Register-Bounded Synthesis Ayrat Khalimov Orna - - PowerPoint PPT Presentation

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register bounded synthesis
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Register-Bounded Synthesis Ayrat Khalimov Orna - - PowerPoint PPT Presentation

Jun 03, 2023 116 likes β€’431 views

Register-Bounded Synthesis Ayrat Khalimov Orna Kupferman Universite libre de Bruxelles Hebrew University Belgium Israel Why study bounded synthesis from universal automata? why

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Register-Bounded Synthesis

Ayrat Khalimov Orna Kupferman Universite libre de Bruxelles Hebrew University Belgium Israel

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Why study bounded synthesis from universal automata?

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why register-bounded synthesis?

  • Not a limitation: designer usually knows the

sensible bound on the number of registers

  • Added benefit: small programs
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SLIDE 4

why universal register automata?

All computations (π‘—π‘œ0, 𝑝𝑣𝑒0) (π‘—π‘œ1, 𝑝𝑣𝑒1) (π‘—π‘œ2, 𝑝𝑣𝑒2) … satisfy a given specification. system

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SLIDE 5

why universal register automata?

All computations (π‘—π‘œ0, 𝑝𝑣𝑒0) (π‘—π‘œ1, 𝑝𝑣𝑒1) (π‘—π‘œ2, 𝑝𝑣𝑒2) … satisfy a given specification. Most specifications are derived from arbiter: βˆ€π‘’ ∈ 𝐸: 𝐇(π‘ π‘“π‘Ÿ ∧ 𝑗 = 𝑒 β†’ 𝐘 𝐆(π‘•π‘ π‘π‘œπ‘’ ∧ 𝑝 = 𝑒)) system

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SLIDE 6

why universal register automata?

All computations (π‘—π‘œ0, 𝑝𝑣𝑒0) (π‘—π‘œ1, 𝑝𝑣𝑒1) (π‘—π‘œ2, 𝑝𝑣𝑒2) … satisfy a given specification. Most specifications are derived from arbiter: βˆ€π‘’ ∈ 𝐸: 𝐇(π‘ π‘“π‘Ÿ ∧ 𝑗 = 𝑒 β†’ 𝐘 𝐆(π‘•π‘ π‘π‘œπ‘’ ∧ 𝑝 = 𝑒)) Universal register automata can express this. Nondeterministic -- cannot. system

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universal register automaton

  • Works on words in Ξ£ Γ— 𝐸 Γ— 𝐸 πœ•
  • Registers 𝑆 = {𝑠

1, … , 𝑠𝑙𝐡}, initialized 𝑀0

  • Transition function

𝑅 Γ— Ξ£ Γ— π‘ˆπ‘‘π‘’π‘— Γ— π‘ˆπ‘‘π‘’π‘ β†’ 2π‘…Γ—π΅π‘‘π‘•π‘œ

Arbiter specification (coBuchi)

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SLIDE 8

universal register automaton

  • Word:

π‘ π‘“π‘Ÿ,1 Β¬π‘•π‘ π‘π‘œπ‘’,0 Β¬π‘ π‘“π‘Ÿ,2 π‘•π‘ π‘π‘œπ‘’,1 Β¬π‘ π‘“π‘Ÿ,3 Β¬π‘•π‘ π‘π‘œπ‘’,1 …

  • Run-graph:

π‘Ÿ0, 0 π‘Ÿ0, 0 π‘Ÿ0, 0 π‘Ÿ1, 1 π‘Ÿ0, 0

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register transducer

  • Reads a letter in Σ𝐽 Γ— 𝐸
  • Outputs a letter in Σ𝑃 Γ— 𝐸
  • Registers 𝑆 = {𝑠

1, … , 𝑠𝑙𝑑}, initialized with 𝑀0

  • Transition function

𝑇 Γ— Σ𝐽 Γ— π‘ˆπ‘‘π‘’π‘— β†’ 𝑇 Γ— Σ𝑃 Γ— 𝑆 Γ— π΅π‘‘π‘•π‘œ

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arbiter

  • Input: π‘ π‘“π‘Ÿ, 1

Β¬π‘ π‘“π‘Ÿ, 2 Β¬π‘ π‘“π‘Ÿ, 3 …

  • Run: (s0, 0)

¬𝑕,0 𝑑1, 1 𝑕,1 𝑑0, 1 ¬𝑕,1 𝑑0, 1 …

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bounded synthesis problem

Given:

  • Σ𝐽, Σ𝑃
  • universal register automaton 𝐡 over Σ𝐽 Γ— Σ𝑃 Γ—

𝐸 Γ— 𝐸

  • the number 𝑙𝑑 of system registers

Return:

  • 𝑙𝑑-register transducer π‘ˆ such that π‘ˆ ⊨ 𝐡, or

β€œunrealizable” Bounded synthesis problem is solvable in EXP in 𝑅 and 𝑙𝑑, and 2EXP in 𝑙𝐡.

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abstraction 𝑩′

π‘ˆ: 𝑇 Γ— Σ𝐽 Γ— π‘ˆπ‘‘π‘’π‘—

𝑑 β†’ 𝑇 Γ— Σ𝑃 Γ— 𝑆𝑑 Γ— π΅π‘‘π‘•π‘œπ‘‘

π‘ˆβ€²: 𝑇 Γ— Σ𝐽

β€² β†’ 𝑇 Γ— Σ𝑃 β€²

We construct register-less automaton 𝐡′ with 𝑅′ Γ— Σ𝐽

β€² Γ— Σ𝑃 β€²

β†’ 2𝑅′ such that π‘ˆβ€² ⊨ 𝐡′ iff π‘ˆ ⊨ 𝐡 for every π‘ˆ or π‘ˆβ€².

πš»π‰ 𝑼𝒕𝒖𝒋

𝒕

𝑩𝒕𝒉𝒐𝒕 πš»π‘· 𝑼′ 𝑺𝒕

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abstracting a single transition

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abstracting a single transition

2 2

2 2 2

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abstracting a single transition

2 2

2 2 2 1 1 1

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abstracting a single transition

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abstracting a single transition

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abstracting a single transition

Two possibilities:

  • 𝑗 = π‘ π‘ˆ
  • 𝑗 β‰  π‘ π‘ˆ
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abstracting a single transition

Two possibilities:

  • 𝑗 = π‘ π‘ˆ
  • 𝑗 β‰  π‘ π‘ˆ
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abstracting a single transition

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  • ne 𝑒𝑑𝑒𝑑 can induce several 𝑒𝑑𝑒𝐡
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bisimulation property of the abstraction

π‘Ÿ, 𝜌 𝑒𝑑𝑒𝑗

𝑑, 𝑠 𝑑, π‘π‘‘π‘•π‘œπ‘‘

π‘Ÿβ€², πœŒβ€² transition

  • f 𝐡′ and

some π‘ˆβ€²

π‘Ÿ, 𝑀𝐡, 𝑀𝑑

𝒋, 𝒑

𝒓′, π’˜π‘©

β€² , π’˜π’• β€²

transition

  • f 𝐡 and

some π‘ˆ

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bisimulation property of the abstraction

π‘Ÿ, 𝜌 𝒖𝒕𝒖𝒋

𝒕, 𝒔𝒕, 𝒃𝒕𝒉𝒐𝒕

𝒓′, 𝝆′ transition

  • f 𝐡′ and

some π‘ˆβ€²

π‘Ÿ, 𝑀𝐡, 𝑀𝑑

𝑗, 𝑝

π‘Ÿβ€², 𝑀𝐡

β€² , 𝑀𝑑 β€²

transition

  • f 𝐡 and

some π‘ˆ

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SLIDE 24
  • For every π‘ˆ or π‘ˆβ€²: π‘ˆβ€² ⊨ 𝐡′ iff π‘ˆ ⊨ 𝐡
  • Recall that synthesis is EXP in |𝑅′|
  • 𝑅′ = 𝑅 Γ— Ξ , where Ξ  is the set of partitions of

𝑆 = 𝑆𝑑 βˆͺ 𝑆𝐡

  • Ξ  is EXP in (𝑙𝑑 + 𝑙𝐡)

=> synthesis is 2EXP in 𝑙𝑑 and 𝑙𝐡 But system partitions behave deterministically => only EXP in 𝒍𝒕

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Part 2

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Environments have their own limits. Let them be register transducers. π‘“π‘œπ‘€||𝑑𝑧𝑑 = π‘—π‘œ0, 𝑝𝑣𝑒0 π‘—π‘œ1, 𝑝𝑣𝑒1 … system

register transducer

environ ment

register transducer

Note: the number of values is at most 𝑙𝑑 + 𝑙𝑓.

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SLIDE 27

env-sys-bounded synthesis problem

Given:

  • Σ𝐽, Σ𝑃
  • universal register automaton 𝐡 with Σ𝐽 and Σ𝑃
  • the number 𝑙𝑑 of system registers
  • the number 𝒍𝒇 of environment registers

Return:

  • 𝑙𝑑-register transducer 𝑑𝑧𝑑 such that

π‘“π‘œπ‘€||𝑑𝑧𝑑 ⊨ 𝐡 for every 𝑙𝑓-register environment,

  • r β€œunrealizable”

Env-sys-bounded synthesis problem is solvable in EXP in |𝑅| and 𝑙𝑑, 2EXP in 𝑙𝐡 and 𝒍𝒇.

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idea of the abstraction

π‘Ÿ0, 𝑠

𝐡 = π‘ π‘ˆ = 𝑠 𝑓

(𝑑𝑒𝑝𝑠𝑓𝑓)

π‘Ÿ0, 𝑠

𝐡 = π‘ π‘ˆ = 𝑠 𝑓

π‘Ÿ0, 𝑠

𝐡 = π‘ π‘ˆ β‰  𝑠 𝑓

similarly…

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conclusion

  • Cleaner algorithm

=> tighter complexity analysis (only EXP in 𝑙𝑑)

  • Solution to the environment-system-bounded

synthesis problem In the full version:

  • Non-determinacy
  • Hierarchy of system and environment power
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