Dyson–Schwinger equations in the theory of computation

Abstract

Following Manin's approach to renormalization in the theory of computation, we investigate Dyson-Schwinger equations on Hopf algebras, operads and properads of flow charts, as a way of encoding self-similarity structures in the theory of algorithms computing primitive and partial recursive functions and in the halting problem.

The Multiresolution Analysis of Flow Graphs

“…This is prima facie cause to restrict consideration to the definition of absorption introduced above. 8 Note that we are not explicitly considering the insertion of loops in this setting.…”

“…One sometimes sees variants of the definition and naming of this particular sort of concept, for the latter most typically as "flowgraph", "flowchart", or "flow chart". Some concepts with the same name are technically quite different but "spiritually" viewed in a similar context, as, e.g., in the work of Manin[8,20].…”

Entropy algebras and Birkhoff factorization

“…A mathematical model of renormalization for perturbative quantum field theories, based on a commutative Hopf algebra H, a Rota-Baxter algebra R and the Birkhoff factorization of morphisms of commutative algebras φ : H → R, was developed in [7], [10], [11]. More recently, an approach to the theory of computation and the halting problem modelled on quantum field theory and renormalization was developed in [15], [16], [17], and further investigated in [9]. In view of applications to the theory of computation, it was observed in §4.6 of [15] that it would be useful to replace characters given by commutative algebra homomorphisms φ : H → R from the Hopf algebra to a Rota-Baxter algebra, with characters ψ : H → S with values in a min-plus semiring, satisfying ψ(xy) = ψ(x) + ψ(y) = ψ(x) ⊙ ψ(y).…”

“…Following §4.6 of [15], we consider a Hopf algebra of "flow charts" for computation, namely graphs endowed with acyclic orientations, so that the flow through the graph, from the input vertices to the output vertices, represents the structure of a computation. Vertices are decorated by elementary operations on partial recursive functions and edges are decorated by partial recursive functions that are inputs and outputs of the vertex operations, see [15], [16], and see also the discussion in [9] on generalizations of Manin's Hopf algebra of flow charts. The computation associated to a graph Γ depends on a set of parameters.…”

Graph Grammars, Insertion Lie Algebras, and Quantum Field Theory

“…This provides a new class of examples of graph languages, in addition to those arising in the context of computer science (such as FFT networks, Petri nets, distributed parallelism), see the articles in [11] for several examples. Relations between the formalism of algebraic renormalization in quantum field theory and aspects of the theory of computation in theoretical computer science have already been investigated in [16], see also the formulation of Dyson-Schwinger equations in the Hopf algebra of flow charts in [7]. it would be interesting to see if a theory of Dyson-Schwinger equations can be formulated for Graph Languages, using the Lie theoretic approach of [9].…”