\h#SrSSKJr SSKJrJrJrJrJrJ r J r SSK J r SSK JrJr SrSrSr04S jrS rS rS rS rSrSrg)a&Implementation of DPLL algorithm Further improvements: eliminate calls to pl_true, implement branching rules, efficient unit propagation. References: - https://en.wikipedia.org/wiki/DPLL_algorithm - https://www.researchgate.net/publication/242384772_Implementations_of_the_DPLL_Algorithm )default_sort_key)OrNot conjuncts disjunctsto_cnf to_int_repr_find_predicates)CNF)pl_trueliteral_symbolcz[U[5(d[[U55nO URnSU;ag[ [ U5[S9n[[S[U5S-55n[X5n[XC05nU(dU$0nUHnURX'S- XW05 M U$)a Check satisfiability of a propositional sentence. It returns a model rather than True when it succeeds >>> from sympy.abc import A, B >>> from sympy.logic.algorithms.dpll import dpll_satisfiable >>> dpll_satisfiable(A & ~B) {A: True, B: False} >>> dpll_satisfiable(A & ~A) False F)key) isinstancer rrclausessortedr rsetrangelenr dpll_int_reprupdate)exprrsymbolssymbols_int_reprclauses_int_reprresultoutputrs S/var/www/auris/envauris/lib/python3.13/site-packages/sympy/logic/algorithms/dpll.pydpll_satisfiabler s dC F4L),, %d+1ABG5CL1$456"74 +r BF  F wQw'56 Mc$[X5up4U(aNURX405 URU5 U(dU)n[X5n[X5up4U(aMN[ X5up4U(aNURX405 URU5 U(dU)n[X5n[ X5up4U(aMN/nUH-n[ Xb5nUSLa gUSLdMUR U5 M/ U(dU$U(dU$UR5nUR5nURUS05 URUS05 USSn [[XS5X5=(d [[U[U55X5$)z Compute satisfiability in a partial model. Clauses is an array of conjuncts. >>> from sympy.abc import A, B, D >>> from sympy.logic.algorithms.dpll import dpll >>> dpll([A, B, D], [A, B], {D: False}) False FTN) find_unit_clauserremoveunit_propagatefind_pure_symbolr appendpopcopydpllr rrmodelPvalueunknown_clausescval model_copy symbols_copys rr*r*1sc /HA  aZ qA ,#G3 ! 1HA  aZ qA ,#G5 !O a %< d?  " "1 %     AJ LL!Tq%j!1:L 3W D T Q8, SUr!c[X5up4U(aNURX405 URU5 U(dU*n[X5n[X5up4U(aMN[ X5up4U(aNURX405 URU5 U(dU*n[X5n[ X5up4U(aMN/nUH-n[ Xb5nUSLa gUSLdMUR U5 M/ U(dU$UR5nUR5nURUS05 URUS05 UR5n [[XS5X5=(d [[XS*5X5$)z Compute satisfiability in a partial model. Arguments are expected to be in integer representation >>> from sympy.logic.algorithms.dpll import dpll_int_repr >>> dpll_int_repr([{1}, {2}, {3}], {1, 2}, {3: False}) False FT) find_unit_clause_int_reprrr$unit_propagate_int_reprfind_pure_symbol_int_reprpl_true_int_reprr'r(r)rr+s rrrbs[)8HA  aZ qA)'5,W< !):HA  aZ qA)'5,W> !O q( %< d?  " "1 %    AJ LL!Tq%j!<<>L 1/Ew V b 1/2F acr!cSnUHEnUS:aURU*5nUbU(+nOURU5nUSLa gUbMCSnMG U$)aF Lightweight version of pl_true. Argument clause represents the set of args of an Or clause. This is used inside dpll_int_repr, it is not meant to be used directly. >>> from sympy.logic.algorithms.dpll import pl_true_int_repr >>> pl_true_int_repr({1, 2}, {1: False}) >>> pl_true_int_repr({1, 2}, {1: False, 2: False}) False FrNT)get)clauser,rlitps rr8r8s^F 7 3$A}E #A 9 YF Mr!c R/nUHnUR[:waURU5 M*URHNnXA):Xa=UR[URVs/sH oUU):wdM UPM sn65 M|XA:XdMM M URU5 M U$s snf)a Returns an equivalent set of clauses If a set of clauses contains the unit clause l, the other clauses are simplified by the application of the two following rules: 1. every clause containing l is removed 2. in every clause that contains ~l this literal is deleted Arguments are expected to be in CNF. >>> from sympy.abc import A, B, D >>> from sympy.logic.algorithms.dpll import unit_propagate >>> unit_propagate([A | B, D | ~B, B], B) [D, B] )funcrr'args)rsymbolrr0argxs rr%r%s"F  66R< MM!  66Cg~ baff"EffW 1f"EFG}  MM!  M #Fs " B$0B$cLU*1nUVs/sHo1U;dM X2- PM sn$s snf)z Same as unit_propagate, but arguments are expected to be in integer representation >>> from sympy.logic.algorithms.dpll import unit_propagate_int_repr >>> unit_propagate_int_repr([{1, 2}, {3, -2}, {2}], 2) [{3}] )rsnegatedr;s rr6r6s/rdG+2 F7vo F 7 FF Fs !!cUHWnSup4UH@nU(dU[U5;aSnU(aM$[U5[U5;dM>SnMB X4:wdMTX#4s $ g)a Find a symbol and its value if it appears only as a positive literal (or only as a negative) in clauses. >>> from sympy.abc import A, B, D >>> from sympy.logic.algorithms.dpll import find_pure_symbol >>> find_pure_symbol([A, B, D], [A|~B,~B|~D,D|A]) (A, True) )FFTNN)rr)rr/sym found_pos found_negr0s rr&r&sa+  A ! !4 9SYq\!9 !  !> ! r!c[5R"U6nURU5nURUVs/sHoD*PM sn5nUHnU*U;dM US4s $ UHnU*U;dM U*S4s $ gs snf)z Same as find_pure_symbol, but arguments are expected to be in integer representation >>> from sympy.logic.algorithms.dpll import find_pure_symbol_int_repr >>> find_pure_symbol_int_repr({1,2,3}, ... [{1, -2}, {-2, -3}, {3, 1}]) (1, True) TFrI)runion intersection)rr/ all_symbolsrKrFrLr=s rr7r7s%++/K((1I((g)>g"g)>?I  2Y d7N 2Y 2u9  *?s A9cUHPnSn[U5H0n[U5nXQ;dMUS- nU[U[5(+pvM2 US:XdMLWW4s $ g)z A unit clause has only 1 variable that is not bound in the model. >>> from sympy.abc import A, B, D >>> from sympy.logic.algorithms.dpll import find_unit_clause >>> find_unit_clause([A | B | D, B | ~D, A | ~B], {A:True}) (B, False) rrrI)rr rr)rr,r;num_not_in_modelliteralrJr-r.s rr#r# sg (G )C A% Jw$< <5 ) q e8O r!c[U5UVs1sHo"*iM sn-nUH9nXC- n[U5S:XdMUR5nUS:aU*S4s $US4s $ gs snf)z Same as find_unit_clause, but arguments are expected to be in integer representation. >>> from sympy.logic.algorithms.dpll import find_unit_clause_int_repr >>> find_unit_clause_int_repr([{1, 2, 3}, ... {2, -3}, {1, -2}], {1: True}) (2, False) rrFTrI)rrr()rr,rJboundr;unboundr=s rr5r5 so J%0%3$%0 0E. w<1  A1ur5y $w 1s A N)__doc__sympy.core.sortingrsympy.logic.boolalgrrrrrr r sympy.assumptions.cnfr sympy.logic.inferencer r r r*rr8r%r6r&r7r#r5rEr!rr\sa0"""%9>.Ub+c`$&6B G..,r!