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

A topological view of compositionality of process algebra Emanuela Merelli University of Camerino OPTC 2017, Wien 26-29th July 2016

We are living in a non-flat world … for some aspects it’s unknown … perhaps some of its symmetries have been missed in concurrency theory Flammarion 1888 Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

25th of PAs, 2005 OPCT 2014 Flavio Corradini’s Conjecture Luca Aceto’s challenge Our Conjecture Introduction The General Setting Equalities Between Programs A menagerie of open problems Abstracting the Problem: Finite Axiomatizations Call to arms Finite, Complete Axiomatizations ! ! The set of regular expressions (without 0) with hnewp is the largest ! language for which bisimulation admits a finite equational ! ! axiomatization. a 1 1 ! The Challenge a Consider: 1 a Given some algebraic signature Σ , and some congruence ∼ over a E= (a + 1)* and 1 (closed) terms a ~ a a Is there a finite set E of Σ -equations s = t such that F= (a(a(…(a(a+1)+1)…)+1)+1)* 1 a p times, p prime a t ∼ u E ` t = u ⇔ 1 a for all (closed) Σ -terms t , u? Dotted lines being for 1 � equivalent states � and 1- labelled arrows mean that E is called a sound and (ground-)complete axiomatization. the source states have the e.w.p. Eindhoven 12.10.2010 Jos Baeten's farewell afternoon 27 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 Luca Aceto (Reykjavik University) Equational Logic of Processes: Open Problems 4 / 19 [1] J. Baeten, F. Corradini, C. A. Grabmayer. A Characterization of Regular Expressions under Bisimulation, Journal of the ACM, 2007 Open Problem: find general sufficient conditions ensuring finite axiomatizability of bisimilarity over process algebras 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. Closed for a universal algebras-like over a metric space (QAs) by: Radu Mardare, Prakash Panangaden, If star expressions do not possess the hnewp then the axiom system is no more complete Gordon Ploktin. On the Axiomatizability of Quantitative Algebras. LICS 2017 Remark 2: Non-existence of a finite equational axiomatization for BPA with Kleene star and the empty process 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 of 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 or 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. Idea : Connect nearby points. Idea : Connect nearby points, build a simplicial Background complex. Example : What is the shape of the data? A simplicial complex is built from points, edges, triangular faces, etc. 1. Choose 2. Connect 1. Choose 2. Connect 3. Fill in a distance pairs of points a distance pairs of points complete example of a ! . that are no 0 -simplex 1 -simplex 2 -simplex 3 -simplex that are no simplices. ! . simplicial complex (solid) further apart further apart than ! . Homology counts components, holds, voids, etc. than ! . ! ! Homology of a simplicial complex is computable Problem : Discrete points have trivial topology. via linear algebra. void 4. Homology detects the hole. hole (contains faces but Problem : How do we choose distance ! ? Problem : A graph captures connectivity, but ignores higher-order empty interior) features, such as holes. 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 set of discreet points e.g. components, graph structure holes ! ! 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 direct transformation A B G p G p G MC < S fiber bundle + field simplicial complex relation data space topological data base space Emanuela Merelli, University of Camerino OPCT 26-29 June 2017

topological data field theory The TFTD consists of four main steps: 1. embedding data space into a combinatorial direct transformation topological object, a simplicial complex ; A B 2. considering the complex as base space of a G p G p (block) fiber bundle ; G MC 3. assuming a field action, which has a free part, < the combinatorial Laplacian over the simplicial S M 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- fiber bundle + field action morphisms modulo isotopy) of the data space. relation patterns Emergent features of data-represented complex topological data field systems were shown to be expressed by the correlation functions of the 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 OPCT 26-29 June 2017

Topological Interpretation of Dynamics of a System • S is the space of states A A C C • Each state is defined by a vector that moves over S driven S S by a dynamical system • If the dynamics moves the vector towards a boundary, we |v> |v> |v’> can say that there is a deadlock • This happens because S has not been defined globally. In B D B D fact the boundary breaks the translational symmetry A C 𝛥 up S • If we allow the boundary to disappear by adding an extra- relation, global in nature, we obtain a global topology that is not trivial C A |v’> |v> S • In the graphical example we add two relations among the generators of the manifold 
 B D |v> |v’> 𝛥 down B D A B C D b 𝛥 up 𝛥 down a Emanuela Merelli, University of Camerino Berkeley, 8 December 2016

Topological Interpretation of Processes 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 ≡ P kQ , 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 )) Emanuela Merelli, University of Camerino OPTC, 26-29 June 2017