We describe how the SgpDec computer algebra package can be used for composing and decomposing permutation groups and transformation semigroups hierarchically by directly constructing substructures of wreath products, the so called cascade products.Final Accepted Versio

C. Wells

H.P. Zeiger

K. Krohn

M. Krasner

English

arXiv

We describe how the SgpDec computer algebra package can be used for composing and decomposing permutation groups and transformation semigroups hierarchically by directly constructing substructures of wreath products, the so called cascade products

Egri-Nagy, Attila (R17046)

Mitchell, James D.

Nehaniv, Chrystopher L.

Western Sydney ResearchDirect

SgpDec : cascade (De)compositions of finite transformation semigroups and permutation groups

Egri-Nagy, Attila

Nehaniv, C.L.

University of Hertfordshire Research Archive

SGPDEC: CASCADE (DE)COMPOSITIONS OF FINITE TRANSFORMATIONSEMIGROUPS AND PERMUTATION GROUPSATTILA EGRI-NAGY1,3, JAMES D. MITCHELL2, AND CHRYSTOPHER L. NEHANIV3Abstract. We describe how the SgpDec computer algebra package can be used for composing anddecomposing permutation groups and transformation semigroups hierarchically by directly constructingsubstructures of wreath products, the so called cascade products. transformation semigroup, permuta-tion group, wreath product, Krohn-Rhodes Theory1. IntroductionWreath products are widely used theoretical constructions in group and semigroup theory wheneverone needs to build a composite structure with hierarchical relations between the building blocks. However,from a computational and engineering perspective they are less useful since wreath products are subjectto combinatorial explosions and we are often interested only in substructures of them. Cascade productsprecisely build these substructures by defining the hierarchical connections explicitly. As input, givena group or a semigroup with unknown internal structure, the goal of cascade decomposition algorithmsis to come up with a list of simpler building blocks and put them together in a cascade product, whichrealizes in some sense the original group or semigroup. Roughly speaking, for permutation groups, cascadeproduct decompositions can be interpreted as putting the inner workings of the Schreier-Sims algorithm(generalized to any subgroup chain) into an external product form, therefore one can build cascadeproducts isomorphic to the group being decomposed. For semigroups, Krohn-Rhodes decompositions [14]can be computationally represented by cascade products of transformation semigroups.In this paper we describe how the Gap [9] package SgpDec [6] implements cascade products anddecomposition algorithms and we also give a few simple example computations. This description of thepackage only focuses on the core functionality of the package.A transformation is a function f : X → X from a set to itself, and a transformation semigroup (X,S)of degree n is a collection S of transformations of X closed under function composition, |X| = n. Incase S is a group of permutations of X, we call (X,S) a permutation group. Using automata theoryterminology sometimes we call X the state set, often represented as a set of integers n = {0, . . . , n− 1}.We write xs to denote the new state resulting from applying a transformation s ∈ S to a state x ∈ X.2. Cascade Product by a Motivating ExampleTo motivate the definition of the cascade product, we consider how the mod-4 counter, the cyclicpermutation group (4,Z4), can be constructed from two mod-2 counters. The direct product Z2 × Z2contains no element of order 4. Since Aut(Z2) is trivial there is only one semidirect product of Z2 andZ2, which equals their direct product. Their wreath product, Z2 o Z2 ∼= D4, the dihedral group of thesquare can be used to emulate a mod-4 counter, since Z4 ↪→ D4. But this construction is not efficient,beyond the required rotations the dihedral group has the flip-symmetry as well, doubling the size of thegroup. However, we would like to have a product construction that is isomorphic to (4,Z4).This motivates the definition of cascade products: efficient constructions of substructures of wreathproducts, induced by explicit dependency functions [5]. Essentially, cascade products are transformationsemigroups glued together by functions in a hierarchical tree. More precisely, let((X1, S1), . . . , (Xn, Sn))be a fixed list of transformation semigroups, and dependency functions of the formdi : X1 × . . .×Xi−1 → Si, for i ∈ {1, . . . , n}.1arXiv:1501.03217v1  [math.GR]  14 Jan 20152 ATTILA EGRI-NAGY1,3, JAMES D. MITCHELL2, AND CHRYSTOPHER L. NEHANIV3s1 = d1(∅) d2 d3s2 = d2(x1)s3 = d3(x1, x2)x1(X1, S1)x2(X2, S2)x3(X3, S3)(X,S)xs11(X1, S1)xs22(X2, S2)xs33(X3, S3)(X,S)Figure 1. Action in a cascade product of components [(X1, S1), (X2, S2), (X3, S3)]. Thecurrent state (x1, x2, x3) (top) is transformed to the new state (xs11 , xs22 , xs33 ) (bottom)by the transformation cascade (d1, d2, d3). The component actions si are calculated byevaluating the dependency functions of (d1, d2, d3) on the states of the components above.The evaluations are highlighted and they happen at the same time. The dependencies,where the state information travels, are denoted by dotted lines.A transformation cascade is then defined to be an n-tuple of dependency functions (d1, . . . , dn), where diis a dependency function of level i. If no confusion arises, on the top level we can simply write d1 ∈ S1instead of d1(∅) ∈ S1. The cascade action is defined coordinatewise by xdi(x1,...,xi−1)i , applying the resultsof the evaluated dependency functions (see Fig. 1), so that the cascade product can be regarded as aspecial transformation representation on the set X1 × . . . × Xn. The hierarchical structure allows usto conveniently distribute computation among the components (Xi, Si), and perform abstractions andapproximations of the system modelled as a cascade product. Then if W is a set of transformationcascades (X1, S1) oW · · · oW (Xn, Sn) denotes the transformation semigroup (X1 × · · ·Xn, 〈W 〉), where〈W 〉 is the semigroup of transformation cascades generated by W .We can construct (4,Z4) exactly by using two copies of (2,Z2). The generator set contains only onepermutation cascade W = {(+1, c)}, where +1 is the generator of Z2 and c is a dependency functionmapping 2 to Z2 with c(0) = 1Z2 , and c(1) = +1. The first dependency is a constant (incrementmodulo 2) while the second dependency implements the carry. Therefore, with fewer dependencies thanrequired by the wreath product, the mod-4 counter can be realized by an isomorphic cascade product:(2,Z2) oW (2,Z2) ∼= (4,Z4), see Fig. 2.SGPDEC 30c(0)1 topc(1)0 1 bottom+11Z2+11Z21Z2 1Z2+1+1Figure 2. Two mod-2 counters cascaded together to build a mod-4 counter.An immediate consequence of the generality of the cascade product is that several well-known con-structions are special cases of the cascade product, and as such they are easy to implement. Directproducts consist of all d = (d1, . . . , dn) with each di constant. Wreath products consist of all possibledependency functions. Direct, cascade, and wreath products constructions for transformation semigroupsare now available in SgpDec, and iterated wreath products for permutation groups also became a bitmore convenient to define.3. FunctionalityThere are two different basic ways of using the SgpDec package. Depending on whether the startingpoint is a complex structure or a set of (simple) building blocks, we can do decomposition or composition.3.1. Composition and Construction. The questions we aim to answer by constructing cascade prod-ucts can be of the following types.(1) What is the (semi)group generated by a given set of transformation cascades?(2) What can be built from a given set of (simple) components?The usual scenario is that for a list of components we give a set of cascades as a generating set. Forinstance, the quaternion group Q = 〈i, j〉 is not a semidirect product, but it embeds into the full cascadeproduct (2,Z2) o (2,Z2) o (2,Z2), a group with 128 elements. Therefore, it can be built from copiesof Z2. The dependency functions can only have two values, thus to define cascade permutations it isenough to give only those arguments that give +1 (the generator of Z2). A cascade permutation realizingi is defined by the dependency functions (d1, d2, d3) where d2(0) = d2(1) = d3(0, 0) = d3(1, 1) = +1and all other arguments map to the identity. Similarly, a cascade realizing j is defined by (d′1, d′2, d′3)where d′1(∅) = d′3(0, 0) = d′3(0, 1) = +1, (see Fig. 3, note that the state values are shifted by 1). Onecan check that these two order 4 elements generate the 8-element quaternion group Q. Therefore byW = {(d1, d2, d3), (d′1, d′2, d′3)} we have(Q,Q) ∼= (2,Z2) oW (2,Z2) oW (2,Z2).4 ATTILA EGRI-NAGY1,3, JAMES D. MITCHELL2, AND CHRYSTOPHER L. NEHANIV3(1,2)1(1,2)2(1,2)1 2 1(1,2)2(1,2)1 2(1,2)1(1,2)2 1 2Figure 3. Generators of a cascade representation of the quaternion group in a tree form.The edge labels are states, while the nodes contain the action. Empty node correspondsto the identity. The gray part of the tree is fixed.[\small]gap> Z2:=CyclicGroup(IsPermGroup,2);Group([ (1,2) ])gap> d:=Cascade([Z2,Z2,Z2],[[[1],(1,2)],[[2],(1,2)],[[1,1],(1,2)],[[2,2],(1,2)]]);<perm cascade with 3 levels with (2, 2, 2) pts, 4 dependencies>gap> dprime:=Cascade([Z2,Z2,Z2],[[[],(1,2)],[[1,1],(1,2)],[[1,2],(1,2)]]);<perm cascade with 3 levels with (2, 2, 2) pts, 3 dependencies>gap> StructureDescription(Group([d,dprime]));"Q8"3.2. Decomposition.(1) What are the basic building blocks of a given (semi)group?(2) How can we represent it as a cascade product?A typical scenario is that for a given composite semigroup or group we choose a decomposition algo-rithm which returns a cascade product.3.2.1. Frobenius-Lagrange Decomposition. In the case of groups the decomposition uses the idea behindinduction in representation theory (see e.g. [1]), so it traces back to Frobenius. Indeed, a special case ofthem comprises the well-known Krasner-Kaloujnine embeddings [13]. All we need here is just standardgroup theory, namely the action on cosets, hence the name Frobenius-Lagrange Decomposition.How would someone come up with the generators cascades of the quaternion group in Section 3.1?The easiest solution is to use this group decomposition.[\small]gap> Q := QuaternionGroup(IsPermGroup,8);Group([ (1,5,3,7)(2,8,4,6), (1,2,3,4)(5,6,7,8) ])gap> CQ := FLCascadeGroup(Q);<cascade group with 2 generators, 3 levels with (2, 2, 2) pts>The actual implementation takes a subgroup chain as input (chief series by default) and form thecomponents by examining the coset space actions derived from the chain. Therefore, the decompositionmethod can be considered as a generalized Schreier-Sims algorithm [12].Coordinatewise calculation in a cascade product can also be thought of as a sequence of refiningapproximate solutions. For instance, each completed step of an algorithm for solving the Rubik’s CubeSGPDEC 5corresponds to calculating the desired value at a hierarchical level of some cascade product representationand it gives a configuration ‘closer’ to the solved state.C21234{1,2,3,4}{2,3,4}[ 1 ]{1,3}[ 2 ]{3,4}[ 2, 1 ]{2,4}[ 1 ]{3}[ 1, 1, 2 ]{1}[ 1, 1, 2, 2 ]{4}[ 2, 1 ]{2}Figure 4. Tiling picture – the internal details of the holonomy decomposition of thetransformation semigroup T generated by t1 and t2. The numbers on the left denote thehierarchical levels (level 4 consists of singleton sets and it is needed by the holonomyalgorithm but not a component of the cascade product). Outer boxes contain subsetsthat are mutually reachable from each other under the semigroup action. The arrowsindicate how a subset is ‘tiled’ by its subsets, the arrow labels contain words (sequencesof generators) that take a subset to one of its tiles. Dotted arrow means the tile is not animage. Roughly, the holonomy algorithm finds the components by checking the actionof the semigroup on a set of tiles.3.2.2. Holonomy Decomposition. For transformation semigroups the holonomy method [4,7,10,11,15–17]is used. The holonomy decomposition works by a close analysis of how the semigroup acts on those subsetsof the state set which are images of the state set. As a small example let’s define T as the transformationsemigroup generated by t1 = ( 1 2 3 43 2 4 4 ) and t2 = (1 2 3 43 3 1 3 ). Calculating its holonomy decomposition anddisplaying some information can be done by the following commands:6 ATTILA EGRI-NAGY1,3, JAMES D. MITCHELL2, AND CHRYSTOPHER L. NEHANIV3[\small]gap> T:=Semigroup([Transformation([3,2,4,4]),Transformation([3,3,1,3])]);<transformation semigroup on 4 pts with 2 generators>gap> HT := HolonomyCascadeSemigroup(T);<cascade semigroup with 2 generators, 3 levels with (2, 2, 4) pts>gap> DisplayHolonomyComponents(SkeletonOf(HT));1: 22: 23: (2,C2) 2The displayed information tells us that this 13-element semigroup can be realized as the cascade productof four copies of the transformation monoid of constant maps of two points and one instance of Z2. Thecomponents are put together in a 3-level cascade product.Holonomy decompositions are useful whenever a finite state-transition model of some process needs tobe analyzed (e.g. [3]).3.3. Visualization. SgpDec uses GraphViz, a widely used graph drawing package [8], for visualisationpurposes. The underlying idea is that a function generates source code in the dot language for the givenmathematical object. Then the actual figure can be generated separately to be included in papers, orusing the Viz package [2] immediately displayed on screen from the Gap command line. Figure 3 and 4were both auto-generated using GraphViz.AcknowledgmentThe work reported in this article was funded in part by the EU project BIOMICS, contract numberCNECT-318202. This support is gratefully acknowledged.References[1] J. L. Alperin and Rowen B. Bell. Groups and Representations. Springer, 1995.[2] Manuel Delgado, Attila Egri-Nagy, James D. Mitchell, and Markus Pfeiffer. VIZ – GAP package for visualisation,2014. https://bitbucket.org/james-d-mitchell/viz.[3] Paolo Dini, Chrystopher L. Nehaniv, Attila Egri-Nagy, and Maria J. Schilstra. Exploring the concept of interactioncomputing through the discrete algebraic analysis of the Belousov-Zhabotinsky reaction. Biosystems, 112(2):145 – 162,2013. Selected papers from the 9th International Conference on Information Processing in Cells and Tissues.[4] Pa´l Do¨mo¨si and Chrystopher L. Nehaniv. Algebraic Theory of Finite Automata Networks: An Introduction, volume 11of SIAM Series on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics, 2005.[5] Attila Egri-Nagy and Chrystopher L. Nehaniv. Cascade Product of Permutation Groups. arXiv:1303.0091v3 [math.GR],2013.[6] Attila Egri-Nagy, Chrystopher L. Nehaniv, and James D. Mitchell. SgpDec – software package for hierarchical decom-positions and coordinate systems, Version 0.7+, 2014. http://sgpdec.sf.net.[7] Samuel Eilenberg. Automata, Languages and Machines, volume B. Academic Press, 1976.[8] John Ellson, Emden R. Gansner, Eleftherios Koutsofios, Stephen C. North, and Gordon Woodhull. Graphviz anddynagraph static and dynamic graph drawing tools. In GRAPH DRAWING SOFTWARE, pages 127–148. Springer-Verlag, 2003.[9] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.7.5, 2014. www.gap-system.org.[10] Abraham Ginzburg. Algebraic Theory of Automata. Academic Press, 1968.[11] W. M. L. Holcombe. Algebraic Automata Theory. Cambridge University Press, 1982.[12] Derek Holt, Bettina Eick, and Eamonn O’Brien. Handbook of Computational Group Theory. CRC Press, 2005.[13] Marc Krasner and Leo Kaloujnine. Produit complet des groupes de permutations et proble`me d’ extension de groupes.Acta Scientiarium Mathematicarum (Szeged), 14:39–66, 1951.[14] Kenneth Krohn and John Rhodes. Algebraic Theory of Machines. I. Prime Decomposition Theorem for Finite Semi-groups and Machines. Transactions of the American Mathematical Society, 116:450–464, April 1965.[15] Charles Wells. A Krohn-Rhodes theorem for categories. Journal of Algebra, 64:37–45, 1980.[16] H. Paul Zeiger. Cascade synthesis of finite state machines. Information and Control, 10:419–433, 1967. plus erratum.[17] H. Paul Zeiger. Cascade Decomposition Using Covers. In Michael A. Arbib, editor, Algebraic Theory of Machines,Languages, and Semigroups, chapter 4, pages 55–80. Academic Press, 1968.SGPDEC 7Centre for Research in Mathematics, School of Computing, Engineering and Mathematics, University ofWestern Sydney, Australia, and School of Mathematics and Statistics, University of St Andrews, UnitedKingdom, and Centre for Computer Science & Informatics Research, University of Hertfordshire, UnitedKingdomE-mail address: A.Egri-Nagy@uws.edu.au,jdm3@st-and.ac.uk,C.L.Nehaniv@herts.ac.uk

A Krohn-Rhodes theorem for categories.

Algebraic Automata Theory.

Algebraic Theory of Automata.

Algebraic Theory of Finite Automata Networks: An Introduction,

Algebraic Theory of Machines. I. Prime Decomposition Theorem for Finite Semigroups and Machines.

Cascade Decomposition Using Covers. In

Cascade Product of Permutation Groups.

Cascade synthesis of finite state machines.

Exploring the concept of interaction computing through the discrete algebraic analysis of the Belousov-Zhabotinsky reaction.

Graphviz and dynagraph static and dynamic graph drawing tools.

Groups and Representations.

Produit complet des groupes de permutations et proble`me d’ extension de groupes. Acta Scientiarium Mathematicarum

SgpDec – software package for hierarchical decompositions and coordinate systems, Version 0.7+,

VIZ – GAP package for visualisation,

SgpDec : Cascade (de)compositions of finite transformation semigroups and permutation groups

Attila Egri-Nagy

James D. Mitchell

Chrystopher L. Nehaniv

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SgpDec: Cascade (De)Compositions of Finite Transformation Semigroups and Permutation Groups

SgpDec : Cascade (de)compositions of finite transformation semigroups and permutation groups

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