Mechanized Refinement of Communication Models with TLA+

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Mechanized Refinement of Communication Models with

TLA+

Florent Chevrou, Aurélie Hurault, Philippe Mauran, Philippe Quéinnec

To cite this version:

Florent Chevrou, Aurélie Hurault, Philippe Mauran, Philippe Quéinnec. Mechanized Refinement of

Communication Models with TLA+. 5th International Conference on Abstract State Machines, Alloy,

B, TLA, VDM, and Z (ABZ 2016), May 2016, Linz, Austria. pp. 312-318. �hal-01535944�

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Eprints ID : 16917

The contribution was presented at ABZ 2016 :

http://www.cdcc.faw.jku.at/ABZ2016/

To cite this version :

Chevrou, Florent and Hurault, Aurélie and Mauran,

Philippe and Quéinnec, Philippe Mechanized Refinement of

Communication Models with TLA+. (2016) In: 5th International

Conference on Abstract State Machines, Alloy, B, TLA, VDM, and Z

(ABZ 2016), 23 May 2016 - 27 May 2016 (Linz, Austria).

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▼❡❝❤❛♥✐③❡❞ ❘❡✜♥❡♠❡♥, ♦❢ ❈♦♠♠✉♥✐❝❛,✐♦♥

▼♦❞❡❧2 ✇✐,❤ ❚▲❆

+ ❋❧♦#❡♥& ❈❤❡✈#♦✉✱ ❆✉#-❧✐❡ ❍✉#❛✉❧&✱ 1❤✐❧✐♣♣❡ ▼❛✉#❛♥✱ ❛♥❞ 1❤✐❧✐♣♣❡ ◗✉-✐♥♥❡❝ ■❘■❚ ✕ ❯♥✐✈❡)*✐+, ❞❡ ❚♦✉❧♦✉*❡ ✷ )✉❡ ❈❛♠✐❝❤❡❧ ❋✲✸✶✵✵✵ ❚♦✉❧♦✉*❡✱ ❋)❛♥❝❡ ❤!!♣✿✴✴✇✇✇✳✐(✐!✳❢( ❆❜"#$❛❝#✳ ■♥ ❞✐*+)✐❜✉+❡❞ *②*+❡♠*✱ ❛*②♥❝❤)♦♥♦✉* ❝♦♠♠✉♥✐❝❛+✐♦♥ ✐* ♦❢+❡♥ ✈✐❡✇❡❞ ❛* ❛ ✇❤♦❧❡ ✇❤❡)❡❛* +❤❡)❡ ❛)❡ ❛❝+✉❛❧❧② ♠❛♥② ❞✐✛❡)❡♥+ ✐♥+❡)❛❝+✐♦♥ ♣)♦+♦❝♦❧* ✇❤♦*❡ ♣)♦♣❡)+✐❡* ❛)❡ ✐♥✈♦❧✈❡❞ ✐♥ +❤❡ ❝♦♠♣❛+✐❜✐❧✐+② ♦❢ ♣❡❡) ❝♦♠✲ ♣♦*✐+✐♦♥*✳ ❆ ❤✐❡)❛)❝❤② ♦❢ ❛*②♥❝❤)♦♥♦✉* ❝♦♠♠✉♥✐❝❛+✐♦♥ ♠♦❞❡❧*✱ ❜❛*❡❞ ♦♥ )❡✜♥❡♠❡♥+*✱ ✐* ❡*+❛❜❧✐*❤❡❞ ❛♥❞ ♣)♦✈❡♥ ✇✐+❤ +❤❡ ❚▲❆+ _{G)♦♦❢ ❙②*+❡♠✳} ❚❤❡ ✇♦)❦ *❡)✈❡* ❛* ❛ ✜)*+ *+❡♣ ✐♥ +❤❡ *+✉❞② ♦❢ +❤❡ *✉❜*+✐+✉❛❜✐❧✐+② ♦❢ +❤❡ ❝♦♠♠✉♥✐❝❛+✐♦♥ ♠♦❞❡❧* ✇❤❡♥ ✐+ ❝♦♠❡* +♦ ❝♦♠♣❛+✐❜✐❧✐+② ❝❤❡❝❦✐♥❣✳

✶ ■♥#$♦❞✉❝#✐♦♥

1#♦♣❡#&✐❡7 ♦❢ ❞✐7&#✐❜✉&❡❞ 7②7&❡♠7 ❛#❡ ❞✐#❡❝&❧② ✐♠♣❛❝&❡❞ ❜② &❤❡ ✐♥&❡#❛❝&✐♦♥ ♣#♦&♦✲ ❝♦❧ ✐♥ ✉7❡✳ ❯♥❧✐❦❡ ✐♥ 7②♥❝❤#♦♥♦✉7 ❝♦♠♠✉♥✐❝❛&✐♦♥✱ &❤❡ ❞❡❝♦✉♣❧✐♥❣ ♦❢ 7❡♥❞ ❛♥❞ #❡✲ ❝❡✐✈❡ ❡✈❡♥&7 ✐♥ ❛7②♥❝❤#♦♥♦✉7 ❝♦♠♠✉♥✐❝❛&✐♦♥ ❛❧❧♦✇7 ❢♦# ♠❛♥② ♦#❞❡#✐♥❣ 7&#❛&❡❣✐❡7 ❛♥❞ &❤✉7✱ ❝♦♠♠✉♥✐❝❛&✐♦♥ ♠♦❞❡❧7✳ ❨❡&✱ &❤❡7❡ ♠♦❞❡❧7 ❛#❡ 7❡❧❞♦♠ ❞✐7&✐♥❣✉✐7❤❡❞✳ ❋♦# ✐♥7&❛♥❝❡✱ &❤❡ ♠✉❧&✐♣❧❡ ✈❛#✐❛&✐♦♥7 ♦❢ ❋■❋❖ ❝♦♠♠✉♥✐❝❛&✐♦♥ ❛#❡ 7❡❡♥ &♦ ❜❡ ✉7❡❞ ✐♥&❡#❝❤❛♥❣❡❛❜❧② ❞❡7♣✐&❡ ♦❢ &❤❡✐# ❢✉♥❞❛♠❡♥&❛❧ ❞✐✛❡#❡♥❝❡7✳ ■♥ ❬❈❍◗✶✺❪✱ &❤❡ ❝♦♥7❡J✉❡♥❝❡7 ♦♥ &❤❡ ❝♦♠♣❛&✐❜✐❧✐&② ♦❢ &❤❡ ❝♦♠♣♦7✐&✐♦♥ ♦❢ ♣❡❡#7 ✉♥❞❡# &❤❡7❡ ❝✐#✲ ❝✉♠7&❛♥❝❡7 ❤❛✈❡ ❜❡❡♥ ❤✐❣❤❧✐❣❤&❡❞ &❤❛♥❦7 &♦ &❤❡ ♠♦❞❡❧✐♥❣ ♦❢ 7✉❝❤ 7②7&❡♠7 ❛♥❞ ❝❧❛77✐❝ ❝♦♠♠✉♥✐❝❛&✐♦♥ ♠♦❞❡❧7 ✐♥ ❚▲❆+_❬_{▲❛♠✵✷}_{❪✳ ❑♥♦✇✐♥❣ ✇❤✐❝❤ 7✉❜7&✐&✉&✐♦♥7 ♦❢}

❝♦♠♠✉♥✐❝❛&✐♦♥ ♠♦❞❡❧7 ♣#❡7❡#✈❡ ❝♦♠♣❛&✐❜✐❧✐&② ✐7 ♦❢ ❣#❡❛& ✐♥&❡#❡7&✳ ❙♦♠❡ ♠♦❞❡❧7 ❤❛✈❡ 7✐♠♣❧❡# 7♣❡❝✐✜❝❛&✐♦♥7 ✇❤✐❝❤ ❡❛7❡ ❢♦#♠❛❧ 7&✉❞✐❡7 ❛♥❞ ♣#♦♦❢7 ♦❢ ❝♦♠♣❛&✐❜✐❧✐&② ✐♥ ♣#❛❝&✐❝❛❧ ❝❛7❡7✳ ❆7 ❛ ✜#7& 7&❡♣ ♦❢ &❤✐7 ✇♦#❦✱ ✇❡ ♣#♦♣♦7❡ ❤❡#❡ &♦ ❡①❤✐❜✐& &❤❡ #❡✜♥❡♠❡♥&7 ❜❡&✇❡❡♥ ❡❛❝❤ ♦❢ &❤❡ ♠♦❞❡❧7✳ ❚❤❡ ❤✐❡#❛#❝❤② ♦❢ #❡✜♥❡♠❡♥&7 ✐7 ❛ ❦❡② #❡7✉❧& ✐♥ &❤❡ ❢✉#&❤❡# 7&✉❞② ♦❢ &❤❡ ❝♦♠♠✉♥✐❝❛&✐♦♥ ♠♦❞❡❧7 ✇❤❡♥ ❝♦♠♣❛&✐❜✐❧✐&② ♦❢ ♣❡❡#7 ✐7 ✐♥✈♦❧✈❡❞✳ ❚❤❡ ♠♦❞❡❧7 ❛♥❞ &❤❡ 7&#✉❝&✉#❡ ♦❢ &❤❡✐# ❚▲❆+_{♠♦❞✉❧❡ ❛#❡ ✐♥&#♦✲}

❞✉❝❡❞ ✐♥ ❙❡❝&✐♦♥✷✳ ■♥ ❙❡❝&✐♦♥ ✸ &❤❡ ❛♣♣#♦❛❝❤ ❜❡❤✐♥❞ &❤❡ ♣#♦♦❢7 ♦❢ #❡✜♥❡♠❡♥& ✇✐&❤ &❤❡ ❚▲❆+_{1#♦♦❢ ❙②7&❡♠ ✐7 ❡①♣♦7❡❞ ❛❧♦♥❣ ✇✐&❤ &❤❡ ♦❜&❛✐♥❡❞ #❡7✉❧&7✳}

✷ ❙♣❡❝✐✜❝❛#✐♦♥

❲❡ ❝♦♥7✐❞❡# ♣♦✐♥&✲&♦✲♣♦✐♥& ♠❡77❛❣❡✲♣❛77✐♥❣ ❝♦♠♠✉♥✐❝❛&✐♦♥ &❤#♦✉❣❤ ❝❤❛♥♥❡❧7✳ ▼❡77❛❣❡7 ❝♦♥7✐7& ♦❢ ❛ ✉♥✐J✉❡ ✐❞ ❛♥❞ ♠❡&❛❞❛&❛✳ ❍✐7&♦#✐❡7 ♦❢ ♣❛7& 7❡♥& ♠❡77❛❣❡7 ❛#❡ ♣❛#& ♦❢ &❤❡ ♠❡&❛❞❛&❛ &♦ ❛❧❧♦✇ ❢♦# &❤❡ 7♣❡❝✐✜❝❛&✐♦♥ ♦❢ ♦#❞❡#✐♥❣ ♣#♦♣❡#&✐❡7✳

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❚❛❜❧❡ ✶✳ ❙♣❡❝✐✜❝❛'✐♦♥ ♦❢ '❤❡ ❈♦♠♠✉♥✐❝❛'✐♦♥ ▼♦❞❡❧2✳ ❚❤❡ ✧❙❡♥❞✧ ❛♥❞ ✧❘❡❝❡✐✈❡✧ ❝♦❧✉♠♥2 ♦♥❧② ❝♦♥'❛✐♥ '❤❡ ♠♦❞❡❧✲2♣❡❝✐✜❝ ❣✉❛;❞2 ♦♥ '❤❡ ❡♥❞ ❛♥❞ $❡❝❡✐✈❡ ❛❝'✐♦♥2 ✭✇❤❡;❡ ♠ ❞❡♥♦'❡2 '❤❡ ♠❡22❛❣❡ '♦ ❜❡ ;❡❝❡✐✈❡❞✮✱ ❛2 ✐♥ '❤❡ ❚▲❆+_{♠♦❞✉❧❡2✳ ❲❤❡♥ ❛♣♣❧✐❝❛❜❧❡✱ '❤❡} ❧❛2' ❝♦❧✉♠♥ ✐♥❢♦;♠❛❧❧② 2②♠❜♦❧✐③❡2 ❛♥ ✐♠♣❧❡♠❡♥'❛'✐♦♥ ❜❛2❡❞ ♦♥ E✉❡✉❡2✳ ▼♦❞❡❧ ❙♣❡❝✐✜❝❛'✐♦♥ ❙❡♥❞ ❘❡❝❡✐✈❡ ◗✉❡✉❡2 ▼❘❙❈ ▼❡22❛❣❡2 ❛;❡ ✐♠♠❡❞✐✲ ❛'❡❧② ❞❡❧✐✈❡;❡❞ ❛❢'❡; '❤❡✐; 2❡♥❞ ❬❈❇▼❚✾✻❪✳ ♥❡* = ∅ ⊤ size 1 ×1 ▼♥−♥ ●❧♦❜❛❧ ♦;❞❡;✐♥❣✳ ▼❡22❛❣❡2 ❛;❡ ❞❡❧✐✈❡;❡❞ ✐♥ '❤❡✐; 2❡♥❞ ♦;❞❡;✳ ⊤ ¬(∃♠_{♠✐❞(♠}2 ∈₂♥❡* :_{) ∈}_{♠❤❣(♠))} n n ×1 ▼1−♥ ▼❡22❛❣❡2 ❢;♦♠ '❤❡ 2❛♠❡ ♣❡❡; ❛;❡ ❞❡❧✐✈❡;❡❞ ✐♥ '❤❡✐; 2❡♥❞ ♦;✲ ❞❡;✳ ⊤ ¬(∃♠2∈♥❡* : ♠♣(♠2) =♠♣(♠) ∧♠✐❞(♠2) ∈♠❤❧(♠)) 1 n ×n ▼♥−1 ❖♥ ❛ ❣✐✈❡♥ ♣❡❡;✱ ♠❡22❛❣❡2 ❛;❡ ;❡❝❡✐✈❡❞ ✐♥ '❤❡✐; 2❡♥❞ ♦;❞❡;✳ ⊤ ¬(∃♠2 ∈♥❡* : ♠❝(♠2) ∈❧✐ *❡♥❡❞ ∧♠✐❞(♠2) ∈♠❤❣(♠)) n 1 ×n ▼❝❛✉'❛❧ ▼❡22❛❣❡2 ❛;❡ ❞❡❧✐✈❡;❡❞ ❛❝❝♦;❞✲ ✐♥❣ '♦ '❤❡ ❝❛✉2❛❧✐'② ♦❢ '❤❡✐; ❡♠✐22✐♦♥ ❬▲❛♠✼✽❪✳ ⊤ ¬(∃♠2∈♥❡* : ♠❝(♠2) ∈❧✐ *❡♥❡❞ ∧♠✐❞(♠2) ∈♠❤❝(♠)) ▼1−1 ▼❡22❛❣❡2 ❜❡'✇❡❡♥ '✇♦ ❞❡2✐❣✲ ♥❛'❡❞ ♣❡❡;2 ❛;❡ ❞❡❧✐✈❡;❡❞ ✐♥ '❤❡✐; 2❡♥❞ ♦;❞❡;✳ ⊤ ¬(∃♠2∈♥❡* : ♠♣(♠2) =♠♣(♠) ∧♠❝(♠2) ∈❧✐ *❡♥❡❞ ∧♠✐❞(♠2) ∈♠❤❧(♠)) 1 1 ×n2 ▼❛'②♥❝ ❋✉❧❧② ❛2②♥❝❤;♦♥♦✉2✳ ◆♦ ♦;❞❡; ♦♥ ♠❡22❛❣❡ ❞❡❧✐✈❡;② ✐2 ✐♠✲ ♣♦2❡❞✳ ⊤ ⊤ bag

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×1 ❚❤✐# ❛❧❧♦✇# ❢♦) ❤♦♠♦❣❡♥❡♦✉# ❞❡#❝)✐♣2✐♦♥# ♦❢ 2❤❡ ♠♦❞❡❧# ❡✈❡♥ 2❤♦✉❣❤ ❛ ♣❛)2✐❝✉❧❛) ♠♦❞❡❧ ♠✐❣❤2 ♥♦2 ♠❛❦❡ ✉#❡ ♦❢ 2❤❡ ✇❤♦❧❡ ✐♥❢♦)♠❛2✐♦♥✳ ❚❤❡ ❝♦♥2❡♥2 ♦❢ ❛ ♠❡##❛❣❡ ✐# ✐))❡❧❡✈❛♥2 2♦ 2❤❡ #♣❡❝✐✜❝❛2✐♦♥ ♦❢ ♦)❞❡)✐♥❣ ♣)♦♣❡)2✐❡# ❛❧2❤♦✉❣❤ ✐2 ❝❛♥ ❜❡ 2❛❦❡♥ ✐♥2♦ ❛❝❝♦✉♥2 ✐♥ ♣)❛❝2✐❝❛❧ ✐♠♣❧❡♠❡♥2❛2✐♦♥#✳ ❆# ♠❡##❛❣❡# ❛)❡ ❡①❝❤❛♥❣❡❞ ♦♥ ❝❤❛♥♥❡❧ ❛♥❞ 2❤❡)❡ ✐# ♥♦ ❡①♣❧✐❝✐2 ♣❡❡) ❞❡#2✐♥❛2✐♦♥✱ ♠✉❧2✐♣❧❡ #❡♥❞❡)# ❛♥❞ )❡❝❡✐✈❡)# ❝❛♥ ✐♥2❡)❛❝2 ✇✐2❤ 2❤❡ #❛♠❡ ❝❤❛♥♥❡❧✳ ❚❤❡ #2❛2❡ ✈❛)✐❛❜❧❡# ✐♥ 2❤❡ ❚▲❆+ _{♠♦❞✉❧❡ ♦❢ ❛} ❝♦♠♠✉♥✐❝❛2✐♦♥ ♠♦❞❡❧ ❛)❡✿ ✕ ♥❡" 2❤❡ ♥❡2✇♦)❦✿ ❛ #❡2 2❤❛2 ❝♦♥2❛✐♥# ♠❡##❛❣❡# ✐♥ 2)❛♥#✐2✳ ✕ ❤❣ 2❤❡ ❣❧♦❜❛❧ ❤✐#2♦)② ❝♦♥2❛✐♥# 2❤❡ ✐❞# ♦❢ ❛❧❧ 2❤❡ ♠❡##❛❣❡# 2❤❡ ♣❡❡)# ❤❛✈❡ #❡♥2✳ ✕ ❤❧ 2❤❡ ❧♦❝❛❧ ❤✐#2♦)✐❡#✿ ❤❧[♣] ✐# ❛ #❡2 2❤❛2 ❤♦❧❞# 2❤❡ ✐❞# ♦❢ ♠❡##❛❣❡# #❡♥2 ❜② ♣✳ ✕ ❤❝ 2❤❡ ❝❛✉#❛❧ ❤✐#2♦)✐❡#✿ ❤❝[♣] ✐# ❛ #❡2 2❤❛2 ❝♦♥2❛✐♥# 2❤❡ ✐❞# ♦❢ 2❤❡ ♠❡##❛❣❡# ✐♥ 2❤❡ ❝❛✉#❛❧ ♣❛#2 ♦❢ ♣ ❜✉✐❧2 ❛❝❝♦)❞✐♥❣ 2♦ ▲❛♠♣♦)2✬# ❝❛✉#❛❧ )❡❧❛2✐♦♥ ❬▲❛♠✼✽❪✳ ❆ ♠❡##❛❣❡ ♠ ♦♥ 2❤❡ ♥❡2✇♦)❦ ✐# ❛ 2✉♣❧❡ h✐❞♠,❝♠,♣♠,❤❧♠,❤❝♠,❤❣♠i ✇❤❡)❡ ✐❞♠ ✐# 2❤❡ ♠❡##❛❣❡✬# ✉♥✐C✉❡ ✐❞✱ ❝♠2❤❡ ❝❤❛♥♥❡❧ ♦♥ ✇❤✐❝❤ ✐2 ❤❛# ❜❡❡♥ #❡♥2✱ ♣♠ 2❤❡ #❡♥❞❡)✱ ❛♥❞ ❤❧♠✱ ❤❝♠✱ ❤❣♠ #♥❛♣#❤♦2# ♦❢ ❤❧(♣)✱ ❤❝(♣)✱ ❛♥❞ ❤❣ ❛2 #❡♥❞ ❡✈❡♥2✳ ❲❡ ❞❡✜♥❡ ♠✐❞✱ ♠❝✱ ♠♣✱ ♠❤❧✱ ♠❤❝✱ ♠❤❣ 2❤❡ ❛##♦❝✐❛2❡❞ ❛❝❝❡##♦)# ✭❡✳❣✳ ♠❝(♠) = ❝♠✮✳ ❈♦♠♠✉♥✐❝❛2✐♦♥ ♠♦❞❡❧# ❛)❡ #♣❡❝✐✜❡❞ ❜② 2✇♦ ❛❝2✐♦♥#✿ +❡♥❞ ❛♥❞ ,❡❝❡✐✈❡✳ ❚❤❡ +❡♥❞(♣❡❡,, ❝❤❛♥) ❛❝2✐♦♥ ❝♦♥#✐#2# ✐♥ #❡♥❞✐♥❣ ❛ ♥❡✇ ♠❡##❛❣❡ ❢)♦♠ ♣❡❡) ♣❡❡, ♦♥

(5)

♠♦❞✉❧❡ ✜❢♦11 ❡①'❡♥❞) ◆❛%✉'❛❧),❉❡❢) ■♥✐% ∆ = ✐❞ = 1 ∧ ♥❡% = {} ∧ ❤❧ = [✐ ∈ 1❡❡' 7→ {}] ∧ ❤❝ = [✐ ∈ 1❡❡' 7→ {}] ∧ ❤❣ = {} )❡♥❞(♣❡❡',❝❤❛♥) ∆ = ❚❤❡ ♣❡❡$ ✏♣❡❡$✑ '❡♥❞' ❛ ♥❡✇ ♠❡''❛❣❡ ♦♥ ❝❤❛♥♥❡❧ ✏❝❤❛♥✑ ∧✐❞′₌_{✐❞ + 1} ∧❧❡' ♠ = {✐❞} × {❝❤❛♥} × {♣❡❡'} × {❤❧[♣❡❡']} × {❤❝[♣❡❡']} × {❤❣}✐♥∆ ♥❡%′₌_{♥❡% ∪ ♠} ∧❤❧′ _{= [❤❧ ❡①❝❡♣' ✦[♣❡❡'] = @ ∪ {✐❞}]} ∧❤❝′_{= [❤❝ ❡①❝❡♣' ✦[♣❡❡'] = @ ∪ {✐❞}]} ∧❤❣′₌_{❤❣ ∪ {✐❞}} ❞❡❧✐✈❡'②❖❦(♠,❧✐)%❡♥❡❞) ∆ = ¬(∃♠2 ∈ ♥❡% : ❖!❞❡!✐♥❣ ♣!♦♣❡!)②✳ ❚❤❡$❡ ✐' ♥♦ 3$❛♥✲ '✐3✐♥❣ ♠❡''❛❣❡ ♠2 ❢$♦♠ 3❤❡ '❛♠❡ ♣❡❡$✱ ✇❤♦'❡ ❝❤❛♥♥❡❧ ✐' ❧✐'3❡♥❡❞✱ ❛♥❞ ✐♥ 3❤❡ ❧♦✲ ❝❛❧ ❤✐'3♦$② ♦❢ ♠ ✭3❤✉' ♣$❡✈✐♦✉'❧② '❡♥3 ❜② 3❤❡ '❛♠❡ ♣❡❡$✮✳ ∧♠♣(♠) = ♠♣(♠2) ∧♠❝(♠2) ∈ ❧✐)%❡♥❡❞ ∧♠✐❞(♠2) ∈ ♠❤❧(♠)) '❡❝❡✐✈❡(♣❡❡',❝❤❛♥,❧✐)%❡♥❡❞) ∆ = ∃♠ ∈ ♥❡% : ❚❤❡ ♣❡❡$ ✏♣❡❡$✑ $❡❝❡✐✈❡' ❛ ♠❡''❛❣❡ ♦♥ ✏❝❤❛♥✑✱ ✇❤✐❧❡ ❧✐'3❡♥✐♥❣ 3♦ ❛ '❡3 ♦❢ ❝❤❛♥♥❡❧' ✏❧✐'3❡♥❡❞✑ ∧❝❤❛♥ = ♠❝(♠) ∧❞❡❧✐✈❡'②❖❦(♠,❧✐)%❡♥❡❞) ∧♥❡%′₌_{♥❡% \ {♠}} ∧✉♥❝❤❛♥❣❡❞ h✐❞,❤❧,❤❣i ∧❤❝′_{= [}_{❤❝ ❡①❝❡♣' ✦[♣❡❡'] = @ ∪ ♠❤❝(♠) ∪ {♠✐❞(♠)}]} ◆❡①%❙❡♥❞ ∆ = ∃♣ ∈ 1❡❡' : ∃ ❝ ∈ ❈❤❛♥♥❡❧ : )❡♥❞(♣,❝) ◆❡①%❘❡❝✈ ∆ = ∃♣ ∈ 1❡❡' : ∃ ❝ ∈ ❈❤❛♥♥❡❧ : ∃ ❧ ∈ )✉❜)❡' ❈❤❛♥♥❡❧ : '❡❝❡✐✈❡(♣,❝,❧) ◆❡①% ∆ = ◆❡①%❙❡♥❞ ∨ ◆❡①%❘❡❝✈ ❙♣❡❝ ∆ = ■♥✐% ∧ ✷[◆❡①%]✈❛"# ❋✐❣✳ ✶✳ ❚▲❆+ _{❙♣❡❝✐✜❝❛3✐♦♥ ♦❢ ▼} 1−1✳ ❚❤❡ ❚▲❆+ ♠♦❞✉❧❡ ❉❡❢) ❞❡✜♥❡' '❡3' ❛♥❞ 3②♣❡ ✐♥✈❛$✐❛♥3' ❢♦$ 3❤❡ '3❛3❡ ✈❛$✐❛❜❧❡'✱ ✐♥✈❛$✐❛♥3' ♦♥ ❤✐'3♦$✐❡' ✭✐♥❝❧✉'✐♦♥ ❜❡3✇❡❡♥ 3❤❡ ❞✐✛❡$❡♥3 ❤✐'3♦$✐❡'✮ ❛♥❞ 3❤❡ ✉♥✐C✉❡♥❡'' ♦❢ ♠❡''❛❣❡ ✐❞❡♥3✐✜❡$'✳ ❝❤❛♥♥❡❧ ❝❤❛♥✳ ■( ✐* ❛❧✇❛②* ❡♥❛❜❧❡❞ ❡①❝❡♣( ✐♥ (❤❡ ❘❙❈ ✭❘❡❛❧✐③❛❜❧❡ ✇✐(❤ ❙②♥✲ ❝❤6♦♥♦✉* ❈♦♠♠✉♥✐❝❛(✐♦♥ ❬❈❇▼❚✾✻❪✮ ♠♦❞❡❧ ✇❤❡6❡ ❛♥ ❡♠♣(② ♥❡(✇♦6❦ ✐* ❡①✲ ♣❡❝(❡❞✳ ❚❤❡ '❡❝❡✐✈❡(♣❡❡', ❝❤❛♥, ❧✐-.❡♥❡❞) ❛❝(✐♦♥ ❝♦♥*✐*(* ✐♥ 6❡❝❡✐✈✐♥❣ ❛ ♠❡**❛❣❡ ♦♥ ♣❡❡6 ♣❡❡'✱ 6❡(6✐❡✈❡❞ ❢6♦♠ ❝❤❛♥♥❡❧ ❝❤❛♥♥❡❧✱ ✇❤✐❧❡ ❜❡✐♥❣ ✐♥(❡6❡*(❡❞ ✐♥ ❝❤❛♥♥❡❧* ✐♥ (❤❡ *❡( ❧✐-.❡♥❡❞✳ ■( ✐* ❡♥❛❜❧❡❞ ✇❤❡♥ ❛ ♠❡**❛❣❡ ♠ ✇✐(❤ ❛ ♠❛(❝❤✐♥❣ ❝❤❛♥♥❡❧ ✐* ✐♥ (6❛♥*✐( ❛♥❞ ♥♦ ♦(❤❡6 ♠❡**❛❣❡ ♦♥ ❛ ❧✐*(❡♥❡❞ ❝❤❛♥♥❡❧ *❤♦✉❧❞ ❜❡ 6❡❝❡✐✈❡❞ ✜6*( ❛❝❝♦6❞✐♥❣ (♦ (❤❡ ♦6❞❡6✐♥❣ ♣6♦♣❡6(② ♦❢ (❤❡ ❝♦♠♠✉♥✐❝❛(✐♦♥ ♠♦❞❡❧✳ ❋♦6 ❡❛❝❤ ❝♦♠✲ ♠✉♥✐❝❛(✐♦♥ ♠♦❞❡❧✱ (❤❡ ♦6❞❡6✐♥❣ ♣♦❧✐❝② ❛♥❞ (❤✐* ❧❛*( ❝♦♥❞✐(✐♦♥ ❛6❡ ✐♥(6♦❞✉❝❡❞ ✐♥ ❚❛❜❧❡✶✳ ❋✐❣✉6❡✶*❤♦✇* ❛ ❝♦♠♣6❡❤❡♥*✐✈❡ ❚▲❆+_{♠♦❞✉❧❡ ♦❢ (❤❡ ❋■❋❖ ✶✲✶ ♠♦❞❡❧✳} ❘❡❝❡✐✈✐♥❣ ❛ ♠❡**❛❣❡ ♦♥ ❛ ♣❡❡6 ❝♦♥*✐*(* ✐♥ 6❡♠♦✈✐♥❣ ✐( ❢6♦♠ (❤❡ ♥❡(✇♦6❦ ❛♥❞ ✉♣❞❛(✐♥❣ (❤❛( ♣❡❡6✬* ❝❛✉*❛❧ ♣❛*( ❛❝❝♦6❞✐♥❣❧②✳ ❙❡♥❞✐♥❣ ❛ ♠❡**❛❣❡ ❝♦♥*✐*(* ✐♥ ❜✉✐❧❞✐♥❣ (❤❡ (✉♣❧❡ h✐❞, ❝❤❛♥, ♣❡❡', ❤❧[♣❡❡'], ❤❝[♣❡❡'], ❤❣i✱ ❛❞❞✐♥❣ ✐( ✐♥ (❤❡ ♥❡(✇♦6❦✱ ❛♥❞ ❛❞❞✐♥❣ (❤❡ ♠❡**❛❣❡ ✐❞ (♦ ❤❣✱ ❤❧[♣❡❡']✱ ❛♥❞ ❤❝[♣❡❡']✳

(6)

Async FIFO 1-1 Causal FIFO n-n RSC FIFO n-1

FIFO 1-n

❋✐❣✳ ✷✳ ❘❡✜♥❡♠❡♥% ♦❢ %❤❡ ❈♦♠♠✉♥✐❝❛%✐♦♥ ▼♦❞❡❧1✳ ❆♥ ❛44♦✇ ♠❡❛♥1 ✧4❡✜♥❡1✧✳ ❆ ❞❛1❤❡❞ ❧✐♥❡ ♠❡❛♥1 %❤❡ ♣4♦♦❢ ✐1 1%✐❧❧ ✐♥ ♣4♦❣4❡11✳

●✐✈❡♥ ❛ &❡' ♦❢ ♣❡❡+& ❡❡" ❛♥❞ ❛ &❡' ♦❢ ❝❤❛♥♥❡❧& ❈❤❛♥♥❡❧✱ '❤❡ &♣❡❝✐✜❝❛'✐♦♥ ♦❢ ❛ ❝♦♠♠✉♥✐❝❛'✐♦♥ ♠♦❞❡❧ ✐& ❙♣❡❝▼ , ■♥✐-▼∧[◆❡①-▼]✈❛#$▼ ✇❤❡+❡ ✈❛"1▼ ❣+♦✉♣& '❤❡ &'❛'❡ ✈❛+✐❛❜❧❡& ♦❢ ▼ ✱ ■♥✐-▼ &♣❡❝✐✜❡& '❤❡✐+ ✐♥✐'✐❛❧ ✈❛❧✉❡& ❛♥❞ [◆❡①-▼]✈❛#$▼ ❛❝❝♦✉♥'& ❢♦+ ❛❧❧ '❤❡ ♣♦&&✐❜❧❡ 1❡♥❞ ❛♥❞ "❡❝❡✐✈❡ ❛❝'✐♦♥& ❛❧♦♥❣ ✇✐'❤ &'✉''❡+✐♥❣ ♦♥ '❤❡ &'❛'❡ ✈❛+✐❛❜❧❡&✳ ◆❡①-▼ , ◆❡①-❙❡♥❞▼ ∨◆❡①-❘❡❝❡✐✈❡▼ ✇❤❡+❡ ◆❡①-❙❡♥❞▼ ,

∃♣ ∈ ❡❡" : ∃❝ ∈ ❈❤❛♥♥❡❧ : 1❡♥❞▼(♣, ❝) ❛♥❞ ◆❡①-❘❡❝❡✐✈❡▼ , ∃♣ ∈ ❡❡" :

∃❝ ∈ ❈❤❛♥♥❡❧ : ∃❧ ⊆ ❈❤❛♥♥❡❧ : "❡❝❡✐✈❡▼(♣, ❝, ❧)✳

✸ ❘❡✜♥❡♠❡♥&

8+♦♦❢& ♦❢ +❡✜♥❡♠❡♥' ❜❡'✇❡❡♥ '❤❡ ❝♦♠♠✉♥✐❝❛'✐♦♥ ♠♦❞❡❧& ❤❛✈❡ ❜❡❡♥ ❝❛++✐❡❞ ✇✐'❤ '❤❡ ❚▲❆+_{8+♦♦❢ ❙②&'❡♠ ❬}_{❈❉▲▼✶✵}_{❪✳ ❚❤❡ +❡&✉❧'✐♥❣ ❤✐❡+❛+❝❤② ✐& &✉♠♠❡❞ ✉♣ ✐♥ ❋✐❣✲}

✉+❡✷✳ ❚❤✐& ❛❞❞& '♦ ❡①✐&'✐♥❣ +❡&✉❧'& ❛❜♦✉' '❤❡ ❝♦♠♣❛+✐&♦♥ ♦❢ ♠♦❞❡❧& ❛& ✐♥ ❬❑❙✶✶❪ ❛♥❞ ❬❈❇▼❚✾✻❪✳

■♥ ❚▲❆+_{✱ ▼}

2 +❡✜♥❡& ▼1 ✐✛ ❙♣❡❝▼2 ⇒ ❙♣❡❝▼1 ✇❤❡+❡ '❤❡ &'❛'❡ ✈❛+✐❛❜❧❡& ♦❢ ▼1 ❛+❡ ♠❛♣♣❡❞ '♦ '❤❡ ✈❛+✐❛❜❧❡& ♦❢ ▼2 ✇❤❡♥ ✐♥&'❛♥'✐❛'✐♥❣ '❤❡ ♠♦❞✉❧❡ ▼1✳

❆❧❧ ♦✉+ ♠♦❞❡❧& ❤❛✈❡ '❤❡ &❛♠❡ &'❛'❡ ✈❛+✐❛❜❧❡& ❛♥❞ ❛❝'✐♦♥& '❤❛' ❡✈♦❧✈❡ ❛❝❝♦+❞✲ ✐♥❣❧②✳ ❋♦+ &♦♠❡ ♠♦❞❡❧&✱ &♦♠❡ ❤✐&'♦+② ✈❛+✐❛❜❧❡& ❛+❡ ❝♦♥&'+✉❝'❡❞ ❜✉' ✉♥✉&❡❞ ✭❡✳❣✳ '❤❡ ❝❛✉&❛❧ ❤✐&'♦+② ✐♥ ▼♥−♥✱ ♦+ ❛❧❧ '❤❡ ❤✐&'♦+✐❡& ✐♥ ▼❘❙❈✮ ❛♥❞ ♣❧❛② '❤❡ +♦❧❡ ♦❢

&❤❛❞♦✇ ✈❛+✐❛❜❧❡&✳ ❚❤✐& &✐♠♣❧✐❢② '❤❡ +❡✜♥❡♠❡♥' ♣+♦♦❢&✱ ❛& '❤❡ ♠❛♣♣✐♥❣ +❡❧❛'✐♦♥ ✐& '❤❡ ✐❞❡♥'✐'②✳ 8+♦✈✐♥❣ '❤❛' ❙♣❡❝▼2 ⇒❙♣❡❝▼1 ❤❡+❡ ❝♦♥&✐&'& ✐♥ +❡✜♥✐♥❣ ❡❛❝❤ ❛❝'✐♦♥✿

∀♣ ∈ ❡❡" : ∀❝ ∈ ❈❤❛♥♥❡❧ : 1❡♥❞▼2(♣, ❝) ⇒ 1❡♥❞▼1(♣, ❝)

∀♣ ∈ ❡❡" : ∀❝ ∈ ❈❤❛♥♥❡❧ : ∀❧ ⊆ ❈❤❛♥♥❡❧ : "❡❝❡✐✈❡▼2(♣, ❝, ❧) ⇒ "❡❝❡✐✈❡▼1(♣, ❝, ❧) ❚❤❡ ♣+♦♦❢& +❡R✉✐+❡ ❤✐❣❤❧✐❣❤'✐♥❣ ✐♥❞✉❝'✐✈❡ ✐♥✈❛+✐❛♥'& ❢♦+ ❡❛❝❤ ♠♦❞❡❧✱ ❡&♣❡✲ ❝✐❛❧❧② '♦ +❡✜♥❡ '❤❡ "❡❝❡✐✈❡ ❛❝'✐♦♥& &✐♥❝❡ '❤❡② ❞✐✛❡+ '❤❡ ♠♦&' ✭&❡❡ ❚❛❜❧❡ ✶✮✳ ❆♠♦♥❣ '❤❡ ✐♥❞✉❝'✐✈❡ ✐♥✈❛+✐❛♥'& '❤❛' ❛+❡ ✐♥'+♦❞✉❝❡❞ '♦ ❣✉✐❞❡ '❤❡ +❡✜♥❡♠❡♥' ♣+♦♦❢&✱ ♠♦&' ❛+❡ ❝♦♠♠♦♥ '♦ ❛❧❧ '❤❡ ♠♦❞❡❧&✳ ❚❤❡ ✉♥✐R✉❡♥❡&& ♦❢ '❤❡ ♠❡&&❛❣❡& ✭❞✐✛❡+❡♥' ♠❡&&❛❣❡ ✐❞&✮ ❛♥❞ +❡❧❛'✐♦♥& ❜❡'✇❡❡♥ '❤❡ ❞✐✛❡+❡♥' ❤✐&'♦+✐❡& ❛+❡ &✉❝❤ ✐♥✲ ✈❛+✐❛♥'&✳ ❋♦+ ✐♥&'❛♥❝❡ ∀♣ ∈ ❡❡" : ❤❧[♣] ⊆ ❤❝[♣] ⊆ ❤❣✿ '❤❡ &❡♥' ♠❡&&❛❣❡& ♦❢ ♣❡❡+ ♣ ✐& ❛ &✉❜&❡' ♦❢ '❤❡ ❦♥♦✇♥ ♠❡&&❛❣❡& ♦❢ '❤✐& ♣❡❡+ ✭'❤❡ ❝❛✉&❛❧ ❤✐&'♦+② ♦❢ ♣❡❡+ ♣✮✱ ✇❤✐❝❤ ✐& ❛ &✉❜&❡' ♦❢ ❛❧❧ &❡♥' ♠❡&&❛❣❡&✳ ❚❤❡ &❛♠❡ ❛♣♣❧✐❡& '♦ ❤✐&'♦+✐❡& ❝❛++✐❡❞ ❜② ♠❡&&❛❣❡& ✐♥ '+❛♥&✐' ✭∀♠ ∈ ♥❡- : ♠❤❧(♠) ⊆ ♠❤❝(♠) ⊆ ♠❤❣(♠)✮✳ ❙♦♠❡ ✐♥✈❛+✐✲ ❛♥'& ❛+❡ &♣❡❝✐✜❝ '♦ ❛ ❝♦♠♠✉♥✐❝❛'✐♦♥ ♠♦❞❡❧✳ ❋♦+ ✐♥&'❛♥❝❡✱ ✐♥ ▼1−♥✱ ♠❡&&❛❣❡&

✐♥ '+❛♥&✐' '❤❛' ❛+❡ ❝❛✉&❛❧❧② +❡❧❛'❡❞ ❛+❡ ❢+♦♠ '❤❡ &❛♠❡ ♣❡❡+ ✭∀♠1,♠2∈♥❡- :

♠✐❞(♠2) ∈♠❤❝(♠1) ⇒♠♣(♠1) =♠♣(♠2)✮✳ ❚❤✐& ❤②♣♦'❤❡&✐& ✐& ❝+✉❝✐❛❧ '♦ ♣+♦✈❡

'❤❛' ▼1−♥ +❡✜♥❡& ▼❝❛✉$❛❧ ✭"❡❝❡✐✈❡ ❛❝'✐♦♥✮✳ ❙✐♠✐❧❛+❧②✱ '❤❡ ♣+♦♦❢ ♦❢ '❤❡ +❡✜♥❡♠❡♥'

♦❢ ▼♥−♥ ❜② ▼❘❙❈ ✭1❡♥❞ ❛❝'✐♦♥✮ +❡R✉✐+❡& ❛♥ ✐♥✈❛+✐❛♥' '❤❛' ✐& &♣❡❝✐✜❝ '♦ ▼❘❙❈

(7)

♠♦❞✉❧❡ ❡✜♥❡♠❡♥% 11 ❝❛✉)❛❧ ❡①'❡♥❞) ❉❡❢) ❝❛✉)❛❧ ∆ = ✐♥)'❛♥❝❡ ❝❛✉)❛❧ ✜❢♦11 ∆ = ✐♥)'❛♥❝❡ ✜❢♦11 '❤❡♦.❡♠ ❘❛✛❙❡♥❞ ∆ = ∀♣ ∈ 3❡❡ : ∀ ❝ ∈ ❈❤❛♥♥❡❧ : ❝❛✉)❛❧ ✦)❡♥❞(♣,❝) ⇒ ✜❢♦11✦)❡♥❞(♣,❝)❜② ❞❡❢ ✜❢♦11✦)❡♥❞,❝❛✉)❛❧ ✦)❡♥❞ '❤❡♦.❡♠ ❘❛✛❘❡❝✈ ∆ = ∀♣ ∈ 3❡❡ : ∀ ❝ ∈ ❈❤❛♥♥❡❧ : ∀ ❧ ∈ )✉❜)❡' ❈❤❛♥♥❡❧ : ❝❛✉)❛❧ ✦✐♥✈❍✐)%♦ ✐❡) ∧ ❝❛✉)❛❧ ✦ ❡❝❡✐✈❡(♣,❝,❧) ⇒ ✜❢♦11✦ ❡❝❡✐✈❡(♣,❝,❧) ❜② ❞❡❢ ✜❢♦11✦ ❡❝❡✐✈❡,❝❛✉)❛❧ ✦ ❡❝❡✐✈❡,❝❛✉)❛❧ ✦❞❡❧✐✈❡ ②❖❦,✜❢♦11✦❞❡❧✐✈❡ ②❖❦, . . . '❤❡♦.❡♠ ❘❡✜♥❡♠❡♥% ∆ = ❝❛✉)❛❧ ✦❙♣❡❝ ⇒ ✜❢♦11✦❙♣❡❝ h1i❛✳ ❝❛✉)❛❧ ✦■♥✐% ∧ ✷([❝❛✉)❛❧ ✦✐♥✈❍✐)%♦ ✐❡) ∧ ❝❛✉)❛❧ ✦◆❡①%]❝❛✉#❛❧✦✈❛ )) ⇒ ✜❢♦11✦❙♣❡❝ h2i✶✳ ❝❛✉)❛❧ ✦■♥✐% ⇒ ✜❢♦11✦■♥✐%❜② ❞❡❢ ❝❛✉)❛❧ ✦■♥✐%,✜❢♦11✦■♥✐% h2i✷✳ ❝❛✉)❛❧ ✦✐♥✈❍✐)%♦ ✐❡) ∧ ❝❛✉)❛❧ ✦◆❡①% ⇒ ✜❢♦11✦◆❡①% ❜② ❘❛✛❙❡♥❞,❘❛✛❘❡❝✈ ❞❡❢ ❝❛✉)❛❧ ✦◆❡①%,✜❢♦11✦◆❡①%,❝❛✉)❛❧ ✦◆❡①%❙❡♥❞, ✜❢♦11✦◆❡①%❙❡♥❞,❝❛✉)❛❧ ✦◆❡①%❘❡❝✈,✜❢♦11✦◆❡①%❘❡❝✈ h2i✸✳ [❝❛✉)❛❧ ✦✐♥✈❍✐)%♦ ✐❡) ∧ ❝❛✉)❛❧ ✦◆❡①%]❝❛✉#❛❧✦✈❛ ) ⇒ [✜❢♦11✦◆❡①%]✜❢♦11✦✈❛ ) ❜② h2i✷ ❞❡❢ ❝❛✉)❛❧ ✦✈❛ ),✜❢♦11✦✈❛ ) h2i2❡❞ ❜② 3❚▲,h2i✶,h2i✸ ❞❡❢ ✜❢♦11✦❙♣❡❝

h1i.2❡❞ ❜② 3❚▲,h1i❛,❝❛✉)❛❧ ✦■♥✈❛ ✐❛♥% ❞❡❢ ❝❛✉)❛❧ ✦❙♣❡❝ ❋✐❣✳ ✸✳ ❚▲❆+_{)*♦♦❢ -❤❛- ▼} ❝❛✉#❛❧ *❡✜♥❡2 ▼1−1✳ ❚❤❡ ♣*♦♣♦2✐-✐♦♥❛❧ -❡♠♣♦*❛❧ ❧♦❣✐❝ -❛❝-✐❝ )❚▲ ✐2 ✉2❡❞ -♦ 2-❡♣ ❢*♦♠ ♦♥❡ -*❛♥2✐-✐♦♥ ✭◆❡①%✮ -♦ -❤❡ 2♣❡❝✐✜❝❛-✐♦♥ ✭✷[◆❡①%]✮✳ ❲❡ ❤❛❞ %♦ ❝❛(❡❢✉❧❧② -❡♣❛(❛%❡ %❤❡ ♣(♦♦❢ -%❡♣- (❡❣❛(❞✐♥❣ ✐♥❞✐✈✐❞✉❛❧ ❛❝%✐♦♥- ❢(♦♠ %❤❡ ♦♥❡- (❡❣❛(❞✐♥❣ %❤❡ ❝♦♠♣❧❡%❡ -♣❡❝✐✜❝❛%✐♦♥✳ ❚❤❡ ❢♦(♠❡( ❛(❡ ❢♦(♠✉❧❛❡ ♦❢ ✜(-%✲ ♦(❞❡( ❧♦❣✐❝ ✇✐%❤ 9✉❛♥%✐✜❡(- ❛♥❞ ❛(❡ ❤❛♥❞❧❡❞ ❜② ❙▼❚ ❜❛❝❦❡♥❞- ✭❈❱❈✸ ❛♥❞ ❩✸ ✐♥ ♦✉( ❝❛-❡✮❀ %❤❡ ❧❛%%❡( ❞❡❛❧ ✇✐%❤ %❡♠♣♦(❛❧ ❧♦❣✐❝ ✭✷ ♦♣❡(❛%♦(✮ ❛♥❞ ❛(❡ ❤❛♥❞❧❡❞ ❜② %❤❡ ▲❙✹ ❜❛❝❦❡♥❞✱ ❛ ♣(♦♣♦-✐%✐♦♥❛❧ %❡♠♣♦(❛❧ ❧♦❣✐❝ ♣(♦✈❡(✳ ❚❤❡ ✐♥❞✉❝%✐✈❡ ✐♥✈❛(✐❛♥%-✇❤✐❝❤ ❛(❡ (❡9✉✐(❡❞ %♦ ♣(♦✈❡ %❤❡ (❡✜♥❡♠❡♥%- ❛(❡ ❧❛(❣❡ ❢♦(♠✉❧❛❡ ✭✶✵ -%❛%❡ ✈❛(✐❛❜❧❡-❛♥❞ ✉♣ %♦ ✷✵ 9✉❛♥%✐✜❡(-✮ ✈❛(✐❛❜❧❡-❛♥❞ ♥❡❡❞ -❡✈❡(❛❧ ♣(♦♦❢ -%❡♣-✳ ❍♦✇❡✈❡(✱ %❤❡② ✇❡(❡ ❣(❛❞✉❛❧❧② ❜✉✐❧% ❛♥❞ ✇❡(❡ ❡❛-✐❧② ❞❡❝♦♠♣♦-❡❞ ✐♥ -✉❝❝❡--✐✈❡ -%(❡♥❣%❤❡♥✐♥❣ ✭%②♣❡ ✐♥✈❛(✐❛♥%-✱ ✐♥✈❛(✐❛♥%- ♦♥ ♣❡❡(-✱ ✐♥✈❛(✐❛♥% ♦♥ ♠❡--❛❣❡-✮ %♦ ❛❧❧♦✇ ❢♦( ✐♥❝(❡♠❡♥%❛❧ ♣(♦♦❢-✳ ❖♥❝❡ %❤✐- ♥❛%✉(❛❧ ❞❡❝♦♠♣♦-✐%✐♦♥ ✇❛- ❞♦♥❡✱ %❤❡ ❚▲❆N❙ ❜❛❝❦❡♥❞- ❤❛✈❡ -❤♦✇♥ %♦ ❜❡ ❡✣❝✐❡♥% ❡♥♦✉❣❤ %♦ ❞✐(❡❝%❧② ♣(♦✈❡ %❤❡ ❢♦(♠✉❧❛❡✱ ✇✐%❤♦✉% ❤❛✈✐♥❣ %♦ ❣♦ ❞♦✇♥ %♦ (❡❛-♦♥✐♥❣ ❜② ❝❛-❡-✳ ❖✉( ♠❛✐♥ ❞✐✣❝✉❧%② ✇❛- ✇✐%❤ %❤❡ (❡♣(❡-❡♥%❛%✐♦♥ ♦❢ ♠❡--❛❣❡-✿ ❛ ♠❡--❛❣❡ ✐-❛ %✉♣❧❡ ♦❢ -✐① ❡❧❡♠❡♥%- ✭♠❡--✐-❛❣❡ ✐❞✱ ❝❤✐-❛♥♥❡❧✱ -❡♥❞❡(✱ ❤✐-%♦(✐❡-✮✳ ■♥ %❤❡ ❝✉((❡♥% -%❛%❡ ♦❢ ❚▲❆N❙✱ %❤❡ ❤❛♥❞❧✐♥❣ ♦❢ %✉♣❧❡- h❡1, . . . ,❡♥i✐- -♦♠❡%✐♠❡- ❛✇❦✇❛(❞✳ ❚❤❡② ❛(❡ ✐♥%❡(♥❛❧❧② ❝♦♥-✐❞❡(❡❞ ❛- ❢✉♥❝%✐♦♥- ♦❢ ❞♦♠❛✐♥ 1..♥✱ ✐♥ ❛❝❝♦(❞❛♥❝❡ ✇✐%❤ %❤❡✐( ❚▲❆+_{-❡♠❛♥%✐❝-✳ ❇✉% ❛ ♣(♦❞✉❝% ♦❢ -❡%- ✐- ❛❧-♦ ❛ -❡% ♦❢ %✉♣❧❡-✱ ❛♥❞ ✇❡ ✇❡(❡ ✉♥❛❜❧❡} %♦ -✇✐%❝❤ ❜❡%✇❡❡♥ ❜♦%❤ ♣♦✐♥%- ♦❢ ✈✐❡✇✳ ❋♦( ✐♥-%❛♥❝❡✱ ✇❡ ❤❛❞ %♦ ❛--✉♠❡ ❛ ❧❡♠♠❛ -✐♠✐❧❛( %♦ {1} × {2} = {h1, 2i} ✭♠♦(❡ ♣(❡❝✐-❡❧②✱ %❤❛% %❤❡ ♣(♦❞✉❝% ♦❢ ◆ -✐♥❣❧❡%♦♥ -❡%- ✭◆ > 0✮ ✐- ❛ -❡% ✇✐%❤ ❛ ✉♥✐9✉❡ %✉♣❧❡✮✳ ❆% %❤✐- ♣♦✐♥%✱ ❛❧❧ %❤❡ (❡✜♥❡♠❡♥%- ❛(❡ ♣(♦✈❡❞ ❡①❝❡♣% ❢♦( %✇♦ -❡❝♦♥❞❛(② ✐♥✈❛(✐✲ ❛♥%-✱ ♦♥❧② (❡9✉✐(❡❞ ❢♦( %❤❡ (❡✜♥❡♠❡♥% ♦❢ ▼❝❛✉$❛❧ ❜② ▼1−♥✳ ❚❤❡-❡ %✇♦

(8)

✐♥✈❛(✐❛♥%-❤❛✈❡ ❜❡❡♥ ♠❛♥✉❛❧❧② ♣+♦✈❡❞✉.✐♥❣ ✐♥❞✉❝2✐♦♥ ❜✉2 2❤❡✐+ ❚▲❆6❙ ♣+♦♦❢ ✐. .2✐❧❧ ❡❧✉.✐✈❡✳ ❆❧❧ 2❤❡ ❚▲❆+_{♠♦❞✉❧❡. 2❤❛2 .♣❡❝✐❢② 2❤❡ ❝♦♠♠✉♥✐❝❛2✐♦♥ ♠♦❞❡❧. ❛♥❞ 2❤❡ ♣+♦♦❢. ♦❢} +❡✜♥❡♠❡♥2 ❛+❡ ❛✈❛✐❧❛❜❧❡ ❛2❤!!♣✿✴✴%✉❡✐♥♥❡❝✳♣❡,-♦✳❡♥-❡❡✐❤!✳❢,✴❆❇❩✷✵✶✻✳

✹ ❘❡❧❛%❡❞ ❲♦)❦

❆.②♥❝❤+♦♥♦✉. ❝♦♠♠✉♥✐❝❛2✐♦♥ ♠♦❞❡❧. ✐♥ ❞✐.2+✐❜✉2❡❞ .②.2❡♠. ❤❛✈❡ ❜❡❡♥ .2✉❞✐❡❞ ❛♥❞ ❝♦♠♣❛+❡❞ ✐♥ ❬❑❙✶✶❪ ✭♥♦2✐♦♥ ♦❢ ♦+❞❡+✐♥❣ ♣❛+❛❞✐❣♠✮ ❛♥❞ ❬❈❇▼❚✾✻❪ ✭♥♦2✐♦♥ ♦❢ ❞✐.2+✐❜✉2❡❞ ❝♦♠♣✉2❛2✐♦♥ ❝❧❛..❡.✮✳ ■♥ ♦✉+ ✇♦+❦✱ ✇❡ ❝♦♥.✐❞❡+ ❛❞❞✐2✐♦♥❛❧ ❞✐.2+✐❜✉2❡❞ ❝♦♠♠✉♥✐❝❛2✐♦♥ ♠♦❞❡❧.✱ ♥❛♠❡❧② ▼♥−♥✱ ▼1−♥ ❛♥❞ ▼♥−1✱ ✇❤✐❝❤ ❛+❡ ♦❢ ✐♥2❡+❡.2 .✐♥❝❡ 2❤❡② ❛+❡ ♥♦2 2♦2❛❧❧② ♦+❞❡+❡❞✳ ▼♥−1 ❢♦+ ✐♥.2❛♥❝❡✱ 2❤❡ ❋■❋❖ ♦+❞❡+ ✇✐2❤ ✐♥.2❛♥2❛♥❡♦✉. ❞❡❧✐✈❡+②✱ ✐. ♦❢2❡♥ ✉.❡❞ ✐♥ 2❤❡ ❧✐2❡+❛2✉+❡ ✇✐2❤♦✉2 ❞✐.2✐♥❝2✐♦♥ ❢+♦♠ 2❤❡ ❝❧❛..✐❝ ❋■❋❖ ♦+❞❡+✳ ❖✉+ ❛♣♣+♦❛❝❤ 2♦ ✐.♦❧❛2❡ 2❤❡ ❝♦♠♠✉♥✐❝❛2✐♦♥ ♠♦❞❡❧ ❛. ❛ 2+❛♥.✐2✐♦♥ .②.2❡♠ ✐. +❡♠✐♥✐.❝❡♥2 ♦❢ ❚❡❧✬. 2❡①2❜♦♦❦ ❬❚❡❧✵✵❪✱ ❜✉2 ❤✐. ❢♦❝✉. ✐. ♦♥ ❞❡.❝+✐❜✐♥❣ ❞✐.2+✐❜✉2❡❞ ❛❧❣♦+✐2❤♠.✱ ✇❤❡+❡❛. ♦✉+. ✐. ♦♥ ❝♦♠♣❛+✐♥❣ 2❤❡ ♠♦❞❡❧.✳

✺ ❈♦♥❝❧✉0✐♦♥

❚❤✐. ♣❛♣❡+ ❡①♣❧❛✐♥. ❤♦✇ ♣+♦♦❢. ♦❢ +❡✜♥❡♠❡♥2 ❜❡2✇❡❡♥ ❝♦♠♠✉♥✐❝❛2✐♦♥ ♠♦❞❡❧. ❤❛✈❡ ❜❡❡♥ ❝♦♥❞✉❝2❡❞ ✇✐2❤ 2❤❡ ❚▲❆+_{6+♦♦❢ ❙②.2❡♠✳ ❆ ✉♥✐✜❡❞ ❚▲❆}+_{.♣❡❝✐✜❝❛2✐♦♥} ♦❢ ❝❧❛..✐❝ ❝♦♠♠✉♥✐❝❛2✐♦♥ ♠♦❞❡❧. ❛❧♦♥❣ ✇✐2❤ ❝♦♠♠♦♥ ❛♥❞ ♠♦❞❡❧✲.♣❡❝✐✜❝ ✐♥✈❛+✐✲ ❛♥2. ✐. 2❤❡ ❦❡② 2♦ ❛❝❤✐❡✈❡ 2❤❡.❡ ❢♦+♠❛❧ ♣+♦♦❢.✳ ❖♥❣♦✐♥❣ ✇♦+❦ ❝♦♥.✐.2. ✐♥ .2✉❞②✐♥❣ ♦2❤❡+ ❞❡.❝+✐♣2✐♦♥. ♦❢ 2❤❡ ♠♦❞❡❧.✳ ❋♦+ ❡①❛♠♣❧❡✱ 2❤❡② ❝❛♥ ❜❡ .♣❡❝✐✜❡❞ ❛. ♣+♦♣❡+2✐❡. ♦♥ ❞✐.2+✐❜✉2❡❞ ❡①❡❝✉2✐♦♥. ✭.❡P✉❡♥❝❡. ♦❢ ❝♦♠♠✉♥✐❝❛2✐♦♥ ❡✈❡♥2.✮✳ 6+❛❝2✐❝❛❧ ✐♠♣❧❡✲ ♠❡♥2❛2✐♦♥. ❜❛.❡❞ ♦♥ P✉❡✉❡. ❛♥❞ ❝♦✉♥2❡+. ❛+❡ ❛❧.♦ ♦❢ ✐♥2❡+❡.2✳ ❚❤❡ ✈❡+✐✜❝❛2✐♦♥ ♦❢ +❡✜♥❡♠❡♥2 +❡❧❛2✐♦♥. ❜❡2✇❡❡♥ 2❤❡.❡ ♠♦❞❡❧. ✐. ✐♥ ♣+♦❣+❡..✳

❘❡❢❡)❡♥❝❡0

❈❇▼❚✾✻✳ ❇❡(♥❛❞❡,,❡ ❈❤❛((♦♥✲❇♦0,✱ ❋(✐❡❞❡♠❛♥♥ ▼❛,,❡(♥✱ ❛♥❞ ●❡(❛(❞ ❚❡❧✳ ❙②♥✲ ❝❤(♦♥♦✉0✱ ❛0②♥❝❤(♦♥♦✉0✱ ❛♥❞ ❝❛✉0❛❧❧② ♦(❞❡(❡❞ ❝♦♠♠✉♥✐❝❛,✐♦♥✳ ❉✐"#$✐❜✉#❡❞ ❈♦♠♣✉#✐♥❣✱ ✾✭✹✮✿✶✼✸✕✶✾✶✱ ❋❡❜(✉❛(② ✶✾✾✻✳ ❈❉▲▼✶✵✳ ❑❛✉0,✉✈ ❈❤❛✉❞❤✉(✐✱ ❉❛♠✐❡♥ ❉♦❧✐❣❡③✱ ▲❡0❧✐❡ ▲❛♠♣♦(,✱ ❛♥❞ ❙,❡♣❤❛♥ ▼❡(③✳ ❱❡(✐❢②✐♥❣ 0❛❢❡,② ♣(♦♣❡(,✐❡0 ✇✐,❤ ,❤❡ ❚▲❆✰ Q(♦♦❢ ❙②0,❡♠✳ ■♥ /$♦❝❡❡❞✐♥❣" ♦❢ #❤❡ ✺#❤ ■♥#❡$♥❛#✐♦♥❛❧ ❈♦♥❢❡$❡♥❝❡ ♦♥ ❆✉#♦♠❛#❡❞ ❘❡❛"♦♥✐♥❣✱ ■❏❈❆❘✬✶✵✱ ♣❛❣❡0 ✶✹✷✕✶✹✽✱ ❇❡(❧✐♥✱ ❍❡✐❞❡❧❜❡(❣✱ ✷✵✶✵✳ ❙♣(✐♥❣❡(✲❱❡(❧❛❣✳ ❈❍◗✶✺✳ ❋❧♦(❡♥, ❈❤❡✈(♦✉✱ ❆✉([❧✐❡ ❍✉(❛✉❧,✱ ❛♥❞ Q❤✐❧✐♣♣❡ ◗✉[✐♥♥❡❝✳ ❆✉,♦♠❛,❡❞ ✈❡(✲ ✐✜❝❛,✐♦♥ ♦❢ ❛0②♥❝❤(♦♥♦✉0 ❝♦♠♠✉♥✐❝❛,✐♥❣ 0②0,❡♠0 ✇✐,❤ ❚▲❆✰✳ ❊❧❡❝#$♦♥✐❝ ❈♦♠♠✉♥✐❝❛#✐♦♥" ♦❢ #❤❡ ❊❆❙❙❚ ✭/♦"#/$♦❝❡❡❞✐♥❣" ♦❢ #❤❡ ✶✺#❤ ■♥#❡$♥❛#✐♦♥❛❧ ❲♦$❦"❤♦♣ ♦♥ ❆✉#♦♠❛#❡❞ ❱❡$✐✜❝❛#✐♦♥ ♦❢ ❈$✐#✐❝❛❧ ❙②"#❡♠"✮✱ ✼✷✿✶✕✶✺✱ ✷✵✶✺✳ ❑❙✶✶✳ ❆❥❛② ❉✳ ❑0❤❡♠❦❛❧②❛♥✐ ❛♥❞ ▼✉❦❡0❤ ❙✐♥❣❤❛❧✳ ❉✐"#$✐❜✉#❡❞ ❈♦♠♣✉#✐♥❣✿ /$✐♥✲ ❝✐♣❧❡"✱ ❆❧❣♦$✐#❤♠"✱ ❛♥❞ ❙②"#❡♠"✳ ❈❛♠❜(✐❞❣❡ ❯♥✐✈❡(0✐,② Q(❡00✱ ▼❛(❝❤ ✷✵✶✶✳ ▲❛♠✼✽✳ ▲❡0❧✐❡ ▲❛♠♣♦(,✳ ❚✐♠❡✱ ❝❧♦❝❦0 ❛♥❞ ,❤❡ ♦(❞❡(✐♥❣ ♦❢ ❡✈❡♥,0 ✐♥ ❛ ❞✐0,(✐❜✉,❡❞ 0②0,❡♠✳ ❈♦♠♠✉♥✐❝❛#✐♦♥" ♦❢ #❤❡ ❆❈▼✱ ✷✶✭✼✮✿✺✺✽✕✺✻✺✱ ❏✉❧② ✶✾✼✽✳ ▲❛♠✵✷✳ ▲❡0❧✐❡ ▲❛♠♣♦(,✳ ❙♣❡❝✐❢②✐♥❣ ❙②"#❡♠"✳ ❆❞❞✐0♦♥ ❲❡0❧❡②✱ ✷✵✵✷✳ ❚❡❧✵✵✳ ●❡(❛(❞ ❚❡❧✳ ■♥#$♦❞✉❝#✐♦♥ #♦ ❉✐"#$✐❜✉#❡❞ ❆❧❣♦$✐#❤♠"✳ ❈❛♠❜(✐❞❣❡ ❯♥✐✈❡(0✐,② Q(❡00✱ 0❡❝♦♥❞ ❡❞✐,✐♦♥✱ ✷✵✵✵✳