The Power of Priority Channel Systems

The Power of

Priority Channel Systems

C. Haase S. Schmitz Ph. Schnoebelen (LSV, ENS Cachan, France)

Feb. 25th, 2014, CMI, Chennai

Outline

priority channel systems a model of computation

priority embedding a well quasi ordering

Contents

Channel Systems with Priorities Priority Embedding

Computational Power

Outline

priority channel systems a model of computation

priority embedding a well quasi ordering

Contents

Channel Systems with Priorities Priority Embedding

Computational Power

Channel Systems (CSs)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

mn n

a

I model for communication protocols

I a.k.a. “queue automata”

I Turing-powerful

Channel Systems (CSs)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

mn n

a

I model for communication protocols

I a.k.a. “queue automata”

I Turing-powerful

Channel Systems (CSs)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

mn n

a

I model for communication protocols

I a.k.a. “queue automata”

I Turing-powerful

Channel Systems (CSs)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

mn n

a

I model for communication protocols

I a.k.a. “queue automata”

I Turing-powerful

Channel Systems (CSs)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

mn n

a

I model for communication protocols

I a.k.a. “queue automata”

I Turing-powerful

Channel Systems (CSs)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

mn n

a

I model for communication protocols

I a.k.a. “queue automata”

I Turing-powerful

Channel Systems (CSs)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

mn n

a

I model for communication protocols

I a.k.a. “queue automata”

I Turing-powerful

Lossy Channel Systems (LCSs)

(Abdulla and Jonsson, 1996; C´ec´eet al., 1996)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

m n n

I applylosingrewriting rules to channel contents:

m→_∗ ε m∈M

I model for imperfect or unreliable communications

I decidable: Fω^ω-complete(Chambart and Schnoebelen, 2008)

Lossy Channel Systems (LCSs)

(Abdulla and Jonsson, 1996; C´ec´eet al., 1996)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

m n n

I applylosingrewriting rules to channel contents:

m→_∗ ε m∈M

I model for imperfect or unreliable communications

I decidable: Fω^ω-complete(Chambart and Schnoebelen, 2008)

Lossy Channel Systems (LCSs)

(Abdulla and Jonsson, 1996; C´ec´eet al., 1996)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

m n n

I applylosingrewriting rules to channel contents:

m→_∗ ε m∈M

I model for imperfect or unreliable communications

I decidable: Fω^ω-complete(Chambart and Schnoebelen, 2008)

Lossy Channel Systems (LCSs)

(Abdulla and Jonsson, 1996; C´ec´eet al., 1996)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

m n n

I applylosingrewriting rules to channel contents:

m→_∗ ε m∈M

I model for imperfect or unreliable communications

I decidable: Fω^ω-complete(Chambart and Schnoebelen, 2008)

Lossy Channel Systems (LCSs)

(Abdulla and Jonsson, 1996; C´ec´eet al., 1996)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

m n n

I applylosingrewriting rules to channel contents:

m→_∗ ε m∈M

I model for imperfect or unreliable communications

I decidable: Fω^ω-complete(Chambart and Schnoebelen, 2008)

Lossy Channel Systems (LCSs)

(Abdulla and Jonsson, 1996; C´ec´eet al., 1996)

c₁?m c₁?n c₂!a

c₁!m c₁!n c₂?a

←c₁

c₂→

m n n

I applylosingrewriting rules to channel contents:

m→_∗ ε m∈M

I model for imperfect or unreliable communications

I decidable: Fω^ω-complete(Chambart and Schnoebelen, 2008)

Priority Channel Systems (PCSs)

c₁?0,m c₁?1,n c₂!1,a

c₁!0,m c₁!1,n c₂?1,a

←c₁

c₂→

0,m0,m1,n 1,n

I applysupersedingrewriting rules to channel contents:

(a,m)(b,n)→_# (b,n) a6b∈N, m,n∈M

I modeling communications with QoS, e.g.

differentiated services (RFC2475)

I decidable: Fε₀-complete

Priority Channel Systems (PCSs)

c₁?0,m c₁?1,n c₂!1,a

c₁!0,m c₁!1,n c₂?1,a

←c₁

c₂→

0,m0,m1,n 1,n

I applysupersedingrewriting rules to channel contents:

(a,m)(b,n)→_# (b,n) a6b∈N, m,n∈M

I modeling communications with QoS, e.g.

differentiated services (RFC2475)

I decidable: Fε₀-complete

Priority Channel Systems (PCSs)

c₁?0,m c₁?1,n c₂!1,a

c₁!0,m c₁!1,n c₂?1,a

←c₁

c₂→

0,m0,m1,n 1,n

I applysupersedingrewriting rules to channel contents:

(a,m)(b,n)→_# (b,n) a6b∈N, m,n∈M

I modeling communications with QoS, e.g.

differentiated services (RFC2475)

I decidable: Fε₀-complete

Priority Channel Systems (PCSs)

c₁?0,m c₁?1,n c₂!1,a

c₁!0,m c₁!1,n c₂?1,a

←c₁

c₂→

0,m0,m1,n 1,n

I applysupersedingrewriting rules to channel contents:

(a,m)(b,n)→_# (b,n) a6b∈N, m,n∈M

I modeling communications with QoS, e.g.

differentiated services (RFC2475)

I decidable: Fε₀-complete

Priority Channel Systems (PCSs)

c₁?0,m c₁?1,n c₂!1,a

c₁!0,m c₁!1,n c₂?1,a

←c₁

c₂→

0,m0,m1,n 1,n

I applysupersedingrewriting rules to channel contents:

(a,m)(b,n)→_# (b,n) a6b∈N, m,n∈M

I modeling communications with QoS, e.g.

differentiated services (RFC2475)

I decidable: Fε₀-complete

Priority Channel Systems (PCSs)

c₁?0,m c₁?1,n c₂!1,a

c₁!0,m c₁!1,n c₂?1,a

←c₁

c₂→

0,m0,m1,n 1,n

I applysupersedingrewriting rules to channel contents:

(a,m)(b,n)→_# (b,n) a6b∈N, m,n∈M

I modeling communications with QoS, e.g.

differentiated services (RFC2475)

I decidable: Fε₀-complete

Priority Channel Systems (PCSs)

c₁?0 c₁?1 c₂!1

c₁!0 c₁!1 c₂?1

←c₁

c₂→

0,m0,m1,n 1,n

I applysupersedingrewriting rules to channel contents:

a b→_# b a6b∈N

I modeling communications with QoS, e.g.

differentiated services (RFC2475)

I decidable: Fε₀-complete

Remark: Alternative Models

strict superseding Turing-powerful ordered channels with rules

a b→_sb a < b∈ N

overtaking Turing-powerful

ordered channels with rules a b →_o b a a6b∈N

priority queues decidable(VASS w. ordered 0-tests)

unordered channels, maximal priority messages read first

Remark: Alternative Models

strict superseding Turing-powerful ordered channels with rules

a b→_sb a < b∈ N

overtaking Turing-powerful

ordered channels with rules a b →_o b a a6b∈N

priority queues decidable(VASS w. ordered 0-tests)

unordered channels, maximal priority messages read first

Losing as an Embedding

I losing rules define a quasi-ordering←−^∗ _∗ over M^∗

I can be restated assubstring embedding:

xv_∗ y⇔^def

x=m₁· · ·m_`,

y=z₁m₁z₂· · ·z_{`</sub>m<sub>`}z_`+1 withz₁, ..∈ M^∗

I examples:

I 201v_∗22011

I 120v_∗10210

I ∀y∈M^∗.εv_∗y

Superseding as an Embedding

I ifd∈N, writeΣ_d =^def {0, . . . ,d}

I superseding rules define a quasi-ordering←−^∗ _# overΣ^∗_d

I can be restated aspriority embedding:

xv_p y⇔^def x=a₁· · ·a`,y=z<sub>1</sub>a<sub>1</sub>z<sub>2</sub>· · ·z`a`,∀i.z_i ∈Σ^∗_a

i I examples:

I 201v_p22011

I 1206v_p10210

I εv_pyiffy=ε

Priority Embedding is Well

c.f. related orderings of Sch ¨utte and Simpson (1985)

Definition (wqo)

A quasi-order(A,6^A)iswell⇔^def in any infinite sequencex₀,x₁, . . . overA, there existi < js.t.

x_i 6A x_j.

Theorem

(Σ^∗_d,v_p) is a wqo.

I proof by induction overd

I nested applications of Higman’s Lemma

PCSs are Well-Structured

(Abdullaet al., 2000; Finkel and Schnoebelen, 2001)

For a PCS with state setQandmchannels:

transition system (Q×(Σ^∗_d)^m,→)with

superseding steps or perfect steps wqo (Q×(Σ^∗_d)^m,v_p) by Dickson’s Lemma monotonicity ∀(p, ¯x),(q, ¯x⁰),(p, ¯y)∈Q×(Σ^∗_d)^m,

if (p, ¯x)→(q, ¯x⁰) and ¯xv_p y,¯ then∃y¯⁰ ∈(Σ^∗_d)^m, ¯yv_py¯ and

(p, ¯y)→(q, ¯y⁰).

Generic Algorithms

for Reachability, Inevitability, etc.

PCSs are Well-Structured

(Abdullaet al., 2000; Finkel and Schnoebelen, 2001)

For a PCS with state setQandmchannels:

transition system (Q×(Σ^∗_d)^m,→)with

superseding steps or perfect steps wqo (Q×(Σ^∗_d)^m,v_p) by Dickson’s Lemma monotonicity ∀(p, ¯x),(q, ¯x⁰),(p, ¯y)∈Q×(Σ^∗_d)^m,

if (p, ¯x)→(q, ¯x⁰) and ¯xv_p y,¯ then∃y¯⁰ ∈(Σ^∗_d)^m, ¯yv_py¯ and

(p, ¯y)→(q, ¯y⁰).

Generic Algorithms

for Reachability, Inevitability, etc.

Fast-Growing Complexity Classes

(Schmitz and Schnoebelen, 2012)

Ordinal-indexed complexity hierarchy inside R:

Elementary

Primitive Recursive Multiply Recursive

ε₀-Ordinal Recursive

F₃

Fω

Fω^ω

Fε₀

Complexity of PCS Problems

Theorem

Reachability and Termination in PCSs are Fε₀-complete.

upper bound usinglength function theorems for applications of Higman’s Lemma

(Schmitz and Schnoebelen, 2011)

lower bound reduction from acceptance of a Turing machine working inH^ε0(n) space

Lower Bound: Hardy Functions

fundamental sequences (λ(x))_x

for limit ordinals λinε₀+1: λ(x)< λ with lim_x→ωλ(x) =λ

Example

ω(x) =x+1 , ω^ω·2(x) =ω^ω+x+1,

(ε₀)(x) =Ωx+1

=def ω^ω···

ω x+1 stackedω’s

Lower Bound: Hardy Functions

Hardy functions (H^α)α6ε₀

H⁰(x)=^def x, H^α+1(x)=^def H^α(x+1), H^λ(x) =^def H^λ(x)(x).

Example

Hⁿ(x) = x+n, H^ω(x) = 2x+1 ,

H^ω2(x) = 2^x+1(x+1) −1 , H^ω3 non elementary, H^ωω Ackermannian,

H^ε0 not provably total in Peano arithmetic

Lower Bound: Hardy Computations

rewrite system over (ε₀+1)×ω:

α+1,x −→^H α,x+1 λ,x −→^H λ(x),x

computations α₀,x₀−→^H α₁,x₁−→ · · ·^H −→^H α_n,x_n

I preserveH^αi(x_i)

I in particular ifα_n=0 then x_n=H^α0(x₀)

Lower Bound: Encoding Ordinals

α∈ Ω_d+1 3 ω²+1 t(α)∈T_d+1

s_d(α)∈Σ^∗_d 222 1122

s_d ω^α1+· · ·+ω^αn

=s_d−1(α₁)d. . .s_d−1(α_n)d

Lower Bound: Encoding Ordinals

α∈ Ω_d+1 3 ω²+1 t(α)∈T_d+1

s_d(α)∈Σ^∗_d 222 1122

s_d ω^α1+· · ·+ω^αn

=s_d−1(α₁)d. . .s_d−1(α_n)d

Proposition (Robustness)

Ifs_d(α)v_ps_d(β), then∀x, H^α(x)6H^β(x).

Lower Bound: Weak Hardy Computations

Implement Hardy stepsα,n→β,mas a PCS:

I work on string encodings: s_d(α),n−→^H _#s_d(β⁰),m⁰

I weak: sd(β⁰) v_p sd(β) andm⁰6m, but the perfect behaviour is possible

I also forinversestepss_d(β),m ^H

-1

−−→_# s_d(α),n⁰ withsd(α⁰)v_p sd(α)andn⁰ 6n

Lower Bound: Weak Hardy Computations

Implement Hardy stepsα,n→β,mas a PCS:

I work on string encodings: s_d(α),n−→^H _#s_d(β⁰),m⁰

I weak: sd(β⁰) v_p sd(β) andm⁰6m, but the perfect behaviour is possible

I also forinversestepss_d(β),m ^H

-1

−−→_# s_d(α),n⁰ withsd(α⁰)v_p sd(α)andn⁰ 6n

Hinting at PCS implementation (1)

o: 3 3 4 5 4 5 $ theordinaltermω^ω2+ω^ω

c: 0 0 0 0 $ thecountervalue 4

t: $ thetemporarystorage

Storing data in channels: $hashighest priority

Hinting at PCS implementation (2)

p o?!x∈P_d o?d o?! $ q r c?! 0

c! 0 c?! $

P_d =^def ε+ (P_d−1)^∗d = allcorrect encodings Implementing Hardy step (α+1,n) −→^H (α,n+1)

ydd, 0ⁿ 7→ yd, 0ⁿ⁺¹

Hinting at PCS implementation (3)

Going froms_d(λ)tos_d(λ(n))

E.g. s₅(ω^ω4) = 333345, also written 333 3^a5 Thens₅(ω^ω3·n) = (3334)ⁿ5

Hinting at PCS implementation (3)

Going froms_d(λ)tos_d(λ(n))

E.g. s₅(ω^ω4) = 333345, also written 333 3^a5 Thens₅(ω^ω3·n) = (3334)ⁿ5

Writexas

y_dy_d−1. . .y_aa(a+1). . .d Thenx(n) is

y_dy_yd−1. . .y_a+1(y_a(a+1))ⁿ(a+2)^ad

Hinting at PCS implementation (4)

p_a

q_a ra

o?!y_d· · ·y_a+1∈P_d· · ·P_a+1

o?y_a∈P_a; t!y_a o?a(a+1); t!(a+1)

t?! $ c?! $ t?u t?! $

o?!(a+2)^ad$

c?! 0 t?!u$; o!u

Implementing Hardy step (λ,n)−→^H (λ(n),n)

Lower Bound: Wrapping Up

reduction from a Turing machineM working in spaceH^ε0(n) =H^Ωd(n)for d=n+1.

simulate Mwith budgetB

q₀ p₀ p_h q_h

Ω_d,n−→^H#· · ·−→^H# 0,B 0,B^0H

-1

−→#· · ·−→^H-1#α,n⁰

I robustness: H^Ωd(n)>B>B⁰>H^α(n⁰)

I coverability: α=Ωd∧n=n⁰

I impliesperfect simulation

Lower Bound: Wrapping Up

reduction from a Turing machineM working in spaceH^ε0(n) =H^Ωd(n)for d=n+1.

simulate Mwith budgetB

q₀ p₀ p_h q_h

Ω_d,n−→^H#· · ·−→^H# 0,B 0,B^0H

-1

−→#· · ·−→^H-1#α,n⁰

I robustness: H^Ωd(n)>B>B⁰>H^α(n⁰)

I coverability: α=Ωd∧n=n⁰

I impliesperfect simulation

Lower Bound: Wrapping Up

reduction from a Turing machineM working in spaceH^ε0(n) =H^Ωd(n)for d=n+1.

simulate Mwith budgetB

q₀ p₀ p_h q_h

Ω_d,n−→^H#· · ·−→^H# 0,B 0,B^0H

-1

−→#· · ·−→^H-1#α,n⁰

I robustness: H^Ωd(n)>B>B⁰>H^α(n⁰)

I coverability: α=Ωd∧n=n⁰

I impliesperfect simulation

Lower Bound: Wrapping Up

reduction from a Turing machineM working in spaceH^ε0(n) =H^Ωd(n)for d=n+1.

simulate Mwith budgetB

q₀ p₀ p_h q_h

Ω_d,n−→^H#· · ·−→^H# 0,B 0,B^0H

-1

−→#· · ·−→^H-1#α,n⁰

I robustness: H^Ωd(n)>B>B⁰>H^α(n⁰)

I coverability: α=Ωd∧n=n⁰

I impliesperfect simulation

Concluding Remarks

model priority channel systems ordering priority embedding

Perspectives

verifying PCSs regular model checking and acceleration

using PCSs reducing problems about other models (e.g. manipulating bounded depth trees and graphs)

Concluding Remarks

model priority channel systems ordering priority embedding

Perspectives

verifying PCSs regular model checking and acceleration

using PCSs reducing problems about other models (e.g. manipulating bounded depth trees and graphs)

References

Abdulla, P.A. and Jonsson, B., 1996. Verifying programs with unreliable channels.Inform. and Comput., 127(2):

91–101. doi:10.1006/inco.1996.0053.

Abdulla, P.A., ˇCer ¯ans, K., Jonsson, B., and Tsay, Y.K., 2000. Algorithmic analysis of programs with well quasi-ordered domains.Inform. and Comput., 160(1–2):109–127. doi:10.1006/inco.1999.2843.

C ´ec ´e, G., Finkel, A., and Purushothaman Iyer, S., 1996. Unreliable channels are easier to verify than perfect channels.Inform. and Comput., 124(1):20–31. doi:10.1006/inco.1996.0003.

Chambart, P. and Schnoebelen, Ph., 2008. The ordinal recursive complexity of lossy channel systems. InLICS 2008, pages 205–216. IEEE Press. doi:10.1109/LICS.2008.47.

Finkel, A. and Schnoebelen, Ph., 2001. Well-structured transition systems everywhere!Theor. Comput. Sci., 256 (1–2):63–92. doi:10.1016/S0304-3975(00)00102-X.

Schmitz, S. and Schnoebelen, Ph., 2011. Multiply-recursive upper bounds with Higman’s lemma. InICALP 2011, volume 6756 ofLNCS, pages 441–452. Springer. doi:10.1007/978-3-642-22012-8 35.

Schmitz, S. and Schnoebelen, Ph., 2012. Algorithmic aspects of WQO theory. Lecture notes.

http://cel.archives-ouvertes.fr/cel-00727025.

Sch ¨utte, K. and Simpson, S.G., 1985. Ein in der reinen Zahlentheorie unbeweisbarer Satz ¨uber endliche Folgen von nat ¨urlichen Zahlen.Arch. Math. Logic, 25(1):75–89. doi:10.1007/BF02007558.

Fundamental Sequences

Definition (Fundamental Sequences)

For limit ordinals inε₀+1:

(γ+ω^β+1)(x)=^def γ+ω^β·(x+1) (γ+ω^λ)(x)=^def γ+ω^λ(x)

(ε₀)(x)=^def Ω_x+1=^def ω^ω···

ω x+1 stackedω’s

Encodings

s_d:T_d+1→Σ^∗_d by induction ond:

s_d(•(t₁· · ·t_n))=^def

ε ifn=0,

s_d−1(t₁)d· · ·s_d−1(t_n)d otherwise.

s_d:Ω_d+1 →Σ^∗_d by induction ond:

s_d

Xⁿ

i=1

γ_i def

=s_d(γ₁)· · ·s_d(γ_n), s_d(ω^α) =^def s_d−1(α)d.