Introducing a syntax-driven and compositional syntax and semantics to hypergraph rewriting system (presented at JSSST 2021)

A slide for a presentation made in jssst2021.

Transcript

1. +4445  ೔໨ 11-  <-> ϋΠύʔάϥϑॻ͖׵͑ܥ΁ͷ ߏจۦಈͰ DPNQPTJUJPOBM ͳ ߏจɾҙຯ࿦ͷఏҊ  ೔ຊιϑτ΢ΣΞՊֶձୈ  ճେձ ૣҴాେֶ جװཧ޻ֶݚڀՊ ৘ใཧ޻ɾ৘ใ௨৴ઐ߈ ࠤ໺ਔ ɹ্ా࿨ل
2.  ݚڀͷഎܠ  'MBU )ZQFS-./UBM ͷந৅ߏจ  'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦  'MBU )ZQFS-./UBM ʹؔͯ͠ߦͳͬͨূ໌  ·ͱΊ 
3. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT എܠʛϋΠύʔάϥϑ 𝜆 ܭࢉͷ߲ɼฒߦϓϩηεͷঢ়ଶɼϙΠϯλͳͲɼ ίϯϐϡʔλαΠΤϯεʹݱΕΔଟ͘ͷ΋ͷ͸ ϋΠύʔάϥϑ<> ͰϞσϧԽͰ͖Δ ล͕ͦΕͧΕ  ͭҎ্ͷ ௖఺Λ݁Ϳ͜ͱ͕Ͱ͖Δ Α͏ʹάϥϑΛ֦ுͨ͠΋ͷ ैͬͯɼϋΠύʔάϥϑΛୈҰڃʹѻ͏؆໌ͳܭࢉϞσϧ ͕͋Δͷ͕๬·͍͠ <> #FSHF $MBVEF )ZQFSHSBQIT $PNCJOBUPSJDT PG 'JOJUF 4FUT /PSUI)PMMBOE  
4. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT എܠʛάϥϑॻ͖׵͑ܥ ʢϋΠύʔʣάϥϑΛୈҰڃʹѻ͏ύϥμΠϜͱͯ͠ ʢϋΠύʔʣάϥϑॻ͖׵͑ܥ<><> ͕͋Δ ߏจۦಈͰͳ͍ҙຯ࿦ʢ࣍εϥΠυ͔Βઆ໌ʣΛ࣋ͭϓϩάϥϛϯάݴޠɾπʔϧɿ (1 <>ɼ%BDUM<> (S(FO<> | ߏจۦಈͳҙຯ࿦Λ࣋ͭܭࢉϞσϧ͔ͭϓϩάϥϛϯάݴޠɿ -./UBM FMFNFOUBM <> <> $ISJTUPQIFS #BL (1  &⒏DJFOU *NQMFNFOUBUJPO PG B (SBQI 1SPHSBNNJOH -BOHVBHF 1I% UIFTJT 6OJWFSTJUZ PG :PSL 4FQU  <> + 3 8 (MBVFSU FU BM %BDUM  BO &YQFSJNFOUBM (SBQI 3FXSJUJOH -BOHVBHF *O Proc. Graph Grammars 1990 7PM  -/$4 4QSJOHFS  QQ r <> 3VCJOP (FJ– FU BM (S(FO " 'BTU 410#BTFE (SBQI 3FXSJUJOH 5PPM *O Proc. ICGT 2006 4QSJOHFS  QQ r <> ,B[VOPSJ 6FEB -./UBM BT B IJFSBSDIJDBM MPHJD QSPHSBNNJOH MBOHVBHF *O 5IFPSFUJDBM $PNQVUFS 4DJFODF   QQ r <> ) &ISJH FU BM 'VOEBNFOUBMT PG "MHFCSBJD (SBQI 5SBOTGPSNBUJPO 4QSJOHFS  <> (S[FHPS[ 3P[FOCFSH )BOECPPL PG (SBQI (SBNNBST BOE $PNQVUJOH CZ (SBQI 5SBOTGPSNBUJPO 8PSME 4DJFOUJpD  
5. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT طଘͷϋΠύʔάϥϑॻ͖׵͑ܥͷҙຯ࿦ͷ໰୊఺ ⟨1/2⟩ ҰൠʹɼϋΠύʔάϥϑॻ׵͑ܥͷҙຯ࿦Ͱ͸ɼϋΠύʔάϥϑ Λ ௖఺ू߹ɼลू߹ɼ௖఺͔Βล΁ͷରԠɼϥϕϦϯάؔ਺ ͳͲͷ૊Ͱఆٛ͢Δ<><> FH ⟨ ௖఺ू߹ɿ {1, 2, 3}, ลू߹ɿ {1}, ௖఺͔Βล΁ͷରԠɿ {1 ↦ 1, 1 ↦ 2, 1 ↦ 3}, ϥϕϦϯάؔ਺ɿ {1 ↦ 𝑎, 2 ↦ 𝑎, 3 ↦ 𝑎} ⟩ ͜Ε͸ DPNQPTJUJPOBM Ͱͳ͍ αϒάϥϑʹ෼ׂ ⇄ ෳ਺ͷαϒάϥϑΛ߹੒ ͷΑ͏ͳߦ͖དྷ͕Ͱ͖ͳ͍ʢͱͯ΋ͮ͠Β͍ʣ <> ) &ISJH FU BM 'VOEBNFOUBMT PG "MHFCSBJD (SBQI 5SBOTGPSNBUJPO 4QSJOHFS  <> (S[FHPS[ 3P[FOCFSH )BOECPPL PG (SBQI (SBNNBST BOE $PNQVUJOH CZ (SBQI 5SBOTGPSNBUJPO 8PSME 4DJFOUJpD  
6. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT طଘͷϋΠύʔάϥϑॻ͖׵͑ܥͷҙຯ࿦ͷ໰୊఺ ⟨2/2⟩ ҰൠʹɼϋΠύʔάϥϑॻ׵͑ܥͷҙຯ࿦Ͱ͸ɼ αϒάϥϑ΁ͷϚονϯά΍ੜ੒ ͸ ௖఺ू߹ͳͲͷ૊Ͱఆٛ͞ΕͨϋΠύʔάϥϑ ΁ͷ ࣹ Λ༻͍ͯఆٛ͞ΕΔ EPVCMFTJOHMFQVTIPVU ʢ͜ΕΒ͸ݍ࿦ͷ༻ޠʣ ͜Ε͸ߏจۦಈͰͳ͍ ߏจཁૉ͔Β௚઀తʹૢ࡞తҙຯ࿦͕ఆٛͰ͖͓ͯΒͣɼ ߴ౓ͳ਺ֶతૉཆΛཁٻɾ࣮ࡍͷϓϩάϥϜͱͷରԠ͕औΓͮΒ͍ 𝜆 ܭࢉɼ𝜋 ܭࢉ<>ɼ*.1 ͳͲͷ ޿͘ීٴ͍ͯ͠Δҙຯ࿦͸ͲΕ΋ߏจۦಈ <> %BWJEF 4BOHJPSHJ FU BM 5IF 1J$BMDVMVT " 5IFPSZ PG .PCJMF 1SPDFTTFT $BNCSJEHF 6OJWFSTJUZ 1SFTT  
7. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ݚڀͷഎܠͱ໨త എܠ ϋΠύʔάϥϑॻ͖׵͑ܥͷҙຯ࿦͸ ҰൠʹߏจۦಈͰͳ͍ ʢݍ࿦ͳͲΛਂ͘ཧղ͍ͯ͠ΔਓͰͳ͍ͱʣ೉͍͠ ໨త 𝜆 ܭࢉɼ𝜋 ܭࢉͳͲͷΑ͏ͳ ߏจۦಈͳҙຯ࿦ Λ࣋ͭܭࢉϞσϧΛఏҊ͢Δ 
8. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ϋΠύʔάϥϑॻ͖׵͑ܥ͕උ͑Δ΂͖ػೳʛลͷ༥߹ ௖఺ͷଟॏू߹΁ͷϚονϯάͱॻ͖׵͕͑Ͱ͖Ε͹ɼͦΕ͚ͩͰྑ͍ʁ %BDUM<> ͸ɼ.BUDIJOHɼ#VJMEJOH ʢॻ͖׵͑ޙͷαϒάϥϑੜ੒ʣͷޙʹ 3FEJSFDUJPO ଞͷ௖఺͕࣋ͭࢀরΛ͋ΔΞυϨεͷํ΁޲͚௚͢ ແ޲άϥϑͰ͸ɼ͋ΔลΛଞͷลͱ༥߹ͤ͞Δ ͜ͱʹରԠ͢Δ Λߦ͏ 3FEJSFDUJPO ͸ίϯϐϡʔλαΠΤϯεʹ͓͍ͯීวతʹݟΒΕΔ֓೦ ϋΠύʔϦϯΫͷ༥߹΋ਫ਼ີʹܗࣜԽ͍ͨ͠ <> + 3 8 (MBVFSU FU BM %BDUM  BO &YQFSJNFOUBM (SBQI 3FXSJUJOH -BOHVBHF *O Proc. Graph Grammars 1990 7PM  -/$4 4QSJOHFS  QQ r 
9. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ؔ࿈ݚڀʛϓϩηε୅਺ ฒߦϓϩηε Λѻ͏ ߏจۦಈͳܭࢉϞσϧ $41 $$4 𝜋 ܭࢉ<> | 'VTJPO DBMDVMVT<> <> %BWJEF 4BOHJPSHJ FU BM 5IF 1J$BMDVMVT " 5IFPSZ PG .PCJMF 1SP DFTTFT $BNCSJEHF 6OJWFSTJUZ 1SFTT  <> + 1BSSPX FU BM 5IF GVTJPO DBMDVMVT FYQSFTTJWFOFTT BOE TZNNF USZ JO NPCJMF QSPDFTTFT *O Proc. LICS 1998  QQ r ຊݚڀͰ͸ϓϩηε୅਺ʹ͓͚Δ ʮ໊લͷӅṭʢείʔϓʣ ʯͷߏจɾҙຯ࿦ Λάϥϑॻ͖׵͑ܥʹ৽ͨʹಋೖͨ͠ 'VTJPO DBMDVMVT 𝜋 ܭࢉͷ೿ੜ νϟωϧΛड৴͢Δࡍʹ ड৴ͨ͠νϟωϧΛ ہॴతʹଋറ͢ΔͷͰ͸ͳ͘ɼ େҬతʹͦΕΒͷ νϟωϧΛಉ͡΋ͷͱ͢Δɼ νϟωϧͷ༥߹ʢGVTJPOʣΛߦ͏ GVTJPO ͸ϋΠύʔϦϯΫͷ༥߹ ͱ௚ʹରԠ͢Δ֓೦ 
10. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ؔ࿈ݚڀʛ֊૚άϥϑॻ͖׵͑ݴޠ -./UBM FMFNFOUBM <> ֊૚άϥϑॻ͖׵͑ʹجͮ͘ ߏจۦಈͳܭࢉϞσϧ͔ͭϓϩάϥϛϯάݴޠ ຊݚڀͰ͸֊૚ԽͷػೳΛল͍ͨ 'MBU -./UBM Λϕʔεʹɼ ֦ு͢ΔܗͰϋΠύʔάϥϑॻ͖׵͑ܥΛܗࣜԽͨ͠ ͨͩ͠ɼ-./UBM ʹ͓͚ΔάϥϑͷลʢϦϯΫʣ͸ϋΠύʔϦϯΫͰ͸ͳ͘ɼ ͦΕΛલఏͱͨ͠ҙຯ࿦ʹͳ͍ͬͯΔ ϋΠύʔϦϯΫ΁ͷ֦ுͷͨΊʹ͸ɼࠜຊ͔ΒखΛೖΕΔඞཁ͕͋Δ <> ,B[VOPSJ 6FEB -./UBM BT B IJFSBSDIJDBM MPHJD QSPHSBNNJOH MBOHVBHF *O 5IFPSFUJDBM $PNQVUFS 4DJFODF   QQ r 
11. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ؔ࿈ݚڀʛ )ZQFS-./UBM<> ϋΠύʔϦϯΫͷػೳΛఏڙ͢Δɼ-./UBM ͷ ࣮૷্ͷ֦ு ʢ ʙʣ ΞϓϦέʔγϣϯ 𝜆 ܭࢉͷ &ODPEJOH<>ɼୈҰڃॻ͖׵͑نଇɾϝλΠϯλϓϦλ<>ɼ $BQBCJMJUZ 5ZQF $IFDLJOH<>ɼ(.BDIJOF ͷ࣮૷<>ɼ| <> " :BTFO FU BM )ZQFSHSBQI 3FQSFTFOUBUJPO PG -BNCEB5FSNT *O TASE 2016 *&&& $PNQVUFS 4PDJFUZ +VMZ  QQ r <> :VUBSP 5TVOFLBXB FU BM *NQMFNFOUBUJPO PG -./UBM .PEFM $IFDLFST " .FUBQSPHSBNNJOH "QQSPBDI *O +PVSOBM PG 0CKFDU 5FDIOPMPHZ  /PW   <> 4UFGBO 8BMUFS $BQBCJMJUZ 5ZQJOH GPS )ZQFS-./UBM .BTUFS`T 5IFTJT 8BTFEB 6OJWFSTJUZ  <> +JO 4BOP *NQMFNFOUJOH (.BDIJOF JO )ZQFS-./UBM IUUQTBSYJWPSHBCT #BDIFMPS`T 5IFTJT 8BTFEB 6OJWFSTJUZ  จݙ <> ʹɼඇܗࣜతʹ࢓༷͕ड़΂ΒΕ͍ͯΔ͚ͩʹཹ·Δ ߏจۦಈͳܭࢉϞσϧͱ͸ͳ͍ͬͯͳ͍ <> ,B[VOPSJ 6FEB FU BM )ZQFS-./UBM "O &YUFOTJPO PG B )JFSBSDIJDBM (SBQI 3FXSJUJOH .PEFM *O ,ÛOTUMJDIF *OUFMMJHFO[   QQ r 
12. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ຊݚڀͷ੒Ռ ߏจۦಈͳϋΠύʔάϥϑॻ͖׵͑ܥͷ ϛχϚϜͳܗࣜతఆٛͱͯ͠ɼ 'MBU )ZQFS-./UBM ΛఏҊ͢Δ 1SPDFTT 𝑃   /VMM | 𝑝(𝑋1 , … , 𝑋𝑚 ) 𝑚 ≥ 0 "UPN | (𝑃, 𝑃) .PMFDVMF | 𝜈𝑋.𝑃 )ZQFSMJOL DSFBUJPO | (𝑃 :- 𝑃) 3VMF fn() = ∅ fn(𝑝(𝑋1 , … , 𝑋𝑚 )) = ⋃𝑚 𝑖=1 {𝑋𝑖 } fn((𝑃, 𝑄)) = fn(𝑃) ∪ fn(𝑄) fn(𝜈𝑋.𝑃) = fn(𝑃) ⧵ {𝑋} fn((𝑃 :- 𝑄)) = ∅ & (, 𝑃) ≡ 𝑃 & (𝑃, 𝑄) ≡ (𝑄, 𝑃) & (𝑃, (𝑄, 𝑅)) ≡ ((𝑃, 𝑄), 𝑅) & 𝑃 ≡ 𝑃′ ⇒ (𝑃, 𝑄) ≡ (𝑃′, 𝑄) & 𝑃 ≡ 𝑄 ⇒ 𝜈𝑋.𝑃 ≡ 𝜈𝑋.𝑄 & 𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝜈𝑋.𝑃⟨𝑌 /𝑋⟩ ʢͨͩ͠ 𝑋 ∈ fn(𝑃) ∨ 𝑌 ∈ fn(𝑃)ʣ & 𝜈𝑋.𝜈𝑌 .𝑋 ⋈ 𝑌 ≡  & 𝜈𝑋. ≡  & 𝜈𝑋.𝜈𝑌 .𝑃 ≡ 𝜈𝑌 .𝜈𝑋.𝑃 & 𝜈𝑋.(𝑃, 𝑄) ≡ (𝜈𝑋.𝑃, 𝑄) ʢͨͩ͠ 𝑋 ∉ fn(𝑄)ʣ 3 𝑃 ⟶ 𝑃′ (𝑃, 𝑄) ⟶ (𝑃′, 𝑄) 3 𝑃 ⟶ 𝑄 𝜈𝑋.𝑃 ⟶ 𝜈𝑋.𝑄 3 𝑄 ≡ 𝑃 𝑃 ⟶ 𝑃′ 𝑃′ ≡ 𝑄′ 𝑄 ⟶ 𝑄′ 3 (𝑃, (𝑃 :- 𝑄)) ⟶ (𝑄, (𝑃 :- 𝑄)) 
13.  ݚڀͷഎܠ  'MBU )ZQFS-./UBM ͷந৅ߏจ  'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦  'MBU )ZQFS-./UBM ʹؔͯ͠ߦͳͬͨূ໌  ·ͱΊ 
14. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ߏจཁૉʛࣝผࢠͱ༧໿໊ 𝑋 ͸ʢϋΠύʔʣϦϯΫ໊Λද͢ 𝑝 ͸ΞτϜ໊ʢ௖఺ͷϥϕϧʣΛද͢ ⋈ ͷΈ༧໿໊ ΞτϜ 𝑋 ⋈ 𝑌 ͸ GVTJPO ͱݺͼɼ ௚ײతʹ͸Ҿ਺ͷ ϋΠύʔϦϯΫಉ࢜Λܨ͛Δ ػೳΛ࣋ͭ 
15. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT 'MBU )ZQFS-./UBM ͷந৅ߏจ 1SPDFTT 𝑃 ∶∶=  /VMM άϥϑ΋ॻ͖׵͑نଇ΋Կ΋ͳ͍ঢ়ଶ | 𝑝(𝑋1 , … , 𝑋𝑚 ) 𝑚 ≥ 0 "UPN άϥϑͷϥϕϧ෇͖௖఺ | (𝑃, 𝑃) .PMFDVMF ௖఺΍ॻ͖׵͑نଇͷଟॏू߹ | 𝜈𝑋.𝑃 )ZQFSMJOL DSFBUJPO ϋΠύʔϦϯΫͷείʔϓ ϓϩηε୅਺͔Β৽ͨʹಋೖ | (𝑃 :- 𝑃) 3VMF αϒάϥϑͷॻ͖׵͑نଇ 
16. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ࣗ༝ϋΠύʔϦϯΫ ϓϩηε 𝑃 ʹ͓͚Δࣗ༝ϋΠύʔϦϯΫͷू߹Λ fn(𝑃) Ͱද͠ɼ ҎԼͷΑ͏ʹؼೲతʹఆٛ͢Δ ϓϩηε୅਺͔Β৽ͨʹಋೖ fn() = ∅ fn(𝑝(𝑋1 , … , 𝑋𝑚 )) = 𝑚 ⋃ 𝑖=1 {𝑋𝑖 } fn((𝑃, 𝑄)) = fn(𝑃) ∪ fn(𝑄) fn(𝜈𝑋.𝑃) = fn(𝑃) ⧵ {𝑋} fn((𝑃 :- 𝑄)) = ∅ 
17. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT 'MBU )ZQFS-./UBM ͷߏจ৚݅ ϧʔϧ (𝑃 :- 𝑄) ͸ҎԼʹࣔ͢৚݅Λຬͨ͢ඞཁ͕͋Δ  ϧʔϧ͸ 𝑃 ʹग़ݱͯ͠͸ͳΒͳ͍ʢ-./UBM ͱಉ༷ʣ  fn(𝑃) ⊇ fn(𝑄) ʢࠓճ৽͘͠ಋೖʣ ௚ײతʹ͸ɼ ʮӈลͰ৽͘͠ग़ݱ͢ΔϋΠύʔϦϯΫ͸ 𝜈 Ͱએݴͯ͠΍Δඞཁ͕͋ΔʯͱಡΊΔ 
18.  ݚڀͷഎܠ  'MBU )ZQFS-./UBM ͷந৅ߏจ  'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦  'MBU )ZQFS-./UBM ʹؔͯ͠ߦͳͬͨূ໌  ·ͱΊ 
19. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ૢ࡞తҙຯ࿦ʛߏ଄߹ಉنଇ άϥϑΛѻ͏ͨΊʹ͸ɼந৅ߏจ໦ + 𝛼 ม׵͚ͩͰ͸ෆ଍ ߏจతʹ͸ҟͳ͍ͬͯͯ΋ɼάϥϑͱͯ͠͸౳͍͠΋ͷ΋͋Δ FH (𝑎(𝑋), 𝑏(𝑋)) ͱ (𝑏(𝑋), 𝑎(𝑋)) ߏ଄߹ಉنଇ ͸ɼ͍͔ͭ͘ͷެཧͰ͜ΕΒͷ౳ՁੑΛఆٛ͢Δ FH (𝑃, 𝑄) ≡ (𝑄, 𝑃) ϓϩηε୅਺ʹ͓͍ͯɼීวతʹ༻͍ΒΕ͍ͯΔ -./UBM ͷҙຯ࿦ʹ͓͍ͯ΋࠾༻͞Ε͍ͯΔ 
20. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ૢ࡞తҙຯ࿦ͷͨΊͷ४උʛϋΠύʔϦϯΫ୅ೖ 𝑃⟨𝑌 /𝑋⟩ ͸ϋΠύʔϦϯΫ୅ೖͰ͋Γɼ શͯͷࣗ༝ग़ݱ͢Δ 𝑋 Λ 𝑌 Ͱஔ͖׵͑Δ ϓϩηε୅਺͔Β৽ͨʹಋೖ ࣄલʹ 𝛼 ม׵͕ඞཁʹͳΔՄೳੑ͕͋Δ 𝜈 ʢ໊લͷӅṭʣ͕ଘࡏ͢ΔͨΊ -./UBM Ͱ͸ 𝜈 ͳͲΛ༻͍ͣʹ ࣗ༝ϦϯΫΛ؅ཧ͢Δ ʢޙͰৄ͘͠ղઆʣ ͨΊɼࣄલͷ 𝛼 ม׵͸ෆཁͩͬͨ ⟨𝑌 /𝑋⟩ def =  𝑝(𝑋1 , … , 𝑋𝑚 )⟨𝑍/𝑌 ⟩ def = 𝑝(𝑋1 ⟨𝑍/𝑌 ⟩, … , 𝑋𝑚 ⟨𝑍/𝑌 ⟩) ͨͩ͠ 𝑋𝑖 ⟨𝑍/𝑌 ⟩ def = { 𝑍 JG 𝑋𝑖 = 𝑌 𝑋𝑖 JG 𝑋𝑖 ≠ 𝑌 (𝑃, 𝑄)⟨𝑌 /𝑋⟩ def = (𝑃⟨𝑌 /𝑋⟩, 𝑄⟨𝑌 /𝑋⟩) (𝜈𝑋.𝑃)⟨𝑍/𝑌 ⟩ def = ⎧ { { ⎨ { { ⎩ 𝜈𝑋.𝑃 JG 𝑋 = 𝑌 𝜈𝑋.𝑃⟨𝑍/𝑌 ⟩ JG 𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 𝜈𝑊.(𝑃⟨𝑊/𝑋⟩)⟨𝑍/𝑌 ⟩ JG 𝑋 ≠ 𝑌 ∧ 𝑋 = 𝑍 ∧𝑊 ∉ fn(𝑃) ∧ 𝑊 ≠ 𝑍 (𝑃 :- 𝑄)⟨𝑌 /𝑋⟩ def = (𝑃 :- 𝑄) 
21. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ʛߏ଄߹ಉنଇ -./UBM ͱಉ༷ ⎧ { { { ⎨ { { { ⎩ & (, 𝑃) ≡ 𝑃 & (𝑃, 𝑄) ≡ (𝑄, 𝑃) & (𝑃, (𝑄, 𝑅)) ≡ ((𝑃, 𝑄), 𝑅) & 𝑃 ≡ 𝑃′ ⇒ (𝑃, 𝑄) ≡ (𝑃′, 𝑄) ϓϩηε୅਺͔Β৽ͨʹಋೖ { & 𝑃 ≡ 𝑄 ⇒ 𝜈𝑋.𝑃 ≡ 𝜈𝑋.𝑄 GVTJPO ͷͨΊͷنଇ ʢຊݚڀͷ֩৺෦෼ʣ ⎧ { { ⎨ { { ⎩ & 𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝜈𝑋.𝑃[𝑌 /𝑋] ʢͨͩ͠ 𝑋 ∈ fn(𝑃) ∨ 𝑌 ∈ fn(𝑃)ʣ & 𝜈𝑋.𝜈𝑌 .𝑋 ⋈ 𝑌 ≡  ϓϩηε୅਺͔Β৽ͨʹಋೖ ⎧ { { { ⎨ { { { ⎩ & 𝜈𝑋. ≡  & 𝜈𝑋.𝜈𝑌 .𝑃 ≡ 𝜈𝑌 .𝜈𝑋.𝑃 & 𝜈𝑋.(𝑃, 𝑄) ≡ (𝜈𝑋.𝑃, 𝑄) ʢͨͩ͠ 𝑋 ∉ fn(𝑄)ʣ 
22. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT -./UBM Ͱ͸ |ʛϦϯΫͷ݁߹ -./UBM Ͱ͸ɼಛघͳΞτϜʢDPOOFDUPSʣ 𝑋 = 𝑌 ͕ ҎԼͷߏ଄߹ಉنଇʹΑͬͯɼ 𝑋 ͱ 𝑌 Λ݁߹͢Δ (𝑋 = 𝑌 , 𝑃) ≡ 𝑃[𝑌 /𝑋] ʢͨͩ͠ 𝑋 ͸ 𝑃 ͷ ࣗ༝ϦϯΫʢ࣍εϥΠυʣ ʣ 𝑃[𝑌 /𝑋] ͸ 𝑃 ʹग़ݱ͢ΔશͯͷϦϯΫ 𝑋 Λ 𝑌 Ͱஔ͖׵͑ΔϦϯΫ୅ೖ -./UBM Ͱ͸ɼ͞Βʹʮ𝑃 ͸ΞτϜʯͱ͍͏৚͕݅͋Δ͕ɼqBU -./UBM ʹ͓͍ͯ͸ॏཁͰͳ͍ͷͰׂѪͨ͠ 
23. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT -./UBM Ͱ͸ |ʛࣗ༝ϦϯΫɾہॴϦϯΫ -./UBM ͷࣗ༝ɾہॴϦϯΫ 𝑃 ʹ  ճग़ݱɿ𝑃 ͷࣗ༝ϦϯΫ 𝑃 ʹ  ճग़ݱɿ𝑃 ͷہॴϦϯΫ -./UBM ͷϦϯΫ͸ϋΠύʔϦϯΫͰ͸ͳ͍ ߴʑ  ճग़ݱ͢Δ ग़ݱճ਺Λ਺͑Δ͚ͩ ͰɼϦϯΫͷہॴੑ͕Θ͔Δ ϋΠύʔϦϯΫ͸ग़ݱճ਺ʹ੍ݶ͕ͳ͍ͨΊɼ ͜ͷΑ͏ʹݟ෼͚Δ͜ͱ͸Ͱ͖ͳ͍ɽ 
24. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ϋΠύʔϦϯΫͷ༥߹ͷૉ๿ͳఆٛ ϋΠύʔϦϯΫ͸ʢ𝜈 ͕ͳ͚Ε͹ʣࣗ༝ɾہॴϦϯΫ͔ݟ෼͚ΒΕͳ͍ ࣗ༝ϦϯΫʹؔ͢Δ෇ଳ৚݅Λͳ͘͢ʁ ʢ࣮͸μϝʣ ϋΠύʔϦϯΫͷ༥߹Λ ϋΠύʔϦϯΫ༥߹نଇ WFS  (𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝑃⟨𝑌 /𝑋⟩ ʢ෇ଳ৚݅ͳ͠ʣ ͷΑ͏ʹఆٛͯ͠͠·͏ͱɼ άϥϑॻ͖׵͑ܥͱͯ͠ҙຯΛͳ͞ͳ͘ͳΔʢ࣍εϥΠυʣ 
25. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ϋΠύʔϦϯΫͷ༥߹ͷૉ๿ͳఆٛͷ໰୊఺ ࠶ܝɿϋΠύʔϦϯΫ༥߹نଇ WFS  (𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝑃⟨𝑌 /𝑋⟩ ʢ෇ଳ৚݅ͳ͠ʣ ͱ͢Δͱ| (𝑎(𝑋), (𝑋 ⋈ 𝑌 , 𝑏(𝑋, 𝑌 ))) ≡ & ༥߹نଇ WFS  (𝑎(𝑋), 𝑏(𝑌 , 𝑌 )) ⏟⏟ ⏟⏟⏟ ⏟⏟ 𝑋 ͱ 𝑌 ͕ग़ݱ ≡ & & (𝑋 ⋈ 𝑌 , (𝑎(𝑋), 𝑏(𝑋, 𝑌 ))) ≡༥߹نଇ WFS  (𝑎(𝑌 ), 𝑏(𝑌 , 𝑌 )) ⏟⏟ ⏟⏟⏟ ⏟⏟ શͯ 𝑌 ʹม׵͞Εͨ ʮϓϩάϥϜʹग़ݱ͢Δ೚ҙͷϋΠύʔϦϯΫ͕౳͍͠ʯ͜ͱ͕ಋग़Մೳ άϥϑॻ͖׵͑ܥͱͯ͠ͷҙຯΛͳ͞ͳ͍ 
26. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT -./UBM Ͱ͸ |ʛϦϯΫͷ݁߹ʹ͓͚ΔϦϯΫͷہॴੑ ࠶ܝɿ-./UBM ͷϦϯΫ݁߹نଇ (𝑋 = 𝑌 , 𝑃) ≡ 𝑃[𝑌 /𝑋] ʢͨͩ͠ 𝑋 ͸ 𝑃 ͷࣗ༝ϦϯΫ ʣ ʹ͓͍ͯ|  𝑋 ͸ 𝑋 = 𝑌 ʹग़ݱ͢Δ  ෇ଳ৚݅ΑΓ 𝑋 ͸ 𝑃 ʹ΋ग़ݱ͢Δ  -./UBM Ͱ͸  ճग़ݱͨ͠ϦϯΫ͸ہॴϦϯΫͰ͋ΔͨΊ 𝑋 ͸ (𝑋 = 𝑌 , 𝑃) ͷ ہॴϦϯΫ ैͬͯɼ(𝑋 = 𝑌 , 𝑃) ͷ֎Ͱ 𝑋 ͕ग़ݱ͠ͳ͍ 
27. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ϋΠύʔϦϯΫ΁ͷہॴੑͷૉ๿ͳಋೖ ϋΠύʔϦϯΫ༥߹نଇ WFS  (𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝑃⟨𝑌 /𝑋⟩ ʢͨͩ͠ (𝑋 ⋈ 𝑌 , 𝑃) ͷ֎Ͱ 𝑋 ͕ग़ݱ͠ͳ͍ ʣ ͜͏͢Ε͹| (𝑎(𝑋), (𝑋 ⋈ 𝑌 , 𝑏(𝑋, 𝑌 ))) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑋 ⋈ 𝑌 , 𝑏(𝑋, 𝑌 )) ͷ֎Ͱ 𝑋 ͕ग़ݱ ≢ & ༥߹نଇ WFS  (𝑎(𝑋), 𝑏(𝑌 , 𝑌 )) ͨͩ͠ɼ͜ͷ··Ͱ͸ʮϓϩάϥϜશମʯʹ͍ͭͯ ݴٴͯ͠͠·͍ͬͯΔͨΊɼ ෦෼͚ͩͰҙຯΛఆٛ͢Δ͜ͱ͕Ͱ͖͓ͯΒͣɼ DPNQPTJUJPOBM Ͱͳ͍ 
28. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT 𝜈 Λಋೖͨ͠ཧ༝ ͦ͜ͰɼຊݚڀͰ͸ɼہॴϦϯΫͱࣗ༝ϦϯΫͱ͍͏֓೦Λ DPNQPTJUJPOBM ʹϋΠύʔϦϯΫʹ࣋ͪࠐΉͨΊʹɼ ໌ࣔతͳείʔϓΛ৽ͨͳߏจ 𝜈𝑋.𝑃 IZQFSMJOL DSFBUJPO ͱͯ͠ಋೖͨ͠ 𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑃) ͕ ʮ(𝑋 ⋈ 𝑌 , 𝑃) ͷ֎Ͱ 𝑋 ͕ग़ݱ͠ͳ͍ʯ อূͷ໾ׂΛՌͨ͢ ͦͯ͠ɼ-./UBM ʹ͓͚ΔϦϯΫͷ݁߹ʹରԠ͢Δنଇ͸ ϋΠύʔϦϯΫ༥߹نଇ WFS  𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝑃⟨𝑌 /𝑋⟩ ʹͳΔʢ|ͱ͍͏ͷ͸΢ιͰ·ͩଓ͘ʣ 
29. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ϋΠύʔϦϯΫ༥߹نଇ & ͷӈลͷ 𝜈 ࠶ܝɿϋΠύʔϦϯΫ༥߹نଇ WFS  𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝑃⟨𝑌 /𝑋⟩ ͱ͢Δͱ|ɼ𝑋 ͱ 𝑌 ͕ͲͪΒ΋ಉ໊͡લͩͬͨͱ͖ʹࠔΔɽྫ͑͹ 𝜈𝑋.(𝑋 ⋈ 𝑋, 𝑎(𝑋)) ⏟⏟ ⏟ ⏟⏟⏟ ⏟ ⏟⏟ ࣗ༝ग़ݱ͢ΔϋΠύʔϦϯΫ͸ͳ͍ ≡༥߹نଇ WFS  𝑎(𝑋) ⏟ 𝑋 ͕ࣗ༝ग़ݱ ͱͳΓɼ≡ ͷ྆ลͰࣗ༝ϋΠύʔϦϯΫͷू߹͕มԽͯ͠͠·͏ ͦ͜Ͱɼ≡ ͷӈลʹ΋ 𝜈𝑋. Λิ͏ ϋΠύʔϦϯΫ༥߹نଇ WFS  𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝜈𝑋.𝑃⟨𝑌 /𝑋⟩ ͜͏͢Ε͹| 𝜈𝑋.(𝑋 ⋈ 𝑋, 𝑎(𝑋)) ⏟⏟ ⏟ ⏟⏟⏟ ⏟ ⏟⏟ ࣗ༝ग़ݱ͠ͳ͍ ≡༥߹نଇ WFS  𝜈𝑋.𝑎(𝑋) ⏟ ⏟ ⏟ ⏟ ⏟ ࣗ༝ग़ݱ͠ͳ͍ 
30. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ϋΠύʔϦϯΫ༥߹نଇ & ͷ෇ଳ৚݅ ࠶ܝɿϋΠύʔϦϯΫ༥߹نଇ WFS  𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝜈𝑋.𝑃⟨𝑌 /𝑋⟩ ͷ··ͩͱ|ɼ𝑋 ͱ 𝑌 ͷͲͪΒ΋ 𝑃 ʹग़ݱ͠ͳ͔ͬͨ࣌ʹࠔΔɽྫ͑͹ 𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑎()) ⏟⏟ ⏟⏟⏟ ⏟⏟ 𝑌 ͕ࣗ༝ग़ݱ͢Δ ≡༥߹نଇ WFS  𝑎() ⏟ ࣗ༝ग़ݱ͢ΔϋΠύʔϦϯΫ͸ͳ͍ ͱͳΓɼ≡ ͷ྆ลͰࣗ༝ϋΠύʔϦϯΫͷू߹͕มԽͯ͠͠·͏ ͦ͜Ͱɼ෇ଳ৚݅ʮ𝑋 ∈ fn(𝑃) ∨ 𝑌 ∈ fn(𝑃)ʯΛิ͏ ߏ଄߹ಉنଇ & 𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝜈𝑋.𝑃⟨𝑌 /𝑋⟩ ʢͨͩ͠ 𝑋 ∈ fn(𝑃) ∨ 𝑌 ∈ fn(𝑃)ʣ ͜͏͢Ε͹| 𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑎()) ⏟⏟⏟⏟⏟⏟⏟ 𝑌 ͕ࣗ༝ग़ݱ͢Δ ≢ & 𝑎() ⏟ ࣗ༝ग़ݱ͠ͳ͍ 
31. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ϋΠύʔϦϯΫ༥߹نଇ & ⟨1/2⟩ ͨͩ͠ɼ ࠶ܝɿߏ଄߹ಉنଇ & 𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝜈𝑋.𝑃⟨𝑌 /𝑋⟩ ʢͨͩ͠ 𝑋 ∈ fn(𝑃) ∨ 𝑌 ∈ fn(𝑃)ʣ ͸෇ଳ৚݅ʮͨͩ͠ 𝑋 ∈ fn(𝑃) ∨ 𝑌 ∈ fn(𝑃)ʯͷͨΊʹɼ 𝑋ɼ𝑌 ͕ 𝑋 ⋈ 𝑌 ʹ͔͠ग़ݱ͠ͳ͍৔߹ʹɼ GVTJPO ΛٵऩͰ͖ͳ͍ɽ ͦ͜Ͱɼ͜ͷέʔεʹରԠ͢ΔͨΊʹɼ͞Βʹ ߏ଄߹ಉنଇ & 𝜈𝑋.𝜈𝑌 .𝑋 ⋈ 𝑌 ≡  Λ༻ҙͨ͠ 
32. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ϋΠύʔϦϯΫ༥߹نଇ & ⟨2/2⟩ ࠶ܝɿߏ଄߹ಉنଇ & 𝜈𝑋.𝜈𝑌 .𝑋 ⋈ 𝑌 ≡  ͷࠨลʹ͓͍ͯ 𝜈𝑌 . ΋ඞཁͳཧ༝͸ɼ 𝜈𝑌 . Λল͍ͯɼ 𝜈𝑋.𝑋 ⋈ 𝑌 ⏟ ⏟ ⏟ ⏟ ⏟ 𝑌 ͕ࣗ༝ग़ݱ͢Δ ≡  ⏟ ࣗ༝ग़ݱ͢ΔϋΠύʔϦϯΫ͸ͳ͍ ͱ͢Δͱɼ≡ ͷ྆ลͰࣗ༝ϋΠύʔϦϯΫͷू߹͕มԽͯ͠͠·͏ͨΊ 
33. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT "ENJTTJCMF ͳنଇ ఏҊͨ͠ 'MBU )ZQFS-./UBM ͷߏ଄߹ಉنଇͰ͸ 'MBU -./UBM ʹ͓͚Δߏ଄߹ಉنଇͱൺֱͯ͠ ہॴϦϯΫͷ 𝛼 ม׵ 𝑃 ≡ 𝑃[𝑌 /𝑋] ʢͨͩ͠ 𝑋 ͸ 𝑃 ͷہॴϦϯΫʣ DPOOFDUPS ͷରশੑ 𝑋 = 𝑌 ≡ 𝑌 = 𝑋 ʹରԠ͢Δ΋ͷ͕͚͍ܽͯΔ ͜Ε͸͜ΕΒͷنଇ͕ଞͷنଇ͔Βಋग़Ͱ͖Δ͜ͱɼ ͢ͳΘͪ BENJTTJCMF Ͱ͋Δ͜ͱ͕Θ͔ͬͨͨΊ ͦΕͧΕఆཧ   ͱͯ͠ূ໌Λߦͳͬͨ 
34. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT 'MBU )ZQFS-./UBM ͷߏ଄߹ಉنଇʛ·ͱΊ & (, 𝑃) ≡ 𝑃 & (𝑃, 𝑄) ≡ (𝑄, 𝑃) & (𝑃, (𝑄, 𝑅)) ≡ ((𝑃, 𝑄), 𝑅) & 𝑃 ≡ 𝑃′ ⇒ (𝑃, 𝑄) ≡ (𝑃′, 𝑄) & 𝑃 ≡ 𝑄 ⇒ 𝜈𝑋.𝑃 ≡ 𝜈𝑋.𝑄 & 𝜈𝑋.(𝑋 ⋈ 𝑌 , 𝑃) ≡ 𝜈𝑋.𝑃⟨𝑌 /𝑋⟩ ʢͨͩ͠ 𝑋 ∈ fn(𝑃) ∨ 𝑌 ∈ fn(𝑃)ʣ & 𝜈𝑋.𝜈𝑌 .𝑋 ⋈ 𝑌 ≡  & 𝜈𝑋. ≡  & 𝜈𝑋.𝜈𝑌 .𝑃 ≡ 𝜈𝑌 .𝜈𝑋.𝑃 & 𝜈𝑋.(𝑃, 𝑄) ≡ (𝜈𝑋.𝑃, 𝑄) ʢͨͩ͠ 𝑋 ∉ fn(𝑄)ʣ ఆٛʹ͋ͨͬͯ͸ɼ ࣗ໌Ͱͳ͍ࢼߦࡨޡ Λཁͨ͠ Կ౓΋ʮίʔφʔέʔεʯΛ ൃݟ͠ɼͦͷͨͼʹमਖ਼Λଓ͚ͨ ॏཁͳੑ࣭͸ఆཧͱͯ͠ূ໌ͨ͠ ͨͩ͠੒Ռ෺͸ ඇৗʹ؆໌ 
35. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ʛભҠنଇ -./UBM ͱಉ༷ { 3 𝑃 ⟶ 𝑃′ (𝑃, 𝑄) ⟶ (𝑃′, 𝑄) ϓϩηε୅਺͔Β৽ͨʹಋೖ { 3 𝑃 ⟶ 𝑃′ 𝜈𝑋.𝑃 ⟶ 𝜈𝑋.𝑃′ -./UBM ͱಉ༷ ⎧ { { ⎨ { { ⎩ 3 𝑄 ≡ 𝑃 𝑃 ⟶ 𝑃′ 𝑃′ ≡ 𝑄′ 𝑄 ⟶ 𝑄′ 3 (𝑃, (𝑃 :- 𝑄)) ⟶ (𝑄, (𝑃 :- 𝑄)) 
36.  ݚڀͷഎܠ  'MBU )ZQFS-./UBM ͷந৅ߏจ  'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦  'MBU )ZQFS-./UBM ʹؔͯ͠ߦͳͬͨূ໌  ·ͱΊ 
37. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT 𝜆 ܭࢉ΍ 𝜋 ܭࢉ͸ɼ 𝛼 ม׵͕ʢߏ଄ʣ߹ಉੑΛอͭ͜ͱ΍ɼ ࣗ༝ม਺ͷू߹͕ߏ଄߹ಉͳϓϩηεؒͰมԽ͠ͳ͍͜ͱ ɼ ࣗ༝ม਺ͷू߹͕ભҠʹ൐͍޿ٛ୯ௐݮগ͢Δ͜ͱ ͱ͍ͬͨੑ࣭Λຬͨ͢ ͜ΕΒ͸ߏจۦಈͳܭࢉϞσϧʹ͓͍ͯීวతͳੑ࣭ͩͱݴ͑Δ ͦ͜Ͱɼ͜ΕΒͷੑ࣭͕ 'MBU )ZQFS-./UBM ʹ͓͍ͯ΋੒Γཱͭ͜ͱΛূ໌ͨ͠  ͨͩ͠ɼ𝜋 ܭࢉͰ͸ɼߏ଄߹ಉنଇ 4D.BU ʹ͓͍ͯͷΈɼߏ଄߹ಉͳϓϩηεؒͰࣗ༝ม਺ͷू߹͕มԽ͢ΔՄೳੑ͕͋Δ 
38. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ূ໌ͨ͠ఆཧ ఆཧ  𝛼 ม׵ 𝜈𝑋.𝑃 ≡ 𝜈𝑌 .𝑃⟨𝑌 /𝑋⟩ XIFSF 𝑌 ∉ fn(𝑃) ఆཧ  ⋈ ͷରশੑ 𝑋 ⋈ 𝑌 ≡ 𝑌 ⋈ 𝑋 ఆཧ  ߏ଄߹ಉͳϓϩηεؒͰͷࣗ༝ϋΠύʔϦϯΫͷू߹ͷ౳Ձੑ fn(𝑃) = fn(𝑄) JG 𝑃 ≡ 𝑄 ఆཧ  ભҠʹ൐͏ࣗ༝ϋΠύʔϦϯΫͷ޿ٛ୯ௐݮগੑ fn(𝑃) ⊇ fn(𝑄) JG 𝑃 ⟶ 𝑄 
39.  ݚڀͷഎܠ  'MBU )ZQFS-./UBM ͷந৅ߏจ  'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦  'MBU )ZQFS-./UBM ʹؔͯ͠ߦͳͬͨূ໌  ·ͱΊ 
40. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ·ͱΊ ຊݚڀͰ͸|  ϋΠύʔάϥϑॻ͖׵͑ܥͷߏจۦಈͳߏจɾҙຯ࿦ΛఏҊͨ͠  ఏҊͨ͠ҙຯ࿦͕ɼߏจۦಈͳܭࢉϞσϧʹ͓͍ͯ ॏཁͳੑ࣭Λຬͨ͢͜ͱΛূ໌ͨ͠  ʢຊൃදͰ͸ׂѪʣ ఏҊͨ͠ҙຯ࿦ΛԠ༻ͯ͠ɼ )ZQFS-./UBMʢ ʮ֊૚άϥϑʯॻ͖׵͑ݴޠ -./UBM ΁ϋΠύʔϦϯΫΛಋೖ֦ͨ͠ுʣ ͷܗࣜԽ΋ߦͬͨ 
41. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ࢀߟจݙ * <> #FSHF $MBVEF )ZQFSHSBQIT $PNCJOBUPSJDT PG 'JOJUF 4FUT /PSUI)PMMBOE  <> ) &ISJH FU BM 'VOEBNFOUBMT PG "MHFCSBJD (SBQI 5SBOTGPSNBUJPO 4QSJOHFS  <> (S[FHPS[ 3P[FOCFSH )BOECPPL PG (SBQI (SBNNBST BOE $PNQVUJOH CZ (SBQI 5SBOTGPSNBUJPO 8PSME 4DJFOUJpD  <> $ISJTUPQIFS #BL (1  &⒏DJFOU *NQMFNFOUBUJPO PG B (SBQI 1SPHSBNNJOH -BOHVBHF 1I% UIFTJT 6OJWFSTJUZ PG :PSL 4FQU  <> + 3 8 (MBVFSU FU BM %BDUM  BO &YQFSJNFOUBM (SBQI 3FXSJUJOH -BOHVBHF *O Proc. Graph Grammars 1990 7PM  -/$4 4QSJOHFS  QQ r <> 3VCJOP (FJ– FU BM (S(FO " 'BTU 410#BTFE (SBQI 3FXSJUJOH 5PPM *O Proc. ICGT 2006 4QSJOHFS  QQ r <> ,B[VOPSJ 6FEB -./UBM BT B IJFSBSDIJDBM MPHJD QSPHSBNNJOH MBOHVBHF *O 5IFPSFUJDBM $PNQVUFS 4DJFODF   QQ r 
42. ݚڀͷഎܠ 'MBU )ZQFS-./UBM ͷந৅ߏจ 'MBU )ZQFS-./UBM ͷૢ࡞తҙຯ࿦ ఆཧ ·ͱΊ 3FGFSFODFT ࢀߟจݙ ** <> %BWJEF 4BOHJPSHJ FU BM 5IF 1J$BMDVMVT " 5IFPSZ PG .PCJMF 1SPDFTTFT $BNCSJEHF 6OJWFSTJUZ 1SFTT  <> + 1BSSPX FU BM 5IF GVTJPO DBMDVMVT FYQSFTTJWFOFTT BOE TZNNFUSZ JO NPCJMF QSPDFTTFT *O Proc. LICS 1998  QQ r <> " :BTFO FU BM )ZQFSHSBQI 3FQSFTFOUBUJPO PG -BNCEB5FSNT *O TASE 2016 *&&& $PNQVUFS 4PDJFUZ +VMZ  QQ r <> :VUBSP 5TVOFLBXB FU BM *NQMFNFOUBUJPO PG -./UBM .PEFM $IFDLFST " .FUBQSPHSBNNJOH "QQSPBDI *O +PVSOBM PG 0CKFDU 5FDIOPMPHZ  /PW   <> 4UFGBO 8BMUFS $BQBCJMJUZ 5ZQJOH GPS )ZQFS-./UBM .BTUFS`T 5IFTJT 8BTFEB 6OJWFSTJUZ  <> +JO 4BOP *NQMFNFOUJOH (.BDIJOF JO )ZQFS-./UBM IUUQTBSYJWPSHBCT #BDIFMPS`T 5IFTJT 8BTFEB 6OJWFSTJUZ  <> ,B[VOPSJ 6FEB FU BM )ZQFS-./UBM "O &YUFOTJPO PG B )JFSBSDIJDBM (SBQI 3FXSJUJOH .PEFM *O ,ÛOTUMJDIF *OUFMMJHFO[   QQ r