OfHNSrSSKJr SSKJr SSKJrJr SSKJ r J r SSK J r J r JrJrJrJrJrJrJrJrJrJrJrJr "SS\5rS r"S S \5r"S S \5r"SS\5r "SS\5r!"SS5r"Sr#Sr$Sr%Sr&Sr'Sr(Sr)\*S:Xa\)"5 gg)z A module to perform nonmonotonic reasoning. The ideas and demonstrations in this module are based on "Logical Foundations of Artificial Intelligence" by Michael R. Genesereth and Nils J. Nilsson. ) defaultdict)reduce)ProverProverCommandDecorator)Prover9Prover9Command)AbstractVariableExpression AllExpression AndExpressionApplicationExpressionBooleanExpressionEqualityExpressionExistsExpression Expression ImpExpressionNegatedExpressionVariableVariableExpressionoperatorunique_variablec\rSrSrSrg)ProverParseError&N)__name__ __module__ __qualname____firstlineno____static_attributes__r=/usr/lib/python3/dist-packages/nltk/inference/nonmonotonic.pyrr&sr rcnUcUnOX*/-n[[RSU5[55$)Nc3@# UHoR5v M g7fN) constants).0as r! get_domain../s H1s)rror_set)goal assumptionsall_expressionss r! get_domainr/*s3 |%%/ (,, H H#% PPr c*\rSrSrSrSrSrSrSrg)ClosedDomainProver2zQ This is a prover decorator that adds domain closure assumptions before proving. cURR5Vs/sHoPM nnURR5n[X25nUVs/sHoPR XT5PM sn$s snfs snfr$)_commandr-r,r/replace_quants)selfr'r-r,domainexs r!r-ClosedDomainProver.assumptions8sj"&--";";"=>"=Qq"= >}}!!#D.:EF+B##B/+FF?Gs A0A5cURR5n[XRR55nUR X5$r$)r4r,r/r-r5)r6r,r7s r!r,ClosedDomainProver.goal>s<}}!!#D--";";"=>""400r c H[U[5(akUVs/sH1o1RRUR[ U55PM3 nnUVs/sHoPR XR5PM nn[SU5$[U[5(aFURUR URU5UR URU55$[U[5(aUR URU5*$[U[5(akUVs/sH1o1RRUR[ U55PM3 nnUVs/sHo0R X25PM nn[SU5$U$s snfs snfs snfs snf)a Apply the closed domain assumption to the expression - Domain = union([e.free()|e.constants() for e in all_expressions]) - translate "exists x.P" to "(z=d1 | z=d2 | ... ) & P.replace(x,z)" OR "P.replace(x, d1) | P.replace(x, d2) | ..." - translate "all x.P" to "P.replace(x, d1) & P.replace(x, d2) & ..." :param ex: ``Expression`` :param domain: set of {Variable}s :return: ``Expression`` c X-$r$rxys r!3ClosedDomainProver.replace_quants..Uqur c X-$r$rr>s r!rArBbrCr ) isinstancer termreplacevariablerr5rr __class__firstsecondrr)r6r8r7d conjunctsc disjunctss r!r5!ClosedDomainProver.replace_quantsCso b- ( (MSMS -?-BCV BKKA,,Q7IK,i8 8 - . .<<##BHHf5##BIIv6 - . .''88 8 , - -MSMS -?-BCV BKKA,,Q7IK,i8 8I'LLs8FF'8F%FrN) rrrr__doc__r-r,r5rrr r!r1r12s G 1 !r r1c\rSrSrSrSrSrg)UniqueNamesProvergzO This is a prover decorator that adds unique names assumptions before proving. cURR5n[[URR 5U55n[ 5nUHYn[ U[5(dMURRnURRnX5RU5 M[ /n[U5HupX(S-SHpn XU;dM [[U5[U 55n [5RX5(aX4RU 5 M^UR!U *5 Mr M X-$)z - Domain = union([e.free()|e.constants() for e in all_expressions]) - if "d1 = d2" cannot be proven from the premises, then add "d1 != d2" N)r4r-listr/r, SetHolderrErrJrHrKadd enumeraterrproveappend) r6r-r7eq_setsr'avbvnew_assumptionsibnewEqExs r!r-UniqueNamesProver.assumptionsms mm//1 j!3!3!5{CD+A!/00WW%%XX&& # f%DAEG_AJ&0*1-/A!/DGyw<< q)(..x8%&,,r rN)rrrrrQr-rrr r!rSrSgs  "-r rSc\rSrSrSrSrSrg)rXz A list of sets of Variables. c[U[5(deUH nX;dM Us $ U1nURU5 U$)z> :param item: ``Variable`` :return: the set containing 'item' )rErr\)r6itemsnews r! __getitem__SetHolder.__getitem__sE $))))Ayf C r rN)rrrrrQrkrrr r!rXrXs  r rXc6\rSrSrSrSrSrSrSrSr Sr g ) ClosedWorldProvera= This is a prover decorator that completes predicates before proving. If the assumptions contain "P(A)", then "all x.(P(x) -> (x=A))" is the completion of "P". If the assumptions contain "all x.(ostrich(x) -> bird(x))", then "all x.(bird(x) -> ostrich(x))" is the completion of "bird". If the assumptions don't contain anything that are "P", then "all x.-P(x)" is the completion of "P". walk(Socrates) Socrates != Bill + all x.(walk(x) -> (x=Socrates)) ---------------- -walk(Bill) see(Socrates, John) see(John, Mary) Socrates != John John != Mary + all x.all y.(see(x,y) -> ((x=Socrates & y=John) | (x=John & y=Mary))) ---------------- -see(Socrates, Mary) all x.(ostrich(x) -> bird(x)) bird(Tweety) -ostrich(Sam) Sam != Tweety + all x.(bird(x) -> (ostrich(x) | x=Tweety)) + all x.-ostrich(x) ------------------- -bird(Sam) c <URR5nURU5n/nUGH`nX$nURU5nUVs/sHn[ U5PM nn/n UR HOn /n [ X5HupU R[X55 M! U R[SU 55 MQ URHCn0n[ XS5H upXU 'M U RUSRU55 ME U (a+URXF5n[SU 5n[UU5nO[URXF55nUSSS2Hn[UU5nM URU5 GMc X-$s snf)Nc X-$r$rr>s r!rA/ClosedWorldProver.assumptions..sQUr rrVc X-$r$rr>s r!rArrsr )r4r-_make_predicate_dict_make_unique_signaturer signatureszipr\rr propertiessubstitute_bindings_make_antecedentrrr )r6r- predicatesr`p predHoldernew_sigv new_sig_exsrOsig equality_exsv1v2propbindings antecedent consequentaccum new_sig_vars r!r-ClosedWorldProver.assumptionssmm//1 ..{; A#J11*=G:AB'Q-a0'KBI",,! !+3FB ''(:2(BC4  (:L!IJ -#--!+Aw7FB#%RL8  a! #$6 B %j*=*$*?*?*KL 'tt} %k59 -  " "5 )GJ,,ECs FcL[S[UR555$)z This method figures out how many arguments the predicate takes and returns a tuple containing that number of unique variables. c36# UHn[5v M g7fr$)r)r&ras r!r(;ClosedWorldProver._make_unique_signature..sP0O1_&&0Os)tuplerange signature_len)r6r~s r!rv(ClosedWorldProver._make_unique_signatures Pj6N6N0OPPPr c>UnUHnU"[U55nM U$)zn Return an application expression with 'predicate' as the predicate and 'signature' as the list of arguments. )r)r6 predicate signaturerrs r!r{"ClosedWorldProver._make_antecedents)  A#$6q$9:Jr cX[[5nUHnURX25 M U$)z Create a dictionary of predicates from the assumptions. :param assumptions: a list of ``Expression``s :return: dict mapping ``AbstractVariableExpression`` to ``PredHolder`` )r PredHolder_map_predicates)r6r-r|r's r!ru&ClosedWorldProver._make_predicate_dicts,!, A   /r c[U[5(aEUR5up4[U[5(aX#R [ U55 gg[U[ 5(a9URURU5 URURU5 g[U[5(GaUR/nURn[U[5(a>URUR5 URn[U[5(aM>[U[5(Ga![UR[5(Ga[UR[5(aURR5upxURR5up[U[5(a[U [5(a|XXV s/sHoRPM sn :Xa\XZV s/sHoRPM sn :Xa<X)R[ U5UR45 X'R!U5 gggggggggs sn fs sn fr$)rEr uncurryr append_sigrr rrJrKr rHrFr\r append_propvalidate_sig_len) r6 expressionpredDictfuncargsrrFfunc1args1func2args2rs r!r!ClosedWorldProver._map_predicatess j"7 8 8#++-JD$ :;;))%+6<  M 2 2  !1!18 <  !2!2H =  M 2 2&&'C??DT=11 4==)yyT=11$ ..djj*?@@ZKK!6FF$(::#5#5#7LE#';;#6#6#8LE"5*DEE&u.HII#>1JJ#>>#>1JJ#>> 33U3Z4LM 88=??JF F@/3 $?#>s &I$I)rN) rrrrrQr-rvr{rurrrr r!rnrns">+-ZQ >r rnc<\rSrSrSrSrSrSrSrSr Sr S r g ) ri/as This class will be used by a dictionary that will store information about predicates to be used by the ``ClosedWorldProver``. The 'signatures' property is a list of tuples defining signatures for which the predicate is true. For instance, 'see(john, mary)' would be result in the signature '(john,mary)' for 'see'. The second element of the pair is a list of pairs such that the first element of the pair is a tuple of variables and the second element is an expression of those variables that makes the predicate true. For instance, 'all x.all y.(see(x,y) -> know(x,y))' would result in "((x,y),('see(x,y)'))" for 'know'. c./Ul/UlSUlgr$rwryrr6s r!__init__PredHolder.__init__?s!r c\URU5 URRU5 gr$)rrwr\r6rs r!rPredHolder.append_sigDs" g& w'r cbURUS5 URRU5 g)Nr)rryr\)r6new_props r!rPredHolder.append_propHs& hqk* x(r cURc[U5UlgUR[U5:wa [S5eg)NzSignature lengths do not match)rlen Exceptionrs r!rPredHolder.validate_sig_lenLs=    %!$WD    3w< /<= =0r cVSURSURSURS3$)N(,)rrs r!__str__PredHolder.__str__Rs.4??#1T__$5Qt7I7I6J!LLr c SU-$)Nz%srrs r!__repr__PredHolder.__repr__Us d{r )ryrrwN) rrrrrQrrrrrrrrr r!rr/s& " ()> Mr rc[RnU"S5nU"S5nU"S5n[X1U/5n[UR 55 [ U5n[S5 UR 5Hn[SU5 M [SUR55 [UR 55 U"S5nU"S5nU"S5nU"S5n[X1X'/5n[UR 55 [ U5n[S5 UR 5Hn[SU5 M [SUR55 [UR 55 U"S5nU"S5nU"S5nU"S5n[X1X'/5n[UR 55 [ U5n[S5 UR 5Hn[SU5 M [SUR55 [UR 55 U"S5nU"S5nU"S 5n[X1U/5n[UR 55 [ U5n[S5 UR 5Hn[SU5 M [SUR55 [UR 55 U"S 5nU"S 5nU"S 5nU"S 5nU"S5n U"S5n[X1X'X/5n[UR 55 [ U5n[S5 UR 5Hn[SU5 M [SUR55 [UR 55 g)Nzexists x.walk(x) man(Socrates)walk(Socrates) assumptions: goal: -walk(Bill)z walk(Bill)z all x.walk(x)z girl(mary)z dog(rover)zall x.(girl(x) -> -dog(x))zall x.(dog(x) -> -girl(x))zchase(mary, rover)z1exists y.(dog(y) & all x.(girl(x) -> chase(x,y))))r fromstringrprintr[r1r-r,) lexprp1p2rNprovercdpr'p3p4p5s r!closed_domain_demorYs  ! !E " #B  B  A ABx (F &,,. V $C . __  eQ '388: #))+ " #B  B ~ B  A AB| ,F &,,. V $C . __  eQ '388: #))+ " #B  B ~ B  A AB| ,F &,,. V $C . __  eQ '388: #))+  !B } B A ABx (F &,,. V $C . __  eQ '388: #))+ } B } B , -B , -B $ %B BCA ABB3 4F &,,. V $C . __  eQ '388: #))+r c[RnU"S5nU"S5nU"S5n[X1U/5n[UR 55 [ U5n[S5 UR 5Hn[SU5 M [SUR55 [UR 55 U"S5nU"S5nU"S 5nU"S 5n[X1X'/5n[UR 55 [ U5n[S5 UR 5Hn[SU5 M [SUR55 [UR 55 g) Nrz man(Bill)zexists x.exists y.(x != y)rrrz!all x.(walk(x) -> (x = Socrates))zBill = Williamz Bill = Billyz-walk(William))rrrrr[rSr-r,)rrrrNrunpr'rs r!unique_names_demors$  ! !E  B | B +,A ABx (F &,,. F #C . __  eQ '388: #))+ 3 4B  !B  B  A AB| ,F &,,. F #C . __  eQ '388: #))+r cd[RnU"S5nU"S5nU"S5n[X1U/5n[UR 55 [ U5n[S5 UR 5Hn[SU5 M [SUR55 [UR 55 U"S5nU"S5nU"S 5nU"S 5nU"S 5n[X1X'U/5n[UR 55 [ U5n[S5 UR 5Hn[SU5 M [SUR55 [UR 55 U"S 5nU"S 5nU"S5nU"S5nU"S5n[X1X'U/5n[UR 55 [ U5n[S5 UR 5Hn[SU5 M [SUR55 [UR 55 g)Nrz(Socrates != Bill)rrrrsee(Socrates, John)see(John, Mary)z(Socrates != John)z(John != Mary)-see(Socrates, Mary)zall x.(ostrich(x) -> bird(x))z bird(Tweety)z -ostrich(Sam)z Sam != Tweetyz -bird(Sam))rrrrr[rnr-r,) rrrrNrcwpr'rrs r!closed_world_demors  ! !E  !B $ %B nA ABx (F &,,. F #C . __  eQ '388: #))+ % &B ! "B $ %B  !B %&A ABB/ 0F &,,. F #C . __  eQ '388: #))+ / 0B  B  B  B mA ABB/ 0F &,,. F #C . __  eQ '388: #))+r cP[RnU"S5nU"S5nU"S5n[X1U/5n[UR 55 [ [ [U555nUR5Hn[U5 M [UR 55 g)Nrrr) rrrrr[r1rSrnr-)rrrrNrcommandr's r!combination_prover_demors  ! !E % &B ! "B %&A ABx (F &,,. !23DV3L!MNG  " a# '--/r cx[Rn/nURU"S55 URU"S55 URU"S55 URU"S55 URU"S55 URU"S55 URU"S55 URU"S55 URU"S 55 URU"S 55 URU"S 55 URU"S 55 URU"S 55 URU"S55 [SU5n[ [ U55nUR 5Hn[U5 M [SU5 [SU5 [SU5 g)Nz'all x.(elephant(x) -> animal(x))z'all x.(bird(x) -> animal(x))z%all x.(dove(x) -> bird(x))z%all x.(ostrich(x) -> bird(x))z(all x.(flying_ostrich(x) -> ostrich(x))z)all x.((animal(x) & -Ab1(x)) -> -fly(x))z(all x.((bird(x) & -Ab2(x)) -> fly(x))z)all x.((ostrich(x) & -Ab3(x)) -> -fly(x))z#all x.(bird(x) -> Ab1(x))z#all x.(ostrich(x) -> Ab2(x))z#all x.(flying_ostrich(x) -> Ab3(x))z elephant(E)zdove(D)z ostrich(O)z-fly(E)zfly(D)z-fly(O)) rrr\rrSrnr-r print_proof)rpremisesrrr's r!default_reasoning_demorsx  ! !EH OOEDEF OOEDEF OOEBCD OOEBCD OOEEFG OO :; OO 9: OO :;  OOE@AB OOE@AB OOE@AB OOE.)* OOE*%& OOE-()D( +F 1& 9:G  " a# 8$(# 8$r c[Rn[U"U5U5n[[ U55n[ XR 5UR 55 gr$)rrrrSrnrr[)r,rrrrs r!rr!sC  ! !E E$K 2F 1& 9:G $  0r ch[5 [5 [5 [5 [ 5 gr$)rrrrrrr r!demor(s r __main__N)+rQ collectionsr functoolsrnltk.inference.apirrnltk.inference.prover9rrnltk.sem.logicr r r r r rrrrrrrrrrrr/r1rSrWrXrnrrrrrrrrrrr r!rs $=:$ y Q2/2j(-.(-V(F>.F>R''TBJ:)X )%X1 zFr