Through the mincing machine with a Boolean layer cake: nonstandard computations over Boolean circuits in the lower-bounds-to-circuit-size complexity proving

Abstract

The purpose of the present paper is threefold. Firstly, it introduces an abstract theory of straight-line (circuit) Boolean computations and complexity based on a strict separation of the syntax and semantics of Boolean circuits; the usual (standard) way to compute a circuit becomes thus only one of many other (nonstandard) operation modes over the circuit. Secondly, we construct and investigate three particular nonstandard operation modes, of which one, the mincing (M-)mode, is the superposition of two others, the injection and ejection modes respectively. The M-mode uses the circuit as a pipe-lining device, channeling simultaneously through each layer of the circuit (a layer comprises all nodes of a given individual depth) information stored in previous layers and renewing the input and storing the output information after every such parallel step; the number of steps is equal to the double depth of the circuit. The function computed by the M-mode is richer as a functional device than the standard functional (S-)representation of a circuit; in particular, it not only detects paths connecting individual inputs and outputs of a circuit, as the S-representation does, but their lengths as well. Thirdly, based on the M-mode, a new (in fact, the first known nontrivial) general lower bound to the relative circuit-size complexity of a Boolean function is presented, in the form of the inequality $$s(f, C') \geqq m(f, C')$$ , which means that the circuit-size (CS-)complexity of a Boolean function f with respect to a fixed subset C′ ⫅ C of the set C of all Boolean circuits is at least as big as the mincing complexity of f with respect to the same subset; choosing appropriate subsets, one arrives at the complete, monotone, synchronous, locally synchronous (and in algebraic computations which can be naturally treated in the same vein, linear, bilinear) or other CS-complexities. The corresponding mincing complexity is defined as the minimum, taken over all circuits from C′ standardly computing f, of the Boolean dimension of the function computing by the M-mode; the Boolean dimension of a function is equal by definition to the 2-logarithm of the cardinality of the function's image. The advantage to work with the new lower bound is that it is defined as a minimum taken over a relatively well defined functional subspace associated with f and C′, and not, as in the CS-complexity case, over a subset of circuits, i.e., a mixed discrete-continuum structure of an obscure nature (the real reason why the lower bound problem is so difficult). To demonstrate how the above inequality works, we prove an Ω(m·log m) lower bound to the locally synchronous CS-complexity of the m-dimensional cyclic convolution. This particular result is important for several reasons: (i) it improves the more than ten-years old non-linear lower bound to the synchronous CS-complexity of the same function, (ii) it can not be, apparently, proved by previously known methods, (iii) there exist functions (an example will be given in this paper) with the linear locally synchronous and a non-linear synchronous CS-complexities.