SATソルバーと制約ソルバー ～基盤技術と最新動向～

2025.12.01 シンポジウム： 制約充足、探索、列挙、最適化 ― よい「答え」を見つける技術の最先端

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1. SAT ιϧόʔͱ੍໿ιϧόʔ ʙج൫ٕज़ͱ࠷৽ಈ޲ʙ ૙ ߶ल (ͦ͏ ͚ͨͻͰ) ໊ݹ԰େֶ 2025 ೥ 12 ݄ 1 ೔ (݄) @ ܚጯٛक़େֶ೔٢Ωϟϯύε γϯϙδ΢Ϝɿ ੍໿ॆ଍ɺ୳ࡧɺྻڍɺ࠷దԽ ʕ Α͍ʮ౴͑ʯΛݟ͚ͭΔٕज़ͷ࠷ઌ୺ ୈ 134 ճਓ޻஌ೳجຊ໰୊ݚڀձ (SIG-FPAI) in ਓ޻஌ೳֶձ߹ಉݚڀձ 2025 (SIGAIs) 1 / 44
2. ߨԋͷ֓ཁ γϯϙδ΢ϜλΠτϧ ੍໿ॆ଍ɺ୳ࡧɺྻڍɺ࠷దԽ ʙΑ͍ʮ౴͑ʯΛݟ͚ͭΔٕज़ͷ࠷ઌ୺ʙ 2 / 44
3. ߨԋͷ֓ཁ γϯϙδ΢ϜλΠτϧ ੍໿ॆ଍ɺ୳ࡧɺྻڍɺ࠷దԽ ʙΑ͍ʮ౴͑ʯΛݟ͚ͭΔٕज़ͷ࠷ઌ୺ʙ ߨԋ಺༰
4. ߨԋͷ֓ཁ γϯϙδ΢ϜλΠτϧ ੍໿ॆ଍ɺ୳ࡧɺྻڍɺ࠷దԽ ʙΑ͍ʮ౴͑ʯΛݟ͚ͭΔٕज़ͷ࠷ઌ୺ʙ ߨԋ಺༰ ૊߹ͤ໰୊ ૊߹ͤ࠷దԽ໰୊
5. ߨԋͷ֓ཁ γϯϙδ΢ϜλΠτϧ ੍໿ॆ଍ɺ୳ࡧɺྻڍɺ࠷దԽ ʙΑ͍ʮ౴͑ʯΛݟ͚ͭΔٕज़ͷ࠷ઌ୺ʙ ߨԋ಺༰ ૊߹ͤ໰୊ ૊߹ͤ࠷దԽ໰୊ ൚༻ιϧόʔ (੍໿ιϧόʔ)
6. ߨԋͷ֓ཁ γϯϙδ΢ϜλΠτϧ ੍໿ॆ଍ɺ୳ࡧɺྻڍɺ࠷దԽ ʙΑ͍ʮ౴͑ʯΛݟ͚ͭΔٕज़ͷ࠷ઌ୺ʙ ߨԋ಺༰ ૊߹ͤ໰୊ ૊߹ͤ࠷దԽ໰୊ ൚༻ιϧόʔ (੍໿ιϧόʔ) ੍໿ιϧόʔͷج൫ٕज़ͱ࠷৽ಈ޲ʹ͍ͭͯઆ໌͢Δɽ 2 / 44
7. ૊߹ͤ໰୊ʹର͢Δ੍໿ιϧόʔ ιϧόʔ CaDiCaL, Kissat Exact, SCIP-NaPS OR-Tools (CP-SAT), ACE CP Optimizer, Sugar Clingo Z3, CVC5 SCIP, Gurobi, CPLEX X ૊߹ͤ໰୊͸͍Ζ͍Ζͳ੍໿ιϧόʔͰղ͘͜ͱ͕Ͱ͖Δɽ X ֤ιϧόʔ͸ҟͳΔೖྗݴޠ (੍໿) Λ͓࣋ͬͯΓಘҙͳ໰୊ ͕ҟͳΔɽ·ͨղ͘͜ͱ͕Ͱ͖Δ໰୊ͷൣғ΋ҟͳΔɽ 3 / 44
8. ૊߹ͤ໰୊ʹର͢Δ੍໿ιϧόʔ ιϧόʔ هड़Ͱ͖Δ੍໿ ໰୊ CaDiCaL, Kissat ໋୊࿦ཧࣜʢCNFʣ SAT Exact, SCIP-NaPS 0-1 ઢܗෆ౳ࣜ PB OR-Tools (CP-SAT), ACE ࢉज़ɼ࿦ཧɼ֎Ԇత੍໿ CSP CP Optimizer, Sugar άϩʔόϧ੍໿ Clingo Ұ֊ड़ޠ࿦ཧࣜ ASP Z3, CVC5 ཧ࿦෇͖࿦ཧࣜ SMT SCIP, Gurobi, CPLEX ࣮਺/੔਺/ઢܗ/ೋ࣍ ਺ཧ࠷దԽ ʢࠞ߹ʣෆ౳ࣜ X ૊߹ͤ໰୊͸͍Ζ͍Ζͳ੍໿ιϧόʔͰղ͘͜ͱ͕Ͱ͖Δɽ X ֤ιϧόʔ͸ҟͳΔೖྗݴޠ (੍໿) Λ͓࣋ͬͯΓಘҙͳ໰୊ ͕ҟͳΔɽ·ͨղ͘͜ͱ͕Ͱ͖Δ໰୊ͷൣғ΋ҟͳΔɽ 3 / 44
9. ૊߹ͤ໰୊ʹର͢Δ੍໿ιϧόʔ ιϧόʔ هड़Ͱ͖Δ੍໿ ໰୊ CaDiCaL, Kissat ໋୊࿦ཧࣜʢCNFʣ SAT Exact, SCIP-NaPS 0-1 ઢܗෆ౳ࣜ PB OR-Tools (CP-SAT) , ACE ࢉज़ɼ࿦ཧɼ֎Ԇత੍໿ CSP CP Optimizer, Sugar άϩʔόϧ੍໿ Clingo Ұ֊ड़ޠ࿦ཧࣜ ASP Z3, CVC5 ཧ࿦෇͖࿦ཧࣜ SMT SCIP, Gurobi, CPLEX ࣮਺/੔਺/ઢܗ/ೋ࣍ ਺ཧ࠷దԽ ʢࠞ߹ʣෆ౳ࣜ X ૊߹ͤ໰୊͸͍Ζ͍Ζͳ੍໿ιϧόʔͰղ͘͜ͱ͕Ͱ͖Δɽ X ֤ιϧόʔ͸ҟͳΔೖྗݴޠ (੍໿) Λ͓࣋ͬͯΓಘҙͳ໰୊ ͕ҟͳΔɽ·ͨղ͘͜ͱ͕Ͱ͖Δ໰୊ͷൣғ΋ҟͳΔɽ ଟ͘ͷιϧόʔ͕ SAT ιϧόʔͷٕज़Λ࢖༻͍ͯ͠Δɽ 3 / 44
10. Q. Կނ࢖ΘΕΔͷ͔? 4 / 44
11. Q. Կނ࢖ΘΕΔͷ͔? Armin Biere ઌੜͷ Twitter (@ArminBiere) ΑΓ (2020 ೥ 7 ݄ 29 ೔) 4 / 44
12. Q. Կނ࢖ΘΕΔͷ͔? Armin Biere ઌੜͷ Twitter (@ArminBiere) ΑΓ (2020 ೥ 7 ݄ 29 ೔) ܧଓతͳߴ଎Խ 4 / 44
13. Q. Կނ࢖ΘΕΔͷ͔? Armin Biere ઌੜͷ Twitter (@ArminBiere) ΑΓ (2020 ೥ 7 ݄ 29 ೔) ܧଓతͳߴ଎Խ ܧଓతͳߴ଎Խ + ֦ுੑ (CNFͱCDCL) 4 / 44
14. SAT ٕज़ͷԠ༻ SAT PB ASP ALLSAT #SAT QBF SMT MaxSAT CSP/COP ج൫ٕज़ DPLL/CDCL 5 / 44
15. SAT ٕज़ͷԠ༻ SAT PB ASP ALLSAT #SAT QBF SMT MaxSAT CSP/COP ج൫ٕज़ DPLL/CDCL 5 / 44
16. ൃද಺༰ X SAT ιϧόʔ (DPLL/CDCL) X SAT ιϧόʔͷ֦ு ž SAT ූ߸Խ ž LCG ιϧόʔ: OR-Tools (CP-SAT) ž SMT ιϧόʔ: Z3, CVC X ૊߹ͤ໰୊ʹ͸ԿΛ࢖͑͹ྑ͍ͷ? ž 2 ࣍ݩۣܗύοΩϯά໰୊Λྫ୊ʹͨ͠ੑೳධՁͷҰྫ ž CPLEX, Gurobi, SCIP ž CP Optimizer, OR-Tools (CP-SAT), ACE ž Z3, CVC5 6 / 44
17. SAT ໰୊ͱ͸ SAT ໰୊ (Boolean Satisability Problem) ༩͑ΒΕ໋ͨ୊࿦ཧࣜΛਅʹ͢Δ஋ׂ౰͕ͯଘࡏ͢Δ͔൑ఆ͢Δ ໰୊ ྫɿCNF ࣜ (p1 ∨ p2 ∨ p3 ) ∧ (¬p1 ∨ ¬p2 ) ∧ (¬p1 ∨ ¬p3 ) ∧ (¬p2 ∨ ¬p3 ) 7 / 44
18. SAT ໰୊ͱ͸ SAT ໰୊ (Boolean Satisability Problem) ༩͑ΒΕ໋ͨ୊࿦ཧࣜΛਅʹ͢Δ஋ׂ౰͕ͯଘࡏ͢Δ͔൑ఆ͢Δ ໰୊ ྫɿCNF ࣜ (p1 ∨ p2 ∨ p3 ) ∧ (¬p1 ∨ ¬p2 ) ∧ (¬p1 ∨ ¬p3 ) ∧ (¬p2 ∨ ¬p3 ) ղ p1 = T, p2 = F, p3 = F 7 / 44
19. SAT ໰୊ͱ͸ SAT ໰୊ (Boolean Satisability Problem) ༩͑ΒΕ໋ͨ୊࿦ཧࣜΛਅʹ͢Δ஋ׂ౰͕ͯଘࡏ͢Δ͔൑ఆ͢Δ ໰୊ ྫɿCNF ࣜ ( p1 ∨ p2 ∨ p3 ) ∧ (¬p1 ∨ ¬p2 ) ∧ (¬p1 ∨ ¬p3 ) ∧ ( ¬p2 ∨ ¬p3 ) ղ p1 = T, p2 = F, p3 = F NP ׬શ໰୊ͷ 1 ͭͰ͋Γɼଟ͘ͷ૊߹ͤ໰୊͸ SAT ໰୊Λ༻͍ͯ දݱͰ͖Δɽ 7 / 44
20. SAT ιϧόʔͱ͸ SAT ໰୊Λࣗಈతʹղ͘ϓϩάϥϜ ೖྗ CNF ܗࣜͷ࿦ཧࣜ ग़ྗ X ॆ଍Մೳ (SAT) + ղ X ॆ଍ෆೳ (UNSAT) 8 / 44
21. SAT ιϧόʔͱ͸ SAT ໰୊Λࣗಈతʹղ͘ϓϩάϥϜ ೖྗ CNF ܗࣜͷ࿦ཧࣜ ग़ྗ X ॆ଍Մೳ (SAT) + ղ X ॆ଍ෆೳ (UNSAT) ओͳΞϧΰϦζϜ X DPLL (1962) ਂ͞༏ઌ୳ࡧ X CDCL (2000 ࠒ) ֶशʹΑΔޮ཰Խ 8 / 44
22. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ 9 / 44
23. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a 9 / 44
24. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b 9 / 44
25. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c 9 / 44
26. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c 9 / 44
27. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c ¬e 9 / 44
28. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c ¬e f 9 / 44
29. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c ¬e f g 9 / 44
30. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c ¬e f g ¬i 9 / 44
31. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c ¬e f g ¬i 9 / 44
32. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c ¬e f g ¬i ໃ६ൃੜ: h ∧ ¬h 9 / 44
33. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c όοΫτϥοΫ 9 / 44
34. DPLL [Davis+, 1962] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c e ஋൓సͯ͠ܧଓ 9 / 44
35. CDCL [Bayardo Jr.+, 1997, Marques-Silva+, 1999] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c ¬e f g ¬i ໃ६ൃੜ: h ∧ ¬h 10 / 44
36. CDCL [Bayardo Jr.+, 1997, Marques-Silva+, 1999] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i LC8 : ¬b ∨ ¬f Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b c ¬e f g ¬i ֶशઅΛੜ੒: ¬b ∨ ¬f 10 / 44
37. CDCL [Bayardo Jr.+, 1997, Marques-Silva+, 1999] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i LC8 : ¬b ∨ ¬f Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b Lv 2 ΁όοΫδϟϯϓ 10 / 44
38. CDCL [Bayardo Jr.+, 1997, Marques-Silva+, 1999] C1 : a ∨ b C2 : c ∨ d C3 : a ∨ e ∨ f C4 : ¬b ∨ ¬f ∨ g C5 : ¬f ∨ h C6 : ¬b ∨ ¬h ∨ i C7 : ¬g ∨ ¬i LC8 : ¬b ∨ ¬f Lv 1 Lv 2 Lv 3 ܾఆ ఻೻ ¬a b ¬f e ୯Ґ఻೻ 10 / 44
39. SAT ιϧόʔͷ୳ࡧͷ༷ࢠ X SATracer [ງԬ+, 2021] MA> 11 / 44
40. SAT ιϧόʔͷ࢓૊Έͷྺ࢙ X 1960 ೥୅ ž DPLL (Davis-Putnam-Logemann-Loveland) [Davis+, 1962] X 1990 ೥୅ ž CDCL (Conict Driven Clause Learning) [Bayardo Jr.+, 1997, Marques-Silva+, 1999] X 2000 ೥Ҏ߱ ž ม਺બ୒ώϡʔϦεςΟοΫ VSIDS [Moskewicz+, 2001] ž 2 Ϧςϥϧ΢Υον [Moskewicz+, 2001] ž Ϧελʔτ [Luby+, 1993, Selman+, 1996, Eén+, 2003] ž Phase Saving [Pipatsrisawat+, 2007] ž ֶशઅͷධՁई౓ [Audemard+, 2009, ುౡ+, 2012] 12 / 44
41. SAT ιϧόʔͷ࢓૊Έͷྺ࢙ X 1960 ೥୅ ž DPLL (Davis-Putnam-Logemann-Loveland) [Davis+, 1962] X 1990 ೥୅ ž CDCL (Conict Driven Clause Learning) [Bayardo Jr.+, 1997, Marques-Silva+, 1999] X 2000 ೥Ҏ߱ ž ม਺બ୒ώϡʔϦεςΟοΫ VSIDS [Moskewicz+, 2001] ž 2 Ϧςϥϧ΢Υον [Moskewicz+, 2001] ž Ϧελʔτ [Luby+, 1993, Selman+, 1996, Eén+, 2003] ž Phase Saving [Pipatsrisawat+, 2007] ž ֶशઅͷධՁई౓ [Audemard+, 2009, ುౡ+, 2012] SAT ιϧόʔͷجຊಈ࡞Λ؆୯ͳྫΛ༻͍ͯઆ໌͢Δɽ ΑΓৄ͘͠஌Γ͍ͨํ͸ҎԼΛࢀর͍ͩ͘͞ɽ X ߴ଎ SAT ιϧόʔͷݪཧ (ղઆ࿦จ) [ುౡ+, 2010] MA> 12 / 44
42. SAT ιϧόʔͷ࢓૊Έͷྺ࢙ X 1960 ೥୅ ž DPLL (Davis-Putnam-Logemann-Loveland) [Davis+, 1962] X 1990 ೥୅ ž CDCL (Conict Driven Clause Learning) [Bayardo Jr.+, 1997, Marques-Silva+, 1999] X 2000 ೥Ҏ߱ ž ม਺બ୒ώϡʔϦεςΟοΫ VSIDS [Moskewicz+, 2001] ž 2 Ϧςϥϧ΢Υον [Moskewicz+, 2001] ž Ϧελʔτ [Luby+, 1993, Selman+, 1996, Eén+, 2003] ž Phase Saving [Pipatsrisawat+, 2007] ž ֶशઅͷධՁई౓ [Audemard+, 2009, ುౡ+, 2012] Ͳͷٕज़͕Ͳͷ͘Β͍ޮ͍͍ͯΔͷ͔? 12 / 44
43. ߴ଎Խٕज़ͷߩݙ౓ (Katebi ଞʹΑΔ) [Katebi+, 2011] X 2WL: 2 Ϧςϥϧ΢Υον, RST: Ϧελʔτ, PHS: Phase Saving ੨ࣈ: DPLL ʹͦͷٕज़Λ௥Ճͨ͠ιϧόʔ 13 / 44
44. ߴ଎Խٕज़ͷߩݙ౓ (Katebi ଞʹΑΔ) [Katebi+, 2011] X 2WL: 2 Ϧςϥϧ΢Υον, RST: Ϧελʔτ, PHS: Phase Saving ੺ࣈ: MiniSat 2.2 ͔Βͦͷٕज़Λ࡟আͨ͠ιϧόʔ 13 / 44
45. Algorithm 1: SAT ιϧόʔ (CDCL) ೖྗ: CNF ܗࣜͷ࿦ཧࣜ ψɼ෦෼ׂ౰ͯ α ग़ྗ: SAT ΋͘͠͸ UNSAT  ܾఆϨϕϧ ← 0; while true do ! (res, α) ← ୯Ґ఻೻(ψ, α); " if res = done then return SAT; # if res = conflict then $ if ܾఆϨϕϧ = 0 then return UNSAT; % (C, bl) ← ໃ६ղੳ(ψ, ν) ; // ֶशઅͱ bl Λܭࢉ & ψ ← ψ ∪ {C} ; // ֶशઅͷ௥Ճ ' ν ← {(xi , vi ) ∈ ν | level(xi ) ≤ bl} ; // backjump  dl ← bl;  else  dl ← dl + 1; ! ະׂ౰ͯม਺ x ͱͦͷ஋ v ΛબͿ; " α ← α ∪ {(x, v)}; 14 / 44
46. ൃද಺༰ X SAT ιϧόʔ (DPLL/CDCL) X SAT ιϧόʔͷ֦ு ž SAT ූ߸Խ ž LCG ιϧόʔ: OR-Tools (CP-SAT) ž SMT ιϧόʔ: Z3, CVC X ૊߹ͤ໰୊ʹ͸ԿΛ࢖͑͹ྑ͍ͷ? ž 2 ࣍ݩۣܗύοΩϯά໰୊Λྫ୊ʹͨ͠ੑೳධՁͷҰྫ ž CPLEX, Gurobi, SCIP ž CP Optimizer, OR-Tools (CP-SAT), ACE ž Z3, CVC5 15 / 44
47. Ͳ͏΍໋ͬͯ୊࿦ࣜҎ֎ͷ੍໿Λѻ͑ΔΑ͏ ʹ͢Δͷ͔? ૊߹ͤ໰୊ͷྫ ੔਺ม਺: x, y ∈ {2, 3, 4, 5, 6} ੍໿: (x + y ≤ 7) ∨ (x − 2y ≥ 3) (x − y ≤ 4) ∨ (2x + y ≤ 13) ූ߸ԽΛ࢖͏ํ๏  ੔਺ม਺Λ໋୊ม਺͓Αͼ࿦ཧࣜʹม׵͢Δ ੍໿Λ໋୊࿦ཧࣜʹม׵͢Δ 16 / 44
48. ௚઀ූ߸Խ [de Kleer, 1989] ΞΠσΞ ֤੔਺ม਺ x ͱͦͷυϝΠϯͷ֤஋ i ʹରͯ͠ɼx = i Λද໋͢ ୊ม਺ px=i Λ༻͍Δɽ ੔਺ม਺ x ∈ {2, 3, 4, 5, 6} ʹରͯ͠༻͍Δ໋୊ม਺ px=2 , px=3 , px=4 , px=5 , px=6 ੔਺ม਺͸ৗʹͪΐ͏Ͳ 1 ͭʹ஋͔͠औΒͳ͍͜ͱΛҎԼͷઅͰ දݱ͢Δɽ (px=2 ∨ px=3 ∨ px=4 ∨ px=5 ∨ px=6 ) ∧ (গͳ͘ͱ΋ 1 ͭ) ∧ 2≤i<j≤6 ¬px=i ∨ ¬px=j (ߴʑ 1 ͭ) 17 / 44
49. ௚઀ූ߸Խ [de Kleer, 1989] px=2 px=3 px=4 px=5 px=6 x = 2 T F F F F x = 3 F T F F F x = 4 F F T F F x = 5 F F F T F x = 6 F F F F T ੔਺ม਺΁ͷ஋ׂ౰ʹԠͯ͡ɼ໋୊ม਺͸ 1 ͚ͭͩ T ͱͳΔɽ͍ ΘΏΔ one-hot දݱͰ͋Δɽ 18 / 44
50. ௚઀ූ߸Խ [de Kleer, 1989, Walsh, 2000] ੍໿ x + y ≤ 7 (x, y ∈ {2, 3, 4, 5, 6}) ͸ɼҧ൓఺ (ਤதͷ × ఺) Λ ྻڍ͢Δ͜ͱͰҎԼͷ 15 અʹූ߸Խ͞ΕΔɽ ¬(px=2 ∧ py=6 ) ¬(px=3 ∧ py=5 ) ¬(px=3 ∧ py=6 ) ¬(px=4 ∧ py=4 ) ¬(px=4 ∧ py=5 ) ¬(px=4 ∧ py=6 ) ¬(px=5 ∧ py=3 ) ¬(px=5 ∧ py=4 ) ¬(px=5 ∧ py=5 ) ¬(px=5 ∧ py=6 ) ¬(px=6 ∧ py=2 ) ¬(px=6 ∧ py=3 ) ¬(px=6 ∧ py=4 ) ¬(px=6 ∧ py=5 ) ¬(px=6 ∧ py=6 ) x y 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 X X X X X X X X X X X X X X X 19 / 44
51. ॱংූ߸Խ [Tamura+, 2009] ΞΠσΞ ֤੔਺ม਺ x ͱͦͷυϝΠϯͷ֤஋ i ʹରͯ͠ɼx ≤ i Λද໋͢ ୊ม਺ px≤i Λ༻͍Δɽ ֤੔਺ม਺ x ʹରͯ͠༻͍Δ໋୊ม਺ px≥3 , px≥4 , px≥5 , px≥6 ੔਺ม਺ͷ஋ͷॱংؔ܎ΛҎԼͷઅͰදݱ͢Δɽ ¬px≥6 ∨ px≥5 (x ≥ 6 ͳΒ͹ x ≥ 5) ¬px≥5 ∨ px≥4 (x ≥ 5 ͳΒ͹ x ≥ 4) ¬px≥4 ∨ px≥3 (x ≥ 4 ͳΒ͹ x ≥ 3) 20 / 44
52. ॱংූ߸Խ [Tamura+, 2009] px≥2 px≥3 px≥4 px≥5 x = 2 F F F F x = 3 T F F F x = 4 T T F F x = 5 T T T F x = 6 T T T T ੔਺ม਺΁ͷ஋ׂ౰ʹԠͯ͡ɼ໋୊ม਺͸ 1 ਐ਺ (aka. thermometer) දݱͱͳΔɽ 21 / 44
53. ॱংූ߸Խ [Tamura+, 2009] ੍໿ x + y ≤ 7 (x, y ∈ {2, 3, 4, 5, 6}) ͸ɼҧ൓͢ΔൣғΛද͢͜ͱ ͰҎԼͷ 5 અʹූ߸Խ͞ΕΔɽ ¬py≥6 ¬(px≥3 ∧ py≥5 ) ¬(px≥4 ∧ py≥4 ) ¬(px≥5 ∧ py≥3 ) ¬px≥6 x y 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 X X X X X X X X X X X X X X X 22 / 44
54. ॱংූ߸Խ [Tamura+, 2009] ੍໿ x + y ≤ 7 (x, y ∈ {2, 3, 4, 5, 6}) ͸ɼҧ൓͢ΔൣғΛද͢͜ͱ ͰҎԼͷ 5 અʹූ߸Խ͞ΕΔɽ ¬py≥6 ¬(px≥3 ∧ py≥5 ) ¬(px≥4 ∧ py≥4 ) ¬(px≥5 ∧ py≥3 ) ¬px≥6 x y 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 X X X X X X X X X X X X X X X 22 / 44
55. ॱংූ߸Խ [Tamura+, 2009] ੍໿ x + y ≤ 7 (x, y ∈ {2, 3, 4, 5, 6}) ͸ɼҧ൓͢ΔൣғΛද͢͜ͱ ͰҎԼͷ 5 અʹූ߸Խ͞ΕΔɽ ¬py≥6 ¬(px≥3 ∧ py≥5 ) ¬(px≥4 ∧ py≥4 ) ¬(px≥5 ∧ py≥3 ) ¬px≥6 x y 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 X X X X X X X X X X X X X X X 22 / 44
56. ॱংූ߸Խ [Tamura+, 2009] ੍໿ x + y ≤ 7 (x, y ∈ {2, 3, 4, 5, 6}) ͸ɼҧ൓͢ΔൣғΛද͢͜ͱ ͰҎԼͷ 5 અʹූ߸Խ͞ΕΔɽ ¬py≥6 ¬(px≥3 ∧ py≥5 ) ¬(px≥4 ∧ py≥4 ) ¬(px≥5 ∧ py≥3 ) ¬px≥6 x y 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 X X X X X X X X X X X X X X X 22 / 44
57. ॱংූ߸Խ [Tamura+, 2009] ੍໿ x + y ≤ 7 (x, y ∈ {2, 3, 4, 5, 6}) ͸ɼҧ൓͢ΔൣғΛද͢͜ͱ ͰҎԼͷ 5 અʹූ߸Խ͞ΕΔɽ ¬py≥6 ¬(px≥3 ∧ py≥5 ) ¬(px≥4 ∧ py≥4 ) ¬(px≥5 ∧ py≥3 ) ¬px≥6 x y 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 X X X X X X X X X X X X X X X 22 / 44
58. ॱংූ߸Խ [Tamura+, 2009] ੍໿ x + y ≤ 7 (x, y ∈ {2, 3, 4, 5, 6}) ͸ɼҧ൓͢ΔൣғΛද͢͜ͱ ͰҎԼͷ 5 અʹූ߸Խ͞ΕΔɽ ¬py≥6 ¬(px≥3 ∧ py≥5 ) ¬(px≥4 ∧ py≥4 ) ¬(px≥5 ∧ py≥3 ) ¬px≥6 x y 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 X X X X X X X X X X X X X X X 22 / 44
59. ૊߹ͤ໰୊ͷSAT ූ߸Խ Ҏ߱ɼDE, OE Λ௚઀ූ߸Խɼॱংූ߸ԽΛදࣸ͢૾ͱ͢Δɽ x, y ∈ {2, 3, 4, 5, 6} (x + y ≤ 7) ∨ (x − 2y ≥ 3) ∧ (x − y ≤ 4) ∨ (2x + y ≤ 13) ݩͷ໰୊ OE(x) OE(y) p ∨ q r ∨ s p ↔ OE(x + y ≤ 7) q ↔ OE(x − 2y ≥ 3) r ↔ OE(x − y ≤ −4) s ↔ OE(2x + y ≤ 13) SAT ໰୊ (ॱংූ߸Խ) ⇐⇒ ͜ͷΑ͏ʹූ߸ԽʹΑΓ CNF ΛಘΔ͜ͱ͕Ͱ͖Ε͹ɼSAT ιϧ όʔͰղ͘͜ͱ͕Ͱ͖Δɽ 23 / 44
60. ൃද಺༰ X SAT ιϧόʔ (DPLL/CDCL) X SAT ιϧόʔͷ֦ு ž SAT ූ߸Խ ž LCG ιϧόʔ: OR-Tools (CP-SAT) ž SMT ιϧόʔ: Z3, CVC X ૊߹ͤ໰୊ʹ͸ԿΛ࢖͑͹ྑ͍ͷ? ž 2 ࣍ݩۣܗύοΩϯά໰୊Λྫ୊ʹͨ͠ੑೳධՁͷҰྫ ž CPLEX, Gurobi, SCIP ž CP Optimizer, OR-Tools (CP-SAT), ACE ž Z3, CVC5 24 / 44
61. LCG ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) ∨ (x − 2y ≥ 3) (x − y ≤ −4) ∨ (2x + y ≥ 13) CNF (SAT ιϧόʔ) OE(x) ∧ OE(y) DE(x) ∧ DE(y) ChannelingCon Lv 1 Lv 2 ܾఆ ఻೻ ੍໿఻೻ث (x + y ≤ 7) ∨ (x − 2y ≥ 3) (x − y ≤ −4) ∨ (2x + y ≥ 13) 25 / 44
62. LCG ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) ∨ (x − 2y ≥ 3) (x − y ≤ −4) ∨ (2x + y ≥ 13) CNF (SAT ιϧόʔ) OE(x) ∧ OE(y) DE(x) ∧ DE(y) ChannelingCon Lv 1 Lv 2 ܾఆ ఻೻ px=2 ੍໿఻೻ث (x + y ≤ 7) ∨ (x − 2y ≥ 3) (x − y ≤ −4) ∨ (2x + y ≥ 13) 25 / 44
63. LCG ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) ∨ (x − 2y ≥ 3) (x − y ≤ −4) ∨ (2x + y ≥ 13) CNF (SAT ιϧόʔ) OE(x) ∧ OE(y) DE(x) ∧ DE(y) ChannelingCon Lv 1 Lv 2 ܾఆ ఻೻ px=2 ੍໿఻೻ث (x + y ≤ 7) ∨ (x − 2y ≥ 3) x = 2 → (y ≤ 5) ∨ ⊥ (x − y ≤ −4) ∨ (2x + y ≥ 13) x = 2 → (y ≥ 6) ∨ ⊥ 25 / 44
64. LCG ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) ∨ (x − 2y ≥ 3) (x − y ≤ −4) ∨ (2x + y ≥ 13) CNF (SAT ιϧόʔ) OE(x) ∧ OE(y) DE(x) ∧ DE(y) ChannelingCon Lv 1 Lv 2 ܾఆ ఻೻ px=2 py=4 ੍໿఻೻ث (x + y ≤ 7) ∨ (x − 2y ≥ 3) x = 2 → (y ≤ 5) ∨ ⊥ (x − y ≤ −4) ∨ (2x + y ≥ 13) x = 2 → (y ≥ 6) ∨ ⊥ 25 / 44
65. LCG ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) ∨ (x − 2y ≥ 3) (x − y ≤ −4) ∨ (2x + y ≥ 13) CNF (SAT ιϧόʔ) OE(x) ∧ OE(y) DE(x) ∧ DE(y) ChannelingCon Lv 1 Lv 2 ܾఆ ఻೻ px=2 py=4 ੍໿఻೻ʹΑΓໃ६ݕग़ ੍໿఻೻ث (x + y ≤ 7) ∨ (x − 2y ≥ 3) x = 2 → (y ≤ 5) ∨ ⊥ (x − y ≤ −4) ∨ (2x + y ≥ 13) x = 2 → (y ≥ 6) ∨ ⊥ 25 / 44
66. LCG ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) ∨ (x − 2y ≥ 3) (x − y ≤ −4) ∨ (2x + y ≥ 13) CNF (SAT ιϧόʔ) OE(x) ∧ OE(y) DE(x) ∧ DE(y) ChannelingCon px=2 → py≤5 px=2 → py≥6 ¬px=2 ∨ ¬py=4 Lv 1 Lv 2 ܾఆ ఻೻ ఻೻અͱֶशઅΛ CNF ʹ௥Ճ͠ bj ੍໿఻೻ث (x + y ≤ 7) ∨ (x − 2y ≥ 3) (x − y ≤ −4) ∨ (2x + y ≥ 13) 25 / 44
67. ൃද಺༰ X SAT ιϧόʔ (DPLL/CDCL) X SAT ιϧόʔͷ֦ு ž SAT ූ߸Խ ž LCG ιϧόʔ: OR-Tools (CP-SAT) ž SMT ιϧόʔ: Z3, CVC X ૊߹ͤ໰୊ʹ͸ԿΛ࢖͑͹ྑ͍ͷ? ž 2 ࣍ݩۣܗύοΩϯά໰୊Λྫ୊ʹͨ͠ੑೳධՁͷҰྫ ž CPLEX, Gurobi, SCIP ž CP Optimizer, OR-Tools (CP-SAT), ACE ž Z3, CVC5 26 / 44
68. SMT ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) ∨ (x − 2y ≥ 3) (x − y ≤ −4) ∨ (2x + y ≥ 13) ໋୊ந৅Խ Lv 1 Lv 2 ܾఆ ఻೻ എܠཧ࿦ιϧόʔ 27 / 44
69. SMT ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) p ∨ (x − 2y ≥ 3) q (x − y ≤ −4) r ∨ (2x + y ≥ 13) s ໋୊ந৅Խ p ∨ q r ∨ s Lv 1 Lv 2 ܾఆ ఻೻ എܠཧ࿦ιϧόʔ 27 / 44
70. SMT ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) p ∨ (x − 2y ≥ 3) q (x − y ≤ −4) r ∨ (2x + y ≥ 13) s ໋୊ந৅Խ p ∨ q r ∨ s Lv 1 Lv 2 ܾఆ ఻೻ q എܠཧ࿦ιϧόʔ x − 2y ≥ 3 27 / 44
71. SMT ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) p ∨ (x − 2y ≥ 3) q (x − y ≤ −4) r ∨ (2x + y ≥ 13) s ໋୊ந৅Խ p ∨ q r ∨ s Lv 1 Lv 2 ܾఆ ఻೻ q r എܠཧ࿦ιϧόʔ x − 2y ≥ 3 x − y ≤ −4 27 / 44
72. SMT ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) p ∨ (x − 2y ≥ 3) q (x − y ≤ −4) r ∨ (2x + y ≥ 13) s ໋୊ந৅Խ p ∨ q r ∨ s Lv 1 Lv 2 ܾఆ ఻೻ q r ཧ࿦఻೻ʹΑΓໃ६ݕग़ എܠཧ࿦ιϧόʔ x − 2y ≥ 3 x − y ≤ −4 27 / 44
73. SMT ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) p ∨ (x − 2y ≥ 3) q (x − y ≤ −4) r ∨ (2x + y ≥ 13) s ໋୊ந৅Խ p ∨ q r ∨ s ¬q ∨ ¬r Lv 1 Lv 2 ܾఆ ఻೻ q ֶशઅ ¬q ∨ ¬r Λ௥Ճ͠ backjump എܠཧ࿦ιϧόʔ x − 2y ≥ 3 27 / 44
74. SMT ιϧόʔ ݩͷ໰୊ x, y ∈ {2, . . . , 6} (x + y ≤ 7) p ∨ (x − 2y ≥ 3) q (x − y ≤ −4) r ∨ (2x + y ≥ 13) s ໋୊ந৅Խ p ∨ q r ∨ s ¬q ∨ ¬r Lv 1 Lv 2 ܾఆ ఻೻ q ¬r ୯Ґ఻೻ എܠཧ࿦ιϧόʔ x − 2y ≥ 3 x − y > −4 27 / 44
75. ൃද಺༰ X SAT ιϧόʔ (DPLL/CDCL) X SAT ιϧόʔͷ֦ு ž SAT ූ߸Խ ž LCG ιϧόʔ: OR-Tools (CP-SAT) ž SMT ιϧόʔ: Z3, CVC X ૊߹ͤ໰୊ʹ͸ԿΛ࢖͑͹ྑ͍ͷ? ž 2 ࣍ݩۣܗύοΩϯά໰୊Λྫ୊ʹͨ͠ੑೳධՁͷҰྫ ž CPLEX, Gurobi, SCIP ž CP Optimizer, OR-Tools (CP-SAT), ACE ž Z3, CVC5 28 / 44
76. ۣܗύοΩϯά໰୊Λྫ୊ʹͨ͠ੑೳධՁ ໨త ૊߹ͤ໰୊ʹ͓͚Διϧόʔੑೳͷ 1 ͭͷࢀߟࢿྉΛ࡞Δ͜ͱ ൺֱͨ͠ιϧόʔ X ਺ཧ࠷దԽιϧόʔ: CPLEX, Gurobi, SCIP, OR-Tools CBC X CP ιϧόʔ: OR-Tools (CP-SAT), ACE X SMT ιϧόʔ: Z3, CVC5 ϕϯνϚʔΫ: Consecutive Square Packing X ೖྗ: ੔਺஋ n X ੍໿: 1x1, 2x2, ..., nxn ͷਖ਼ํܗΛॏͳΒͣʹ഑ஔ͢Δ X ग़ྗ: શͯͷਖ਼ํܗΛ഑ஔ͢Δͷʹඞཁͳ࠷খͷਖ਼ํܗίϯ ςφͷล௕ 10 ͔Β 100 ·Ͱͷ 90 ໰͓Αͼ 500, 1000 ͷ 2 ໰ͷ߹ܭ 92 ໰ɽ 29 / 44
77. ܭࢉػ࣮ݧ (ଓ͖) ࣮ݧઃఆ X CPU: Intel(R) Xeon(R) Gold 6346 CPU @ 3.10GHz X MEM: 128 GB X ੍࣌ؒݶ: 1800 ඵ X ࣮ߦ͸֤ιϧόʔͷઃఆͰγϯάϧεϨουʹ੍ݶ ධՁࢦඪ  ࠷ద஋Λݟ͚ͭͨ໰୊਺ͱܭࢉ࣌ؒ (ΧΫλεϓϩοτ) ੍ݶ࣌ؒ಺ʹݟ͚ͭͨ࠷ྑղͷ඼࣭ (ώʔτϚοϓ) ँࣙ ͜ͷൺֱͰ͸ Gurobi ࣾɼIBM ͔ࣾΒି༩͞Εֶͨज़ϥΠηϯεΛ ࢖༻͠·ͨ͠ɽޚྱਃ্͛͠·͢ɽ 30 / 44
78. 1) ࠷ద஋Λݟ͚ͭͨ໰୊਺ͱܭࢉ࣌ؒ 2 4 6 8 10 12 14 16 Number of Instances Solved 0 200 400 600 800 1000 1200 1400 Wall Clock Time (seconds) ace cplex cpoptimizer cvc5 gurobi_indicator_int_all gurobi_indicator_int_obj gurobi_indicator_mip ortools_cbc ortools_cpsat scip z3 31 / 44
79. 2) ੍ݶ࣌ؒ಺ʹݟ͚ͭͨ࠷ྑղͷ඼࣭ X ԣ࣠: ΠϯελϯεͷೖྗαΠζ n X ॎ࣠: ιϧόʔ X ৭: =࠷ྑ, =ղ͸ग़͕ͨ࠷ѱ, =ղ͕ग़ͳ͔ͬͨ ⋆=࠷ద஋ 10 20 30 40 50 60 70 80 90 100 Z3 SCIP GRB-IO CPOpt OR-SAT 32 / 44
80. SAT ٕज़ͷԠ༻ SAT PB ASP ALLSAT #SAT QBF SMT MaxSAT CSP/COP ج൫ٕज़ DPLL/CDCL 33 / 44
81. SAT ٕज़ͷԠ༻ SAT PB ASP ALLSAT #SAT QBF SMT MaxSAT CSP/COP ج൫ٕज़ DPLL/CDCL 33 / 44
82. ϋΠϒϦουූ߸Խ [Soh+, 2017] X ॱংූ߸Խ [Tamura+, 2009] (࣮૷: Sugar MA> ) ž ੔਺ͷॱংؔ܎Λ༻͍͓ͯΓɼઢܗൺֱ ∑ aixi ≥ b ͷූ߸Խ ʹద͍ͯ͠Δɽ ž ॱংූ߸ԽΛ࣮૷ͨ͠ Sugar ͸ 2008 ೥ɼ 2009 ೥ʹ CSP ιϧ όʔڝٕձͷάϩʔόϧ෦໳Ͱ༏উͨ͠ɽ ž ཧ࿦తʹ΋ଞͷූ߸Խʹ͸ͳ͍ྑ͍ੑ࣭Λ͍࣋ͬͯΔ͜ͱ͕ূ ໌͞Ε͍ͯΔ [Petke+, 2011]ɽ ž ͔͠͠ɼ໰୊ͷن໛͕େ͖͘ͳΔͱੑೳ͕ѱ͘ͳΔɽ X ର਺ූ߸Խ [Iwama+, 1994] ž ੔਺ͷ 2 ਐ਺දݱΛ༻͍͓ͯΓίϯύΫτͳ SAT ූ߸Խ͕Մ ೳ͕ͩɼઢܗൺֱʹ͍ͭͯҰൠʹॱংූ߸ԽΑΓੑೳ͕ѱ͍ɽ ž ໰୊ͷن໛͕େ͖ͯ͘΋ղ͘͜ͱ͕Ͱ͖Δɽ X ϋΠϒϦουූ߸Խ [Soh+, 2017] (࣮૷: Fun-sCOP MA> ) ž ॱংූ߸Խͱର਺ූ߸ԽΛ༥߹ͨ͠ SAT ූ߸Խ๏ɽ ž ֤੔਺ม਺͸ॱংූ߸Խ΋͘͠͸ର਺ූ߸ԽͷͲͪΒ͔ҰͭͰ ූ߸Խ͞Εɼ੍֤໿͸྆ํͷූ߸Խม਺ΛؚΉ͜ͱ͕Ͱ͖Δɽ 34 / 44
83. ϋΠϒϦουූ߸Խ: ྫ ੔਺ม਺ x1 , x2 ∈ {0, 1, 2, 3} Λߟ͑Δ. ੍໿ 2x1 + x2 ≥ 5 ͸ τL {x2} ◦ τO {x1} ʹΑͬͯҎԼͷΑ͏ʹූ߸Խ͞ΕΔ. (τL {x2} ◦ τO {x1} )(2x1 + x2 ≥ 5) ⇐⇒ τO {x1} (2x1 + px2,0 + 2px2,1 ≥ 5) ⇐⇒ ∧ d1∈{0,1,2,3} (x1 ≥ d1 + 1) ∨ (px2,0 + 2px2,1 ≥ 5 − 2d1 ) ⇐⇒ (px1≤1 ) ∧ (px1≤2 ∨ (px2,0 + 2px2,1 ≥ 3)) ∧ (px1≤3 ∨ (px2,0 + 2px2,1 ≥ 1)) 35 / 44
84. ։ൃͨ͠CSP ιϧόʔFun-sCOP ͷ৆ྺ ࠃࡍ CSP ιϧόʔڝٕձ XCSP3 ͸ 2005 ೥ʹͦͷલ਎͕࢝·ͬͨྺ࢙ͷ͋Δࠃࡍ CSP ιϧ όʔڝٕձͰ͋Γɼੈք֤ࠃͷڭҭݚڀػؔͰ։ൃ͞Ε͍ͯΔ CSP ιϧόʔ͕ଟ਺ࢀՃ͍ͯ͠Δ. X Fun-sCOP ͸ 2018 ೥Ҏ߱ XCSP3 ʹ͓͚Δ࠷΋ॏཁͳ෦໳Ͱ ͋Δ Main CSP ʹຖճࢀՃ͠ҎԼͷ੒੷Λ͓͞Ί͍ͯΔɽ X 2018 ೥༏উ X 2019 ೥४༏উ X 2022 ೥४༏উ X 2023 ೥४༏উ X 2024 ೥ 3 Ґೖ৆ ࠃࡍձٞ CP18 ͷ݁Ռൃද 36 / 44
85. γεςϜੜ෺ֶͱҨ఻ࢠ੍ޚωοτϫʔΫ X γεςϜੜ෺ֶͱ͸ɼੜ෺ͷ׆ಈΛҨ఻ࢠɾλϯύΫ࣭ɾࡉ ๔ͳͲ͕૬ޓʹ࡞༻͢ΔෳࡶͳγεςϜͱͯ͠ଊ͑ɼͦͷશ ମతͳڍಈ΍ػೳΛཧղ͠Α͏ͱ͢Δֶ໰෼໺Ͱ͋Δɽ The Nobel Committee for Physiology or Medicine / Illustration: Annika Röhl Ҩ఻ࢠ a λϯύΫ࣭ a Ҩ఻ࢠ b λϯύΫ࣭ b X Ҩ఻ࢠ੍ޚωοτϫʔΫͱ͸ɼҨ఻ࢠͷ૬ޓ࡞༻Λ௖఺ΛҨ ఻ࢠɼลΛௐઅҼࢠͱͨ͠άϥϑͰදݱͨ͠΋ͷͰ͋Δɽ 37 / 44
86. ϒʔϦΞϯωοτϫʔΫͱෆಈ఺ ϒʔϦΞϯωοτϫʔΫ (BN) BN ͸Ҩ఻ࢠ੍ޚωοτϫʔΫͷ୅දతͳ਺ཧϞσϧͷ 1 ͭͰ͋ ΔɽҨ఻ࢠͷঢ়ଶΛ 1 ͱ 0 ͷೋ஋ɼௐઅҼࢠ͸໋୊࿦ཧࣜͰද͢ɽ v1 v2 v3 vt+1 1 = vt 2 vt+1 2 = ¬vt 3 vt+1 3 = ¬vt 1 ∧ vt 2 ঢ়ଶભҠਤͱෆಈ఺ 000 010 011 001 100 110 111 101 े෼௕͘ભҠΛଓ͚Δͱ 110 (ෆ ಈ఺) ʹ౸ୡ͠ભҠ͠ͳ͘ͳΔ (≓ ࡉ๔ͷఆৗ (࠷ऴ) ঢ়ଶ)ɽ 38 / 44
87. SAT ܕBN ෆಈ఺ܭ਺ιϧόʔ [Higuchi+, 2025] ͜Ε·Ͱͷख๏Λෆಈ఺ͷܭ਺ʹԠ༻͢Δ͜ͱͰطଘݚڀΛେ͖ ্͘ճΔੑೳΛୡ੒ͨ͠ɽ 0.01 0.1 1 10 100 1000 0 50 100 150 200 250 300 CPU Time [s] Solved Instances PyBoolNet fASP-conj SAF fASP-src Hybrid Enum. AEON Direct Count. Indirect Count. Hybrid Count. 39 / 44
88. ະ஌ͷෆಈ఺ͷܭ਺ [Higuchi+, 2025] ෆಈ఺ͷ਺͕ະ஌Ͱ࣮͋ͬͨੈքͷωοτϫʔΫʹର͠ɼͦͷ਺ Λਖ਼֬ʹܭ਺͢Δ͜ͱʹ੒ޭͨ͠ (IJCAI 2025 Ͱൃද). Instance #Vars #Attractors #113 ER-STRESS 182 1168455003694263561093120 #122 NSP14 168 33278627362665583108034953216 #124 NSP9-PROTEIN 252 13611294676837538538534984297270728458240 #144 SNF1-AMPK-PATHWAY 202 10096027719780900754667077632 #220 H.-RESPONSE-IN-L. 342 2656331146614175432704000 #221 MYCOBACTERIAL-L. 317 2473901162496 Alzheimer 762 1355318094474400392445140020586319209- -7103960354330270737143428036029317120 Cholocystokinin 383 47935169240579835005239296 Yeast-Pheromone 246 5711631030629640192 40 / 44
89. ASP ͷ࣮༻γεςϜྫ: ASPITAL ࢁསେֶපӃͰӡ༻͞Ε͍ͯΔNS γεςϜ 41 / 44
90. ASPITAL σϞ ΫϦοΫͯ͠ಈըΛ࠶ੜ 42 / 44
91. ·ͱΊ X SAT ιϧόʔͱ੍໿ιϧόʔͷجૅٕज़ͱ࠷৽ಈ޲Λ঺հ ž DPLL/CDCL ΞϧΰϦζϜ͕ج൫ٕज़ X SAT ιϧόʔͷ֦ுͱԠ༻ ž SAT ූ߸Խ (௚઀ූ߸Խɾॱংූ߸Խ) ʹΑΔ໰୊ม׵ ž LCG ιϧόʔ (OR-Tools CP-SAT): ஗Ԇූ߸ԽͰޮ཰Խ ž SMT ιϧόʔ (Z3, CVC5): ཧ࿦ͱ૊Έ߹Θͤͨਪ࿦ ž 2 ࣍ݩۣܗύοΩϯά໰୊ͷੑೳධՁ X ۙ೥ͷݚڀ੒Ռ ž ϋΠϒϦουූ߸Խ (Fun-sCOP): ॱংූ߸Խͱର਺ූ߸Խͷ༥߹ ž ϒʔϧ੍ޚωοτϫʔΫͷෆಈ఺ܭ਺ (IJCAI 2025) ž ASP: એݴతͳ໰୊هड़ͱ࣮༻γεςϜ (ASPITAL) 43 / 44
92. ݚڀϓϩδΣΫτ X ࠓ೔঺հͨ͠ݚڀ͸൪ݪກଇઌੜ (໊ݹ԰େֶ) ͱುౡӳ஌ઌ ੜ (ࢁསେֶ) ͱߦ͖ͬͯͨɽ X ࠓ೥͔Β 3 ਓͰ৽͘͠ KRR (Knowledge Representation and Reasoning) ϓϩδΣΫτΛ࢝Ίͨɽ X େֶͰ͸൪ݪઌੜͱೋਓͰݚڀࣨΛӡӦ͍ͯ͠Δɽ X ڵຯͷ͋Δํ͸ͥͻ࿈བྷΛʂ 44 / 44
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