U Zʹj#@sdZddlmZddlmZmZmZmZmZm Z m Z ddl m Z ddl mZmZddZdd Zd d Zifd d ZddZddZddZddZddZddZdS)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_symbolcCst|tstt|}n|j}d|kr*dStt|td}tt dt |d}t ||}t ||i}|sn|Si}|D]}| ||d||iqv|S)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)exprrsymbolsZsymbols_int_reprZclauses_int_reprresultoutputr rZ/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/logic/algorithms/dpll.pydpll_satisfiables   rc CsFt||\}}|rN|||i|||s4|}t||}t||\}}qt||\}}|r|||i|||s|}t||}t||\}}q\g}|D].}t||}|dkrdS|dk r||q|s|S|s|S|}|}||di||di|dd} t t||||pDt t|t || |S)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_clausescvalZ model_copyZ symbols_copyrrrr%1sF        r%c Cs8t||\}}|rN|||i|||s4| }t||}t||\}}qt||\}}|r|||i|||s| }t||}t||\}}q\g}|D].}t||}|dkrdS|dk r||q|s|S|}|}||di||di|} t t||||p6t t|| | |S)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_reprrrunit_propagate_int_reprfind_pure_symbol_int_reprpl_true_int_reprr"r#r$rr&rrrrbsB       rcCsZd}|D]L}|dkr0|| }|dk r:| }n ||}|dkrHdS|dkrd}q|S)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'rZlitprrrr0s   r0csvg}|D]h}|jtkr"||q|jD]<}|krX|tfdd|jDq|kr(qq(||q|S)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] csg|]}|kr|qSrr).0xsymbolrr s z"unit_propagate..)funcrr"args)rr7rr+argrr6rr s     r cs hfdd|DS)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}] csg|]}|kr|qSrr)r4r2Znegatedsrrr8sz+unit_propagate_int_repr..r)rr=rr<rr.s r.cCs`|D]V}d\}}|D]0}|s,|t|kr,d}|st|t|krd}q||kr||fSqdS)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_negr+rrrr!s r!cCsptj|}||}|dd|D}|D]}| |kr.|dfSq.|D]}| |krN| dfSqNdS)a 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) cSsg|] }| qSrr)r4r=rrrr8sz-find_pure_symbol_int_repr..TFr>)runion intersection)rr*Z all_symbolsr@rAr3rrrr/s    r/cCs^|D]T}d}t|D].}t|}||kr|d7}|t|t }}q|dkr||fSqdS)a 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) rrr>)rr rr)rr'r2Znum_not_in_modelliteralr?r(r)rrrr s  rcCsbt|dd|DB}|D]B}||}t|dkr|}|dkrP| dfS|dfSqdS)a 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) cSsh|] }| qSrr)r4r?rrr +sz,find_unit_clause_int_repr..rrFTr>)rrr#)rr'boundr2unboundr3rrrr- s  r-N)__doc__Zsympy.core.sortingrZsympy.logic.boolalgrrrrrrr Zsympy.assumptions.cnfr Zsympy.logic.inferencer r rr%rr0r r.r!r/rr-rrrrs $ 10 !