LANDSCAPE of COMPUTATIONAL COMPLEXITY Spring 2008 State University of New York at Buffalo Department of Computer Science & Engineering Mustafa M

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LANDSCAPE OF COMPUTATIONAL COMPLEXITY Spring 2008 State University of New York at Buffalo Department of Computer Science & Engineering Mustafa M. Faramawi, MBA Dr. Kenneth W. Regan A complete language for EXPSPACE: PIM, “Polynomial Ideal Arithmetical Hierarchy (AH) Membership”—the simplest natural completeness level that is known PIM not to have polynomial‐size circuits. TOT = {M : M is total, K(2) TOT e e ∑n Πn i.e. halts for all inputs} EXPSPACE Succinct 3SAT Unknown ∑2 Π2 but Commonly Believed: RECRE L ≠ NL …………………….. L ≠ PH NEXP co‐NEXP = ∃∀.REC = ∀∃.REC P ≠ NP ∩ co‐NP ……… P ≠ PSPACE nxn Chess p p K D NP ≠ ∑ 2 ∩Π 2 ……… NP ≠ EXP D = {Turing machines M : EXP e Best Known Separations: Me does not accept e} = RE co‐RE the complement of K. AC0 ⊂ ACC0 ⊂ PP, also TC0 ⊂ PP QBF REC (“Diagonal Language”) For any fixed k, NC1 ⊂ PSPACE, …, NL ⊂ PSPACE = ∃.REC = ∀.REC there is a PSPACE problem in this P ⊂ EXP, NP ⊂ NEXP BQP: Bounded‐Error Quantum intersection that PSPACE ⊂ EXPSPACE Polynomial Time. Believed larger can NOT be Polynomial Hierarchy (PH) than P since it has FACTORING, solved by circuits p p k but not believed to contain NP. of size O(n ) ∑ 2 Π 2 L WS5 p p ∑ 2 Π 2 The levels of AH and poly. poly. BPP: Bounded‐Error Probabilistic TAUT = ∃∀ P = ∀∃ P SAT Polynomial Time. Many believe BPP = P. NC1 PH are analogous, NLIN except that we believe NP co‐NP 0 NP ∩ co‐NP ≠ P and NTIME [n2] TC QBF PARITY ∑p ∩Πp NP PP Probabilistic NP FACT 2 2 ≠ P , which NP ACC0 stand in contrast to P PSPACE Polynomial Time MAJ‐SAT CVP RE ∩ co‐RE = REC and 0 AC RE P P ∑2 ∩Π2 = REC GAP NP co‐NP REG NL poly 0 0 ∃ . AC ≠ ACC = P poly p p UGAP . = ∀ P ∑ 2 Π 2 L etc WS Low‐Level Classes 3 5 P DTIME [n ] NP co‐NP DLIN REG BQP DTIME [n2] FACTORING is not REG ≠ L believed to be FACT complete for BQP WS , the word problem for the symmetric group S , is a regular 5 5 Analogy between or for NP ∩ co‐NP. language that is complete for NC1 under AC0 many‐one reductions. P Deterministic and Nondeterministic Arithmetical and BPP Time Hierarchies Within NP Polynomial Hierarchies Complexity “Main Sequence” Realm of Feasibility? .

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Week 1: an Overview of Circuit Complexity 1 Welcome 2
Topics in Circuit Complexity (CS354, Fall’11) Week 1: An Overview of Circuit Complexity Lecture Notes for 9/27 and 9/29 Ryan Williams 1 Welcome The area of circuit complexity has a long history, starting in the 1940’s. It is full of open problems and frontiers that seem insurmountable, yet the literature on circuit complexity is fairly large. There is much that we do know, although it is scattered across several textbooks and academic papers. I think now is a good time to look again at circuit complexity with fresh eyes, and try to see what can be done. 2 Preliminaries An n-bit Boolean function has domain f0; 1gn and co-domain f0; 1g. At a high level, the basic question asked in circuit complexity is: given a collection of “simple functions” and a target Boolean function f, how efficiently can f be computed (on all inputs) using the simple functions? Of course, efficiency can be measured in many ways. The most natural measure is that of the “size” of computation: how many copies of these simple functions are necessary to compute f? Let B be a set of Boolean functions, which we call a basis set. The fan-in of a function g 2 B is the number of inputs that g takes. (Typical choices are fan-in 2, or unbounded fan-in, meaning that g can take any number of inputs.) We define a circuit C with n inputs and size s over a basis B, as follows. C consists of a directed acyclic graph (DAG) of s + n + 2 nodes, with n sources and one sink (the sth node in some fixed topological order on the nodes).
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