General Description of Network Systems  Ugo Montanari  Dipartimento di Informatica, University of Pisa CINA: Compositionality, Interaction, Negotiation, Autonomicity, MIUR PRIN Project Final General Meeting, January 19-21, 2016, Civitanova joint work with Roberto Bruni (and others)

Networks  hypergraphs with labels, structure and observation interface  labels/buffer content may change with executions  structure usually changes only via explicit reconfiguration operations CINA, Civitanova Marche, January Ugo Montanari 2

Roadmap Networks Milner flowgraph algebras Denotational process algebras (Soft) constraint networks Networks as components & connectors Petri nets Signal flow graphs Electric circuits PROPS: product permutation categories From graphs to categories Petri nets Rewriting logic Process calculi Conclusions CINA, Civitanova Marche, January Ugo Montanari 3

Roadmap Networks Milner flowgraph algebras Denotational process algebras (Soft) constraint networks Networks as components & connectors Petri nets Signal flow graphs Electric circuits PROPS: product permutation categories From graphs to categories Petri nets Rewriting logic Process calculi Conclusions CINA, Civitanova Marche, January Ugo Montanari 4

Ugo Montanari Communication networks & routing Neural networks & connectivism Computer networks Stochastic models of the internet (Barabasi, Simon) Social networks and concept mining, complex systems Computation, semantics, economics of networks Networks of connectors & buffers 5 A Variety of Networks

Flow Graphs and Flow Algebras Robin Milner, Flow Graphs and Flow Algebras, JACM 1979 CINA, Civitanova Marche, January Ugo Montanari 6

Flow Graphs and Flow Algebras Robin Milner, Flow Graphs and Flow Algebras, JACM 1979 CINA, Civitanova Marche, January Ugo Montanari 7

CHARM, Chemical Abstract with Restriction Machine Corradini, Montanari, Rossi, An Abstract Machine for Concurrent Modular Systems: CHARM, TCS 1994 Parallel composition fuses visible nodes and hyperarcs with the same name Modelling concurrent constraint programming Modelling P/T Petri nets CINA, Civitanova Marche, January Ugo Montanari 8

Denotational Semantics CINA, Civitanova Marche, January Ugo Montanari 9  George Milne, Robin Milner, Concurrent Processes and Their Syntax, JACM 1979 Tony Hoare, Communicating Sequential Processes, CACM 1978  Interpreting the algebra in a semantic domain

Networks of (soft) Constraints CINA, Civitanova Marche, January Ugo Montanari 10  Bistarelli, Montanari, Rossi, Semiring-Based Constraint Satisfaction and Optimization, JACM 1997  Semiring definition of constraint composition  c: (V→D)→S  (c 1 x c 2 )η = c 1 η x c 2 η

11 Seite Courcelle Graph Algebras Graphs have an interface with a list of visible nodes called sources. There are basic graphs consisting of a single vertex or edge and three classes of operations on graphs:  disjoint union where source lists are concatenated  fusion of sources  permutation, duplication and restriction of sources every operation is indexed by the arities of the involved sources: thus there are infinite operations The sets of graphs of bounded treewidth are exactly those that can be constructed using only a finite number of operations (but possibly unboundedly many times). Alternatively, the sets of graphs of bounded treewidth are subsets of graph languages generated by hyperedge replacement graph grammars CINA, Civitanova Marche, January Ugo Montanari 11

12 Seite Structural Recursion on Graph Operations Inductive families of evaluations on Courcelle operations (or on graph productions) can be defined on several discrete and continuous domains. In practice, a tuple of properties must be defined on graph syntactic trees by structural recursion. It should be proved that the result of the evaluation does not depend on the particular syntactic tree of the given graph. CINA, Civitanova Marche, January Ugo Montanari 12

13 Seite Terminal Reliability An example: probability of a multiparty connection in a network all the arcs have independent probabilities of failure A subgraph connected to the rest of the graph by a set V of nodes a probability distribution assigns to each partition P of V the probability that exactly the nodes in each class of P are actually connected Inductively, given a graph production and the probability distributions for all the nonterminals in its right hand side compute the connection probability for the right hand side. Fratta, L., and Montanari, U., Terminal Reliability in a Communication Network: An Efficient Algorithm, Second International Symposium on Network Theory, Herceg-Novi, July 3-7, 1972, pp CINA, Civitanova Marche, January Ugo Montanari 13

14 Seite Back to (Almost) Milner + structural congruence: commutative monoidality of ||, α-conversion, swapping of restrictions + nominal structure CINA, Civitanova Marche, January Ugo Montanari Problem made of two subproblems p and q Assignment for x in p has already been determined Atomic subproblem of a single constraint A Empty subproblem 14 Terms up-to structural congruence are (hyper)graphs with hidden nodes: A(X) is a graph consisting of a single hyperedge and its distinct nodes

15 Seite Nominal structure CINA, Civitanova Marche, January Ugo Montanari bijective with axioms + distributivity over all other operators Permutations come with a notion of “untyped” free variables, independent on the specific algebra Signature includes permutations 15

16 Seite Secondary Optimization Problem CINA, Civitanova Marche, January Ugo Montanari Scope extension allows us to choose the order of variable elimination (secondary optimization problem) … Normal form (x)(y)(z)(v)(f 1 (x,y)+f 2 (y,z)+f 3 (z,v)+f 4 (v,x)) Canonical forms (x)(y)(f 1 (x,y)+(z)(f 2 (y,z)+(v)(f 3 (z,v)+f 4 (v,x)))) (y)(z)(f 2 (y,z)+(v)(f 3 (z,v)+(x)(f 4 (v,x)+f 1 (x,y)))) … Apply from left to right as much as you can => assign variables as soon as you can 16

Roadmap Networks Milner flowgraph algebras Denotational process algebras (Soft) constraint networks Networks as components & connectors Petri nets Signal flow graphs Electric circuits PROPS: product permutation categories From graphs to categories Petri nets Rewriting logic Process calculi Conclusions CINA, Civitanova Marche, January Ugo Montanari 17

Ugo Montanari 18 Two decades ago:  Sw systems seen as a collection of cooperating reusable components emerged as a trend Sw architectures are centered around three main kinds of elements:  processing elements (components),  data elements, and  connecting elements (connectors) Components, Data, and Connectors

Ugo Montanari Some Connectors from Literature 19

Ugo Montanari Reo by Arbab at al. channels and bounded buffers Some Connectors from Literature 20

Ugo Montanari Some Connectors from Literature BIP by Sifakis at al. 21

Ugo Montanari Span(Graph) by Katis, Sabadini and Walters Some Connectors from Literature 22

Ugo Montanari One-place buffers Some Connectors from Literature 23

Ugo Montanari Connectors can be seen as black boxes input interface output interface admissible signals on interfaces Abstract semantics is just a matrix n inputs  2 n rows m outputs  2 m columns sequential composition is matrix multiplication parallel composition is matrix expansion …  0101  0010 … …111001… Abstract Semantics

Ugo Montanari An Example: Symmetries  11  10  01  = connectors boxes are immaterial

Ugo Montanari Tiles Combine horizontal and vertical structures through interfaces initial configuration final configuration trigger effect

Ugo Montanari Tiles Compose tiles horizontally

Ugo Montanari Tiles Compose tiles horizontally (also vertically and in parallel)

Ugo Montanari 29 Sobocinski’s Nets with Boundaries Over-simplified graphical notation: transitions are not drawn Boundaries = attach points for pending arcs Composition = can fuse and multiply transitions +

PROPS: PRoduct Permutation categories The basic structure: symmetric monoidal categories sequential and parallel composition permutation of wires axiomatization = string diagram isomorphism additional connectors with axioms Petri nets: 4 binary connectors: AND, OR, coAnd, coOR and buffers most general combination of synchronization and nondeterminism coalgebraic theory for F(X) = P(A x X) bialgebraic theory: operations preserve bisimilarity standard representatives of equivalence classes for finite Petri nets Signal flow graphs => Electric circuits => CINA, Civitanova Marche, January Ugo Montanari 30

Signal Flow Graphs  Foundations of control theory  S. J. Mason. Feedback Theory: I. Some Properties of Signal Flow Graphs. Massachusetts Institute of Technology, Research Laboratory of Electronics,  Coalgebraic theory for F(X) = A x X  J. J. M. M. Rutten. A tutorial on coinductive stream calculus and signal flow graphs, TCSSci.,  PROP treatment  Bonchi, Sobocinski, Zanasi, A Categorical Semantics of Signal Flow Graphs, CONCUR 2014  a sound and complete graphical theory of vector subspaces over the field of polynomial fractions, with relational composition  buffers are derivatives in the operational calculus (e.g. via Laplace transforms)  deterministic functional vs deadlock-prone relational CINA, Civitanova Marche, January Ugo Montanari 31

Electric Circuits  John C. Baez, Brendan Fong, A Compositional Framework for Passive Linear Networks, arXiv.org  resistor, reactors and capacitors  buffers represent derivatives of the operational calculus  nodes of the network with two values  a potential (voltage of the node)  and a current (current entering/exiting the node)  again coalgebraic theory for F(X) = A x X  the temporal series of the final coalgebra  represent the temporal behavior of the circuit  similar to the functional definition  vs. temporal behavior of Fibonacci. CINA, Civitanova Marche, January Ugo Montanari 32

Roadmap Networks Milner flowgraph algebras Denotational process algebras (Soft) constraint networks Networks as components & connectors Petri nets Signal flow graphs Electric circuits PROPS: product permutation categories From graphs to categories Petri nets Rewriting logic Process calculi Conclusions CINA, Civitanova Marche, January Ugo Montanari 33

Algebraic “ Petri nets are monoids ” by Meseguer, Montanari –Algebra of (concurrent) computations via the lifting of the monoidal structure of markings to steps and computations sequential composition “ ; ” (of computations) plus identities (idle steps) plus parallel composition  (of markings, steps and computations) plus functoriality of  (concurrency!) leads to a (strictly) symmetric monoidal category of computations Collective Token Philosophy (CTPh) –T (_) (commutative processes) Individual Token Philosophy (ITPh) –P (_) (concatenable processes) CINA, Civitanova Marche, January Ugo Montanari 34

Collective vs. Token View Best-Devillers vs. Goltz-Reisig processes CINA, Civitanova Marche, January Ugo Montanari 35

The ITPh Story, I PTNetsDecOcc U (_) (_) + D (_) F (_) Sassone ’ s chain of adjunctions SafeOccPES Winskel ’ s chain of coreflections U (_) E (_) N (_) L (_) Pr (_) Petri * SMonCat * PreOrd P (N) Dom (_)  CINA, Civitanova Marche, January Ugo Montanari 36

Pre-Nets Under the CTPh, the construction T (_) is completely satisfactory –T (_) is left adjoint to the forgetful functor from CMonCat  to Petri –T (_) can be conveniently expressed at the level of (suitable) theories (e.g. in PMEqtl) But the CTPh does not model concurrency We argue that, under the ITPh, all the difficulties are due to the multiset (marking) view of states Pre-nets (Roberto Bruni et al.) are the natural implementation of P/T nets under the ITPh –pre-sets and post-sets are strings, not multisets! –for each transition t: u  v, just one implementation t p,q : p  q is considered CINA, Civitanova Marche, January Ugo Montanari 37

Pre-Nets, Algebraically Under the ITPh, the construction Z (_) is completely satisfactory –Z (_) is left adjoint to the forgetful functor from SMonCat  to PreNets –Z (_) can be conveniently expressed at the level of (suitable) theories (e.g. in PMEqtl) –All the pre-nets implementations R of the same P/T net N have isomorphic Z (R) –P (N) can be recovered from (any) Z (R) PreNetsSMonCat  Z (_) G (_) CINA, Civitanova Marche, January Ugo Montanari 38

The Pre-Net Picture PreNetsPreOccPESDom U (_) E (_) ? L (_) Pr (_) PreOrd (_)  SMonCat  Z (_) G (_) Functorial diagram reconciling all views Algebraic semantics via adjunction A missing link in the unfolding PTNets A (_)  CINA, Civitanova Marche, January Ugo Montanari 39

Operational Concurrency Quite similar developments: Term rewriting (2-categories), Jose Meseguer Logic programming (double categories), Andrea Corradini Graphs (DPO / SPO) Paolo Baldan Process Calculi (Tiles, monoidal double categories), Fabio Gadducci, Roberto Bruni CINA, Civitanova Marche, January Ugo Montanari 40

Roadmap Networks Milner flowgraph algebras Denotational process algebras (Soft) constraint networks Networks as components & connectors Petri nets Signal flow graphs Electric circuits PROPS: product permutation categories From graphs to categories Petri nets Rewriting logic Process calculi Conclusions CINA, Civitanova Marche, January Ugo Montanari 41

Conclusion  Additional kinds of networks  Computational fields  Synchronized hyperedge replacement  Neural networks  Bayesian networks  Additional interpreted domains  Cyberphysical systems  Hardware and software architectures  Heterogeneous systems CINA, Civitanova Marche, January Ugo Montanari 42