Long multiplication by instruction sequences with backward jump instructions

Abstract

For each function on bit strings, its restriction to bit strings of any 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. Backward jump instructions are not necessary for this, but instruction sequences can be significantly shorter with them. We take the function on bit strings that models the multiplication of natural numbers on their representation in the binary number system to demonstrate this by means of a concrete example. The example is reason to discuss points concerning the halting problem and the concept of an algorithm.

On Algorithmic Equivalence of Instruction Sequences for Computing Bit String Functions

“…In [9], where instruction sequences are considered which contain backward jump instructions in addition to instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction, it is shown that the function on bit strings that models the multiplication of natural numbers on their representation in the binary number system can be computed according to a minor variant of the long multiplication algorithm by quadratic-length instruction sequences without backward jump instructions and by linear-length instruction sequences with backward jump instructions.…”

“…In the case that we restrict ourselves to instruction sequences for computing partial functions from {0, 1} n to {0, 1} m , taking into account the experience gained in [7,8,9] with expressing algorithms by instruction sequences, we consider the following to be a first rough approximation of a definition of the concept of an algorithm: "an algorithm is an equivalence class of instruction sequences from IS n,m br with respect to an equivalence relation that completely captures the intuitive notion that two instruction sequences express the same algorithm". However, the equivalence relation to be defined in Section 8 is likely to incompletely capture this notion of algorithmic sameness.…”

“…Then the concept of a PNoriented algorithm is essentially the same as the concept of a basic algorithm in the sense that there exists a surjection ψ from L to IS n,m br such that, for all A ∈ L/≡ PN sa , A is a PN -oriented algorithm iff {ψ(P ) | P ∈ A} is a basic algorithm. In [9], instruction sequences without backward jump instructions and instruction sequences with backward jump instructions were given which were assumed to express the same minor variant of the long multiplication algorithm. If we take the program notation used as PN , then according to the definition of ≡ PN sa given above, the instruction sequences without backward jump instructions and the instruction sequences with backward jump instructions are structurally algorithmically equivalent, which indicates that they express the same algorithm.…”

“…In [10], it is among other things shown that a total function on bit strings whose result is a bit string of length 1 belongs to P/poly iff it can be computed by polynomial-length instruction sequences that contain only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. In [9], where instruction sequences are considered which contain backward jump instructions in addition to the above-mentioned instructions, it is among other things shown that there exist a total function on bit strings and an algorithm for computing the function such that the function can be computed according to the algorithm by quadratic-length instruction sequences without backward jump instructions and by linear-length instruction sequences with backward jump instructions.…”