Is a two-party function, possibly with randomized output, securely computable? We provide a finite procedure to answer this question, thereby settling a foundational, three-decade-old open problem in secure computation and information complexity.Beaver-Chor-Kushilevitz [11], [22], [8] answered this question for deterministic output functions. Basu et al. [3] recently gave a geometric characterization of randomized functions securely computable with bounded communication complexity. Randomized functions can have arbitrarily high communication complexity, even for fixed input-output sets [5]. Without an upper bound on the communication complexity, the decidability of the question of whether a given two-party function with randomized output is securely computable was a formidable challenge.We reduce answering this question to proving specific lamination hulls are semi-algebraic. Lamination hulls are an infinite union of recursively defined sets independently motivated by the hydrodynamics literature. We connect this technical objective to solving a system of linear inequalities over convex sets in high dimensions, where inequalities represent the natural containment relation. We present a Gaussian elimination-inspired algorithm to compute the smallest simultaneous solutions to such systems. After that, using these solutions, we prove that our lamination hulls are semi-algebraic.Our technical solution introduces a novel set operator called positive geometric join. In our application context, it characterizes algebraically well-behaved sets that generalize polytopes, which we call hemihedra. The positive geometric join operator and hemihedral sets should interest the broader mathematics and computer science community. These advancements should help further information complexity investigations more broadly via the recently established connection by Basu et al. [3].

What is the round and communication complexity of secure computation? The seminal results of Chor-Kushilevitz-Beaver (STOC-1989, FOCS-1989, DIMACS-1989) answer this question for computations with deterministic output. However, this question has remained unanswered for computations with randomized output. Our work answers this question for two-party secure function evaluation functionalities. We introduce a geometric encoding of all candidate secure protocols for a given computation as points in a high-dimensional space. The following results follow by analyzing the properties of these sets of points.1)It is decidable to determine if a given computation has a secure protocol within round or communication constraints.2)We construct one such protocol if it exists.3)Otherwise, we present an obstruction to achieving security.Our technical contributions imply new information complexity bounds for secure computation.

We introduce harmonic persistent homology spaces for filtrations of finite simplicial complexes. As a result we can associate concrete subspaces of cycles to each bar of the barcode of the filtration. We prove stability of the harmonic persistent homology subspaces under small perturbations of functions defining them. We relate the notion of “essential simplices,” introduced in an earlier work to identify simplices which play a significant role in the birth of a bar, with that of harmonic persistent homology. We prove that the harmonic representatives of simple bars maximizes the “relative essential content” amongst all representatives of the bar, where the relative essential content is the weight a particular cycle puts on the set of essential simplices.

We introduce a robust mathematical definition of the notion of essential elements in a basis of the homology space and prove that these elements are unique. Next we give a novel visualization of the essential elements of the basis of the homology space through a rainfall-like plot (RFL). This plot is data-centric, i.e., is associated with the individual samples of the data, as opposed to the structure-centric barcodes of persistent homology. The proof-of-concept was tested on data generated by SimRA that simulates different admixture scenarios. We show that the barcode analysis can be used not just to detect the presence of admixture but also estimate the number of admixed populations. We also demonstrate that data-centric RFL plots have the potential to further disentangle the common history into admixture events and relative timing of the events, even in very complex scenarios.

Toda proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P#P, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of BlumShub-Smale real Turing machines) has been missing so far. In this paper we formulate and prove a real analogue of Toda's theorem. Unlike Toda's proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature. As a consequence of our techniques we are also able to relate the computational hardness of two extremely well-studied problems in algorithmic semi-algebraic geometry namely the problem of deciding sentences in the first order theory of the reals with a constant number of quantifier alternations, and that of computing Betti numbers of semi-algebraic sets. We obtain a polynomial time reduction of the compact version of the first problem to the second. This latter result might be of independent interest to researchers in algorithmic semi-algebraic geometry.

In this paper we prove tight bounds on the combinatorial and topological complexity of sets dened in terms of n denable sets belonging to some fixed denable family of sets in an o-minimal structure. This generalizes the combinatorial parts of similar bounds known in the case of semi-algebraic and semi-Pfaffian sets, and as a result vastly increases the applicability of results on combinatorial and topological complexity of arrangements studied in discrete and computational geometry. As a sample application, we extend a Ramsey-type theorem due to Alon et al. [3], originally proved for semi-algebraic sets of fixed description complexity to this more general setting.

For any fixed l > 0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P1 ≥ 0,...,Ps ≥ 0, where each Pi ∈ R[X1,...,Xk] has degree ≤ 2, and computes the top l Betti numbers of S, bk-1(S), ..., bk-l(S), in polynomial time. The complexity of the algorithm, stated more precisely, is Σi=0l+2 (si k2O((l,s)). For fixed l, the complexity of the algorithm can be expressed as sl+2 k2O(l), which is polynomial in the input parameters s and k. To our knowledge this is the first polynomial time algorithm for computing non-trivial topological invariants of semi-algebraic sets in Rk defined by polynomial inequalities, where the number of inequalities is not fixed and the polynomials are allowed to have degree greater than one. For fixed s, we obtain by letting l = k, an algorithm for computing all the Betti numbers of S whose complexity is k2O(s)

In this paper we describe the first singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. We also describe algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincaré characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zero-th one.

We study the sparsity of real polynomials that sign represent parity on n variables, each of which takes values from some finite subset A of integers. While the degree of such polynomials has been well studied by M. Minsky and S. Papert (1968) and J. Aspnes (1994), relatively little is known about their sparsity. We study this problem using Descartes rule of signs, a classical result in algebra, relating the sparsity of a polynomial to its number of real roots. We show that sign representing parity over {0,1,..., m - 1}/sup n/ with the degree in each variable at most m - 1 requires sparsity at least m/sup n/. We show a bound of (m - l)/sup n/ for weak representations. We show that a tradeoff exists between sparsity and degree, by constructing a sign representation that has higher degree but lower sparsity. In some cases, the difference in sparsities is exponential. We show a lower bound of n(m - 2) + 1 on the sparsity of polynomials of any degree representing parity over {0, 1, ..., m -1 }/sup n/. We prove exact bounds on the sparsity of such polynomials for any two element subset A. We show that for depth-two and-or-not circuits with a threshold gate at the top, the minimum circuit size for a function f equals the minimum sparsity of a polynomial sign representing f over a certain basis. We use this to give a simple proof that such circuits need size (3/2)/sup n/ to compute parity, which improves on previous bounds by M. Goldmann (1997). We also show a tight lower bound of 2/sup n/ for the inner product function over {0,1}/sup n/ /spl times/ {0,1}/sup n/. The main technical tool used is Descartes rule of signs. Our bounds hold for various bases where Descartes sign rule is valid.

(MATH) In this paper, we consider the problem of computing the Betti numbers of an arrangement of $n$ compact semi-algebraic sets, $S_1,\ldots,S_n \subset \R^k$, where each $S_i$ is described using a constant number of polynomials with degrees bounded by a constant. Such arrangements are ubiquitous in computational geometry. We give an algorithm for computing $\ell$-th Betti number, $\beta_\ell(\cup_i S_i), 0 \leq \ell \leq k-1$, using $O(n^{\ell+2})$ algebraic operations. Additionally, one has to perform linear algebra on matrices of size bounded by $O(n^{\ell+1})$. All previous algorithms for computing the Betti numbers of arrangements, triangulated the arrangement giving rise to a complex of size $O(n^{2^k})$ in the worst case. To our knowledge this is the first algorithm for computing $\beta_\ell(\cup_i S_i)$ that does not rely on such a global triangulation, and has a graded complexity which depends on $\ell$.

A classic result in real algebraic geometry due to Oleinik-Petrovsky, Thom and Milnor, bounds the {\em topological complexity} (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers.

In this paper, we give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms [Basu et al. 1996; Renegar 1992; Heintz et al. 1990], the combinatorial part of the complexity (the part depending on the number of polynomials in the input) of this new algorithm is independent of the number of free variables. Moreover, under the assumption that each polynomial in the input depends only on a constant number of the free variables, the algebraic part of the complexity(the part depending on the degrees of the input polynomials) can also be made independent of the number of free variables. This new feature of our algorithm allow us to obtain a new algorithm for a variant of the quantifier elimination problem. We give an almost optimal algorithm for this new problem, which we call theuniform quantifier elimination problem. Using tthe uniform quantifier elimination algorithm, we give an algorithm for solving a problem arising in the field of constraint databases with real polynomial constraints. We give an algorithm for converting a query with natural domain semantics to an equivalent one with active domain semantics. A nonconstructive version of this result was proved in Benedikt et al. [1998]. Very recently, a constructive proof was also given independently in Benedikt and Libkin [1997]. However, complexity issues were not considered and no algorithm with a reasonable complexity bound was known for this latter problem till now. We also point out interesting logical consequences of this algorithmic result, concerning the expressive power of a constraint query language over the reals. This leads to simpler and constructive proofs for these inexpressibility results than the ones known before. Moreover, our improved algorithm for performing quantifier elimination immediately leads to improved algorithms for several problems for which quantifier elimination is a basic step, for example, the problem of computing the closure of a given semi-algebraic set.

The problem of bounding the combinatorial complexity of a single connected component (a single cell) of the complement of a set of a geometric objects in R/sup k/, each object of constant description complexity, is an important problem in computational geometry which has attracted much attention over the past decade. It has been conjectured that the combinatorial complexity of a single cell is bounded by a function much closer to O(n/sup k-1/) rather than O(n/sup k/) which is the bound for the combinatorial complexity of the whole arrangement. Till now, this was known to be rule only for k/spl les/3 and only for some special cases in higher dimensions. A classic result in real algebraic geometry due to Oleinik-Petrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. However, till now no better bounds were known if we restricted attention to a single connected component of a basic semi-algebraic set. In this paper, we show how these two problems are related. We prove a new bound on the sum of the Betti numbers of one connected component of a basic semi-algebraic set which is an improvement over the Oleinik-Petrovsky-Thom-Milnor bound. This also implies that the topological complexity of a single cell, measured by the sum of the Betti numbers, is bounded by O(n/sup k-1/).

Given Q 2 R[X1 ; : : : ; Xk ] with deg(Q) d; we give an algorithm that outputs a semi-algebraic description for each of the semi-algebraically connected components of Z(Q) ae R k : The complexity of the algorithm as well as the size of the output are bounded by d O(k 3 ) : More generally, given any semi-algebraic set S defined by a quantifier-free formula involving a family of polynomials, P = fP1 ; : : : ; Psg ae R[X1 ; : : : ; Xk ] whose degrees are at most d; we give an algorithm that outputs a semi-algebraic description for each of the semialgebraically connected components of S: The complexity of the algorithm as well as the size of the output is bounded by s k+1 d O(k 3 ) : This improves the previously best known bound of (sd) k O(1) for this problem due to Canny, Grigor'ev, Vorobjov and Heintz, Roy and Solern`o [9, 14]. 1 Introduction Let R be a real closed field. A semi-algebraic set in R k is the set of points which satisfy a boolean combination of polynom...

We give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms the combinatorial part of the complexity of this new algorithm is independent of the number of free variables. Moreover, under the assumption that each polynomial in the input depend only on a constant number of the free variables, the algebraic part of the complexity can also be made independent of the number of free variables. This new feature of our algorithm allows us to obtain a new algorithm for a variant of the quantifier elimination problem. We give an almost optimal algorithm for this new problem, which we call the uniform quantifier elimination problem and apply it to solve a problem arising in the field of constraint databases. No algorithm with reasonable complexity bound was known for this latter problem till now. We also point out interesting logical consequences of this algorithmic result, concerning the expressive power of a constraint query language over the reals. Moreover, our improved algorithm for performing quantifier elimination immediately leads to improved algorithms for several problems for which quantifier elimination is a basic step, for example, the problem of computing the closure of a given semi-algebraic set.

Inthis paperwe introduce avariant of the quantifier elimination problem for the first order theory of real closed fields.Instead of considering a single quantified formula, we consider a uniform sequence of such formulas and eliminate quantifiers to obtain another uniform sequence.Our immediate motivation comes from a problem in the theory of constraint databases with real polynomial constraints.Using the uniform quantifier elimination algorithm, we give an algorithm for converting a query with natural domain semantics to an equivalent one with active domain semantics.A non-constructive version of this result was proved in [2].Very recently, a constructive proof was also given independently in [5].In the last part of the paper, using some elementary tools from first order logic and results described above, we prove that parity is not expressible in the constraint query language over the reals.This was proved by Benedikt et al [2].However unlike ours, their proof uses difficult techniques from non-standard analysis and model theory of ordered struct7mes."

We consider a family ofspolynomials, P = {P1, …,Ps}, inkvariables with coefficients in a real closed fieldR, each of degree at mostd, and an algebraic varietyVof real dimensionk′ which is defined as the zero set of a polynomialQof degree at mostd. The number of semi-algebraically connected components of all non-empty sign conditions on P overVis bounded bysk′(O(d))k. In this paper we present a new algorithm to compute a set of points meeting every semi-algebraically connected component of each non-empty sign condition of P overV. Its complexity issk′ + 1dO(k). This interpolates a sequence of results between the Ben-Or–Kozen–Reif algorithm which is the casek′ = 0, in one variable, and the Basu–Pollack–Roy algorithm which is the casek′ =k. It improves the results where the same problem was solved in timesk′ + 1dO(k′k).

Article Free Access Share on On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets Author: Saugata Basu Courant Institute of Mathematical Sciences, New York University, New York, NY Courant Institute of Mathematical Sciences, New York University, New York, NYView Profile Authors Info & Claims STOC '96: Proceedings of the twenty-eighth annual ACM symposium on Theory of ComputingJuly 1996 Pages 408–417https://doi.org/10.1145/237814.237988Published:01 July 1996Publication History 7citation345DownloadsMetricsTotal Citations7Total Downloads345Last 12 Months13Last 6 weeks2 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

We consider a semi-algebraic set S defined by s polynomials of degree d in k variables.We present a new algorithm for computing a semi-algebraic path in S comecting two points if they happen to lie in the same connected component of S.This algorithm, which works in time s ~tldo(~z) improves the complexity of the fastest algorithm solving this problem known to this date.

In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this data. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the imput polynomials) and the combinatorial part (the dependence on the number of polynomials) are sparated. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that is output, are independent of the number of input polynomials. As special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.

In this paper we give a new algorithm for performing quantifier elimination from first order formulae over real closed fields. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of our algorithm is that the role of the algebraic part (the dependence on the degrees of the input polynomials) and the combinatorial part (the dependence on the number of polynomials) are separated, making possible our improved complexity bound. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that we output, are independent of the number of input polynomials. As special cases of this algorithm, we obtain new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields. Using the theory developed in this paper, we also give an improved bound on the radius of a ball centered at the origin, which is guaranteed to intersect every connected component of the sign partition induced by a family of polynomials. We also use our methods to obtain algorithms for solving certain decision problems in real and complex geometry which improves the complexity of the currently known algorithms for these problems.>

A new scheme for designing error detecting and error correcting codes around cellular automata (CA) is reported. A simple and efficient scheme for generating SEC-DED codes is presented which can also be extended for generating codes with higher distances. A CA-based hardware scheme for very fast decoding (and correcting) of the codewords is also reported.>