Related papers: Long multiplication by instruction sequences with …
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…
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,…
Each Boolean function can be computed by a single-pass instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. Auxiliary Boolean…
Instruction sequences with direct and indirect jump instructions are as expressive as instruction sequences with direct jump instructions only. We show that, in the case where the number of instructions is not bounded, we are faced with…
The secure hash function SHA-256 is a function on bit strings. This means that its restriction to the bit strings of any given length can be computed by a finite instruction sequence that contains only instructions to set and get the…
We investigate the expressiveness of backward jumps in a framework of formalized sequential programming called program algebra. We show that - if expressiveness is measured in terms of the computability of partial Boolean functions - then…
Large language models (LLMs) are typically trained to acquire behaviours from demonstrations or experience, yet much of their training data is declarative: instructions, rules, and descriptions that specify behaviours without showing how to…
We motivate and study an infinite sequence of binary operations on the ordinal numbers, extending the standard arithmetic on the ordinals to higher degrees of iteration. Connections to the hyperoperations on the natural numbers are…
Aligned instruction following models can better fulfill user requests than their unaligned counterparts. However, it has been shown that there is a length bias in evaluation of such models, and that training algorithms tend to exploit this…
We discuss how to implement backjumping (or intelligent backtracking) in Prolog programs by means of exception handling. This seems impossible in a general case. We provide two solutions. One works for binary programs; in a general case it…
Language models are increasingly capable, yet still fail at a seemingly simple task of multi-digit multiplication. In this work, we study why, by reverse-engineering a model that successfully learns multiplication via \emph{implicit…
We study sequential programs that are instruction sequences with jump-shift instructions in the setting of PGA (ProGram Algebra). Jump-shift instructions preceding a jump instruction increase the position to jump to. The jump-shift…
For most deep learning algorithms training is notoriously time consuming. Since most of the computation in training neural networks is typically spent on floating point multiplications, we investigate an approach to training that eliminates…
Training a neural network using backpropagation algorithm requires passing error gradients sequentially through the network. The backward locking prevents us from updating network layers in parallel and fully leveraging the computing…
We study sequential programs that are instruction sequences with direct and indirect jump instructions. The intuition is that indirect jump instructions are jump instructions where the position of the instruction to jump to is the content…
As robots become increasingly capable of manipulation and long-term autonomy, long-horizon task and motion planning problems are becoming increasingly important. A key challenge in such problems is that early actions in the plan may make…
Combinatorial optimisation problems framed as mixed integer linear programmes (MILPs) are ubiquitous across a range of real-world applications. The canonical branch-and-bound algorithm seeks to exactly solve MILPs by constructing a search…
Learning long-term dependencies in extended temporal sequences requires credit assignment to events far back in the past. The most common method for training recurrent neural networks, back-propagation through time (BPTT), requires credit…
Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, by F\"{u}rer, shows that two $n$-bit numbers can be multiplied via a boolean circuit of size $O(n \lg…
In previous work carried out in the setting of program algebra, including work in the area of instruction sequence size complexity, we chose instruction sets for Boolean registers that contain only instructions of a few of the possible…