This paper presents a complete algorithmic study of the decision Boolean Satisfiability Problem under the classical computation and quantum computation theories. The paper depicts deterministic and probabilistic algorithms, propositions of their properties and the main result is that the problem has not an efficient algorithm (NP is not P). Novel quantum algorithms and propositions depict that the complexity by quantum computation approach for solving the Boolean Satisfiability Problem or any NP problem is lineal time.
This paper presents a complete algorithmic study of the decision Boolean Satisfiability Problem under the classical computation and quantum computation theories. The paper depicts deterministic and probabilistic algorithms, propositions of their properties and the main result is that the problem has not an efficient algorithm (NP is not P). Novel quantum algorithms and propositions depict that the complexity by quantum computation approach for solving the Boolean Satisfiability Problem or any NP problem is lineal time.
We connect both discrete and algebraic complexity theory with the satisfiability problem in certain non-Boolean lattices. Specifically, quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables: in the 1D case it coincides with Boolean propositional logic but, starting with dimension two, violates the distributive law. We introduce the weak and strong satisfiability problem for quantum logic propositional formulae. It turns out that in dimension two, both are also NP --complete. For higher-dimensional spaces ℝ d and ℂ d with d ≥ 3 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale (BSS) model of real computation. This provides a unified view on both Turing and real BSS complexity theory, and extends the (still relatively scarce) list of problems established NP ℝ --complete with one, perhaps, closest in spirit to the classical Cook-Levin Theorem. More precisely, strong satisfiability of ∧ ∨ ∧ --terms is complete, while that of ∧ ∨--terms (i.e., those in conjunctive form) can be decided in polynomial time in dimensions d ≥ 2. The decidability of the infinite-dimensional case being still open, we proceed to investigate the case of indefinite finite dimensions. Here, weak satisfiability still belongs to NP R and strong satisfiability is still hard; the latter, in fact, turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations over matrix rings.
This paper proposes a quantum algorithm for solving the tautology and the satisfiability problems for a Boolean formula. Let’s say we are given a Boolean formula. The variables of the Boolean formula can take only two values—TRUE or FALSE. The tautology problem asks whether the Boolean formula always evaluates to TRUE for all values of its Boolean variables. The satisfiability problem asks whether the Boolean formula can evaluate to TRUE for at least one set of values of its Boolean variables. This paper proposes that both problems can be solved using the proposed quantum algorithm. The tautology problem is known to be co-NP-complete. The satisfiability problem is known to be NP-complete.
Quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables. It generalizes, and on 1-dimensional Hilbert space coincides with, Boolean propositional logic. We introduce the weak and strong satisfiability problem for quantum logic terms. It turns out that in dimension two both are also NP-complete. For higher-dimensional spaces R^d and C^d with d>2 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale model of real computation. This provides a unified view on both Turing and real BSS complexity theory; and extends the still relatively scarce family of NP_R-complete problems with one perhaps closest in spirit to the classical Cook-Levin Theorem. Our investigations on the dimensions a term is weakly/strongly satisfiable in lead to satisfiability problems in indefinite finite and finally in infinite dimension. Here, strong satisfiability turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations
This study explores how well Grover's Algorithm performs in solving the Boolean Satisfiability Problem (SAT) using quantum circuits. The algorithm is implemented with IBM's Qiskit framework and compared to classical brute-force methods. Experiments focus on 3-SAT, 4-SAT, and 5-SAT problems, using quantum simulators and IBM quantum hardware. The results show that Grover's Algorithm is more efficient, offering a theoretical quadratic speedup over classical methods. However, practical issues like limited qubit availability, hardware noise, and optimization challenges impact its current performance. The data highlights the potential for quantum computing to scale and solve NP-complete problems. This research shows how quantum computing can improve problem-solving and lays the groundwork for future studies on more complex SAT problems and advanced quantum hardware.
This study explores how well Grover's Algorithm performs in solving the Boolean Satisfiability Problem (SAT) using quantum circuits. The algorithm is implemented with IBM's Qiskit framework and compared to classical brute-force methods. Experiments focus on 3-SAT, 4-SAT, and 5-SAT problems, using quantum simulators and IBM quantum hardware. The results show that Grover's Algorithm is more efficient, offering a theoretical quadratic speedup over classical methods. However, practical issues like limited qubit availability, hardware noise, and optimization challenges impact its current performance. The data highlights the potential for quantum computing to scale and solve NP-complete problems. This research shows how quantum computing can improve problem-solving and lays the groundwork for future studies on more complex SAT problems and advanced quantum hardware.
Quantum logic generalizes, and in dimension one coincides with, Boolean propositional logic. We introduce the weak and strong satisfiability problem for quantum logic formulas, and show both NP-complete in dimension two as well. For higher-dimensional spaces Rdand Cdwith d≥3 fixed, on the other hand, we show the problem to be complete for the nondeterministic Blum-Shub-Smale model of real computation. This provides a unified view on both Turing and real BSS complexity theory, and adds (a perhaps more natural and combinatorially flavoured) one to the still sparse list of NPR-complete problems, mostly pertaining to real algebraic geometry. Our proofs rely on (a careful examination of) works by John von Neumannas well as contributions by Hagge et. al (2005,2007,2009). We finally investigate the problem over Indefinite finite dimensions and relate it to NON-commutative semi algebraic geometry.