Lower Bounds for Monotone Counting Circuits

A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by a given 0-1 input vector (with multiplicities given by their coefficients). A circuit decides $f$ if it has the same 0-1 roots as f. We first show that some multilinear polynomials can be exponentially easier to count than to compute them, and can be exponentially easier to decide than to count them. Then we give general lower bounds on the size of counting circuits.Comment: 20 page

Circuits with arbitrary gates for random operators

We consider boolean circuits computing n-operators f:{0,1}^n --> {0,1}^n. As gates we allow arbitrary boolean functions; neither fanin nor fanout of gates is restricted. An operator is linear if it computes n linear forms, that is, computes a matrix-vector product y=Ax over GF(2). We prove the existence of n-operators requiring about n^2 wires in any circuit, and linear n-operators requiring about n^2/\log n wires in depth-2 circuits, if either all output gates or all gates on the middle layer are linear.Comment: 7 page

Formulas vs. Circuits for Small Distance Connectivity

We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance $k(n)$ Connectivity, which asks whether two specified nodes in a graph of size $n$ are connected by a path of length at most $k(n)$. This problem is solvable (by the recursive doubling technique) on {\bf circuits} of depth $O(\log k)$ and size $O(kn^3)$. In contrast, we show that solving this problem on {\bf formulas} of depth $\log n/(\log\log n)^{O(1)}$ requires size $n^{\Omega(\log k)}$ for all $k(n) \leq \log\log n$. As corollaries: (i) It follows that polynomial-size circuits for Distance $k(n)$ Connectivity require depth $\Omega(\log k)$ for all $k(n) \leq \log\log n$. This matches the upper bound from recursive doubling and improves a previous $\Omega(\log\log k)$ lower bound of Beame, Pitassi and Impagliazzo [BIP98]. (ii) We get a tight lower bound of $s^{\Omega(d)}$ on the size required to simulate size-$s$ depth-$d$ circuits by depth-$d$ formulas for all $s(n) = n^{O(1)}$ and $d(n) \leq \log\log\log n$. No lower bound better than $s^{\Omega(1)}$ was previously known for any $d(n) \nleq O(1)$. Our proof technique is centered on a new notion of pathset complexity, which roughly speaking measures the minimum cost of constructing a set of (partial) paths in a universe of size $n$ via the operations of union and relational join, subject to certain density constraints. Half of our proof shows that bounded-depth formulas solving Distance $k(n)$ Connectivity imply upper bounds on pathset complexity. The other half is a combinatorial lower bound on pathset complexity

Graphs and Circuits: Some Further Remarks

We consider the power of single level circuits in the context of graph complexity. We first prove that the single level conjecture fails for fanin-$2$ circuits over the basis ${oplus,land,1}$. This shows that the (surpisingly tight) phenomenon, established by Mirwald and Schnorr (1992) for quadratic functions, has no analogon for graphs. We then show that the single level conjecture fails for unbounded fanin circuits over ${lor,land,1}$. This partially answers the question of Pudl\u27ak, R"odl and Savick\u27y (1986). We also prove that $Sigma_2 eq Pi_2$ in a restricted version of the hierarhy of communication complexity classes introduced by Babai, Frankl and Simon (1986). Further, we show that even depth-$2$ circuits are surprisingly powerful: every bipartite $n imes n$ graph of maximum degree $Delta$ can be represented by a monotone CNF with $O(Deltalog n)$ clauses. We also discuss a relation between graphs and $ACC$-circuits

Investor Protection : Segregation of Assets

The article is devoted to the contemporary problems of the ownership rights on intermediated securities and investor protection. Authors provide brief introduction to the major aspects of the Latvian securities law, then they study peculiarities of US approach to the ownership rights on intermediated securities as well US techniques of investor protection in the context of segregation and separation of assets of investors as well as briefly review the insolvency regimes and return of the financial assets to the investors, customers of insolvent US regulated firms. In conclusion authors provide brief comparative summarizing of Latvian and US approaches. Objectives: The study aims at continuing development of the securities law theory, while its task is to characterise the problematic of investor protection especially in insolvency proceedings on example of US as well as to discuss some peculiarities ownership rights on intermediated securities from the point of view of US law and practice. Methods/Approach Scientific research methods - both comparative and analytical - is used in the process of drawing up of this article. Results: Authors come to conclusion that US investor protection system is more complex than Latvian, it uses the separation and segregation approaches, while Latvian system only segregation approach. Both systems do not excluding situations of shortfall in customer assets. Both systems recognize the pro-rata distribution of damages, but US system recognises various pro-rata approaches to the distribution of damages (from the defined amount of assets; from all the assets; from all the assets belonging to a specific category of customers), while Latvian law stays silent regarding this topic. Authors believe that modernisation of the law is required for the better protection of the customers in case of insolvency of investment service providers and obtaining of the greater certainty of the legal outcomes. Author suggest should to determine the prohibition to meet the claims of one class of assets at the expense of another class of assets during insolvency proceedings of investment service providers in case of shortfall in assetspublishersversionPeer reviewe