+ i#Rt^RIHt^RIHtHtHtHtHtH t H t ^RI H t ^RI HtHtRtRtRt/3RltR tR tR tR tR tRtR#)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_symbolc\V\4'g\\V44pM VPpRV9dR#\ \ V4\R7p\\^\V4^,44p\W4p\WC/4pV'gV#/pVF)pVPW'^, ,WW,/4K+ V#)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& Y/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/logic/algorithms/dpll.pydpll_satisfiablers dC F4L),, %d+1ABG5CL1$456"74 +r BF  F wQw'56 Mc$\W4wr4V'dIVPW4/4VPV4V'gV(p\W4p\W4wr4KP\ W4wr4V'dIVPW4/4VPV4V'gV(p\W4p\ W4wr4KP.pVF0p\ Wb4pVRJdR#VRJgKVP V4K2 V'gV#V'gV#VP4pVP4pVPVR/4VPVR/4VR,p \\WS4W4;'g \\V\V44W4#)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 FT:NNN) find_unit_clauserremoveunit_propagatefind_pure_symbolr appendpopcopydpllr rrmodelPvalueunknown_clausescval model_copy symbols_copys &&& rr(r(1s[ /HA  aZ qA ,#G351HA  aZ qA ,#G55O a %< d?  " "1 %     AJ LL!Tq%j!1:L 3W D T T Q8, SUrc \W4wr4V'dIVPW4/4VPV4V'gV)p\W4p\W4wr4KP\ W4wr4V'dIVPW4/4VPV4V'gV)p\W4p\ W4wr4KP.pVF0p\ Wb4pVRJdR#VRJgKVP V4K2 V'gV#VP4pVP4pVPVR/4VPVR/4VP4p \\WS4W4;'g\\WS)4W4#)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 &&& rrrbsS)8HA  aZ qA)'5,W<5(:HA  aZ qA)'5,W>5O q( %< d?  " "1 %    AJ LL!Tq%j!<<>L 1/Ew V b b 1/2F acrcRpVFJpV^8dVPV)4pVeV'*pMVPV4pVRJdR#VeKHRpKL V#)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 FNT)get)clauser*rlitps&& rr6r6s^F 7 3$A}E #A 9 YF Mrc Z.pVFpVP\8wdVPV4K+VPFQpWA(8Xd?TP\VPUu.uFqUV(8wgK VNK up!4KWA8XgKPK VPV4K V#uupi)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)rsymbolrr.argxs&& rr#r#s"F  66R< MM!  66Cg~ baff"EffW 11f"EFG}  MM!  M #Fs $ B(2B(cTV)0pVUu.uFq1V9gK W2, NK up#uupi)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}] )rsnegatedr9s&& rr4r4s2rdG+2 F7vo F  7 FF Fs% %cVF[pRRrCVFDpV'gV\V49dRpV'dK'\V4\V49gKBRpKF W48wgKXW#3u# R#)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) FTNN)rr)rr-sym found_pos found_negr.s&& rr$r$sc$e9 A ! !4 9SYq\!9 !  !> ! rc\4P!V!pVPV4pTPVUu.uFqD)NK up4pVFpV)V9gK VR3u# VFpV)V9gK V)R3u# R#uupi)z Same as find_pure_symbol, but arguments are expected to be in integer representation >>> from sympy.logic.algorithms.dpll import find_pure_symbol_int_repr >>> find_pure_symbol_int_repr({1,2,3}, ... [{1, -2}, {-2, -3}, {3, 1}]) (1, True) TFrG)runion intersection)rr- all_symbolsrIrDrJr;s&& rr5r5s%++/K((1I((g)>g"g)>?I  2Y d7N 2Y 2u9  *?s A<cVFVp^p\V4F5p\V4pWQ9gKV^, pT\V\4'*rvK7 V^8XgKRXX3u# R#)z A unit clause has only 1 variable that is not bound in the model. >>> from sympy.abc import A, B, D >>> from sympy.logic.algorithms.dpll import find_unit_clause >>> find_unit_clause([A | B | D, B | ~D, A | ~B], {A:True}) (B, False) rG)rr rr)rr*r9num_not_in_modelliteralrHr+r,s&& rr!r! sj (G )C A% Jw$< <5 ) q e8O rc\V4VUu0uFq")kK up,pVF?pWC, p\V4^8XgKVP4pV^8dV)R3u#VR3u# R#uupi)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) FTrG)rrr&)rr*rHboundr9unboundr;s&& rr3r3 sr J%0%3$%0 0E. w<1  A1ur5y $w 1s A*N)__doc__sympy.core.sortingrsympy.logic.boolalgrrrrrrr sympy.assumptions.cnfr sympy.logic.inferencer r rr(rr6r#r4r$r5r!r3rCrrrZsa0"""%9>.Ub+c`$&6B G..,r