Graph Representation of Sessions and Pipelines for Structured Service Programming

Transcription

1 Graph Representation of Sessions and Pipelines for Structured Service Programming Liang Zhao 1,2 with Roberto Bruni 1 and Zhiming Liu 2 1 University of Pisa, Italy 2 UNU-IIST, Macao SR, China FCS 2010 Zhao, et al. Graph Representation of CaSPiS 2010

2 Introduction Traditional process calculi successful in modeling concurrent systems, mobile systems modeling service systems? Problem: low level communication primitives, complexity of analysis Calculus of Sessions and Pipelines (CaSPiS) spects: service autonomy, client-service interaction, orchestration Key notions: session (define interactions between two sides), pipeline (orchestrate the flow of data) Sessions and pipelines can be nested. Operational semantics: transition rules, silent transitions (reductions) Zhao, et al. Graph Representation of CaSPiS 2010

3 Introduction Motivation hierarchical service system S D I P 1 s P 2 P 3 r S P 4 vs. textual expression: s.p 1 r (s.p 2 P 3 ) r P 4 Zhao, et al. Graph Representation of CaSPiS 2010

4 Outline 1 The calculus CaSPiS Syntax Operational semantics (reduction) 2 lgebra of hierarchical graphs Grammar and semantic model Graph transformation by DPO 3 Graph representation of CaSPiS Processes as designs Graph transformation rules Soundness and completeness Zhao, et al. Graph Representation of CaSPiS 2010

5 Outline 1 The calculus CaSPiS Syntax Operational semantics (reduction) 2 lgebra of hierarchical graphs Grammar and semantic model Graph transformation by DPO 3 Graph representation of CaSPiS Processes as designs Graph transformation rules Soundness and completeness Zhao, et al. Graph Representation of CaSPiS 2010

6 Syntax Syntax of CaSPiS Process P,Q ::= M P Q s.p s.p r P (νn)p P > Q Sum M ::= 0 (?x)p V P V P M + M Value V ::= x c Example: P 0 = time. T time.(?x) x Structural Congruence (P 1 P 2 ) P 3 c P 1 (P 2 P 3 ) M 1 + M 2 c M 2 + M 1 (νn)0 c 0 r (νn)p c (νn)(r P) if n r... Syntax Zhao, et al. Graph Representation of CaSPiS 2010

7 Semantics Contexts Dynamic operators: (?x)[ ], [ ] + M, s.[ ], P > [ ] Static contexts: [ ] Q, r [ ], (νx)[ ], [ ] [ ],... Basic Reductions (Sync): C[s.P, s.q] (νr)c[r P, r Q] (S-Sync): C[r (P 0 y P), r (?x)q] C[r (P 0 P), r Q[y/x]] (P-Sync): C[(P 0 y P) > (?x)q] C[Q[y/x] ((P 0 P) > (?x)q)]... Example P 0 = time. T time.(?x) x P 1 = (νr)(r T r (?x) x ) P 2 = (νr)(r 0 r T ) Operational semantics (reduction) Zhao, et al. Graph Representation of CaSPiS 2010

8 Outline 1 The calculus CaSPiS Syntax Operational semantics (reduction) 2 lgebra of hierarchical graphs Grammar and semantic model Graph transformation by DPO 3 Graph representation of CaSPiS Processes as designs Graph transformation rules Soundness and completeness Zhao, et al. Graph Representation of CaSPiS 2010

9 Graph Grammar Terms Graph G ::= 0 x l( x) G G (νx)g D x Design D ::= L y [G] Free Node Free node: not restricted or exposed Example: L y [(νx)l(x,y)] z Type Fixed Type: each node x, edges label l, design label L Well-typedness: l( x), L y [G] x Grammar and semantic model Zhao, et al. Graph Representation of CaSPiS 2010

10 Semantic Model Interpretation of Terms G 1 = a(y 1 ) y 2 y1 a y2 Order of Tentacles of (Hyper-)edges 2 1 l l Grammar and semantic model Zhao, et al. Graph Representation of CaSPiS 2010

11 Semantic Model Interpretation of Terms G 2 = L (y1,y 2 )[a(y 1 ) y 2 ] x 1,x 2 L 1 1 x1 a L 1 2 x2 G 3 = L (y1,y 2 )[a(y 1 ) y 2 ] x 1,x 2 L (y1,y 2 )[y 1 a(y 2 )] x 1,x 2 L 1 1 x1 L 2 1 a a L 1 2 x2 L 2 2 L a x1 L a x2 Grammar and semantic model Zhao, et al. Graph Representation of CaSPiS 2010

12 Semantic Model Flat Design Edge L (y1,y 2 )[a(y 1 ) y 2 ] x 1,x 2 vs. F (y1,y 2 )[a(y 1 ) y 2 ] x 1,x 2 L a x1 x2 x1 a x2 Node Sharing K x [b(x,y)] z K x [b(x,y)] z vs. K x [(νy)b(x,y)] z K x [(νy)b(x,y)] z K b z K b y K b z K b Grammar and semantic model Zhao, et al. Graph Representation of CaSPiS 2010

13 Graph Transformation Morphism and Pushout Morphism: a mapping (m :G 1 G 2 ) that preserves types of nodes, labels and tentacles of edges Pushout: a square of (four) morphisms that commute Double Pushout (DPO) Rules R :GL m l GI m r GR, or simply GL GI GR m l mr GL GI GR m 1 m 2 m 3 m l G G m r G Direct derivation: G R G Derivation: a sequence of direct derivations, G G Graph transformation by DPO Zhao, et al. Graph Representation of CaSPiS 2010

14 Outline 1 The calculus CaSPiS Syntax Operational semantics (reduction) 2 lgebra of hierarchical graphs Grammar and semantic model Graph transformation by DPO 3 Graph representation of CaSPiS Processes as designs Graph transformation rules Soundness and completeness Zhao, et al. Graph Representation of CaSPiS 2010

15 Graph Representation Nil, bstraction and Parallel composition P Nil P bs P P P Par Q 0 (?x)p P Q 0 (?x)p P Q def = P (p,i,o,t) [i o t Nil(p)] def = P (p,i,o,t) [(ν{p 1,x}) ( bs(p,x,p 1,i) P p 1,i,o,t ) ] def = P (p,i,o,t) [(ν{p ( 1,p 2 }) Par(p,p1,p 2 ) P p 1,i,o,t Q p 2,i,o,t ) ] Processes as designs Zhao, et al. Graph Representation of CaSPiS 2010

16 Graph Representation Session and Pipeline P S r Ses P r P P P R Pip Q P > Q r P P > Q def = P (p,i,o,t) [i t S ( (p,t) [(ν{p 1,i 1,o 1 }) Ses(p,r,p1,i 1,o 1 ) P p 1,i 1,o 1,t ) ] p,o ] def = P (p,i,o,t) [(ν{p 1,p 2,o 1 }) ( Pip(p,p 1,p 2,o 1,i,o,t) P p 1,i,o 1,t R p [(ν{i,o,t})q p,i,o,t ] p 2 ) ] Processes as designs Zhao, et al. Graph Representation of CaSPiS 2010

17 Graph Representation n Example p D time T Def Con Nil I Inv bs Ret Par P 0 p,i,o,t i o t Nil P 0 = time. T time.(?x) x Processes as designs Zhao, et al. Graph Representation of CaSPiS 2010

18 Graph Representation Tagged Graph p D time T Def Con Nil I Inv bs Ret Par P 0 p,i,o,t i o t Nil P 0 = time. T time.(?x) x Processes as designs Zhao, et al. Graph Representation of CaSPiS 2010

19 Graph Transformation Rules Do congruence processes have the same graph representation? P 1 P 2 vs. P 2 P 1 C : Rules for congruence What is the relation between P and Q with P Q? R : Rules for reduction uxiliary rules (tagging, garbage collection) Rule for Congruence Par 3 1 Par 2 C[P 1 P 2 ] C[P 2 P 1 ] Graph transformation rules Zhao, et al. Graph Representation of CaSPiS 2010

20 Graph Transformation Rules Rule for Congruence (Cont.) Par Par Par Par C[(P 1 P 2 ) P 3 ] C[P 1 (P 2 P 3 )] Rule for Congruence (Cont.) Par Res Res Par C[P 1 (νn)p 2 ] C[(νn)(P 1 P 2 )] Graph transformation rules Zhao, et al. Graph Representation of CaSPiS 2010

21 Graph Transformation Rules p D time T Def Con Nil I Inv bs Ret Par i o t Nil D Def I Inv S rv Ses S Ses Graph transformation rules Zhao, et al. Graph Representation of CaSPiS 2010

22 Graph Transformation Rules p S rv time T Ses Con Nil Par S Ses bs Ret i o t Nil Garbage Collection Rule Graph transformation rules Zhao, et al. Graph Representation of CaSPiS 2010

23 Graph Transformation Rules p S rv Ses Con Nil Par T S Ses bs Ret i o t Nil Tagging Rule S Ses S Ses S Ses Graph transformation rules Zhao, et al. Graph Representation of CaSPiS 2010

24 Graph Transformation Rules p S rv Ses Con Nil Par T S Ses bs Ret i o t Nil P 1 p,i,o,t P 1 = (νr)(r T r (?x) x ) Graph transformation rules Zhao, et al. Graph Representation of CaSPiS 2010

25 Graph Transformation Rules Rule for Reduction S Ses S Ses Con bs S Ses S Ses n p p n q q p/p p, q/q q;n/n n S Ses S p Ses q n Graph transformation rules Zhao, et al. Graph Representation of CaSPiS 2010

26 Graph Transformation Rules p S rv Ses Par S Ses Nil i o t T Ret Nil P 2 p,i,o,t P 2 = (νr)(r 0 r T ) Graph transformation rules Zhao, et al. Graph Representation of CaSPiS 2010

27 Soundness and Completeness Theorem (soundness w.r.t. congruence) P C G implies G d Q for some Q c P Theorem (completeness w.r.t. congruence) P c Q implies P C Q and Q c Q for some Q Conjecture (soundness w.r.t. reduction) P Q implies P Q difficulty: intermediate states Conjecture (completeness w.r.t. reduction) P Q implies P Q for some Q c Q difficulty: replications Soundness and completeness Zhao, et al. Graph Representation of CaSPiS 2010

28 Summary What we have done a graph algebra: grammar, semantic model graph representation of CaSPiS processes (tagged graphs) graph transformation rules (DPO): congruence, reduction, auxiliary (tagging, garbage collection) soundness and completeness of DPO rules w.r.t. congruence Future Work soundness and completeness of DPO rules w.r.t. reduction case study and implementation Zhao, et al. Graph Representation of CaSPiS 2010