Introduction ≈ REG and ≃ REG for tnPA Remarks Regularity Problems of Process Rewrite Systems Fei Yang 1 (Based on joint work with Yuxi Fu 2 ) 1 Department of Mathematics and Computer Science Eindhoven University of Technology 2 BASICS, Department of Computer Science and Engineering Shanghai Jiao Tong University FSA Colloquium November 11, 2014 Fei Yang Regularity Problems of Process Rewrite Systems 1/33
Introduction ≈ REG and ≃ REG for tnPA Remarks Outline Introduction 1 Overview Process Rewrite System ≈ REG and ≃ REG for tnPA 2 Semi-Decidability Witness of Infinity Finite Witness Remarks 3 Related Work Open Problems Fei Yang Regularity Problems of Process Rewrite Systems 2/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Outline Introduction 1 Overview Process Rewrite System ≈ REG and ≃ REG for tnPA 2 Semi-Decidability Witness of Infinity Finite Witness Remarks 3 Related Work Open Problems Fei Yang Regularity Problems of Process Rewrite Systems 3/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks A hardware system is always a finite state system. A software system has potentially infinite states, no matter how simple its functionality may be. Fei Yang Regularity Problems of Process Rewrite Systems 4/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Three Verification Problems Equivalence processes α, β , equivalence relation ∼ = . Input: α ∼ Problem: = β ? Finiteness process α , FS γ , equivalence relation ∼ Input: = . α ∼ = γ ? Problem: Regularity process α , equivalence relation ∼ Input: = . ∃ FS γ . α ∼ Problem: = γ ? Fei Yang Regularity Problems of Process Rewrite Systems 5/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Equivalence Checking Is an implementation correct w.r.t. a specification? Finiteness Checking Is a hardware design correct w.r.t. a specification? Regularity Checking Is a specification implementable by hardware? Fei Yang Regularity Problems of Process Rewrite Systems 6/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Several Bisimulation Relations ∼ Strong Bisimulation [Par81] 1 ≈ Weak Bisimulation [Mil89] 2 ≃ Branching Bisimulation [vGW96] 3 Fei Yang Regularity Problems of Process Rewrite Systems 7/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Several Bisimulation Relations ∼ Strong Bisimulation [Par81] 1 ≈ Weak Bisimulation [Mil89] 2 ≃ Branching Bisimulation [vGW96] 3 Fei Yang Regularity Problems of Process Rewrite Systems 7/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Several Bisimulation Relations ∼ Strong Bisimulation [Par81] 1 ≈ Weak Bisimulation [Mil89] 2 ≃ Branching Bisimulation [vGW96] 3 Fei Yang Regularity Problems of Process Rewrite Systems 7/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Syntax Definition Act = { a , b , . . . } is a set of atomic actions; Const = { ǫ } ∪ { X , Y , Z , . . . } is a set of process constants. S = { α 1 , α 2 , . . . } is the set of process terms, which describe the states of the system, are generated from the following BNF: α ::= ǫ | X | α 1 .α 2 | α 1 � α 2 where ǫ is the empty process; α 1 .α 2 is a sequential process; α 1 � α 2 is a parallel process. We shall write α, β, γ, . . . for process terms. Fei Yang Regularity Problems of Process Rewrite Systems 8/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Operational Semantics The transition relation → is generated from a finite set ∆ of transition a − → β . rules of the form α a For every a ∈ Act , the transition relation − → is the smallest relation constructed from the following inference rules: a a − → β ∈ ∆ α − → α ′ α a a α − → β α.β − → α ′ .β a a − → β ′ β − → α ′ α a → α ′ � β a α � β − α � β − → α � β ′ The parallel composition α � β is different from concurrent composition. No communication between α and β is admitted. Fei Yang Regularity Problems of Process Rewrite Systems 9/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Classification of Process Term Definition We distinguish four classes of process term: 1 : Terms consisting of a single process constant like X . S : Terms consisting of a single constant or a sequential composition of process constants like X . Y . Z . P : Terms consisting of a single constant or a parallel composition of process constants like X � Y � Z . G : Terms with arbitrary sequential and parallel composition like ( X . ( Y � Z )) � W . Fei Yang Regularity Problems of Process Rewrite Systems 10/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Classification of PRS Definition Let Ξ , Π ∈ { 1 , S , P , G } . A (Ξ , Π) -PRS is a finite set of rules ∆ a − → β ∈ ∆ satisfying the following: for every rewrite rule α Ξ ⊆ Π , α ∈ Ξ \ { ǫ } , β ∈ Π , the initial state is given as a term α 0 ∈ Ξ . A ( G , G ) -PRS is simply called a PRS. W.l.o.g. it can be assumed that the initial state α 0 of a PRS is a single constant. Fei Yang Regularity Problems of Process Rewrite Systems 11/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Example of BPA Recursive specification: X def B def = a . X . B + c .ǫ = b .ǫ The following is a ( 1 , S ) -PRS transition system with initial state X : a − → X . B X c − → ǫ X b − → ǫ B This process recognises the language { a n cb n : n ≥ 0 } . Fei Yang Regularity Problems of Process Rewrite Systems 12/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks PRS Hierarchy PRS ( G , G ) PAD ( S , G ) PAN ( P , G ) PDA ( S , S ) PA ( 1 , G ) PN ( P , P ) BPA ( 1 , S ) BPP ( 1 , P ) FS ( 1 , 1 ) Fei Yang Regularity Problems of Process Rewrite Systems 13/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks Normed Process Definition A process is normed if it can reach ǫ after a finite number of transition steps. We say a process is totally normed if it is normed and can reach ǫ after at least one non- τ transition step. We write for example nBPA for normed BPA and tnBPA for totally normed BPA. Fei Yang Regularity Problems of Process Rewrite Systems 14/33
Introduction Overview ≈ REG and ≃ REG for tnPA Process Rewrite System Remarks PA PA nPA tnPA ? NL [Kuˇ c96] NL [Kuˇ c96] ∼ REG PSPACE-H [Srb02a] NL-H [Srb02a] NL-H [Srb02a] ? ? P ≃ REG EXPT-H [May03] PSPACE-H [Srb03] NL-H [Srb02a] ? ? P ≈ REG EXPT-H [May03] PSPACE-H [Srb03] NL-H [Srb02a] Table: PA Regularity Fei Yang Regularity Problems of Process Rewrite Systems 15/33
Introduction Semi-Decidability ≈ REG and ≃ REG for tnPA Witness of Infinity Remarks Finite Witness Outline Introduction 1 Overview Process Rewrite System ≈ REG and ≃ REG for tnPA 2 Semi-Decidability Witness of Infinity Finite Witness Remarks 3 Related Work Open Problems Fei Yang Regularity Problems of Process Rewrite Systems 16/33
Introduction Semi-Decidability ≈ REG and ≃ REG for tnPA Witness of Infinity Remarks Finite Witness A PA Process X a − → X � X → ∗ X . ( X � ( X . X )) X − X b → X . X − → ∗ X � X � ( X . ( X � X )) X a X − − → ǫ X τ − → ǫ Fei Yang Regularity Problems of Process Rewrite Systems 17/33
Introduction Semi-Decidability ≈ REG and ≃ REG for tnPA Witness of Infinity Remarks Finite Witness A PA Process X a − → X � X → ∗ X . ( X � ( X . X )) X − X b → X . X − → ∗ X � X � ( X . ( X � X )) X a X − − → ǫ X τ − → ǫ Fei Yang Regularity Problems of Process Rewrite Systems 17/33
Introduction Semi-Decidability ≈ REG and ≃ REG for tnPA Witness of Infinity Remarks Finite Witness ∼ REG for nPA is in polynomial time [Kuˇ c96]. How about ≈ REG and ≃ REG ? Why not adding totally normed constraint? Fei Yang Regularity Problems of Process Rewrite Systems 18/33
Introduction Semi-Decidability ≈ REG and ≃ REG for tnPA Witness of Infinity Remarks Finite Witness ∼ REG for nPA is in polynomial time [Kuˇ c96]. How about ≈ REG and ≃ REG ? Why not adding totally normed constraint? Fei Yang Regularity Problems of Process Rewrite Systems 18/33
Introduction Semi-Decidability ≈ REG and ≃ REG for tnPA Witness of Infinity Remarks Finite Witness Lemma (Semi-Decidability) Given a process ( α, ∆) , if ∼ = or ∼ = FS is decidable, then ∼ = REG is semi-decidable. Fei Yang Regularity Problems of Process Rewrite Systems 19/33
Introduction Semi-Decidability ≈ REG and ≃ REG for tnPA Witness of Infinity Remarks Finite Witness Infinite Transition Path Lemma (Infinite �∼ = Path) Given a process ( α, ∆) , it is non-regular w.r.t. ∼ = , iff there exists an infinite transition path a 0 a 1 a 2 − → α 1 − → α 2 − → . . . α with α i �∼ = α j , for any i � = j. Fei Yang Regularity Problems of Process Rewrite Systems 20/33
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