Process identification through modular neural networks and rule extraction

Monolithic neural networks may be trained from measured data to establish knowledge about the process. Unfortunately, this knowledge is not guaranteed to be found and - if at all - hard to extract. Modular neural networks are better suited for this purpose. Domain-ordered by topology, rule extraction is performed module by module. This has all the benefits of a divide-and-conquer method and opens the way to structured design. This paper discusses a next step in this direction by illustrating the potential of base functions to design the neural model. \ud [Full paper published as: Berend Jan van der Zwaag, Kees Slump, and Lambert Spaanenburg. Process identification through modular neural networks and rule extraction. In Proceedings FLINS-2002, Ghent, Belgium, 16-18 Sept. 2002.

Variable binding, symmetric monoidal closed theories, and bigraphs

This paper investigates the use of symmetric monoidal closed (SMC) structure for representing syntax with variable binding, in particular for languages with linear aspects. In our setting, one first specifies an SMC theory T, which may express binding operations, in a way reminiscent from higher-order abstract syntax. This theory generates an SMC category S(T) whose morphisms are, in a sense, terms in the desired syntax. We apply our approach to Jensen and Milner's (abstract binding) bigraphs, which are linear w.r.t. processes. This leads to an alternative category of bigraphs, which we compare to the original.Comment: An introduction to two more technical previous preprints. Accepted at Concur '0

Modular invariants from subfactors

In these lectures we explain the intimate relationship between modular invariants in conformal field theory and braided subfactors in operator algebras. A subfactor with a braiding determines a matrix $Z$ which is obtained as a coupling matrix comparing two kinds of braided sector induction ("alpha-induction"). It has non-negative integer entries, is normalized and commutes with the S- and T-matrices arising from the braiding. Thus it is a physical modular invariant in the usual sense of rational conformal field theory. The algebraic treatment of conformal field theory models, e.g. $SU(n)_k$ models, produces subfactors which realize their known modular invariants. Several properties of modular invariants have so far been noticed empirically and considered mysterious such as their intimate relationship to graphs, as for example the A-D-E classification for $SU(2)_k$. In the subfactor context these properties can be rigorously derived in a very general setting. Moreover the fusion rule isomorphism for maximally extended chiral algebras due to Moore-Seiberg, Dijkgraaf-Verlinde finds a clear and very general proof and interpretation through intermediate subfactors, not even referring to modularity of $S$ and $T$. Finally we give an overview on the current state of affairs concerning the relations between the classifications of braided subfactors and two-dimensional conformal field theories. We demonstrate in particular how to realize twisted (type II) descendant modular invariants of conformal inclusions from subfactors and illustrate the method by new examples.Comment: Typos corrected and a few minor changes, 37 pages, AMS LaTeX, epic, eepic, doc-class conm-p-l.cl

Modular Invariants from Subfactors: Type I Coupling Matrices and Intermediate Subfactors

A braided subfactor determines a coupling matrix Z which commutes with the S- and T-matrices arising from the braiding. Such a coupling matrix is not necessarily of "type I", i.e. in general it does not have a block-diagonal structure which can be reinterpreted as the diagonal coupling matrix with respect to a suitable extension. We show that there are always two intermediate subfactors which correspond to left and right maximal extensions and which determine "parent" coupling matrices Z^\pm of type I. Moreover it is shown that if the intermediate subfactors coincide, so that Z^+=Z^-, then Z is related to Z^+ by an automorphism of the extended fusion rules. The intertwining relations of chiral branching coefficients between original and extended S- and T-matrices are also clarified. None of our results depends on non-degeneracy of the braiding, i.e. the S- and T-matrices need not be modular. Examples from SO(n) current algebra models illustrate that the parents can be different, Z^+\neq Z^-, and that Z need not be related to a type I invariant by such an automorphism.Comment: 25 pages, latex, a new Lemma 6.2 added to complete an argument in the proof of the following lemma, minor changes otherwis

Generalized Connectives for Multiplicative Linear Logic

In this paper we investigate the notion of generalized connective for multiplicative linear logic. We introduce a notion of orthogonality for partitions of a finite set and we study the family of connectives which can be described by two orthogonal sets of partitions. We prove that there is a special class of connectives that can never be decomposed by means of the multiplicative conjunction ? and disjunction ?, providing an infinite family of non-decomposable connectives, called Girard connectives. We show that each Girard connective can be naturally described by a type (a set of partitions equal to its double-orthogonal) and its orthogonal type. In addition, one of these two types is the union of the types associated to a family of MLL-formulas in disjunctive normal form, and these formulas only differ for the cyclic permutations of their atoms

Integrating Evolutionary Computation with Neural Networks

There is a tremendous interest in the development of the evolutionary computation techniques as they are well suited to deal with optimization of functions containing a large number of variables. This paper presents a brief review of evolutionary computing techniques. It also discusses briefly the hybridization of evolutionary computation and neural networks and presents a solution of a classical problem using neural computing and evolutionary computing technique

Exactly Solvable Lattice Models with Crossing Symmetry

We show how to compute the exact partition function for lattice statistical-mechanical models whose Boltzmann weights obey a special "crossing" symmetry. The crossing symmetry equates partition functions on different trivalent graphs, allowing a transformation to a graph where the partition function is easily computed. The simplest example is counting the number of nets without ends on the honeycomb lattice, including a weight per branching. Other examples include an Ising model on the Kagome' lattice with three-spin interactions, dimers on any graph of corner-sharing triangles, and non-crossing loops on the honeycomb lattice, where multiple loops on each edge are allowed. We give several methods for obtaining models with this crossing symmetry, one utilizing discrete groups and another anyon fusion rules. We also present results indicating that for models which deviate slightly from having crossing symmetry, a real-space decimation (renormalization-group-like) procedure restores the crossing symmetry