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A Type System for CHR

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(1)A Type System for CHR Emmanuel Coquery, Francois Fages. To cite this version: Emmanuel Coquery, Francois Fages. A Type System for CHR. [Research Report] RR-5565, INRIA. 2005, pp.17. �inria-00070441�. HAL Id: inria-00070441 https://hal.inria.fr/inria-00070441 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A Type System for CHR Emmanuel Coquery — François Fages. N° 5565 Mai 2005. ISSN 0249-6399. ISRN INRIA/RR--5565--FR+ENG. Thème SYM. apport de recherche.

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(28) < . GQZ[jCsEFg©hoZ]Pkdj=Ff²ORrFEjCPTZ[X'j=XmŽ k¤@_X8PeRYnkdj_€d^Ckd€dF'«dRr@_F)ReNOp!FhPTNQPeRrFED ¬F)p_arXdp!X'PTFZ]Pip6kmalkmDFfRrFEarZ[´/F/q OV N|Rr@_FReN8p!F­PSNQPSRTF/D «dj_XmRrF/q ` «'XmŽ_Rr@_FA@_X'PSR4Ynkdj_€d^Ckd€dF'¯ ‹ FA¬­ZnY[YQD kd¹dFiRT@_FŽXdYnYnX¬­Z[j_€|kdPrPT^_DpQRTZnXdjCP X'j ` ¯ ‹ FHPT^_p_p!X'PTFRT@CkmR Z]P­VCk'PSFUq Xdjyk+ReNOp6FWkmYn€dFEVCarkC«ORT@_FHPTFfRhXmŽ ReN8p!F/P)V6F/Z[jC€ j_XmRrF/q ¯A?bNOp!F/P kdaTFj_XdRTFUq¸^6PSZnj_€ÊRT@CF=Y[`FERSRrFEa τ ¯Ê?bNOp_Znj_€·F/j8®OZnaTX'j_DFEj8RrP/«#j_XdRTFUq Γ « k'PTPTXQsfZ]kRTF+ReNOp!F/P|RrX·T®5kdaTZ]kmVCY[FUPE¯  Zn®dF/j kdj’FE²Qp_aTFUPTPTZnXdj kmj6q kWReNOp_Znj_€FEjO®OZ[arXdjCD+F/j8R « Z]P­^CPTF/q’RTX qQFUqQ^CsfF|ReNOp_Znj_€Kce^CqQ€'DFEj8RrP XdŽ3RT@CFWŽX'aTD Γ ` t :tτ ¯GOZnD+ZnY]kmarY[N'« ` ZnP|^6PSFUq·RTXÊΓqQFU`qQ^CsfF -b¦E œ0  /»–  ¦lŸ¸sfXdj6PeRrarkdZ[j8RrPhZnj¸kƒReN8pCZ[j_€ F/jO®8ZnarXdj_DFEj8R Γ « k·sfX'jce^_jCsfRTZnXdj C ∧ . . . ∧ C XmŽAjCkRrZ[®'F=sfX'jCPeRrarkdZ[j8RlPhV!FEZnj_€·¬FEYnY±™ReNOp6FUqZ[j Γ Z[Ž ŽX'aF/k'sl@ i ∈ {1, « Z]P¬FEYnY±™ReNOp6FUqwZnj ¯ ‹ F j_XmRrF «žRT@_FŽ k's—R|RT@CkmR Z]P ¬FEYnY[±<ReNOp!F/q Z[j Rr@_F. .Re.NO,p_n}Znj_€ CFEjO®OZ[arXdjCD+F/j8R Γ ¯ ‹ FHΓkmY]PTXk'PTPT^_DF|ΓRT@6`kR­RTC@CF¤Atom F/\8^CkdY[Z[ReN’sEXdjCPSRTalkmZnj8R sC= t V!FfRe¬FEF/j s kdjCq t ZnP ¬FEYnY±™ReNOp6FUq¤Znj Γ Z[ŽCRT@CFEarFiFE²OZ]PSRrP%khReN8p!F τ PT^Csl@¤Rr@CkR Γ ` s : τ kmjCq Γ ` t : τ ¯ ‹ F’k'PTPT^_DF RT@6kR¤Rr@_Fƒ^_jCZ[X'j XmŽAReNOp6F’FEjO®OZ[arXdjCD+F/j8RrPX®'FEa¤qQZ]PceX'Z[j8RPSFERrP¤XdŽ­®5kdaTZ]kmVCY[FUPHsEkdj¼V6F ŽX'aTDFUq ¬­ZÂRr@ƒkmjƒXdp!FEalkRTZnXdj jCXmRTFUq ] PT^Csl@RT@6kRAZŽ Γ ` t : τ Rr@_FEj Γ ] Γ ` t : τ ŽXdaAkmjONHReNOp_Znj_€ F/jO®8ZnarXdj_DFEj8R Γ qQZ]PceX'Z[j8R­ŽarXdD Γ ¯ ‹ F¤kmY]PTX=kdPrPS^_DFKRT@6kR)Z[Ž3k sfXdjmce^_jCs—RrZ[X'jÊXmŽ4jCkRrZ[®'FsfXdj6PeRrarkdZ[j8RrP Z]P|¬iF/Y[Y[±<ReNOp!F/q¸Z[j¼k ReNOp_Znj_€ÊFEjO®OZ[arXdjCD+F/j8R kmj6q «#Rr@_FEjRT@CFEarF Ff²QZnPSRrPk’ReNOp_Znj_€ C F/jO®8ZnarXdj_DFEj8R Γ «CPT^Csl@ƒRr@CkR­Rr@_F¤sfX'jce^_jCsfRTZnXdjÊΓXmŽ%sfX'jCCTPeRrarkd|=Z[j8RlCP D⇔Z]P­D¬FEYnY±™ReNOp6FUqƒZnj Γ ] Γ ¯ N. N. N. N. N. N. n. 1. i. N. N. 0. N. N. N. 0. 0.  . . 8;:. 0. G *:ED.   0. ? <. 

(29). ?A@CFhReNOp!F¤PSNQPSRTF/D ¬iF|p_arXdp!X'PTFKŽXdahg©ho qQF#)Cj_FUPk+j_XmRrZ[X'j’XdŽ ¬FEYnY±™ReNOp6FUqQj_FUPTPiŽXdahg©hoÀaT^_YnF/P/¯4?žX FUkdsl@Hg©)o sfXdj6PeRrarkdZ[j8R#PSNODWV6X'Y c/n Z]Pžk'PTPTXQsfZ]kRTFUq)X'j_Fd«kdjCq|Xdj_YnN)X'j_Fd«UReNOp!F3XmŽORT@CF3ŽXdarD τ ×. . .×τ ¯ ?A@CZnP=ReN8p!FyZ]Pƒk'PTPT^_DF/q RrX V6F )_²QF/q «­ŽX'a=Ff²_kdD+pCY[Fy^6PSZnj_€ PTXdDFyqQFUsfY]kmalkRTZnXdj6P p_arX®8Z]qQFUqVONRT@_F pCaTX'€dalkmDDFEaU¯$"hjCF=sEkmj arFED kmar¹·Rr@CkRRr@_F=YnZ[DZ[RrkRrZ[X'j RTX·X'j_FReNOp6FŽX'a¤F/k'sl@sEXdjCPSRTalkmZnj8RPTNODHV!XdY pCaTF/®dFEj8RlP4RT@CFh^CPTF)XdŽ ReNOp6F|PTsl@CFEDF/P/«8PT^Csl@ƒk'P3Rr@_X'PTF)^6PSFUqŽXdapCkmalkmDFERTarZns­p!XdYnNODXdarp_@_Z]PS: D [} ¨#¯ _Xda FE²_kmDp_YnFd«OZÂRZ]PŽXdarV_Z]q_qQF/j=RTX€'Z[®'F)Rr@_FKReN8p!F ∀α.list(α) × list(α) × list(α)) RrX+k=g©)oÀsfX'jCPSRTalkmZnj'R «V!F/s/km^CPTFiZ[R%ZnP F/\8^_Zn®kmYnFEj8RžRrXh€'Z[®'F3Z[R4kdY[Y'RT@CFbReNOp!F/P XmŽ_Rr@_FiŽXdarD ¯ ?A(@C%ZnPi)*aTFU2 Pe/3RraTZ]s—RrZ[X'j=Z]PiDXdRTZn®5kmRTFUq=VON+RT@_FKFf²_kmDp_Yn.F ¤€dZn®dF/j=V!FEYnX¬«8¬­@CZnsl@ƒPT@_list(τ X¬PbRT)×list @CkmRiRr@_(τF|)×list kmYnY[X¬Akm(τjCsE)F XdŽžReNOp!F¤PTsl@_F/DF/PAZnjyg©hoÀ¬iX'^_Ynq’D km¹'F)Rr@_F|ReNOp6F¤PTNQPeRrFED ZnjCsfX'jCPTZnPSRTF/j'RU¯ o kdaTF €dZn®dF/j Znj RrkdV_YnF}'6¯ 5Œg©ho sEXdjCPSRTalkmZnj8R H ZnP -X'ba|¦E œFf0 ?A²Q/»– @_p_FÊar F/¦larPrŸ¸^_PSZnYnXdZnF/j jCPWPKΓXmkmŽZ[pCŽiRrp6@_RrFU@_F kmF|arReZ[N8cej_^Cp!€ qQFʀ'kdPSD+P|NQF/PSkmRTj8arFE€dR D ^CΓD+Ž`X'F/j8aƒHRrPKg©)XmAtom sEkdj¸V!F qQFEarZ[®'F/qwŽaTX'D RT@CFWReNOp_Znj_€Êar^_Y[F'¯ ?žF/aTD P Ž3Rr@_F=sfX'jCPSRTalkmZnj'RlPhkdaTFWReNOp6FUqw^6PSZnj_€’RT@_F+ReNOp6F PTNQPeRrFED ŽX'abRr@_F|@_X'PSRiY]kmjC€d^Ckd€dFd¯%?A@_FKaT^CY[F FE²OpCaTFUPTPTF/PbRT@CkmRAk g©)o aT^CY[FKZnP -b¦E œ0  /»–  ¦lŸ Z[Ž ` Z[RrP)@_F/k'q «CZ[RrP)€d^CkdarqÊkdjCq ZÂRlPV!XOq_N kdaTF(Simpl ¬FEYnY±™ReNOCHR) p6FUqÊZ[jÊRr@_FHPrkmDF|ReNOp_Znj_€=FEjO®OZnaTX'j_DFEj8R Γ ¯ 5 g©ho pCaTX'€dalkmD ZnP¬iF/Y[Y[±<ReNOp!F/q’ZŽ4kmYnY Z[RrPsEYnkd^CPSFUP­kmarFh¬FEYnY[±<ReNOp!F/q ¯ ?A@_FŽXdYnY[X¬­Znj_€|Y[F/DDk¤Ff²Qp_arF/PrPSFUP RT@CkmR3RT@_F)¬iF/Y[Y[±<ReNOp!F/qQjCF/PrP%XmŽ €dX8kmY]P%ZnP4pCaTFUPSF/aT®'F/qWV8NWFf²ORTF/jCPTZ[X'j XdŽžRr@_FFEjO®OZ[arXdjCD+F/j8R'* iÄ   ˆ  Kž¦E– G 6 ¦Wš+¢8“/šm %šm”!Ÿ Γ šƒ–  ›œ”8¢Ê¦f” /›œ˜r“” ¦E”C–i•l¥ 4(y-– (Cš– Γ ` G Goal :K¦f– Γ 6 ¦Wš –  ›œ”O¢’¦E” U›œ˜T“m” ¦f”6–3•—¥  (·-– (Cš– Γ ] Γ ›]•¤Ÿ'¦ ;b”!¦l2Ÿ :  (_¦E” Γ ] Γ ` G Goal : n. 1. N. . 0. 0. 0. 8. @)A B@-C.

(30)  ¦  •—–¨¦

(31) f“m˜ ‘#)£ (Native). Γ `N C Atom Γ ` C Atom. (CHR Atom). Γ ` N t1 : τ 1 . . . Γ ` N tn : τ n Γ ` c(t1 , . . . , tn ) Atom. Z[Ž C Z]P­kWj6kRTZn®dFsEXdjCPSRTalkmZnj8R Z[Ž k=g©)o°sEXdjCPSRTalkmZnj8R kdjCq ZŽ c/n c/n @CkdPReNOp!F τ × . . . × τ n. 1. (Goal ). Γ ` B1 Atom . . . Γ ` Bn Atom Γ ` B1 , . . . , Bn Goal. (Simpl CHR). Γ ` H1 , . . . , Hi Goal Γ ` G1 , . . . , Gj Goal Γ ` B1 , . . . , Bk Goal ` H1 , . . . , Hi =0> ? G1 , . . . , Gj | B1 , . . . , Bk Rule. ? kmV_YnF }+*4?iN8p!F¤PTNOPSRTF/D ŽXdahg©ho ‡ È#È4S Š N ZnjCqQ^CsfRTZnXdj XdjƒRT@_FqQF/aTZn®kRTZnXdj kmjCq’VON ^CPTZ[j_€WRT@CF|k'PTPT^_DpQRrZ[X'j=RT@6kR­ZŽ Γ ` t : τ Rr@_FEj ŽX'akmjON=FE²OpCaTFUPTPTZ[X'j t ¯ Γ]Γ ` t:τ  ?A@_F¤sEXdjCPTZ]PeRrFEjCsEN XmŽžRr@_F|ReNOp6FPTNQPeRrFED ¬¯ a/¯ R/¯%Rr@_FXdp!FEalkRrZ[X'jCkmY:PTFED kmj8RTZ]sEPAXdŽ4g©ho Z]PA€dZn®dF/jƒVON Rr@_FhŽXdYnY[X¬­Znj_€WPT^_VQceFUs—RAaTFUqQ^CsfRTZnXdj=Rr@_FEX'aTF/DÊ«8¬­@CZnsl@’Ff²Qp_arF/PrPSFUP4RT@6kRART@_FK¬FEYnY±™ReNOp6FUqQj_F/PrPXmŽž€'X'kdYnPbZnP pCaTFUPSF/aT®'F/q=VON RrarkdjCPTZÂRrZ[X'jC P * š -b¦f œ0  /»–  ¦rŸ‘ h£  ˜T“r¢m˜r4š :JKž¦E– E, Di š”:Ÿ hF , E , D i 6 ¦y– -b“  ÄOÈ ‡ Ä  K¦f– P ¦w7 •—–»š–»¦—•K•—¥  (·-– (Cš– hF, E,6 Di 7−→ hF , E , D i :  |)– (_¦E˜T¦W¦ '›]•—– •¤hF, š –  ›œ”O¢’¦E” U›œ˜T“” ¦E”C– Γ •—¥  (·-– (Cš–  -– (C¦f”¸-– (C¦f˜r¦W¦ '›]•l–<•¤š –  ›œ”8¢Ê¦E •—¥ 4(Ê)– (_šm– : ” U› œ T ˜  “ ” . E ¦ C ” – Γ ` F, E,  D Goal. w“m˜T¦l4“ ¦f˜ › ¤-– (C¦¤–™˜Tš”C•—›œ–<› “m”y˜—¥Q [¦ r“m”C–»š›œ”C•¤šW¢m¥Cšm˜TŸ G )– (_¦E” ΓΓ ` G Goal : Γ ` F , E , D Goal ‡ È#4È S Š  Nƒs/kdPTF|Xdj’RT@_F|RrarkdjCPSZ[RTZnXdjž¯ O È -QP Ä  N¸@ONOp6XdRT@_FUPSZ]P/« Γ ` D Goal kdjCq Γ ` C Atom ¯¸GOZnjCsfF CT |= (C ∧ D) ⇔ D «4kmj6q VONkdPrPT^_DpQRTZnXdj Xdj ` «bRr@_FEarF·Ff²QZ]PeRlP kReN8pCZ[j_€ F/j8®OZnaTX'j_DFEj8R Γ PT^Csl@RT@CkmR Γ ] Γ ` ¯  Np!X'PTZ[j_€ Γ = Γ ] Γ kdjCqV8NYnFEDD k·v_« ¬iF=X'VQRrkdZ[j Γ ` F, E Goal «žRr@8^6P D Goal ¯ Γ ` F, E, D Goal R+$ †E‡ È É

(32) ‰ Ä~?A@_Z]P)RTalkmjCPTZÂRrZ[X'jyXdjCY[N·DX®dFUP)k sfX'jCPeRrarkdZ[j8RŽarXdDŒRr@_F+€dX'kdY#RrXƒRr@_F g©hosEXdjCPSRTalkmZnj8R PeRrXdarFd«ORr@8^6PART@_FarF/PT^_Y[RTZnj_€PeRlkRTFZ]P­kmY]PTX+¬FEYnY[±<ReNOp!F/q’Z[j Γ ¯ O % 9- %TS™Å  N@ONOp6XdRT@_FUPSZ]PE«5Rr@_FAaT^CY[F (H ZnP ¬FEYnY±™ReNOp6FUq ¯?A@O^CP/«Rr@_FEarFiFE²QZnPSRrP kReNOp_Znj_€ | B)  FEjO®OZ[arXdjCD+F/j8R Γ PS^6sl@WRr@CkR Γ ` H,=0G,> ? BGGoal ¯ NHY[F/D+D kvC«dkmjCq+VONHp!X'PTZnj_€ Γ = Γ ] Γ « ¬iF+XdV_RrkmZnj kmjCq ¯Wº¨RKarFED kmZnjCPhRTX p_arX®dFHRT@CkmR H = H ZnPh¬FEYnY±™ReNOp6FUΓq·Z[`j F,Γ E,¯ HDZ]PhGoal XmŽ4RT@CFHŽX'aTΓD `c G,(t B, . .Goal kmjCq XdŽ4RT@_F ŽXdarD c (s , . . . , s ), . . . , c (s , . . . , s ) ¯ ¾#FfR .l, ∈t {1,), .. .. .. ,, ci}(tkdjC, q. . .p, ∈t {1,) . . . , Hm } ¯¾#FfR V!FhRT@_FKReNOp6FhXmŽ ¯3GOZ[j6sfF «Q¬iFh@Ck5®dF ¯ τ ×...×τ (t , . . . , t ) Atom GOZnD+ZnY]kmarY[N'« Γ ` s : τ ¯?A@O^CcP Γ ` t Γ= s` cAtom ¯­?A@O^CP/«_¬iFHXdVQRlkmZnj Γ ` ΓB,`F, E,t H: τ= kmjCq Γ ` H = H , G Goal ¯ H , D Goal  N. 0. N. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 00. N. 0. 0. 0. 00. 00. N. 0. 0. 00. 00. 0. 0. 0. 0. 1. 1. 1 1. 1 m1. ml 0. 1. 0. 88:94;<<=<. 00. 0. i 1. i. i mi. l. N. l p 0. 0. p. 0. l p. 1 1. 1 m1. i. i 1. 0. i mi. l. 0. l. l p. 1 l. 0. l ml. 0. N. l p. p.

(33) }U{ .  š”6¥C¦f 4‘3“ f¥C¦E˜  ˜Tš” U“m›]• š/¢'¦f• . . . . ?A@_F=ŽXdYnY[X¬­Znj_€·Ff²_kdD+pCY[F=ZnYnY[^CPSRTalkRrF/P@CX¬Rr@_F=^CPTF=XmŽk ReNOp!F’Prsl@_FEDFŽXdaWk¸g©)o sEXdjCPSRTalkmZnj8R PTNODHV!XdY D k5N=YnF/k'q=RrXsl@CF/sl¹ƒkdj’ZnY[Y[±™ReN8p!F/qÊsEXdjce^_j6s—RTZnXdjÊXdŽ sfX'jCPSRTalkmZnj'RlPE¯  ˆ  - Ä  K¦f–K¥O•ƒš5•l•—¥ ¦’)– (_š–|)– (_ ¦ l“”_•—–™˜Tš›œ”6– (_š5•=)– (_¦’–  ¦ƒ+• 4(_¦ ¦ : K¦f–K¥O• ”:Ÿ=)– (_ךm–iα)– (_¦f•E¦W– -b“ •—¥   “•f¦W)– (_šm– ( š”:Ÿ  (C4š ¦+–  ¦ string  -– (Cš– > Ê= š”:Ÿ  (_š 5¦W–  ¦ int  š∀α.α –  ¦—•Hš˜r¦¤›œ7” r“  š–™› 6  [¦ : K¦E–b¥O• r“m”_•—› Ÿd¦E˜¤)– (_¦H˜—¥Q [¦ *. " ()+""0 > ='- > =1 >3> ?  >=1 & šm”!ŸH–-(C¦­•—–»š–»¦ htrue, ( > =  ∧  > =  , truei : (Q›]•­•l–»š–»¦)›]• b¦f œ  »– ¦rŸ0:iW“ b¦ 5¦E˜)› )“m”!¦)–<˜—› ¦f•–»“ š   –)(_¦H˜—¥Q [¦ 4›œ–-( !#"%$6&$'  “”:¦H”:¦l¦rŸ•–¨“  (C¦  ƒ–-(Cš–A–)(_¦ r“m”_•—–™˜Tš›œ”6–b•—–»“˜T¦ ›  Â› ¦—•  > ( ∧"3 >()+*∧ >  "∧ 1#>   (Q›  (’›]•)”:“–3š b¦f œ  »– ¦lŸr“m”_•—–™˜Tš›œ”6–  6 ¦4lš¥O•E¦  +¥O•—–B(Cš ¦ š”:Ÿ : “m–-(·– ¦—•. ! +   6. . - -. . string. - / . - 0/ . 4- . 4. int.  

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(43) }5z . (Var ). X:τ ∈Γ Γ`X:τ.  š”6¥C¦f 4‘3“ f¥C¦E˜  ˜Tš” U“m›]• š/¢'¦f• . . . . Γ ` t : τ τ ≤ τ0 Γ ` t : τ0. (Sub). Z]P­k+ReN8p!FPS^CVCPeRrZÂRr^QRTZnXdj. (Func). Γ ` t 1 : τ1 ρ . . . Γ ` t n : τn ρ Γ ` f (t1 , . . . , tn ) : τ ρ. ρ τ1 × . . . τn → τ ∈ types(f /n). (Atom). Γ ` t 1 : τ1 ρ . . . Γ ` t n : τn ρ Γ ` p(t1 , . . . , tn ) Atom. ρ τ1 × . . . τn ∈ types(p/n). (Head ). Γ ` t 1 : τ1 ρ . . . Γ ` t n : τn ρ Γ ` p(t1 , . . . , tn ) Head τ1 ×...×τn. ρ τ1 × . . . τn ∈ types(p/n). Z]P­k+ReN8p!FPS^CVCPeRrZÂRr^QRTZnXdj. Z]PkWReNOp6FarFEj6kmDZ[jC€. Γσ ` p(t1 , . . . , tn ) Head σ Γσ ` B1 Atom . . . Γσ ` Bk Atom ` p(t1 , . . . , tn )  B1 , . . . , Bk Clause. ∀σ ∈ types(p/n) (Clause). Γ ` H Atom Γ ` G Goal ` label with H if G Label with. (Label with). ? kmV_YnF¤z*3?iN8p!FPSNQPSRTF/D ŽXdahg¾ž`ÁkdjCq. label with. Rr@_F)k'q_qQZ[RTZnXdjXmŽ PS^CVQReNOp_Z[jC€¤kmjCq+X®dF/aTYnX'k'qQZnj_€C¯ 5ÁpCaTFUqQZns/kRrFsEkmYnY p(t , . . . , t ) §arF/PTp#¯ k|j6kRTZn®dFsEXdjQ± PSRTalkmZnj8RlªžZnP -b¦E œ0  /»–  ¦lŸHZ[jkReNOp_Znj_€KFEjO®OZ[arXdjCD+F/j8R ZŽ s/kmjWV6F­qQF/aTZn®dFUq|ŽaTX'D , . . . , t ) Atom Rr@_FAaT^CY[FUPE3¯ 5ÁsfY]km^CPTF H  B ZnP -b¦E œ0  /»–  ¦lŸWZŽ ` ΓH Γ `Bp(tClause sEkmj+V!FqQFEarZ[®'F/q¤ŽaTX'D1RT@_FAar^_YnF/P/¯ 5½s/kmjYnkdV!V6FhF/Y[qQZnj_F/€=aTZn®dqQFUFUq sfY]¯km?Aalk@_RTF ZnXdj labelar^_Y[with Z]P­¬FEYnY±™ReNOp6FUq ZŽ HP4RTif@_F)GPTFED kdj'RrZns/P4XmŽ#PT^_VQReNO`p_Znlabel ) F d € n Z d ® / F j_€VONWwith FE²Qp_aTFUHPTPTZnifj_€KGRT@CLabel kmRbZŽžkKRrwith FEarD (Sub) @t CkdPReNOp!F τ «_Rr@_FEjÊZ[R@CkdP­kdY[Y:ReNOp6FUPRT@CkmR)kmarF|€darF/kRrFEaRT@6kmj τ ¯ ?A@_FWPSFERXmŽ%aT^_YnF/PAXdŽ%RrkmVCY[FUP|}¤kmjCq·zqQ#F )Cj_F|RT@CFReNOp6F¤PTNQPeRrFED ŽX'aKg¾#`A¿g©)o¯ 5 g¾#`¿g©ho pCaTX'€dalkmDµZ]P¤¬FEYnY±™ReNOp6FUq ZŽ)kdY[YiZÂRlPg©ho aT^CY[FUPE«4kmYnYbZ[RrP+sfY]km^6PSFUP¤kmjCq¼kdY[YiZÂRlPHY]kmV!FEYnZnj_€wqQFUsfY]kmalkRTZnXdj6P kdaTF ¬FEYnY[±<ReNOp!F/q ¯·?A@_F’aT^_YnF (CHR Atom) sEkmj¼V!FƒPTFEF/j¼@CkdPWk·pCkmaTRTZ]sf^CYnkda¤sEkdPTF=XdŽAar^_YnF/P (Atom) kdjCq «O¬­@_F/aTFRT@_F|PSNODWV6X'Y @6kdPiXdj_YnNWX'j_FReNOp!FhPrsl@_FEDF¬­Z[RT@CXd^QR­kmjON+pCkmalkmDFfRrFEaU«d¬­@_Z]sl@ sEXdaraTFU(Head PSp!Xdj6q_)P3RrXRT@_FarF/PSRTarZ]s—RTZnXdjƒRTp/n @CkmR)k=g©)o°sfX'jCPeRrarkdZ[j8R­@Ck'PAXdj_YnN=XdjCFhReNOp!Fd¯ ?A@_F¤q_ZnPSRTZnjCs—RrZ[X'jÊV6FERe¬iF/FEj aT^_YnF/P (Atom) kdjCq (Head ) FE²Qp_aTFUPTPTF/PiRr@_Fp_arZ[jCsEZ[pCY[F|XmŽŸd¦ ;3”6›œ–<› “m”!šm  ¢8¦E”!¦E˜—› f›œ–  n'} (¨«d¬­@_Z]sl@F/PSRrkdV_Y[Z]PT@_F/P Rr@CkR4Rr@_F­ReNOp6F­XdŽ6Rr@_F­@_FUkdq+XmŽ:ksfY]km^CPTFADH^CPSR3V6F­FU\8^_Z[®kdY[F/j'R4RrXC« ^CpWRrXaTF/jCkmDZnj_€|V_^QRbjCXmRbkmj+ZnjCPeRlkmjCsEF­XmŽe«dRT@_FqQFUsfY]kmarF/qHReNOp6FXdŽ6Rr@_F­p_arF/qQZ]sEkmRTF'¯ ?A@_F­ar^_YnF (Clause) ZnDp6X8PSFUPWRT@CkmR k sEYnkd^CPSF DH^6PeRV6F·¬FEYnY[±<ReNOp!F/q¼ŽX'a kmYnY­p6X8PTPTZ[VCY[F’ReNOp6FUPXmŽRr@_F·qQF )CjCF/q p_arF/qQZ]sEkmRTFd« ¬­@CZnsl@¼sEkdj V6FƒPTFEF/j kdP¤k·sEXdjCq_ZÂRrZ[X'j PTZnD+ZnY]kmaRrXyqQF )CjCZÂRrZ[X'jCkmYi€dFEjCFEarZnsEZÂReN·ŽX'a¤X®dF/aTYnX'k'qQZnj_€C¯=?A@_FUPSF Re¬X·sfX'jCqQZ[RTZnXdjCPkdaTF^CPTFfŽ^_Y3ŽXdaRr@_FŽXdYnY[X¬­Znj_€·PT^_VQceF/sfR¤aTFUqQ^CsfRTZnXdj¸RT@_F/XdarFEDµ¬­@_Z]sl@ Ff²Qp_arF/PrPSFUPKRT@_F sEXdjCPTZ]PeRrFEjCsENHXmŽ:Rr@_F­ReNOp6FhPSNQPSRTFED ¬­Z[RT@ arFfŽFEarFEj6sfFARTX¤RT@_FX'p6F/arkmRTZnXdjCkdY_PSF/D kmj8RTZ]sEPbXmŽžg¾#`¿g©ho¯'º¨R Z]Pip_aTFUsfFUqQF/qVONk¤YnFEDD k¤¬­@_Z]sl@ Ff²Qp_arF/PrPSFUP4Rr@CkRZ[jƒkHqQFEarZ[®kmRTZnXdj=kmp6kmaTR3ŽaTX'D (Head ) Xda (Clause) « Rr@_F|ReNOp6FUP­sEkdj’V!F¤kmarV_Z[RTalkmarZ[YnN Z[j6PeRlkmj8RTZ]kRrF/q ¯ n. 1. 1. n. 8. @)A B@-C.

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(45) f“m˜ ‘#)£. }%. iÄ   ˆ  “m˜ š”  –  ›œ”8¢ ¦f” /›œ˜T“m” ¦f”6– Γ  f“˜ š”  M—¥CŸ/¢ ¦E”C– š”:Ÿ’šm”  –  ¦¤•—¥ 6 •—–™›œ–<¥Q–™› “” ρ  › Γ ` R –)(_¦E” Γρ ` Rρ : Clause ‡ È#4È S Š  N=ZnjCqQ^6s—RTZnXdjÊX'jƒRr@_F¤qQFEarZn®5kmRTZnXdj#¯ . R. Ÿm› )¦f˜r¦f”6– l˜T“4. “˜. Head. . ÄOÈ ‡ Ä  & Kž¦E–:¥O• l“”_•—› Ÿ'¦f˜Aš -b¦E œ 0/»–  ¦lŸ+‘#)£‘EK  ˜T“r¢d˜Tš4 :'K¦f– šm”!Ÿ ¦6 +– -b“’•—–¨šm–¨¦f•Wš”:Ÿ Γ 6 ¦=š’–  ›œ”O¢y¦f” /›œ˜r“” ¦E”C–A•—¥  (w–)(_šm– Γ ` F, E, D hF,GoalE,: Di hF, E,hFDi, E7−,→D i  -– (C¦f” –)(_¦E˜T¦|¦('›]•—– •)šW–  ›œ”8¢¦E”U›œ˜T“” ¦E”C– •—¥ 4( –)(_šm– : w“˜r¦/ hF , E “5¦E˜  › |, D)– (_¦Hi –<˜rš”_•—›œ–™› “”¸˜—¥Q [¦—•|˜T¦4E¥Q›œ˜T¦—•HšW¢m¥Cšm˜TŸ G –)(_¦E” ΓΓ ` G GoalΓ: ` F , E , D Goal ‡ È#4È S Š "hjCFKs/kmjÊsl@_F/sl¹Rr@CkRART@CF|k'PTPT^_DpQRrZ[X'jCPXmŽ PTF/sfRTZnXdj _¯n}KkdaTF|sfX'aTarF/sfRbŽX'aART@_FKRT@CF|PTNQPeRrFED XdŽ RlkmV_YnFKz_¯4J·XdarFEX®'FEaU«dkmjƒkmRTX'D sEXdaraTFUPSp!Xdj6qQZ[jC€|RTXWkHp_arF/qQZ]sEkmRTFhs/kmYnY!kdjCq=kdjƒkRrXdD sfX'aTarF/PTp6X'jCqQZnj_€|RTX k+jCkmRTZn®dFsfX'jCPSRTalkmZnj'RkdaTFhReNOp6FUq’Znj RT@_F¤PrkmDF|¬Ak5Nd¯4?A@_FEarFfŽX'aTF'«QVON RT@_F/XdarFED¶Q«_Z[ŽRr@_F|RTalkmjCPTZ[RTZnXdj ZnP X'j_FXdŽ O È -QP Ä!« R $ †/‡ È É

(46) ‰ Ä+X'a O %  - % S<Åb«RT@_F/j RT@CFEarFFf²QZ]PeRlPbk|ReNOp_Znj_€HF/jO®8ZnarXdj_DFEj8R te PS^Csl@Rr@CkR kdjCq Znj·sEkdPTF|XmŽ j_FEFUq ¯ Γ `¾žF FfR),^CEPhsf, X'DjCPSZ]Goal qQF/a­RT@_F Γ$ S¨`È -<GÉ RrGoal arkdjCPSZ[RTZnXdjž¯ ‹ FHsEkdjykdPrPS^CD+F'«_¬­Z[RT@_X'^QRhYnX'PrP­XmŽ4€'FEj_F/arkdY[Z[ReNd«QRr@CkR kmjCq ¯GOZ[j6sfF «QRT@_F/aTFFE²QZnPSRrP­k+ReNOp!F¤PTsl@_F/DF kdjCq H = p(s) kƒPS^_V6PeRrZÂRr^QRTZnXdj ρHPT^C=sl@Êp(t)RT@CkmR Γ ` sΓ:`τ ρp(s) ¯KGOZ[Atom j6sfF¤Rr@_FHpCaTX'€dalkmD Z]P¬FEYnY[±<ReNOp!F/q « ` Hτ ∈types(p) « B Clause Rr@O^CP4Rr@_FEarFFf²QZ]PeRlPbk|ReNOp_Znj_€F/j8®OZnaTX'j_DFEj8R Γ PS^Csl@Rr@CkR Γ ` B Goal kdjCq Γ ` p(t) Head «dRr@CkR Z]P Γ ` t : τ ρ ¬­@_FEarF ρ ZnPik|aTF/jCkmDZnj_€|XmŽ τ ¯  NWp6X8PSZnj_€ ρ = ρ ρ «Okmj6qWVONWY[F/D+D k C«d¬F­XdVQRlkmZnj kdjCq ¯  Nyp!X'PTZ[j_€ Γ = Γ ρ ] Γ «¬iF X'VQRrkdZ[j Γ ` t = s Atom ¯ Γ ρ ` t : τρ Goal ?A@O^CP Γ ` B, F, E, Γs =ρ t,`DBGoal ¯ ¾žFfR4^CP )CjCkdY[YnN¤sfX'jCPTZnqQF/ažRr@_F­sEk'PSFXmŽ:k ˆQ‚ Ä - RrarkdjCPTZÂRrZ[X'j#¯ GOZnDZ[Y]kmarYnN|RTXKRT@_F­s/kdPTFbXmŽ6RT@_F $ S¨È - É RrarkdjCPTZÂRrZ[X'j#«RT@CFEarFFE²QZnPSRrP3khReNOp_Znj_€FEjO®OZnaTX'j_DFEj8R Γ «8PS^6sl@WRr@CkR Γ ` B, F, E, t = s, D ¯%GQZ[jCsEF H ZnP kg©)o sfXdj6PeRrarkdZ[j8R/« qQXOFUPbj_XdRsEXdj8RrkdZ[jƒkdjONWpCkdarkdDFfRTF/a/«'RT@6kRAZnP Z]P3Rr@_FhZ]qQF/j'RrZÂReN PT^_VCPSRTZ[RT^QRrZ[X'j#¯ ‹ F­@Ck5®'F H = p(u)τ ŽX'abPSX'DFiRrFEarD u ¯%GOZnjCsEF ` label with H ifρ G Label with «mRr@_FEarF­Ff²QZ]PeRlP4k ReNOp_Znj_€|FEjO®OZnaTX'j_DFEj8R Γ PT^Csl@WRT@CkmR Γ ` G Goal kmjCq Γ ` H Atom «dRT@CkmR3ZnP Γ ` u : τ ¯  N p!X'PTZnj_€ Γ = Γ ] Γ «Q¬FKX'VQRrkdZ[j Γ ` s = u Atom « Γ ` B, F, E, s = t, D kmjCq Γ ` G Goal « kdjCqƒRT@O^CP Γ ` s = u, G Goal ¯  . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. . 0. τ. τ. τ 0. r. τ. τ. τ. 0. r 0. 0. τ. 0. −1 r. τ. 0. 0. . . 0. 0. 0. 00. 00. lw. 00. 00. 0. lw. 00. lw. lw 00. 00. lw. 00. 00.

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