas computing monotone Boolean functions and the size of firstorder and nonrecursive Datalog rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower bounds and separation results from circuit complexity to prove similar results for the size of rewritings that do not use nonsignature constants. For example, we show that, in the worst case, positive existential and nonrecursive Datalog rewritings are exponentially longer than the original queries; nonrecursive Datalog rewritings are in general exponentially more succinct than positive existential rewritings; while firstorder rewritings can be superpolynomially more succinct than positive existential rewritings.
Noncommutative circuits and the sumofsquares problem
 J. Amer. Math. Soc
"... 1.1. Noncommutative computation. Arithmetic complexity theory studies the computation of formal polynomials over some field or ring. Most of this theory is concerned with the computation of commutative polynomials. The basic model of computation is that of an arithmetic circuit. Despite decades of ..."
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1.1. Noncommutative computation. Arithmetic complexity theory studies the computation of formal polynomials over some field or ring. Most of this theory is concerned with the computation of commutative polynomials. The basic model of computation is that of an arithmetic circuit. Despite decades of work, the best
OntologyBased Data Access with Databases: A Short Course
"... Ontologybased data access (OBDA) is regarded as a key ingredient of the new generation of information systems. In the OBDA paradigm, an ontology defines a highlevel global schema of (already existing) data sources and provides a vocabulary for user queries. An OBDA system rewrites such queries an ..."
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Ontologybased data access (OBDA) is regarded as a key ingredient of the new generation of information systems. In the OBDA paradigm, an ontology defines a highlevel global schema of (already existing) data sources and provides a vocabulary for user queries. An OBDA system rewrites such queries and ontologies into the vocabulary of the data sources and then delegates the actual query evaluation to a suitable query answering system such as a relational database management system or a datalog engine. In this chapter, we mainly focus on OBDA with the ontology language OWL 2 QL, one of the three profiles of the W3C standard Web Ontology Language OWL 2, and relational databases, although other possible languages will also be discussed. We consider different types of conjunctive query rewriting and their succinctness, different architectures of OBDA systems, and give an overview of the OBDA system Ontop.
Averagecase lower bounds for formula size
 Electronic Colloquium on Computational Complexity (ECCC
, 2012
"... We give an explicit function h: {0, 1} n → {0, 1} such that any deMorgan formula + ɛ fraction of the inputs, where ɛ is of size O(n 2.499) agrees with h on at most 1 2 exponentially small (i.e. ɛ = 2−nΩ(1)). We also show, using the same technique, that any boolean formula of size O(n1.999) over the ..."
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We give an explicit function h: {0, 1} n → {0, 1} such that any deMorgan formula + ɛ fraction of the inputs, where ɛ is of size O(n 2.499) agrees with h on at most 1 2 exponentially small (i.e. ɛ = 2−nΩ(1)). We also show, using the same technique, that any boolean formula of size O(n1.999) over the complete basis, agrees with h on at most 1 2 + ɛ fraction of the inputs, where ɛ is exponentially small (i.e. ɛ = 2−nΩ(1)). Our construction is based on Andreev’s Ω(n2.5−o(1) ) formula size lower bound that was proved for the case of exact computation [And87]. 1
Tight Bounds on Computing ErrorCorrecting Codes by BoundedDepth Circuits with Arbitrary Gates
, 2012
"... We bound the minimum number w of wires needed to compute any (asymptotically good) errorcorrecting code C: {0, 1} Ω(n) → {0, 1} n with minimum distance Ω(n), using unbounded fanin circuits of depth d with arbitrary gates. Our main results are: (1) If d = 2 then w = Θ(n(lg n / lg lg n) 2). (2) If ..."
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We bound the minimum number w of wires needed to compute any (asymptotically good) errorcorrecting code C: {0, 1} Ω(n) → {0, 1} n with minimum distance Ω(n), using unbounded fanin circuits of depth d with arbitrary gates. Our main results are: (1) If d = 2 then w = Θ(n(lg n / lg lg n) 2). (2) If d = 3 then w = Θ(n lg lg n). (3) If d = 2k or d = 2k + 1 for some integer k ≥ 2 then w = Θ(nλk(n)), where λ1(n) = ⌈lg n⌉, λi+1(n) = λ ∗ i (n), and the ∗ operation gives how many times one has to iterate the function λi to reach a value at most 1 from the argument n. (4) If d = lg ∗ n then w = O(n). For depth d = 2, our Ω(n(lg n / lg lg n) 2) lower bound gives the largest known lower bound for computing any linear map. Using a result by Ishai, Kushilevitz, Ostrovsky, and Sahai [17], we also obtain similar bounds for computing pairwiseindependent hash functions. Our lower bounds are based on a superconcentratorlike condition that the graphs of circuits computing good codes must satisfy. This condition is provably intermediate between superconcentrators and their weakenings considered before.
Query Rewriting over Shallow Ontologies
"... Abstract. We investigate the size of conjunctive query rewritings over OWL 2 QL ontologies of depth 1 and 2 by means of a new formalism, called hypergraph programs, for computing Boolean functions. Both positive and negative results are obtained. All conjunctive queries over ontologies of depth 1 ha ..."
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Abstract. We investigate the size of conjunctive query rewritings over OWL 2 QL ontologies of depth 1 and 2 by means of a new formalism, called hypergraph programs, for computing Boolean functions. Both positive and negative results are obtained. All conjunctive queries over ontologies of depth 1 have polynomialsize nonrecursive datalog rewritings; treeshaped queries have polynomialsize positive existential rewritings; however, for some queries and ontologies of depth 1, positive existential rewritings can only be of superpolynomial size. Both positive existential and nonrecursive datalog rewritings of conjunctive queries and ontologies of depth 2 suffer an exponential blowup in the worst case, while firstorder rewritings can grow superpolynomially unless NP ⊆ P/poly. 1
NonCommutative Circuits and the SumofSquares Problem [Extended Abstract] ∗
"... We initiate a direction for proving lower bounds on the size of noncommutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of noncommutative arithmetic circuits and a problem about commutative degree four polynomials, the classical sumofsquares pr ..."
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We initiate a direction for proving lower bounds on the size of noncommutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of noncommutative arithmetic circuits and a problem about commutative degree four polynomials, the classical sumofsquares problem: find the smallest n such that there exists an identity (x 2 1+x 2 2+ · · ·+x 2 k)·(y 2 1+y 2 2+ · · ·+y 2 k) = f 2 1 +f 2 2 + · · ·+f 2 n, (1) where each fi = fi(X, Y) is bilinear in X = {x1,..., xk} and Y = {y1,..., yk}. Over the complex numbers, we show that a sufficiently strong superlinear lower bound on n in (1), namely, n ≥ k 1+ɛ with ε> 0, implies an exponential lower bound on the size of arithmetic circuits computing the noncommutative permanent. More generally, we consider such sumofsquares identities for any biquadratic polynomial h(X, Y), namely h(X, Y) = f 2 1 + f 2 2 + · · · + f 2 n. (2) Again, proving n ≥ k 1+ɛ in (2) for any explicit h over the complex numbers gives an exponential lower bound for the noncommutative permanent. Our proofs relies on several new structure theorems for noncommutative circuits, as well as a noncommutative analog of Valiant’s completeness of the permanent. We proceed to prove such superlinear bounds in some restricted cases. We prove that n ≥ Ω(k 6/5) in (1), if f1,..., fn are required to have integer coefficients. Over the real numbers, we construct an explicit biquadratic polynomial h such that n in (2) must be at least Ω(k 2). Unfortunately, these results do not imply circuit lower bounds.
Notes on Complexity Theory Last updated: December, 2011
"... Recall that one motivation for studying nonuniform computation is the hope that it might be easier to prove lower bounds in that setting. (This is somewhat paradoxical, as nonuniform algorithms are more powerful than uniform algorithms; nevertheless, since circuits are more “combinatorial” in natu ..."
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Recall that one motivation for studying nonuniform computation is the hope that it might be easier to prove lower bounds in that setting. (This is somewhat paradoxical, as nonuniform algorithms are more powerful than uniform algorithms; nevertheless, since circuits are more “combinatorial” in nature than uniform algorithms, there may still be justification for such hope.) The ultimate goal here would be to prove that N P ̸ ⊂ P /poly, which would imply P ̸ = N P. Unfortunately, after over two decades of attempts we are unable to prove anything close to this. Here, we show one example of a lower bound that we have been able to prove; we then discuss one “barrier ” that partly explains why we have been unable to prove stronger bounds.
1 NonUniform Complexity
"... We have seen that there exist “very hard ” languages (i.e., languages that require circuits of size (1 − ε)2n /n). If we can show that there exists a language in N P that is even “moderately hard” (i.e., requires circuits of superpolynomial size) then we will have proved P ̸ = N P. (In some sense, ..."
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We have seen that there exist “very hard ” languages (i.e., languages that require circuits of size (1 − ε)2n /n). If we can show that there exists a language in N P that is even “moderately hard” (i.e., requires circuits of superpolynomial size) then we will have proved P ̸ = N P. (In some sense, it would be even nicer to show some concrete language in N P that requires circuits of superpolynomial size. But mere existence of such a language is enough.) Here we show that for every c there is a language in Σ2 ∩ Π2 that is not in size(nc). Note that this does not prove Σ2 ∩ Π2 ̸ ⊆ P /poly since, for every c, the language we obtain is different. (Indeed, using the time hierarchy theorem, we have that for every c there is a language in P that is not in time(nc).) What is particularly interesting here is that (1) we prove a nonuniform lower bound and (2) the proof is, in some sense, rather simple. Theorem 1 For every c, there is a language in Σ4 ∩ Π4 that is not in size(n c). Proof Fix some c. For each n, let Cn be the lexicographically first circuit on n inputs such that (the function computed by) Cn cannot be computed by any circuit of size at most n c. By the nonuniform hierarchy theorem (see [1]), there exists such a Cn of size at most n c+1 (for n large
. Let
"... (Note that even though we believe Σi ̸ = Πi, oracle access to Σi gives the same power as oracle access to Πi. Do you see why?) We show that this leads to an equivalent definition. For this section only, let ΣO i refer to the definition in terms of oracles. We prove by induction that Σi = ΣO i. (Sinc ..."
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(Note that even though we believe Σi ̸ = Πi, oracle access to Σi gives the same power as oracle access to Πi. Do you see why?) We show that this leads to an equivalent definition. For this section only, let ΣO i refer to the definition in terms of oracles. We prove by induction that Σi = ΣO i. (Since ΠOi = coΣOi, this proves it for Πi, ΠO i as well.) For i = 1 this is immediate, as Σ1 = N P = N PP = ΣO 1. Assuming Σi = ΣO i, we prove that Σi+1 = ΣO i+1. Let us first show that Σi+1 ⊆ ΣO i+1 L ∈ Σi+1. Then there exists a polynomialtime Turing machine M such that x ∈ L ⇔ ∃w1∀w2 · · · Qi+1wi+1 M(x, w1,..., wi+1) = 1. In other words, there exists a language L ′ ∈ Πi such that x ∈ L ⇔ ∃w1 (x, w1) ∈ L ′.