A topological view of compositionality of process algebra Emanuela - - PowerPoint PPT Presentation

A topological view of compositionality

• f process algebra

Emanuela Merelli University of Camerino

OPTC 2017, Wien 26-29th July 2016

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

We are living in a non-flat world

Flammarion 1888

… for some aspects it’s unknown … perhaps some

• f its

symmetries have been missed in concurrency theory

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

25th of PAs, 2005

Flavio Corradini’s Conjecture

Open Problem: find general sufficient conditions ensuring finite axiomatizability of bisimilarity over process algebras

Introduction The General Setting A menagerie of open problems Call to arms Equalities Between Programs Abstracting the Problem: Finite Axiomatizations

Finite, Complete Axiomatizations

The Challenge Given some algebraic signature Σ, and some congruence ∼ over (closed) terms Is there a finite set E of Σ-equations s = t such that t ∼ u ⇔ E ` t = u for all (closed) Σ-terms t, u? E is called a sound and (ground-)complete axiomatization.

Luca Aceto (Reykjavik University) Equational Logic of Processes: Open Problems 4 / 19

27 Jos Baeten's farewell afternoon Eindhoven 12.10.2010

Our Conjecture

! ! ! ! ! !

The set of regular expressions (without 0) with hnewp is the largest language for which bisimulation admits a finite equational axiomatization. Consider: E= (a + 1)* and F= (a(a(…(a(a+1)+1)…)+1)+1)*

p times, p prime 1 1 1 1 1 1 1 a a a a a a a a a a

~

Dotted lines being for equivalent states and 1- labelled arrows mean that the source states have the e.w.p. Main result: In [1], Flavio et al. define a “class of recursive specifications” that corresponds exactly to the “class of regular expressions”, the well behave specifications [1] J. Baeten, F. Corradini, C. A. Grabmayer. A Characterization of Regular Expressions under Bisimulation, Journal of the ACM, 2007 Remark 1: well-behaved specifications may be useful for proving completeness of Salomaa’s inference systems without axioms X(Y+Z)=XY+XZ and X0=0 but with axiom 0X=0 up to bisimulation. If star expressions do not possess the hnewp then the axiom system is no more complete Remark 2: Non-existence of a finite equational axiomatization for BPA with Kleene star and the empty process

OPCT 2014

Luca Aceto’s challenge

Closed for a universal algebras-like over a metric space (QAs) by: Radu Mardare, Prakash Panangaden, Gordon Ploktin. On the Axiomatizability of Quantitative Algebras. LICS 2017

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

Open problems

• 1. Can a topological setting help to find a

complete axiomatisation for a formal theory of regular events modulo bisimulation?

• 2. Can a topological setting help to find general

sufficient conditions ensuring finite axiomatizability

• f bisimilarity over process algebras?

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

What is topology?

Topology is the geometry of a shape, it deals with with qualitative geometric information of a space, such as connectivity, classification of loops and higher dimensional manifolds, invariants. Algebraic topology is a branch of mathematics that uses algebraic tools to study topological spaces, a set of points and for each point a set neighbourhoods, both satisfying a set of axioms.
 Its goals is to find algebraic invariants that classify topological spaces up to some homeomorphism

• r homotopy equivalences.

Two topological spaces are homeomorphic if a continuous map can deform, without cuts and strains, one to the other maintaining same topological invariants (e.g. coffe mag and donut) In a discrete setting a full information about topological spaces is inherent in their simplicial representation, a piece-wise linear, combinatorially complete, discrete realization of functoriality.

Example: What is the shape of the data? Problem: Discrete points have trivial topology.

Background

A simplicial complex is built from points, edges, triangular faces, etc. Homology counts components, holds, voids, etc.

0-simplex 1-simplex 2-simplex 3-simplex (solid) example of a simplicial complex hole void

Homology of a simplicial complex is computable via linear algebra.

! Idea: Connect nearby points, build a simplicial complex.

• 1. Choose

a distance !.

Problem: How do we choose distance !?

• 2. Connect

pairs of points that are no further apart than !.

• 3. Fill in

complete simplices.

• 4. Homology detects the hole.

! Idea: Connect nearby points.

• 1. Choose

a distance !.

Problem: A graph captures connectivity, but ignores higher-order features, such as holes.

• 2. Connect

pairs of points that are no further apart than !.

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

Persistent Homology

Persistent homology is an algebraic method for discerning topological features of space of data

e.g. components, graph structure holes set of discreet points

movie by Matthew L. Wright

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

The Group of symmetries

We apply the label symmetric to anything which remains invariant under some transformations.

• Flavio’s conjecture can be formulated as the statement for which bisimulation is a process that manipulated data

characterised by a particular symmetry: a binary relation between possible state transitions of two systems, such that if one system simula the other, also the viceversa holds.

• The incompleteness of the set of axioms leads to the Goedel theorem, propositions that are not decidable can be

evaluated by extending the set of operators. This implies the extension of the way to construct the process algebra.

• We propose to represent process algebra as path algebra using “quivers” (a direct multigraph). In such a way they

can be exponentiated to a group, Gp, that in this case is finite with a finite presentation and can be interpreted as an automata (more general a Turing machine).

• In the topological field theory, this group is an element of a fiber of a fiver bundle, a G-bundle with a symmetry

imposed and defined by a gauge group, G

• To construct a gauge theory we need 4 ingredients: 1) a base space, the space of object involved in the dynamics;

2) the group G of transformations, a sort of coordinate system that for each point associate both global and local properties; 3) a representation of G through the fiber, in fact a point in the fiber represents the field in the point where the finer is attached; 4) a field action generates the dynamics of the system

data space simplicial complex base space

S

B A

direct transformation

GMC Gp

Gp

topological data relation fiber bundle + field

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

The TFTD consists of four main steps:

• 1. embedding data space into a combinatorial

topological object, a simplicial complex;

• 2. considering the complex as base space of a

(block) fiber bundle;

• 3. assuming a field action, which has a free part,

the combinatorial Laplacian over the simplicial complex, and an interaction part depending on the process algebra;

• 4. constructing the gauge group as semi-direct

product of the group generated by the algebra of processes (the fibers) and the group of (simplecio- morphisms modulo isotopy) of the data space. Emergent features of data-represented complex systems were shown to be expressed by the correlation functions of the field theory.

SM

B A

direct transformation

GMC Gp

Gp

topological data field relation patterns fiber bundle + field action

topological data field theory

• M. Rasetti, E. Merelli Topological field theory of data: mining data beyond complex networks, Cambridge University Press, 2016

Emanuela Merelli, University of Camerino Berkeley, 8 December 2016

• S is the space of states
• Each state is defined by a vector that moves over S driven

by a dynamical system

• If the dynamics moves the vector towards a boundary, we

can say that there is a deadlock

• This happens because S has not been defined globally. In

fact the boundary breaks the translational symmetry

• If we allow the boundary to disappear by adding an extra-

relation, global in nature, we obtain a global topology that is not trivial

• In the graphical example we add two relations among the

generators of the manifold


A C B D

𝛥down 𝛥up

b a

A C D B

S

|v>

|v’>

A C D B

S

|v>

A C D B

S

|v> |v’>

Topological Interpretation of Dynamics of a System

A C B D

S

𝛥up 𝛥down

|v> |v’>

Emanuela Merelli, University of Camerino OPTC, 26-29 June 2017

The ′process interpretation′ scheme of P in P, a topological space in the sense of Groethendick, is indeed nothing but a quiver Q (or, more generally, a set of quivers, over some arbitrary ring κ). Associate to quiver Q its ′natural′ path algebra A ≡ PkQ, i.e., the path algebra of which Q is the basis. The structure is simpler and elegant because space P has an underlying natural formal language (that generates in general a subgroup of the much wilder group of all possible homeomorphisms of P(P))

Topological Interpretation of Processes

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

A quiver Q is a direct graph. Q =(Q0, Q1, s, e), where Q0 is a set

• f vertices (states) and Q

1 a set of arrows (transitions) and

s,e:Q

1➝Q 0, are maps. 


Given an arrow a∈Q

1 with a: i➝j for i,j∈Q 0


When s(a)=e(a), arrow a is said to be a loop.

The process quiver Q represents a system behavior.
 Behavioural equivalences is characterized by homotopy: two paths in process quivers, are distinguished by some homotopic equivalence on the set of continuous maps.

Q: a b

1

b b a a Q:

1 2

Q: a b

1 2 3

A quiver with relations is a pair (Q,R),

An path in a quiver Q is either an ordered composition of arrows 
 p = a1 a2 …an with e(at) = s(at+1) for 1≤ t <n

• r the symbol vi for i∈Q0

A path p that starts and ends at the same vertex is a cycle. Loops are cycles.

paths: v1, v2, v3, a, b, a b paths: v1, a*, b*, (a+b)* paths: v1, v2, a*, b*, (a+b)*

• 1. define a processes as Quivers Q

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

• 2. associate a natural path algebra PkQ to given Q
• Let k be a field. The path

algebra PkQ of the quiver Q is defined to be the k-vector space generated by all paths in Q. The composition (product) of two paths is induced by simple concatenation of paths if it exists, and zero otherwise. Q is the basis of PkQ.

• PkQ is unitary if Q0 is finite

Q: a b

1 2 3

Q=({1,2,3},{a,b},s,e)

{v1,v2,v3, a, b, ab} is the basis for the PkQ Q: a b

1

{v1, a*, b*, a*b*, (a+b)*} is the basis for the PkQ

PQR is a path co-algebra of quivers with relations

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

• 4. define L in Hopf algebra H

Turn L into a Hopf algebra H, equipping it in particular with a a coalgebra

• 5. redefine the theorem G in H and prove that holds in H and not in P
• 3. identify the Lie algebra L given by P

Identify the Lie algebra L arising from PkQ

Theorem G:
 The SHC [Star Height Conjecture] is a topological application to the space P, generated by the formal representation

• f a (any) process P.

Corradini’s Star Heigh conjecture:
 the set of regular expressions (without 0) with hnewp is the largest language for which bisimulation admits a finite equational axiomatization.

• Def. hnewp structural property:

1. each *-behaviour must avoid to enter in a pure cycle, 2. each cycle must be of the form

E*=1+EE* 
 E*F ➝ X=EX + F

3. in *-behaviour a*a≄ aa*

Regular Epressions

Well-Behaved Specifications Recursive Specifications

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

Save our History and Research After the recent earthquakes in Central Italy, the research and historical heritage of the University of Camerino, one of the worlds’ oldest research institutions, is in danger. With your help we can save our history, art, and research.

Emanuela Merelli, University of Camerino OPTC, 26-29 June 2017 Oltre in confine, Salone Int. del libro, Torino 2017

thanks!