On Algorithmic Equivalence of Instruction Sequences for Computing Bit String Functions

Abstract

Every partial function from bit strings of a given length to bit strings of a possibly different given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. We look for an equivalence relation on instruction sequences of this kind that captures to a reasonable degree the intuitive notion that two instruction sequences express the same algorithm.

Instruction Sequence Size Complexity of Parity

“…It is still an open question whether there exist a k ≥ 1 and an f : N → N with f (n) < 2 · n + 3 for all n > 0 such that PAR ∈ IS k br \B(f (n)). According to the view taken in [16], differences in the number of auxiliary Boolean registers whose use contributes to computing the function at hand always go with algorithmic differences. This view is supported by the instruction sequences PARIS 0 n and PARIS 1 n : in addition to having different lengths, they express undeniably quite different algorithms to compute PAR n (for n > 1).…”

Untitled

“…The preliminaries to the work presented in this paper (Sections 2 and 3) are almost the same as the preliminaries to the work presented in [7] and earlier papers. For this reason, there is some text overlap with those papers.…”

“…For example, we have introduced instruction sequence based counterparts of the complexity classes P/poly and NP/poly and we have formulated an instruction sequence based counterpart of the well-known complexity-theoretic conjecture that NP ⊆ P/poly. 7 However, for many a question that arises naturally with the approach to complexity based on instruction sequence size, it is far from obvious whether a comparable question can be raised in classical complexity theory based on Turing machines or Boolean circuits. In particular, this is far from obvious for questions concerning instruction sets for Boolean registers.…”

“…7 The non-uniform complexity classes P/poly and NP/poly, as well as the conjecture that NP ⊆ P/poly, are treated in many textbooks on classical complexity theory (see e.g. [1,10,14]).…”

On the complexity of the correctness problem for non-zeroness test instruction sequences

“…Like in the work on computational complexity (see [3,5]) and the work on algorithmic equivalence of programs (see [4]) referred to above, in the work presented in this paper, use is made of the fact that, for each n > 0, each function from {0, 1} n to {0, 1} can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. Program algebra is parameterized by a set of uninterpreted basic instructions.…”