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Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem domain. Major logic programming language families include Prolog, answer set programming (ASP) and Datalog. In all of these languages, rules are written in the form of clauses:
Mathematical logic, also called formal logic, is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, philosophy, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.
In mathematics, more precisely in mathematical logic, model theory is the study of the relationship between formal theories, and their models, taken as interpretations that satisfy the sentences of that theory. The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. The relative emphasis placed on the class of models of a theory as opposed to the class of definable sets within a model fluctuated in the history of the subject, and the two directions are summarised by the pithy characterisations from 1973 and 1997 respectively:
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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, boolean satisfiability problem (SAT), the 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.
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In mathematics, given a collection of subsets of a set X, an exact cover is a subcollection of such that each element in is contained in exactly one subset in . One says that each element in is covered by exactly one subset in . An exact cover is a kind of cover.
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In constraint satisfaction, local consistency conditions are properties of constraint satisfaction problems related to the consistency of subsets of variables or constraints. They can be used to reduce the search space and make the problem easier to solve. Various kinds of local consistency conditions are leveraged, including node consistency, arc consistency, and path consistency.
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.
Constraint logic programming is a form of constraint programming, in which logic programming is extended to include concepts from constraint satisfaction. A constraint logic program is a logic program that contains constraints in the body of clauses. An example of a clause including a constraint is A(X,Y):-X+Y>0,B(X),C(Y). In this clause, X+Y>0 is a constraint; A(X,Y), B(X), and C(Y) are literals as in regular logic programming. This clause states one condition under which the statement A(X,Y) holds: X+Y is greater than zero and both B(X) and C(Y) are true.
The complexity of constraint satisfaction is the application of computational complexity theory on constraint satisfaction. It has mainly been studied for discriminating between tractable and intractable classes of constraint satisfaction problems on finite domains.
Abductive logic programming (ALP) is a high-level knowledge-representation framework that can be used to solve problems declaratively based on abductive reasoning. It extends normal logic programming by allowing some predicates to be incompletely defined, declared as abducible predicates. Problem solving is effected by deriving hypotheses on these abducible predicates as solutions of problems to be solved. These problems can be either observations that need to be explained or goals to be achieved. It can be used to solve problems in diagnosis, planning, natural language and machine learning. It has also been used to interpret negation as failure as a form of abductive reasoning.
In mathematics, leximin order is a total preorder on finite-dimensional vectors. A more accurate, but less common term is leximin preorder. The leximin order is particularly important in social choice theory and fair division.
References
- ↑ Mackworth, Alan K.; Mulder, Jan A.; Havens, William S. (1985-01-01). "Hierarchical arc consistency: exploiting structured domains in constraint satisfaction problems". Computational Intelligence. 1 (1): 118–126. doi:10.1111/j.1467-8640.1985.tb00064.x. ISSN 1467-8640.
- ↑ Wilson, Molly; Borning, Alan (1993-07-01). "Hierarchical constraint logic programming". The Journal of Logic Programming. 16 (3–4): 277–318. doi: 10.1016/0743-1066(93)90046-J . ISSN 0743-1066.
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