100 lines
3.0 KiB
Markdown
100 lines
3.0 KiB
Markdown
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# Constraint Satisfaction Problems
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Book pg. 164 to 169
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We have 3 components
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- $\mathcal{X}$: Set of variables
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- $\mathcal{D}$: Set of domains, one (domain) for each variable
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- $\mathcal{C}$: Set of constraints
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Here we try to assign a value to variables, such that each variable holds a value contained in their
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domain, without violating any constraint.
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A solution to this problem must be:
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- Consistent: Each assignment is legal
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- Complete: All variables have a value assigned
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## Inference in CSP
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We can inference a solution in these ways
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- Generate successors and if not consistent assignments, prune and continue
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- Constraint Propagation:\
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Use contraints to reduce possible assignments and mix with search, if needed
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The idea is to treat each assignment as a **vertex** and each contraint as an **edge**, we our tree would not have inconsistent values, making
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the search faster.
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## Consistency
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### Node consistency
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A domain already satisfies constraints
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### Arc Consistency
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A variable is arc consistent wrt another variable id its domain respects all constraints
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> [!WARNING]
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> Arco Consistency is not useful for every problem
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> [!NOTE]
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> AC-3 is an algorithm to achieve Arc Consistency (alorithm on page 171 book)
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### Path Consistency
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It is achievable on 3 variables and says that a set of 2 variable is path consistent wrt a 3rd only if for each assignment
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consistent with constraints between these variables makes it possible to make a legal assignment on the 3rd
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### K-Consistency
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basically the same as the one above, but for k-variables, technically we need $O(n^2 d)$ to check if a problem is
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total k-consistent (consistent from 1 up to k), however it is expensive and only 2-Consistency is usually computed.
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## Global Contraints
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Constraints that take an arbitrary number of variables (but not necessarily all).
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### Checking for Aldiff Constraint
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Aldiff constraint says that all variables must have different values.
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A way to check for inconsistencies is to remove variables with single value domains and remove such value
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from all domains, then iterate until there are no more single value domain variables.
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If an empty domain is produced, or there are more variables than domain values, congrats!! You found an inconsistency
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### Checking for Atmost Constraint
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Atmost constraint says that we can assign max tot resources
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Sum all minimum resources for each problem, and if it goes over the max, we have an inconsistency
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### Bound Consistency
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We check that for each variable X and for both the lower and higher bound of X there exists a value of U that satisfies the constraints between X and each Y.
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## Backtracking Search in CSP
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Book pg 175 to 178
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After finishing the constraint propagation, reducing our options, we still need to search for a solution.
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- The order of values is not relevant (we can save up on computation and memoru)
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### Forward Checking
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<!-- TODO -->
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### Intelligent Backtracking (Backjumping)
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<!-- TODO -->
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## The structure of Problems
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<!-- TODO -->
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Book from 183 to 187
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