Boolean satisfiability problem in the context of "Hilary Putnam"

Knowledge representation (KR) aims to model information in a structured manner to formally represent it as knowledge in knowledge-based systems whereas knowledge representation and reasoning (KRR, KR&R, or KR²) also aims to understand, reason, and interpret knowledge. KRR is widely used in the field of artificial intelligence (AI) with the goal to represent information about the world in a form that a computer system can use to solve complex tasks, such as diagnosing a medical condition or having a natural-language dialog. KR incorporates findings from psychology about how humans solve problems and represent knowledge, in order to design formalisms that make complex systems easier to design and build. KRR also incorporates findings from logic to automate various kinds of reasoning.

Traditional KRR focuses more on the declarative representation of knowledge. Related knowledge representation formalisms mainly include vocabularies, thesaurus, semantic networks, axiom systems, frames, rules, logic programs, and ontologies. Examples of automated reasoning engines include inference engines, theorem provers, model generators, and classifiers.

Martin David Davis (March 8, 1928 – January 1, 2023) was an American mathematician and computer scientist who contributed to the fields of computability theory and mathematical logic. His work on Hilbert's tenth problem led to the MRDP theorem. He also advanced the Post–Turing model and co-developed the Davis–Putnam–Logemann–Loveland (DPLL) algorithm, which is foundational for Boolean satisfiability solvers.

Davis won the Leroy P. Steele Prize, the Chauvenet Prize (with Reuben Hersh), and the Lester R. Ford Award. He was a fellow of the American Academy of Arts and Sciences and a fellow of the American Mathematical Society.

In logic and computer science, the Davis–Putnam–Logemann–Loveland (DPLL) algorithm is a complete, backtracking-based search algorithm for deciding the satisfiability of propositional logic formulae in conjunctive normal form, i.e. for solving the CNF-SAT problem.

It was introduced in 1961 by Martin Davis, George Logemann and Donald W. Loveland and is a refinement of the earlier Davis–Putnam algorithm, which is a resolution-based procedure developed by Davis and Hilary Putnam in 1960. Especially in older publications, the Davis–Logemann–Loveland algorithm is often referred to as the "Davis–Putnam method" or the "DP algorithm". Other common names that maintain the distinction are DLL and DPLL.

In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation-complete theorem-proving technique for sentences in propositional logic and first-order logic. For propositional logic, systematically applying the resolution rule acts as a decision procedure for formula unsatisfiability, solving the (complement of the) Boolean satisfiability problem. For first-order logic, resolution can be used as the basis for a semi-algorithm for the unsatisfiability problem of first-order logic, providing a more practical method than one following from Gödel's completeness theorem.

The resolution rule can be traced back to Davis and Putnam (1960); however, their algorithm required trying all ground instances of the given formula. This source of combinatorial explosion was eliminated in 1965 by John Alan Robinson's syntactical unification algorithm, which allowed one to instantiate the formula during the proof "on demand" just as far as needed to keep refutation completeness.

In computational complexity theory, NP-complete problems are the hardest of the problems to which solutions can be verified quickly.Somewhat more precisely, a problem is NP-complete when:

1. It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no".
2. When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) solution.
3. The correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions.
4. The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. Hence, if we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified.

The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a deterministic algorithm to check a single solution, or for a nondeterministic Turing machine to perform the whole search. "Complete" refers to the property of being able to simulate everything in the same complexity class.

In computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete. That is, it is in NP, and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem.

The theorem is named after Stephen Cook and Leonid Levin. The proof is due to Richard Karp, based on an earlier proof (using a different notion of reducibility) by Cook.

Constraint satisfaction problems (CSPs) are mathematical questions defined as a set of objects whose state must satisfy a number of constraints or limitations. CSPs represent the entities in a problem as a homogeneous collection of finite constraints over variables, which is solved by constraint satisfaction methods. CSPs are the subject of research in both artificial intelligence and operations research, since the regularity in their formulation provides a common basis to analyze and solve problems of many seemingly unrelated families. CSPs often exhibit high complexity, requiring a combination of heuristics and combinatorial search methods to be solved in a reasonable time. Constraint programming (CP) is the field of research that specifically focuses on tackling these kinds of problems. Additionally, the Boolean satisfiability problem (SAT), satisfiability modulo theories (SMT), mixed integer programming (MIP) and answer set programming (ASP) are all fields of research focusing on the resolution of particular forms of the constraint satisfaction problem.

Examples of problems that can be modeled as a constraint satisfaction problem include: