Instruction Set and Language for Symbolic Regression

A fundamental but largely unaddressed obstacle in Symbolic regression (SR) is structural redundancy: every expression DAG with admits many distinct node-numbering schemes that all encode the same expression, each occupying a separate point in the search space and consuming fitness evaluations without adding diversity. We present IsalSR (Instruction Set and Language for Symbolic Regression), a representation framework that encodes expression DAGs as strings over a compact two-tier alphabet and computes a pruned canonical string -- a complete labeled-DAG isomorphism invariant -- that collapses all the equivalent representations into a single canonical form.

Instruction set for the representation of graphs

We present IsalGraph, a method for representing the structure of any finite, simple graph as a compact string over a nine-character instruction alphabet. The encoding is executed by a small virtual machine comprising a sparse graph, a circular doubly-linked list (CDLL) of graph-node references, and two traversal pointers. Instructions either move a pointer through the CDLL or insert a node or edge into the graph. A key design property is that every string over the alphabet decodes to a valid graph, with no invalid states reachable. A greedy \emph{GraphToString} algorithm encodes any connected graph into a string in time polynomial in the number of nodes; an exhaustive-backtracking variant produces a canonical string by selecting the lexicographically smallest shortest string across all starting nodes and all valid traversal orders. We evaluate the representation on five real-world graph benchmark datasets (IAM Letter LOW/MED/HIGH, LINUX, and AIDS) and show that the Levenshtein distance between IsalGraph strings correlates strongly with graph edit distance (GED). Together, these properties make IsalGraph strings a compact, isomorphism-invariant, and language-model-compatible sequential encoding of graph structure, with direct applications in graph similarity search, graph generation, and graph-conditioned language modelling

Enhanced QKNorm normalization for neural transformers with the Lp norm

The normalization of query and key vectors is an essential part of the Transformer architecture. It ensures that learning is stable regardless of the scale of these vectors. Some normalization approaches are available. In this preliminary work, a generalization of the QKNorm normalization scheme is proposed. The approach is based on the Lp norm, allowing non-Euclidean norms to be employed. Experimental results demonstrate the suitability of the method for a simple problem.

Alternative positional encoding functions for neural transformers

A key module in neural transformer-based deep architectures is positional encoding. This module enables a suitable way to encode positional information as input for transformer neural layers. This success has been rooted in the use of sinusoidal functions of various frequencies, in order to capture recurrent patterns of differing typical periods. In this work, an alternative set of periodic functions is proposed for positional encoding. These functions preserve some key properties of sinusoidal ones, while they depart from them in fundamental ways. Some tentative experiments are reported, where the original sinusoidal version is substantially outperformed. This strongly suggests that the alternative functions may have a wider use in other transformer architectures.

Representation of the structure of graphs by sequences of instructions

The representation of graphs is commonly based on the adjacency matrix concept. This formulation is the foundation of most algebraic and computational approaches to graph processing. The advent of deep learning language models offers a wide range of powerful computational models that are specialized in the processing of text. However, current procedures to represent graphs are not amenable to processing by these models. In this work, a new method to represent graphs is proposed. It represents the adjacency matrix of a graph by a string of simple instructions. The instructions build the adjacency matrix step by step. The transformation is reversible, i.e., given a graph the string can be produced and vice versa. The proposed representation is compact, and it maintains the local structural patterns of the graph. Therefore, it is envisaged that it could be useful to boost the processing of graphs by deep learning models. A tentative computational experiment is reported, demonstrating improved classification performance and faster computation times with the proposed representation.