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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,

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Introduction ≈REG and ≃REG for tnPA Remarks

Regularity Problems of Process Rewrite Systems

Fei Yang1 (Based on joint work with Yuxi Fu2)

1Department of Mathematics and Computer Science

Eindhoven University of Technology

2BASICS, Department of Computer Science and Engineering

Shanghai Jiao Tong University

FSA Colloquium November 11, 2014

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Introduction ≈REG and ≃REG for tnPA Remarks

Outline

1

Introduction Overview Process Rewrite System

2

≈REG and ≃REG for tnPA Semi-Decidability Witness of Infinity Finite Witness

3

Remarks Related Work Open Problems

Fei Yang Regularity Problems of Process Rewrite Systems 2/33

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

Outline

1

Introduction Overview Process Rewrite System

2

≈REG and ≃REG for tnPA Semi-Decidability Witness of Infinity Finite Witness

3

Remarks Related Work Open Problems

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

A hardware system is always a finite state system. A software system has potentially infinite states, no matter how simple its functionality may be.

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

Three Verification Problems

Equivalence Input: processes α, β, equivalence relation ∼ =. Problem: α ∼ = β ? Finiteness Input: process α, FS γ, equivalence relation ∼ =. Problem: α ∼ = γ ? Regularity Input: process α, equivalence relation ∼ =. Problem: ∃ FS γ. α ∼ = γ ?

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

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?

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

Several Bisimulation Relations

1

∼ Strong Bisimulation [Par81]

2

≈ Weak Bisimulation [Mil89]

3

≃ Branching Bisimulation [vGW96]

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

Several Bisimulation Relations

1

∼ Strong Bisimulation [Par81]

2

≈ Weak Bisimulation [Mil89]

3

≃ Branching Bisimulation [vGW96]

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

Several Bisimulation Relations

1

∼ Strong Bisimulation [Par81]

2

≈ Weak Bisimulation [Mil89]

3

≃ Branching Bisimulation [vGW96]

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

Syntax

Definition Act = {a, b, . . .} is a set of atomic actions; Const = {ǫ} ∪ {X, Y, Z, . . .} is a set of process constants. S = {α1, α2, . . .} is the set of process terms, which describe the states

  • f the system, are generated from the following BNF:

α ::= ǫ | X | α1.α2 | α1 α2 where ǫ is the empty process; α1.α2 is a sequential process; α1 α2 is a parallel process. We shall write α, β, γ, . . . for process terms.

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

Operational Semantics

The transition relation → is generated from a finite set ∆ of transition rules of the form α

a

− → β. For every a ∈ Act, the transition relation

a

− → is the smallest relation constructed from the following inference rules: α

a

− → β ∈ ∆ α

a

− → β α

a

− → α′ α.β

a

− → α′.β α

a

− → α′ α β

a

− → α′ β β

a

− → β′ α β

a

− → α β′ The parallel composition α β is different from concurrent

  • composition. No communication between α and β is admitted.

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

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.

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

Classification of PRS

Definition Let Ξ, Π ∈ {1, S, P, G}. A (Ξ, Π)-PRS is a finite set of rules ∆ satisfying the following: for every rewrite rule α

a

− → β ∈ ∆ Ξ ⊆ Π, α ∈ Ξ \ {ǫ}, β ∈ Π, the initial state is given as a term α0 ∈ Ξ. A (G, G)-PRS is simply called a PRS. W.l.o.g. it can be assumed that the initial state α0 of a PRS is a single constant.

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

Example of BPA

Recursive specification: X def = a.X.B + c.ǫ B def = b.ǫ The following is a (1, S)-PRS transition system with initial state X: X

a

− → X.B X

c

− → ǫ B

b

− → ǫ This process recognises the language {ancbn : n ≥ 0}.

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

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)

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

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.

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Introduction ≈REG and ≃REG for tnPA Remarks Overview Process Rewrite System

PA

PA nPA tnPA ∼REG ? NL [Kuˇ c96] NL [Kuˇ c96] PSPACE-H [Srb02a] NL-H [Srb02a] NL-H [Srb02a] ≃REG ? ? P EXPT-H [May03] PSPACE-H [Srb03] NL-H [Srb02a] ≈REG ? ? P EXPT-H [May03] PSPACE-H [Srb03] NL-H [Srb02a]

Table: PA Regularity

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

Outline

1

Introduction Overview Process Rewrite System

2

≈REG and ≃REG for tnPA Semi-Decidability Witness of Infinity Finite Witness

3

Remarks Related Work Open Problems

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

A PA Process

X a − → X X X b − → X.X X a − → ǫ X τ − → ǫ X− → ∗ X.(X (X.X)) X− → ∗ X X (X.(X X))

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

A PA Process

X a − → X X X b − → X.X X a − → ǫ X τ − → ǫ X− → ∗ X.(X (X.X)) X− → ∗ X X (X.(X X))

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

∼REG for nPA is in polynomial time [Kuˇ c96]. How about ≈REG and ≃REG? Why not adding totally normed constraint?

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

∼REG for nPA is in polynomial time [Kuˇ c96]. How about ≈REG and ≃REG? Why not adding totally normed constraint?

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

Lemma (Semi-Decidability) Given a process (α, ∆), if ∼ = or ∼ =FS is decidable, then ∼ =REG is semi-decidable.

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity 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 α

a0

− → α1

a1

− → α2

a2

− → . . . with αi ∼ = αj, for any i = j.

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

Lemma (Infinite ≡ Path) Given a process (α, ∆), it is non-regular w.r.t. ∼ =, then there exists an infinite transition path α

a0

− → α1

a1

− → α2

a2

− → . . . with αi ≡ αj, for any i = j. Lemma (Infinite Norm Increasing Path) Given a process (α, ∆), it is non-regular w.r.t. ∼ =, if there exists an infinite transition path α

a0

− → α1

a1

− → α2

a2

− → . . . s.t., for any i, there exists j, with norm(αi) < norm(αj).

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

Lemma (Infinite ≡ Path) Given a process (α, ∆), it is non-regular w.r.t. ∼ =, then there exists an infinite transition path α

a0

− → α1

a1

− → α2

a2

− → . . . with αi ≡ αj, for any i = j. Lemma (Infinite Norm Increasing Path) Given a process (α, ∆), it is non-regular w.r.t. ∼ =, if there exists an infinite transition path α

a0

− → α1

a1

− → α2

a2

− → . . . s.t., for any i, there exists j, with norm(αi) < norm(αj).

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

Finite Witness

Fire(α) =        ∅ if α = ǫ {X} if α = X Fire(β1) if α = β1.β2 Fire(β1) ∪ Fire(β2) if α = β1 β2

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

Finite Witness

Definition Given a tnPA process (α, ∆), a process constant X ∈ Var(∆) is a growing constant if there exists β s.t., X− → ∗β, with X ∈ Fire(β) and length(β) ≥ 2. Lemma Given a tnPA process (α, ∆), it is ≃reg and ≈reg, iff it cannot activate any growing constant.

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

Finite Witness

Definition Given a tnPA process (α, ∆), a process constant X ∈ Var(∆) is a growing constant if there exists β s.t., X− → ∗β, with X ∈ Fire(β) and length(β) ≥ 2. Lemma Given a tnPA process (α, ∆), it is ≃reg and ≈reg, iff it cannot activate any growing constant.

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Introduction ≈REG and ≃REG for tnPA Remarks Semi-Decidability Witness of Infinity Finite Witness

Polynomial Time Decidability

Theorem There is a polynomial time algorithm deciding whether there is a growing constant. The time complexity is O(n3 + mn), where n is the number of transition rules, and m is the maximum length of the rules. Theorem ≈REG and ≃REG for tnPA can be decided in polynomial time.

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Introduction ≈REG and ≃REG for tnPA Remarks Related Work Open Problems

Outline

1

Introduction Overview Process Rewrite System

2

≈REG and ≃REG for tnPA Semi-Decidability Witness of Infinity Finite Witness

3

Remarks Related Work Open Problems

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Introduction ≈REG and ≃REG for tnPA Remarks Related Work Open Problems

BPA

BPA nBPA ∼REG Decidable [BCS95, BCS96] PSPACE-H [Srb02b] NL-C [Kuˇ c96, Srb02b] ≃REG ? EXPT-H [May03] Decidable [Fu13] NP-H [Srb03, Stˇ r98] ≈REG ? EXPT-H [May03] ? NP-H [Srb03, Stˇ r98]

Table: BPA Regularity

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Introduction ≈REG and ≃REG for tnPA Remarks Related Work Open Problems

BPP

BPP nBPP ∼REG PSPACE-C [Kot05, JE96, Srb02a] NL [Kuˇ c96] NL-H [Srb02a] ≃REG ? ? ≈REG ? PSPACE-H [Srb03] ? PSPACE-H [Srb03]

Table: BPP Regularity

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Introduction ≈REG and ≃REG for tnPA Remarks Related Work Open Problems

PDA

PDA nPDA ∼REG ? EXPT-H [KM02, Srb02b] P [EHRS00] NL-H [Srb02b] ≃REG ? ? ≈REG ? EXPT-H [KM02, Srb02b] ? EXPT-H [KM02, Srb02b]

Table: PDA Regularity

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Introduction ≈REG and ≃REG for tnPA Remarks Related Work Open Problems

PN

PN nPN ∼REG Decidable [JE96] PSPACE-H [Srb02a] EXPSAPCE [Rac78] EXPSPACE-H [CLM76] ≃REG Undecidable ? EXPSPACE-H [CLM76] ≈REG Undecidable [JE96] EXPSPACE-H [CLM76] ? EXPSPACE-H [CLM76]

Table: PN Regularity

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Open Problems

Decidability for of ≃ for nBPP is proved in [CHL11]. Thus we have Corollary ≃REG for nBPP is semi-decidable. Is ≃REG decidable for nBPP?

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Open Problems

Decidability for of ≃ for nBPP is proved in [CHL11]. Thus we have Corollary ≃REG for nBPP is semi-decidable. Is ≃REG decidable for nBPP?

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Equivalence Checking of a Model with its Sub-model

For example: PN ∼ BPP

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Complexity Lower Bound

Is ∼REG for BPA EXPT-hard? Is the proof for ≈REG valid for ≃REG?

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Complexity Lower Bound

Is ∼REG for BPA EXPT-hard? Is the proof for ≈REG valid for ≃REG?

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Thank You!

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