Circuit Complexity
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Circuit Complexity: Arithmetic Circuit Complexity

Circuit Complexity: Arithmetic Circuit Complexity


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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In computational complexity theory, arithmetic circuits are the standard model for computing polynomials. Informally, an arithmetic circuit takes as inputs either variables or numbers, and is allowed to either add or multiply two expression it already computed. Arithmetic circuits give us a formal way for understanding the complexity of computing polynomials. The basic type of question in this line of research is what is the most efficient way for computing a given polynomial f?'. An arithmetic circuit C over the field F and the set of variables x1,...,xn is a directed acyclic graph as follows. Every node in it with indegree zero is called an input gate and is labeled by either a variable xi or a field element in F. Every other gate is labeled by either + or ; in the first case it is a sum gate and in the second a product gate. An arithmetic formula is a circuit in which every gate has outdegree one (and so the underlying graph is a directed tree). A circuit has two complexity measures associated with it: size and depth. The size of a circuit is the number of gates in it, and the depth of a circuit is the length of the longest directed path in it. For example, the circuit in the figure has size six and depth two. An arithmetic circuit computes a polynomial in the following natural way. An input gate computes the polynomial it is labeled by. A sum gate v computes the sum of the polynomials computed by its children (a gate u is a child of v if the directed edge (u,v) is in the graph). A product gate computes the product of the polynomials computed by its children. Consider the circuit in the figure, for example: the input gates compute (from right to left) x1,x2 and 1, the sum gates compute x1 + x2 and x2 + 1, and the product gate computes (... More: http://booksllc.net/?id=21890324


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Product Details
  • ISBN-13: 9781156454275
  • Publisher: Books LLC
  • Publisher Imprint: Books LLC
  • Height: 152 mm
  • No of Pages: 46
  • Spine Width: 3 mm
  • Weight: 82 gr
  • ISBN-10: 1156454271
  • Publisher Date: 31 May 2010
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Sub Title: Arithmetic Circuit Complexity
  • Width: 229 mm


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