Equivalence and Independence in Controlled Graph-Rewriting Processes - PowerPoint PPT Presentation

Introduction Unmarked Processes Marked Processes Abstraction Independence Equivalence and Independence in Controlled Graph-Rewriting Processes ICGT@STAF 2018, Toulouse ar 1 , Andrea Corradini 2 , Malte Lochau 1 G´ eza Kulcs´ 1 Real-Time Systems Lab, TU Darmstadt 2 Department of Computer Science, University of Pisa geza.kulcsar@es.tu-darmstadt.de June 26, 2018 1 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Motivation • Graph-rewriting systems are inherently non-deterministic 2 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Motivation • Graph-rewriting systems are inherently non-deterministic • When designing algorithms, need to add control structures: 2 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Motivation • Graph-rewriting systems are inherently non-deterministic • When designing algorithms, need to add control structures: • sequentialization 2 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Motivation • Graph-rewriting systems are inherently non-deterministic • When designing algorithms, need to add control structures: • sequentialization • conditionals 2 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Motivation • Graph-rewriting systems are inherently non-deterministic • When designing algorithms, need to add control structures: • sequentialization • conditionals • iteration and recursion 2 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Motivation • Graph-rewriting systems are inherently non-deterministic • When designing algorithms, need to add control structures: • sequentialization • conditionals • iteration and recursion • parallelism 2 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Motivation • Graph-rewriting systems are inherently non-deterministic • When designing algorithms, need to add control structures: • sequentialization • conditionals • iteration and recursion • parallelism • ⇒ Controlled graph rewriting [Bunke, Sch¨ urr, Habel & Plump, Plump & Steinert, Kreowski et al.; PROGRES, Fujaba, eMoflon] 2 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Motivation • Graph-rewriting systems are inherently non-deterministic • When designing algorithms, need to add control structures: • sequentialization • conditionals • iteration and recursion • parallelism • ⇒ Controlled graph rewriting [Bunke, Sch¨ urr, Habel & Plump, Plump & Steinert, Kreowski et al.; PROGRES, Fujaba, eMoflon] • However, mostly I/O semantics so far, not considering: 2 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Motivation • Graph-rewriting systems are inherently non-deterministic • When designing algorithms, need to add control structures: • sequentialization • conditionals • iteration and recursion • parallelism • ⇒ Controlled graph rewriting [Bunke, Sch¨ urr, Habel & Plump, Plump & Steinert, Kreowski et al.; PROGRES, Fujaba, eMoflon] • However, mostly I/O semantics so far, not considering: • Concurrency in graph-rewriting algorithms 2 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Motivation • Graph-rewriting systems are inherently non-deterministic • When designing algorithms, need to add control structures: • sequentialization • conditionals • iteration and recursion • parallelism • ⇒ Controlled graph rewriting [Bunke, Sch¨ urr, Habel & Plump, Plump & Steinert, Kreowski et al.; PROGRES, Fujaba, eMoflon] • However, mostly I/O semantics so far, not considering: • Concurrency in graph-rewriting algorithms • Reactive (non-terminating) specifications 2 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence In this paper... • Process-algebraic (CCS-like) syntax and semantics for controlled graph rewriting 3 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence In this paper... • Process-algebraic (CCS-like) syntax and semantics for controlled graph rewriting • Transition systems both for control processes and for their executions on graphs 3 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence In this paper... • Process-algebraic (CCS-like) syntax and semantics for controlled graph rewriting • Transition systems both for control processes and for their executions on graphs • Expressiveness of our control language (encoding graph programs by Habel & Plump) 3 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence In this paper... • Process-algebraic (CCS-like) syntax and semantics for controlled graph rewriting • Transition systems both for control processes and for their executions on graphs • Expressiveness of our control language (encoding graph programs by Habel & Plump) • Handling of parallel rules and derivations by synchronization 3 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence In this paper... • Process-algebraic (CCS-like) syntax and semantics for controlled graph rewriting • Transition systems both for control processes and for their executions on graphs • Expressiveness of our control language (encoding graph programs by Habel & Plump) • Handling of parallel rules and derivations by synchronization • An abstract semantics, with graphs up to isomorphism 3 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence In this paper... • Process-algebraic (CCS-like) syntax and semantics for controlled graph rewriting • Transition systems both for control processes and for their executions on graphs • Expressiveness of our control language (encoding graph programs by Habel & Plump) • Handling of parallel rules and derivations by synchronization • An abstract semantics, with graphs up to isomorphism • Equivalence and congruence notions of CCS are reflected in our setting 3 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Graph-Rewriting Actions • DPO rules: p : ( L ← K → R ) • . . . Parallel composition of DPO rules: p 1 | p 2 : ( L 1 + L 2 ← K 1 + K 2 → R 1 + R 2 ) • R is the set of rule names, R ∗ the set of parallel rule names ranged over by ρ • Actions are pairs ( ρ, N ) ∈ Act where ρ ∈ R ∗ and N ⊆ R is a set of non-applicability conditions 4 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Unmarked Processes • Unmarked processes specify control over rule applications, having a CCS-like syntax Definition (Unmarked Process Term Syntax) P , Q ::= 0 | γ. P | A | P + Q | P || Q where γ ranges over Act and A := P. 5 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Unmarked Transition System (UTS) • The semantics of unmarked processes is an LTS where states are processes and transitions are labeled with actions PRE γ γ. P − → P γ → P ′ P − STOP CHOICE γ 0 � → P ′ P + Q − − → 0 γ α → P ′ → P ′ P − A := P P − PAR REC → P ′ || Q γ A α → P ′ − P || Q − ( ρ 1 , N 1 ) ( ρ 2 , N 2 ) → P ′ → Q ′ P − − − − Q − − − − SYNC ( ρ 1 | ρ 2 , N 1 ∪ N 2 ) → P ′ || Q ′ P || Q − − − − − − − − 6 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Rule Applications ρ @ m • G = = ⇒ H ρ L K r R l ( PO ) ( PO ) m n k g G D H f • A linear derivation from G 0 is a sequence of rule applications p 1 @ m 1 p n @ m n = = = ⇒ . . . = = = = ⇒ G n with no parallel rules G 0 • A parallel derivation from G 0 is a sequence of rule applications ρ 1 @ m 1 ρ n @ m n = = = ⇒ . . . = = = = ⇒ G n with (potentially) parallel rules G 0 ρ i = p i 1 | . . . | p ik 7 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Marked Transition System (MTS) ( ρ, N ) ρ @ m p → P ′ P − − − G = = ⇒ H ∀ p ∈ N : G � = ⇒ MARK ( ρ,δ, N ) → M ( P ′ , H ) ( P , G ) − − − − ρ δ := L K r R l ( PO ) ( PO ) m n k g G D H f � → P ′ − P STOP ( P , G ) � → M ( P ′ , G ) − 8 / 18

Introduction Unmarked Processes Marked Processes Abstraction Independence Equivalence of Graph-Rewriting Processes • A trace of a state P is a sequence of transition labels starting from it, e.g., ( ρ 1 , δ 1 , N 1 )( ρ 2 , δ 2 , N 2 ) . . . ( ρ n , δ n , N n ) • P and Q are trace equivalent ( P ≃ T Q ) if they have the same set of traces • P and Q are bisimilar ( P ≃ BS Q ) if for each P α → M P ′ , there − → M Q ′ such that P ′ ≃ BS Q ′ (and vice versa) α is Q − Proposition For any unmarked processes P , Q and graph G, • P ≃ BS Q implies ( P , G ) ≃ BS M ( Q , G ) • P ≃ T Q implies ( P , G ) ≃ T M ( Q , G ) 9 / 18