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Exploiting Past and Future: Pruning by Inconsistent Partial State
Exploiting Past and Future: Pruning by Inconsistent Partial State

10-3
10-3

... Solving Equations with 10-3 Variables on Both Sides Additional Example 3 Continued First solve for the price of one doughnut. Let d represent the price 1.25 + 2d = 0.50 + 5d of one doughnut. ...
Constraint Programming - What is behind?
Constraint Programming - What is behind?

UNIT 7 Systems of Equations
UNIT 7 Systems of Equations

CAHOOTS: A SOFTWARE PLATFORM FOR ENHANCING
CAHOOTS: A SOFTWARE PLATFORM FOR ENHANCING

... Although conceived of as bi-directional activity, current visual aids focus only on unidirectional activity (Greene et al., 2011; Nielsen, 2012). Top-down problem framing refines the expression of the goal. In our approach, bottom-up problem solving involves two sub-processes: (1) uncovering the fea ...
INTRODUCTION TO Al AND PRODUCTION SYSTEMS 9
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On the connections between backdoors, restarts, and heavy
On the connections between backdoors, restarts, and heavy

... than 400 variables are out of reach of most complete solvers. See also analysis in [1].) Backdoor variables are related to the notion of independent variables [20, 5]. In fact, a set of independent variables of a problem also forms a backdoor set (provided the propagation mechanism of the SAT solver ...
A WK-Means Approach for Clustering
A WK-Means Approach for Clustering

... data. It has been used for storing and representing large amounts of information as data. Cluster analysis can be defined as discovering natural hidden groups of objects. It has been used for assigning the same objects to the same groups [22], where different objects are in different groups. Cluster ...
Systems of linear equations, Gaussian elimination
Systems of linear equations, Gaussian elimination

Potential Search: a Bounded
Potential Search: a Bounded

10-3
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Structural Types for the Factorisation Calculus
Structural Types for the Factorisation Calculus

SOLVING SYSTEMS BY GRAPHING INTRODUCTION The objective
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... When we are solving an equation, what does it mean to find one solution? The solution is the value that when substituted back into the equation will make the statement true. Using the equations we began with, let’s substitute some values on the lines into the equations to see if they are solutions. ...
11-3 Solving Equations with Variables on Both Sides
11-3 Solving Equations with Variables on Both Sides

... Some problems produce equations that have variables on both sides of the equal sign. Solving an equation with variables on both sides is similar to solving an equation with a variable on only one side. You can add or subtract a term containing a variable on both sides of an equation. ...
PDF - 1.4 MB - Massachusetts Institute of Technology
PDF - 1.4 MB - Massachusetts Institute of Technology

... useful situations, which we will discuss later. Note that because we are not dealing with consistency constraints (all negative literals) we will not be able to deal with negative facts either. ...
2-Solving Multi-Step Equations
2-Solving Multi-Step Equations

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2-2 PPT - My eCoach

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... problems, involving linear equations and linear inequalities in one variable and provide justification for each step. ...
5. Constraint Satisfaction Problems CSPs as Search Problems
5. Constraint Satisfaction Problems CSPs as Search Problems

... The arc-consistency algorithm AC-3. After applying AC-3, either every arc is arcFoundations of AI Mai 11, 2012 be solved. 28 / 39 The variable has an empty domain, indicating that the CSP cannot name “AC-3” was used by the algorithm’s inventor (?) because it’s the third version developed in the ...
A Fast Arc Consistency Algorithm for n-ary Constraints Olivier Lhomme Jean-Charles R´egin
A Fast Arc Consistency Algorithm for n-ary Constraints Olivier Lhomme Jean-Charles R´egin

... With our notations, we can simplify the result of (Bessière & Régin 1999) on which their filtering algorithms are based. Property 1 generalizes their first result and their second result can be rewritten (and generalized) as: Property 3 Let x ∈ X with D(x) = {a} and y 6= x be any variable of X and ...
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On the Learnability of Description Logic Programs
On the Learnability of Description Logic Programs

... These language constructs can be used to build complex terms, e.g. the term train u ∀ has car.(car u ≤ 0 has load) can be used to define empty trains (all cars do not have a load), in Michalski’s well-known train domain. An statement not expressible as a logic program, if we are restricted to the pr ...
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Unification (computer science)

Unification, in computer science and logic, is an algorithmic process of solving equations between symbolic expressions.Depending on which expressions (also called terms) are allowed to occur in an equation set (also called unification problem), and which expressions are considered equal, several frameworks of unification are distinguished. If higher-order variables, that is, variables representing functions, are allowed in an expression, the process is called higher-order unification, otherwise first-order unification. If a solution is required to make both sides of each equation literally equal, the process is called syntactical unification, otherwise semantical, or equational unification, or E-unification, or unification modulo theory.A solution of a unification problem is denoted as a substitution, that is, a mapping assigning a symbolic value to each variable of the problem's expressions. A unification algorithm should compute for a given problem a complete, and minimal substitution set, that is, a set covering all its solutions, and containing no redundant members. Depending on the framework, a complete and minimal substitution set may have at most one, at most finitely many, or possibly infinitely many members, or may not exist at all. In some frameworks it is generally impossible to decide whether any solution exists. For first-order syntactical unification, Martelli and Montanari gave an algorithm that reports unsolvability or computes a complete and minimal singleton substitution set containing the so-called most general unifier.For example, using x,y,z as variables, the singleton equation set { cons(x,cons(x,nil)) = cons(2,y) } is a syntactic first-order unification problem that has the substitution { x ↦ 2, y ↦ cons(2,nil) } as its only solution.The syntactic first-order unification problem { y = cons(2,y) } has no solution over the set of finite terms; however, it has the single solution { y ↦ cons(2,cons(2,cons(2,...))) } over the set of infinite trees.The semantic first-order unification problem { a⋅x = x⋅a } has each substitution of the form { x ↦ a⋅...⋅a } as a solution in a semigroup, i.e. if (⋅) is considered associative; the same problem, viewed in an abelian group, where (⋅) is considered also commutative, has any substitution at all as a solution.The singleton set { a = y(x) } is a syntactic second-order unification problem, since y is a function variable.One solution is { x ↦ a, y ↦ (identity function) }; another one is { y ↦ (constant function mapping each value to a), x ↦ (any value) }.The first formal investigation of unification can be attributed to John Alan Robinson, who used first-order syntactical unification as a basic building block of his resolution procedure for first-order logic, a great step forward in automated reasoning technology, as it eliminated one source of combinatorial explosion: searching for instantiation of terms. Today, automated reasoning is still the main application area of unification.Syntactical first-order unification is used in logic programming and programming language type system implementation, especially in Hindley–Milner based type inference algorithms.Semantic unification is used in SMT solvers and term rewriting algorithms.Higher-order unification is used in proof assistants, for example Isabelle and Twelf, and restricted forms of higher-order unification (higher-order pattern unification) are used in some programming language implementations, such as lambdaProlog, as higher-order patterns are expressive, yet their associated unification procedure retains theoretical properties closer to first-order unification.
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