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In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable.
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.
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.
In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time.
The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic.
In computational complexity theory, the PCP theorem states that every decision problem in the NP complexity class has probabilistically checkable proofs of constant query complexity and logarithmic randomness complexity.
In computational complexity, problems that are in the complexity class NP but are neither in the class P nor NP-complete are called NP-intermediate, and the class of such problems is called NPI. Ladner's theorem, shown in 1975 by Richard E. Ladner, is a result asserting that, if P ≠ NP, then NPI is not empty; that is, NP contains problems that are neither in P nor NP-complete. Since it is also true that if NPI problems exist, then P ≠ NP, it follows that P = NP if and only if NPI is empty.
In graph theory and computer science, the lowest common ancestor (LCA) of two nodes v and w in a tree or directed acyclic graph (DAG) T is the lowest node that has both v and w as descendants, where we define each node to be a descendant of itself.
ACC0, sometimes called ACC, is a class of computational models and problems defined in circuit complexity, a field of theoretical computer science. The class is defined by augmenting the class AC0 of constant-depth "alternating circuits" with the ability to count; the acronym ACC stands for "AC with counters". Specifically, a problem belongs to ACC0 if it can be solved by polynomial-size, constant-depth circuits of unbounded fan-in gates, including gates that count modulo a fixed integer. ACC0 corresponds to computation in any solvable monoid. The class is very well studied in theoretical computer science because of the algebraic connections and because it is one of the largest concrete computational models for which computational impossibility results, so-called circuit lower bounds, can be proved.
In database theory, a conjunctive query is a restricted form of first-order queries using the logical conjunction operator. Many first-order queries can be written as conjunctive queries. In particular, a large part of queries issued on relational databases can be expressed in this way. Conjunctive queries also have a number of desirable theoretical properties that larger classes of queries do not share.
In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification over sets. It is particularly important in the logic of graphs, because of Courcelle's theorem, which provides algorithms for evaluating monadic second-order formulas over graphs of bounded treewidth. It is also of fundamental importance in automata theory, where the Büchi–Elgot–Trakhtenbrot theorem gives a logical characterization of the regular languages.
Mihalis Yannakakis is professor of computer science at Columbia University. He is noted for his work in computational complexity, databases, and other related fields. He won the Donald E. Knuth Prize in 2005.
Georg Gottlob FRS is an Austrian-Italian computer scientist who works in the areas of database theory, logic, and artificial intelligence and is Professor of Informatics at the University of Calabria. He was Professor at the University of Oxford.
In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching to 3-partite hypergraphs, which consist of hyperedges each of which contains 3 vertices.
In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of the form
In computer science, an enumeration algorithm is an algorithm that enumerates the answers to a computational problem. Formally, such an algorithm applies to problems that take an input and produce a list of solutions, similarly to function problems. For each input, the enumeration algorithm must produce the list of all solutions, without duplicates, and then halt. The performance of an enumeration algorithm is measured in terms of the time required to produce the solutions, either in terms of the total time required to produce all solutions, or in terms of the maximal delay between two consecutive solutions and in terms of a preprocessing time, counted as the time before outputting the first solution. This complexity can be expressed in terms of the size of the input, the size of each individual output, or the total size of the set of all outputs, similarly to what is done with output-sensitive algorithms.
In database theory, the query evaluation problem is the problem of determining the answers to a query on a database. Research in database theory aims at determining the computational complexity of answering different kinds of queries over databases, in particular over relational databases.
The Yannakakis algorithm is an algorithm in database theory for computing the output of an (alpha-)acyclic conjunctive query. The algorithm is named after Mihalis Yannakakis.
References
- 1 2 3 Hemaspaandra, Lane A.; Ogihara, Mitsunori (2002), The Complexity Theory Companion, Texts in Theoretical Computer Science: An EATCS Series, Springer, p. 280, doi:10.1007/978-3-662-04880-1, ISBN 9783662048801
- ↑ Kapron, Bruce M., ed. (2023), Logic, Automata, and Computational Complexity: The Works of Stephen A. Cook, ACM Books, Morgan & Claypool, p. 124, ISBN 9798400707803
- 1 2 Gottlob, Georg; Leone, Nicola; Scarcello, Francesco (2001), "The complexity of acyclic conjunctive queries", Journal of the ACM, 48 (3): 431–498, doi:10.1145/382780.382783
- ↑ Vortmeier, Nils; Kokkinis, Ioannis (2021), "The dynamic complexity of acyclic hypergraph homomorphisms", in Kowalik, Lukasz; Pilipczuk, Michal; Rzazewski, Pawel (eds.), Graph-Theoretic Concepts in Computer Science - 47th International Workshop, WG 2021, Warsaw, Poland, June 23-25, 2021, Revised Selected Papers, Lecture Notes in Computer Science, vol. 12911, Springer, pp. 232–244, arXiv: 2107.06121 , doi:10.1007/978-3-030-86838-3_18