17_pspace_qbf_reality_gates.py

#Vurt Reality Gates

'''In the dystopian world of Vurt, there exist mysterious gates known as Reality Gates that lead to alternate realities. Each Reality Gate has a unique combination lock mechanism consisting of Boolean switches, which must be set in a specific pattern to activate the gate and access the desired reality.

The goal is to determine if it is possible to set the Boolean switches of a given Reality Gate in a way that satisfies a set of logical constraints.

The constraints are represented as a Quantified Boolean Formula (QBF) in prenex normal form. A QBF consists of a sequence of quantifiers followed by a Boolean formula. The quantifiers can be either existential (∃) or universal (∀), indicating whether the variables are to be existentially or universally quantified. The Boolean formula can contain variables, logical operators (AND, OR, NOT), and parentheses.

Your task is to determine whether there exists a valid assignment of Boolean values to the switches of the Reality Gate that satisfies the given QBF constraints.

To illustrate this concept, let's consider an example code that demonstrates the solution for the Vurt Reality Gates problem using the Z3 Theorem Prover library, which provides an efficient way to solve QBFs:
'''
from z3 import *

def check_reality_gate(qbf_formula):
    # Create a Z3 solver instance
    solver = Solver()

    # Parse the QBF formula
    qbf = parse_qbf(qbf_formula)

    # Assert the QBF formula to the solver
    solver.add(qbf)

    # Check the satisfiability of the QBF formula
    result = solver.check()

    # Return the result
    return result

def parse_qbf(qbf_formula):
    # Create a Z3 solver instance to parse the QBF formula
    solver = SolverFor('QF_UF')

    # Parse the QBF formula using Z3's parse_smt2_string function
    qbf = parse_smt2_string(qbf_formula, solver)

    # Return the parsed QBF formula
    return qbf

# Example usage
qbf_formula = "(∀x1 ∃x2 ∃x3) ((x1 ∧ x2) ∨ (¬x2 ∧ x3))"
result = check_reality_gate(qbf_formula)

if result == sat:
    print("There exists a valid assignment of Boolean values.")
elif result == unsat:
    print("No valid assignment of Boolean values exists.")
else:
    print("The satisfiability of the QBF formula is unknown.")




'''In this code, the check_reality_gate() function takes a QBF formula as input and uses the Z3 Theorem Prover library to determine the satisfiability of the formula. The QBF formula is parsed using Z3's parse_smt2_string() function, and then added to the solver using the add() method. The check() method is used to check the satisfiability of the formula, and the result is returned.
    
    The parse_qbf() function is used to parse the QBF formula using Z3's parse_smt2_string() function, ensuring compatibility with the Z3 solver.
    
    The example usage demonstrates how to check the satisfiability of a QBF formula represented as a string. The provided QBF formula "(∀x1 ∃x2 ∃x3) ((x1 ∧ x2) ∨ (¬x2 ∧ x3))" is a simple example where three variables, x1, x2, and x3, are quantified with the universal and existential quantifiers. The formula states that for all possible x1 values, there exist x2 and x3 values that satisfy the given Boolean formula.
    
    The code utilizes the Z3 solver to determine the satisfiability of the QBF formula. If the result is sat, it means that there exists a valid assignment of Boolean values that satisfies the given QBF constraints. If the result is unsat, it means that no valid assignment of Boolean values exists. If the result is neither sat nor unsat, it indicates that the satisfiability of the QBF formula is unknown.'''