Copyless Cost Register Automata Bounded Ambiguity vs Determinism

(1)

Copyless Cost Register Automata Bounded Ambiguity vs Determinism

Benjamin Monmege

joint work with Th´eodore Lopez and Jean-Marc Talbot

Laboratoire d’Informatique et Syst`emes — Aix-Marseille University

December 10, 2019

1/21

(2)

Quantitative models

Classical Framework Quantitative Framework Q: ”Is it possible ?” Q: ”How much does it cost?”

A: Yes or No A: Weight Automaton

MSO logic

Weighted Automaton Weighted logic

Cost Register Automaton (CRA) Weights on a semiring:

Natural (N,+,×) Count ambiguity Tropical (N,min,+) Shortest path Boolean ({>,⊥},∨,∧) Classical setting

(3)

Cost Register Automaton (CRA) [Alur et al. LICS’13]

x, y←0 A y 0

x←2x y←4y

1

x←2x+1 y←2x+1

I States: A I Registers: x, y I Initial valuation I Final outputs

I Transitions: update registers I Deterministic

On input10100: x=0

y =0

−1

→ x=1 y=1

−0

→ x=2 y=4

−1

→ x=5 y=5

−0

→ x=10 y =20

−0

→ x=20 y=80

3/21

(4)

Cost Register Automaton (CRA) [Alur et al. LICS’13]

x, y←0 A y 0

x←2x y←4y

1

x←2x+1 y←2x+1

I States: A I Registers: x, y I Initial valuation I Final outputs

I Transitions: update registers I Deterministic

On input10100:

x=0 y =0

−1

→ x=1 y=1

−0

→ x=2 y=4

−1

→ x=5 y=5

−0

→ x=10 y =20

−0

→ x=20 y=80

(5)

Copyless CRA

In updates and final outputs, each register is used at most once.

x, y←0 A y 0

x←2x y←4y

1

x←2x+1 y←2x+1

7

x←0 A

t←1 xt

0

x←2x t←2t

1

x←2x+1 t←1

3

4/21

(6)

Known results on (copyless) CRA

I CRA over (+,×c) ≡WA over semiring(+,×)

I Equivalence undecidable for copyless CRA over the tropical semiring(N,min,+) [Almagor, Cadilhac, Mazowiecki, P´erez, 2018]

I Register minimisation for CRA over (Z,+c) [Alur,

Raghothaman, 2013] or (G,×c) with Gan infinitary group [Daviaud, Reynier, Talbot, 2016]

I Copyless CRA (unlike WA) are not closed under reverse over commutative semirings [Mazowiecki, Riveros, 2016]: subclass of bounded-alternation copyless CRAclosed under reverse, and for which deterministic look-ahead do not increase expressiveness

I Copyless CRA (WA over commutative semirings [Mazowiecki, Riveros, 2016]

(7)

Known results on (copyless) CRA

5/21

(8)

Known results on (copyless) CRA

(9)

Known results on (copyless) CRA

5/21

(10)

Known results on (copyless) CRA

I Copyless CRA (WA over commutative semirings

(11)

Unambiguous Non-Deterministic CRA

NCRA with at most one run for each input word.

A

x←0 B

x←0 x 0|x←2x

1|x←2x+1

0|x←4x

I Modelling via regular look-ahead

Open problem [Mazowiecki, Riveros 2016] copyless CRA vs unambiguous

copyless NCRA ?

Deterministic WA(Unamb. WA (Finitely-amb. WA( WA

6/21

(12)

Unambiguous Non-Deterministic CRA

A

x←0 B

x←0 x 0|x←2x

1|x←2x+1

0|x←4x

Open problem [Mazowiecki, Riveros 2016]

copyless CRA vs unambiguous copyless NCRA ?

(13)

Unambiguous Non-Deterministic CRA

A

x←0 B

x←0 x 0|x←2x

1|x←2x+1

0|x←4x

Open problem [Mazowiecki, Riveros 2016]

copyless CRA vs unambiguous copyless NCRA ?

6/21

(14)

Main Result

Theorem

Unambiguous copyless NCRA are as expressive as copyless CRA

Proof scheme

Unambiguous

Copyless NCRA -less CRA Copyless CRA

(15)

Flow Graph

Flow of registers of a CRA along a run.

x, y←0 A y 0

x←2x y←4y

1

x←2x+1 y←2x+1 On input word10100:

x=0 y =0

−1

→ x=1 y=1

−0

→ x=2 y=4

−1

→ x=5 y=5

−0

→ x=10 y =20

−0

→ x=20 y=80

x

y

Ω

0 1 2 3 4 5

layer

8/21

(16)

Flow Graph

Flow of registers of a CRA along a run.

x, y, z←ε A y 0

x←xy y ←a z←zya

1

x←ε y←zb z←ε

On input word1010:

x y z

Ω

0 1 2 3 4

(17)

-less Flow Graph

Contains no diamonds.

Ω 3

Ω 7

9/21

(18)

-less CRA

Flow graph of each run is-less.

x, y, z←ε A y 0

x←xy y←a z←zya

1

x←ε y ←zb z←ε

3 x y z

· · ·

0 1 2 3 4 5

(19)

-less CRA

Flow graph of each run is-less.

x, y, z←ε A y 0

x←xy y←a z←zya

1

x←ε y←xzb z←ε

7 x y z

· · ·

0 1 2 3 4 5

10/21

(20)

Main Result

Theorem

Proof scheme

Unambiguous

Determinisation Unfolding + Reduction

(21)

Main Result

Theorem

Proof scheme

Unambiguous

Determinisation Unfolding + Reduction

11/21

(22)

Determinisation

x←0 A B

x←0 x 0|x←2x

1|x←2x+1

0|x←4x

(23)

Determinisation

x←0 A B

x←0 x 0|x←2x

1|x←2x+1

0|x←4x

{A, B}

x_A, x_B←0

0

xA←2xA

xB ←4xB

1

xA←2xA+1 x_B←2x_A+1

12/21

(24)

Determinisation

x←0 A B

x←0 x 0|x←2x

1|x←2x+1

0|x←4x

{A, B}

x_A, x_B←0 0

xA←2xA

x_B ←4x_B

1

(25)

Determinisation

x←0 A B

x←0 x 0|x←2x

1|x←2x+1

0|x←4x

{A, B}

x_A, x_B←0 0

xA←2xA

x_B ←4x_B

1

12/21

(26)

Determinisation

x←0 A B

x←0 x 0|x←2x

1|x←2x+1

0|x←4x

{A, B}

x_A, x_B←0 xB

0

xA←2xA

x_B ←4x_B

1

(27)

Determinisation

x←0 A B

x←0 x 0|x←2x

1|x←2x+1

0|x←4x

x, y←0 A y 0

x←2x y←4y

1

x←2x+1 y←2x+1

12/21

(28)

Flow graph with terms

x, y←0 A y 0

x←2x y←4y

1

x←2x+1 y←2x+1

On input word10100:

x

y Ω

0 1 2 3 4 5

×

2 +

× 2 0

1

×

4 ×

4 +

× 2 ×

2 +

× 2 0

1 1

(29)

Flow graph with terms

x, y←0 A y 0

x←2x y←4y

1

x←2x+1 y←2x+1

On input word10100:

x

y Ω

0 1 2 3 4 5

×

4 ×

4 +

× 2 ×

2 +

× 2 0

1 1

13/21

(30)

Unfolding

x, y←0 A y

0

x←2x y←4y

1

x←2x+1 y←2x+1

0 0

0

2x+1 2x+1

2·0+1

2x 4x

2x+1 2x+1

2·(2·0+1) +1

2x 4x

4·(2·0 + 1)

· · ·

1

0

1

0

(31)

Unfolding

x, y←0 A y

0

x←2x y←4y

1

x←2x+1 y←2x+1

ωx ωy

ωy

ωx, ωy←0

2x+1 2x+1

2·0+1

2x 4x

4·0

2x+1 2x+1

2·(2·0+1) +1

2x 4x

4·(2·0 + 1)

· · ·

1

0

1

0

14/21

(32)

Unfolding

x, y←0 A y

0

x←2x y←4y

1

x←2x+1 y←2x+1

ωx ωy

ωy

ωx, ωy←0

α₁x+γ₁ α₂x+γ₂

α2ωx+γ2

2x 4x

2x+1 2x+1

2·(2·0+1) +1

2x 4x

4·(2·0 + 1)

· · ·

1

α¹, α²←2 γ¹, γ²←1

0

1

0

(33)

Unfolding

A

x, y←0 y

0

x←2x y←4y

1

x←2x+1 y←2x+1

ωx ω_y

ωy

ωx, ωy←0

α₁x+γ₁ α2x+γ2

α2ωx+γ2

α₁x+γ₁ α₂y+γ₂

α2ωy+γ2

2x+1 2x+1

2·(2·0+1) +1

2x 4x

4·(2·0 + 1)

· · ·

1

α¹, α²←2 γ¹, γ²←1

0

α₁←2 α₂←4 γ₁, γ₂←0

1

0

14/21

(34)

Unfolding

x, y←0 A y

0

x←2x y←4y

1

x←2x+1 y←2x+1

ωx ωy

ωy

ωx, ωy←0

α1x+γ1 α₂x+γ₂

α2ωx+γ2

α₁x+γ₁ α2y+γ2

α₃x+γ₃ α₄x+γ₄

α4(α1ωx+γ1)+γ4

2x 4x

4·(2·0 + 1)

· · ·

1

α¹, α²←2 γ¹, γ²←1

0

α₁←2 α₂←4 γ₁, γ₂←0

1

α³, α⁴←2 γ³, γ⁴←1

0

(35)

Unfolding

x, y←0 A y

0

x←2x y←4y

1

x←2x+1 y←2x+1

ωx ωy

ωy

ωx, ωy←0

α1x+γ1 α₂x+γ₂

α2ωx+γ2

α₁x+γ₁ α2y+γ2

α2ωy+γ2

α₃x+γ₃ α₄x+γ₄

α4(α1ωx+γ1)+γ4

α3x+γ3 α₄x+γ₄

α4(α2ωx+γ2)+γ4

· · ·

1

α¹, α²←2 γ¹, γ²←1

0

α₁←2 α₂←4 γ₁, γ₂←0

1

α³, α⁴←2 γ³, γ⁴←1

0

α3←2 α4←4 γ3, γ4←0

14/21

(36)

Removal of Copyless Layers

x y

0 1 2

+

× α3 x

γ₃

+

× α₁ x

γ1

x y

0 1

+

× α3 +

× α₁ x

γ₁

γ₃ +

× α⁰₁ x

γ₁⁰

≡

α⁰₁=α₃α₁ γ₁⁰ =α3γ1+γ3

(37)

Removal of Copyless Layers

x y

0 1 2

+

× α3 x

γ₃

+

× α₁ x

γ1

x y

0 1

+

× α3 +

× α₁ x

γ₁ γ₃

+

× α⁰₁ x

γ₁⁰

≡

α⁰₁=α₃α₁ γ₁⁰ =α3γ1+γ3

15/21

(38)

Removal of Copyless Layers

x y

0 1 2

+

× α3 x

γ₃

+

× α₁ x

γ1

x y

0 1

+

× α3 +

× α₁ x

γ₁

γ₃ +

× α⁰₁ x

γ₁⁰

≡

α⁰₁=α₃α₁ γ⁰ =α γ +γ

(39)

Reduced Unfolding

A

x, y←0 y

0

x←2x y←4y

1

x←2x+1 y←2x+1

ωx ωy

ωy

α2ωx+γ2

α2ωy+γ2

α₃x+γ₃ α4x+γ4

α4(α1ωx+γ1)+γ4

α4(α2ωx+γ2)+γ4

· · ·

1

α¹, α²←2 γ¹, γ²←1

0

α₁←2 α₂←4 γ₁, γ₂←0

1

α³, α⁴←2 γ³, γ⁴←1

0

α₃←2 α₄←4 γ3, γ4←0

1

ωx←α1ωx+γ1

α1, α2←2 γ1, γ2←1

0

α1←2α1

γ1←2γ1

α2←4α2

γ2←4γ2

ωx, ωy←0

16/21

(40)

Reduced Unfolding

A

x, y←0 y

0

x←2x y←4y

1

x←2x+1 y←2x+1

ωx ωy

ωy

α2ωx+γ2

α4(α1ωx+γ1)+γ4

α4(α2ωx+γ2)+γ4

· · ·

1

α¹, α²←2 γ¹, γ²←1

0

α₁←2 α₂←4 γ₁, γ₂←0

1

α³, α⁴←2 γ³, γ⁴←1

0

α₃←2 α₄←4 γ3, γ4←0 1

ωx←α1ωx+γ1

α1, α2←2 γ1, γ2←1

0

α1←2α1

γ1←2γ1

α2←4α2

γ2←4γ2

ωx, ωy←0

(41)

Reduced Unfolding

A

x, y←0 y

0

x←2x y←4y

1

x←2x+1 y←2x+1

ωx ωy

ωy

α2ωx+γ2

α2ωy+γ2

α4(α1ωx+γ1)+γ4

α4(α2ωx+γ2)+γ4

· · ·

1

α¹, α²←2 γ¹, γ²←1

0

α₁←2 α₂←4 γ₁, γ₂←0

1

α³, α⁴←2 γ³, γ⁴←1

0

α₃←2 α₄←4 γ3, γ4←0 1

ωx←α1ωx+γ1

α1, α2←2 γ1, γ2←1

0

α1←2α1

γ1←2γ1

α2←4α2

γ2←4γ2

ωx, ωy←0

16/21

(42)

Reduced Unfolding

A

x, y←0 y

0

x←2x y←4y

1

x←2x+1 y←2x+1

ωx ωy

ωy

α2ωx+γ2

α3x+γ3 α₄x+γ₄

α4(α1ωx+γ1)+γ4

α4(α2ωx+γ2)+γ4

· · ·

1

α¹, α²←2 γ¹, γ²←1

0

α₁←2 α₂←4 γ₁, γ₂←0

1

α³, α⁴←2 γ³, γ⁴←1

0

α₃←2 α₄←4 γ3, γ4←0

1

ωx←α1ωx+γ1

α1, α2←2 γ1, γ2←1

0

α1←2α1

γ1←2γ1

α2←4α2

γ2←4γ2

ωx, ωy←0

(43)

Reduced Unfolding

x, y←0 A y

0

x←2x y←4y

1

x←2x+1 y←2x+1

ω_x ωy

ωx, ωy←0

ωy

ωx ω_y

α2ωx+γ2

1

α1, α2←2 γ1, γ2←1 0

ωx←2ωx

ωy←2ωy

1

ωx←α1ωx+γ1

α1, α2←2 γ1, γ2←1

0

α1←2α1

γ1←2γ1

α2←4α2

γ2←4γ2

16/21

(44)

Problem 1: Removal of layers make copies

α⁰₁ =α₃α₁ γ₁⁰ =α₃γ₁+γ₃

... but each coefficient is copied only a bounded number of times (proof not too difficult)

Unambiguous

Copyless NCRA -less CRA Bounded-copy CRA Copyless CRA

Determinisation Unfolding + Reduction Initiate enough copies

(45)

Problem 1: Removal of layers make copies

α⁰₁ =α₃α₁ γ₁⁰ =α₃γ₁+γ₃

Unambiguous

17/21

(46)

Problem 1: Removal of layers make copies

α⁰₁ =α₃α₁ γ₁⁰ =α₃γ₁+γ₃

Unambiguous

(47)

Bounded copyness is not trivial...

x, y, z←ε A y 0

1

On input word1010· · ·:

x y z

· · ·

0 1 2 3 4

18/21

(48)

Problem 2: Abstracted terms with multiple registers

More intricate reshaping into normal forms... more copies

s(x, L+1)

α1(y, L)β1+γ1

αtβ+γ s(y, L)

α⁰1(z, L)β⁰1+γ1⁰

ω

s(z, L)

α2tβ2+γ2 ω2

α×+γ α+ +γ

α x+γ ω

×

α y+γ α+ +γ α×+γ α y+γ α x+γ

α z+γ

α z+γ α×+γ α s+γ α r+γ

α+ +γ α r+γ

ω

α s+γ ω

(49)

Problem 2: Abstracted terms with multiple registers

More intricate reshaping into normal forms... more copies α×+γ

α+ +γ α x+γ

ω

×

α y+γ α+ +γ α×+γ α y+γ α x+γ

α z+γ

α z+γ α×+γ α s+γ α r+γ

α+ +γ α r+γ

ω

α s+γ ω

19/21

(50)

Problem 3: Removing copyless layers is not sufficient

x, y, z←ε A y 0

1

x y z

· · ·

0 1 2 3

I The new copy of y teaches us that the previous copies of y cannot be copied many times anymore, because of the -less hypothesis...

I We find a notion of primarily-copyless layers, that we remove, then proving (much more delicate) that this again creates only a bounded number of copies of coefficients

(51)

Problem 3: Removing copyless layers is not sufficient

x, y, z←ε A y 0

1

x y z

· · ·

0 1 2 3

I We find a notion of primarily-copyless layers, that we remove, then proving (much more delicate) that this again creates only a bounded number of copies of coefficients

20/21

(52)

Problem 3: Removing copyless layers is not sufficient

x, y, z←ε A y 0

1

x y z

· · ·

0 1 2 3

I We find a notion of primarily-copyless layers, that we remove,

(53)

Summary

Theorem: Results

copyless CRA = Unambiguous copyless NCRA

Theorem: More results

= finitely-amb. NCRA(linearly-amb. NCRA Perspectives:

I We build a bounded-copy CRA with an exponential number of copies (and an exponential number of states): this leads to a copyless CRA with a doubly-exponential number of registers... Improvement/optimal?

I Minimisation of registers for copyless CRA in a semiring I Adapt the approach for copyless CRA over (+,×c)?

I Biggest subclass of copyless CRA that is closed under reverse Thank you!

21/21

(54)

Summary

= finitely-amb. NCRA(linearly-amb. NCRA

Perspectives:

I We build a bounded-copy CRA with an exponential number of copies (and an exponential number of states): this leads to a copyless CRA with a doubly-exponential number of registers... Improvement/optimal?

(55)

Summary

I We build a bounded-copy CRA with an exponential number of copies (and an exponential number of states): this leads to a copyless CRA with a doubly-exponential number of registers...

Improvement/optimal?

21/21

(56)

Summary

I Minimisation of registers for copyless CRA in a semiring

I Adapt the approach for copyless CRA over (+,×c)?

(57)

Summary

21/21

(58)

Summary

I Biggest subclass of copyless CRA that is closed under reverse

Thank you!

(59)

Summary

21/21