Singular tuples of matrices is not a null cone (and the symmetries of algebraic varieties)

The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: SING n , m {{\rm SING}_{n,m}} , consisting of all m-tuples of n × n {n\times n} complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in SING n , m {{\rm SING}_{n,m}} will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation. A sequence of recent works suggests such efficient algorithms for memberships in a general class of algebraic varieties, namely the null cones of linear group actions. Can this be used for the problem above? Our main result is negative: SING n , m {{\rm SING}_{n,m}} is not the null cone of any (reductive) group action! This stands in stark contrast to a non-commutative analog of this variety, and points to an inherent structural difficulty of SING n , m {{\rm SING}_{n,m}} . To prove this result, we identify precisely the group of symmetries of SING n , m {{\rm SING}_{n,m}} . We find this characterization, and the tools we introduce to prove it, of independent interest. Our work significantly generalizes a result of Frobenius for the special case m = 1 {m=1} , and suggests a general method for determining the symmetries of algebraic varieties.

Exploring Minds: Modes of Modeling and Simulation in Artificial Intelligence

The aim of this paper is to grasp the relevant distinctions between various ways in which models and simulations in Artificial Intelligence (AI) relate to cognitive phenomena. In order to get a systematic picture, a taxonomy is developed that is based on the coordinates of formal versus material analogies and theory-guided versus pre-theoretic models in science. These distinctions have parallels in the computational versus mimetic aspects and in analytic versus exploratory types of computer simulation. The proposed taxonomy cuts across the traditional dichotomies between symbolic and embodied AI, general intelligence and symbol and intelligence and cognitive simulation and human/non-human-like AI. According to the taxonomy proposed here, one can distinguish between four distinct general approaches that figured prominently in early and classical AI, and that have partly developed into distinct research programs: first, phenomenal simulations (e.g., Turing’s “imitation game”); second, simulations that explore general-level formal isomorphisms in pursuit of a general theory of intelligence (e.g., logic-based AI); third, simulations as exploratory material models that serve to develop theoretical accounts of cognitive processes (e.g., Marr’s stages of visual processing and classical connectionism); and fourth, simulations as strictly formal models of a theory of computation that postulates cognitive processes to be isomorphic with computational processes (strong symbolic AI). In continuation of pragmatic views of the modes of modeling and simulating world affairs, this taxonomy of approaches to modeling in AI helps to elucidate how available computational concepts and simulational resources contribute to the modes of representation and theory development in AI research—and what made that research program uniquely dependent on them.

Linguistics Then and Now: Some Personal Reflections

By mid-twentieth century, a working consensus had been reached in the linguistics community, based on the great achievements of preceding years. Synchronic linguistics had been established as a science, a “taxonomic” science, with sophisticated procedures of analysis of data. Taxonomic science has limits. It does not ask “why?” The time was ripe to seek explanatory theories, using insights provided by the theory of computation and studies of explanatory depth. That effort became the generative enterprise within the biolinguistics framework. Tensions quickly arose: The elements of explanatory theories (generative grammars) were far beyond the reach of taxonomic procedures. The structuralist principle that language is a matter of training and habit, extended by analogy, was unsustainable. More generally, the mood of “virtually everything is known” became “almost nothing is understood,” a familiar phenomenon in the history of science, opening a new and exciting era for a flourishing discipline. Expected final online publication date for the Annual Review of Linguistics, Volume 7 is January 14, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Prolegomena to an Operator Theory of Computation

Defining computation as information processing (information dynamics) with information as a relational property of data structures (the difference in one system that makes a difference in another system) makes it very suitable to use operator formulation, with similarities to category theory. The concept of the operator is exceedingly important in many knowledge areas as a tool of theoretical studies and practical applications. Here we introduce the operator theory of computing, opening new opportunities for the exploration of computing devices, processes, and their networks.