We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switching-and-rectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networks, bounded-width devices , oblivious devices and read-k times only devices. 1 Introduction The main goal of the Boolean complexity theory is to prove lower bounds on the complexity of computing "explicitly given" Boolean functions in interesting computational models. By "explicitly given" researchers usually mean "belonging to the class NP ". This is a very plausible interpretation since on the one hand this class contains the overwhelming majority of interesting Boolean functions and on the other hand it is small enough to prevent us from the necessity to take into account counting arguments. To illustrate the second point, let me remind the reader that already the class \Delta p 2 ,...

Traditionally, computational complexity has considered only static problems. Classical Complexity Classes such as NC, P, and NP are defined in terms of the complexity of checking -- upon presentation of an entire input -- whether the input satisfies a certain property. For many applications of computers it is more appropriate to model the process as a dynamic one. There is a fairly large object being worked on over a period of time. The object is repeatedly modified by users and computations are performed. We develop a theory of Dynamic Complexity. We study the new complexity class, Dynamic First-Order Logic (Dyn-FO). This is the set of properties that can be maintained and queried in first-order logic, i.e. relational calculus, on a relational database. We show that many interesting properties are in Dyn-FO including multiplication, graph connectivity, bipartiteness, and the computation of minimum spanning trees. Note that none of these problems is in static FO, and this f...

We obtain the first non-trivial time-space tradeoff lower bound for functions f : {0, 1}^n → {0, 1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1 + ε)n, for some constant ε > 0. We also give the first separation result between the syntactic and semantic read-k models [BRS93] for k > 1 by showing that polynomial-size semantic read-twice branching programs can compute functions that require exponential size on any syntactic read-k branching program. We also show...

We prove the first time-space lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by Ajtai [Ajt99a, Ajt99b] in his time-space tradeoffs for deterministic RAM algorithms computing element distinctness and for Boolean branching programs computing a natural quadratic form. Ajtai's bounds were of the following form...

We prove the first time-space lower bound tradeoffs for randomized computation of decision problems.

We first consider so-called (1; +s)-branching programs in which along every consistent path at most s variables are tested more than once. We prove that any such program computing a characteristic function of a linear code C has size at least 2\Omega\Gamma2/1 , where d 1 and d 2 are the minimal distances of C and its dual C : We apply this criterion to explicit linear codes and obtain a super-polynomial lower bound for s = o(n= log n): Then we introduce a natural generalization of read-k-times and (1; +s)- branching programs that we call semantic branching programs. These programs correspond to corrupting Turing machines which, unlike eraser machines, are allowed to read input bits even illegally, i.e. in excess of their quota on multiple readings, but in that case they receive in response an unpredictably corrupted value. We generalize the above-mentioned bound to the semantic case, and also prove exponential lower bounds for semantic read-once nondeterministic branching programs.

We extend recent techniques for time-space tradeoff lower bounds using multiparty communication complexity ideas. Using these arguments, for inputs from large domains we prove larger tradeoff lower bounds than previously known for general branching programs, yielding time lower bounds of the form T = n) when space S = n , up from T = n log n) for the best previous results. We also prove the first unrestricted separation of the power of general and oblivious branching programs by proving that 1GAP , which is trivial on general branching programs, has a time-space tradeoff of the form T = (n=S)) on oblivious Finally, using time-space tradeoffs for branching programs, we improve the lower bounds on query time of data structures for nearest neighbor problems in d dimensions from d= log n), proved in the cell-probe model [8, 5], to d) or log d= log log d) or even d log d) (depending on the metric space involved) in slightly less general but more reasonable data structure models.

Almost the same types of restricted branching programs (or binary decision diagrams BDDs) are considered in complexity theory and in applications like hardware verification. These models are read-once branching programs (free BDDs) and certain types of oblivious branching programs (ordered and indexed BDDs with k layers). The complexity of the satisfiability problem for these restricted branching programs is investigated and tight hierarchy results are proved for the classes of functions representable by k layers of ordered or indexed BDDs of polynomial size. Keywords: branching programs, binary decision diagrams, communication complexity, hierarchies, lower bounds. Supported in part by DFG grant We 1066/7. 1 1. INTRODUCTION Branching programs are a well established computation model for discrete functions. Definition 1: A branching program G for a function f : A n ! B, where A = f0; : : : ; a \Gamma 1g and B = f0; : : : ; b \Gamma 1g is a directed acyclic graph with on...

Branching programs (b. p.'s) or decision diagrams are a general graph-based model of sequential computation. The b. p.'s of polynomial size are a nonuniform counterpart of LOG. Lower bounds for different kinds of restricted b. p.'s are intensively investigated. An important restriction are so called k-b. p.'s, where each computation reads each input bit at most k times. Although, for more restricted syntactic k-b.p.'s, exponential lower bounds are proven and there is a series of exponential lower bounds for 1-b. p.'s, this is not true for general (nonsyntactic) k-b.p.'s, even for k = 2. Therefore, so called (1; +k)-b. p.'s are investigated. For some explicit functions, exponential lower bounds for (1; +k)-b. p.'s are known. Investigating the syntactic (1; +k)-b. p.'s, Sieling has found functions f n;k which are polynomially easy for syntactic (1; +k)-b. p.'s, but exponentially hard for syntactic (1; +(k \Gamma 1))-b. p.'s. In the present paper, a similar hierarchy with res...

Almost the same types of restricted branching programs (or binary decision diagrams BDDs) are considered in complexity theory and in applications like hardware verification. These models are read-once branching programs (free BDDs) and certain types of oblivious branching programs (ordered and indexed BDDs with k layers). The complexity of the satisfiability problem for these restricted branching programs is investigated and tight hierarchy results are proved for the classes of functions representable by k layers of ordered or indexed BDDs of polynomial size. Supported in part by DFG grant We 1066/7--1 and 2. 1. INTRODUCTION Branching programs are a well established computation model for discrete functions. Definition 1: A branching program G for a function f : A n ! B m , where A = f0; : : : ; a \Gamma 1g and B = f0; : : : ; b \Gamma 1g is a directed acyclic graph. The sink nodes are labeled by constants from B. The inner nodes are labeled by variables from X = fx 1 ; : : :...