136 research outputs found
Graph Grammars, Insertion Lie Algebras, and Quantum Field Theory
Graph grammars extend the theory of formal languages in order to model
distributed parallelism in theoretical computer science. We show here that to
certain classes of context-free and context-sensitive graph grammars one can
associate a Lie algebra, whose structure is reminiscent of the insertion Lie
algebras of quantum field theory. We also show that the Feynman graphs of
quantum field theories are graph languages generated by a theory dependent
graph grammar.Comment: 19 pages, LaTeX, 3 jpeg figure
Syntax-semantics interface: an algebraic model
We extend our formulation of Merge and Minimalism in terms of Hopf algebras
to an algebraic model of a syntactic-semantic interface. We show that methods
adopted in the formulation of renormalization (extraction of meaningful
physical values) in theoretical physics are relevant to describe the extraction
of meaning from syntactic expressions. We show how this formulation relates to
computational models of semantics and we answer some recent controversies about
implications for generative linguistics of the current functioning of large
language models.Comment: LaTeX, 75 pages, 19 figure
Mathematical Structure of Syntactic Merge
The syntactic Merge operation of the Minimalist Program in linguistics can be
described mathematically in terms of Hopf algebras, with a formalism similar to
the one arising in the physics of renormalization. This mathematical
formulation of Merge has good descriptive power, as phenomena empirically
observed in linguistics can be justified from simple mathematical arguments. It
also provides a possible mathematical model for externalization and for the
role of syntactic parameters
Prospects for Declarative Mathematical Modeling of Complex Biological Systems
Declarative modeling uses symbolic expressions to represent models. With such
expressions one can formalize high-level mathematical computations on models
that would be difficult or impossible to perform directly on a lower-level
simulation program, in a general-purpose programming language. Examples of such
computations on models include model analysis, relatively general-purpose
model-reduction maps, and the initial phases of model implementation, all of
which should preserve or approximate the mathematical semantics of a complex
biological model. The potential advantages are particularly relevant in the
case of developmental modeling, wherein complex spatial structures exhibit
dynamics at molecular, cellular, and organogenic levels to relate genotype to
multicellular phenotype. Multiscale modeling can benefit from both the
expressive power of declarative modeling languages and the application of model
reduction methods to link models across scale. Based on previous work, here we
define declarative modeling of complex biological systems by defining the
operator algebra semantics of an increasingly powerful series of declarative
modeling languages including reaction-like dynamics of parameterized and
extended objects; we define semantics-preserving implementation and
semantics-approximating model reduction transformations; and we outline a
"meta-hierarchy" for organizing declarative models and the mathematical methods
that can fruitfully manipulate them
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
A hierarchy of languages, logics, and mathematical theories
We present mathematics from a foundational perspective as a hierarchy in which each tier consists of a language, a logic, and a mathematical theory. Each tier in the hierarchy subsumes all preceding tiers in the sense that its language, logic, and mathematical theory generalize all preceding languages, logics, and mathematical theories. Starting from the root tier, the mathematical theories in this hierarchy are: combinatory logic restricted to the identity I, combinatory logic, ZFC set theory, constructive type theory, and category theory. The languages of the first four tiers correspond to the languages of the Chomsky hierarchy: in combinatory logic Ix = x gives rise to a regular language; the language generated by S, K in combinatory logic is context-free; first-order logic is context-sensitive; and the typed lambda calculus of type theory is recursively enumerable. The logic of each tier can be characterized in terms of the cardinality of the set of its truth values: combinatory logic restricted to I has 0 truth values, while combinatory logic has 1, first-order logic 2, constructive type theory 3, and categeory theory omega_0. We conjecture that the cardinality of objects whose existence can be established in each tier is bounded; for example, combinatory logic is bounded in this sense by omega_0 and ZFC set theory by the least inaccessible cardinal.
We also show that classical recursion theory presents a framework for generating the above hierarchy in terms of the initial functions zero, projection, and successor followed by composition and m-recursion, starting with the zero function I in combinatory logic
This paper begins with a theory of glossogenesis, i.e. a theory of the origin of language, since this theory shows that natural language has deep connections to category theory and since it was through these connections that the last tier and ultimately the whole hierarchy were discovered. The discussion covers implications of the hierarchy for mathematics, physics, cosmology, theology, linguistics, extraterrestrial communication, and artificial intelligence
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