CP 2011 Poster
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The document discusses Boolean equi-propagation, an approach for optimizing SAT encodings of constraint satisfaction problems (CSPs). It involves inferring new equalities from constraints and simplifying the model. This allows representing problem structures directly and compactly in conjunctive normal form (CNF). Examples show how equi-propagation simplifies models by removing variables for all-different and bit-sum constraints. Experiments demonstrate the approach generates small CNFs for balanced incomplete block designs and Nonogram puzzles.
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![Boolean Equi-Propagation for Optimized SAT EncodingAmitMetodi, Michael Codish, Vitaly Lagoon, and Peter J. Stuckey Order encoding Binary Unary Direct encoding xi ↔ (X ≥ i) (X = 3) = [1,1,1,0,0] Number Xwith a domain {0,1,2, …, d} will be represented using n=𝑂(𝑑)Boolean variables. xi ↔ (X = i) (X = 3) = [0,0,0,1,0,0] b=c e=f=g Problems: ,[object Object]](https://image.slidesharecdn.com/poster-13159250389978-phpapp01-110913094543-phpapp01/85/CP-2011-Poster-1-320.jpg)
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1
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
Triangular Probability Distribution
The triangular probability distribution (also called: “a lack of knowledge distribution”) is a
simplistic continuous model that is mainly used in situations when there is only limited sample
data and information about a population. It is based on the knowledge of a minimum (a lower
value), a maximum (an upper value), and a mode (peak) between those two values. For this
reason, this distribution is very popular in simulation processes related to business decision
models, project management models, financial models, and for modeling noises in digital audio
and video data.
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𝒇(𝒙) = {
𝟐
(𝒃−𝒂)(𝒄−𝒂)
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𝟐
(𝒃−𝒂)(𝒃−𝒄)
(𝒃 − 𝒙) 𝒊𝒇 𝒙 > 𝒄
(1)
The following are some of the important numerical characteristics of the triangular distribution:
2
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
𝒎𝒆𝒂𝒏 = 𝑬(𝒙) = 𝝁 =
𝒂+𝒃+𝒄
𝟑
(2)
𝒎𝒆𝒅𝒊𝒂𝒏 = 𝒎 =
{
𝒂 + √𝟎.𝟓 (𝒃 − 𝒂)(𝒄 − 𝒂) 𝒊𝒇 𝒄 <
𝒂+𝒃
𝟐
𝒄 𝒊𝒇 𝒄 =
𝒂+𝒃
𝟐
𝒃 − √𝟎.𝟓 (𝒃 − 𝒂)(𝒃 − 𝒄) 𝒊𝒇 𝒄 ≥
𝒂+𝒃
𝟐
(3)
𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆 = 𝝈𝟐 =
𝒂𝟐+𝒃𝟐+𝒄𝟐−𝒂𝒃−𝒂𝒄−𝒃𝒄
𝟏𝟖
(4)
The 𝒄𝒅𝒇 (Cumulative density function) 𝑷(𝑿 ≤ 𝒙) of the triangular random variable is:
𝑭(𝒙) =
{
𝟏
(𝒃−𝒂)(𝒄−𝒂)
(𝒙 − 𝒂)𝟐 𝒊𝒇 𝒙 < 𝒄
𝒄−𝒂
𝒃−𝒂
𝒊𝒇 𝒙 = 𝒄
𝟏 −
𝟏
(𝒃−𝒂)(𝒃−𝒄)
(𝒃 − 𝒙)𝟐 𝒊𝒇 𝒙 > 𝒄
(5)
For example, the following is a display of the cumulative density function of a triangular random variable
with a minimum value of 2, a maximum of 8, and with a peak at 4.
3
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
Random Number Generation of Triangular Random Variables:
The CDF expression given by formula (5) can be used to generate random values according to a specific
triangular distribution. In this method, first a standard uniform random value 𝒓 is created. This value is
then used as a cumulative probability and replaces 𝑭(𝒙) in formula (5). The formula is then solved for
the random variable 𝒙. The following rule describes this random number generation:
𝒙 = {
𝒂 + √𝒓 (𝒃 − 𝒂)(𝒄 − 𝒂) 𝒊𝒇 𝒓 ≤
𝒄−𝒂
𝒃−𝒂
𝒃 − √(𝟏 − 𝒓)(𝒃 − 𝒂)(𝒃 − 𝒄) 𝒊𝒇 𝒓 >
𝒄−𝒂
𝒃−𝒂
(6)
Example:
In this example, we will simulate ten million triangular random values in R. We will then compare the
numerical characteristics of this randomly generated set with the expected values.
1. Specify the specification of the triangular distribution:
> a<-2
>
1 PROBABILITY DISTRIBUTIONS R. BEHBOUDI Triangu.docx
1
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
Triangular Probability Distribution
The triangular probability distribution (also called: “a lack of knowledge distribution”) is a
simplistic continuous model that is mainly used in situations when there is only limited sample
data and information about a population. It is based on the knowledge of a minimum (a lower
value), a maximum (an upper value), and a mode (peak) between those two values. For this
reason, this distribution is very popular in simulation processes related to business decision
models, project management models, financial models, and for modeling noises in digital audio
and video data.
The probability density function (𝒑𝒅𝒇) of the triangular random variable 𝑿 is given by:
𝒇(𝒙) = {
𝟐
(𝒃−𝒂)(𝒄−𝒂)
(𝒙 − 𝒂) 𝒊𝒇 𝒙 ≤ 𝒄
𝟐
(𝒃−𝒂)(𝒃−𝒄)
(𝒃 − 𝒙) 𝒊𝒇 𝒙 > 𝒄
(1)
The following are some of the important numerical characteristics of the triangular distribution:
2
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
𝒎𝒆𝒂𝒏 = 𝑬(𝒙) = 𝝁 =
𝒂+𝒃+𝒄
𝟑
(2)
𝒎𝒆𝒅𝒊𝒂𝒏 = 𝒎 =
{
𝒂 + √𝟎.𝟓 (𝒃 − 𝒂)(𝒄 − 𝒂) 𝒊𝒇 𝒄 <
𝒂+𝒃
𝟐
𝒄 𝒊𝒇 𝒄 =
𝒂+𝒃
𝟐
𝒃 − √𝟎.𝟓 (𝒃 − 𝒂)(𝒃 − 𝒄) 𝒊𝒇 𝒄 ≥
𝒂+𝒃
𝟐
(3)
𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆 = 𝝈𝟐 =
𝒂𝟐+𝒃𝟐+𝒄𝟐−𝒂𝒃−𝒂𝒄−𝒃𝒄
𝟏𝟖
(4)
The 𝒄𝒅𝒇 (Cumulative density function) 𝑷(𝑿 ≤ 𝒙) of the triangular random variable is:
𝑭(𝒙) =
{
𝟏
(𝒃−𝒂)(𝒄−𝒂)
(𝒙 − 𝒂)𝟐 𝒊𝒇 𝒙 < 𝒄
𝒄−𝒂
𝒃−𝒂
𝒊𝒇 𝒙 = 𝒄
𝟏 −
𝟏
(𝒃−𝒂)(𝒃−𝒄)
(𝒃 − 𝒙)𝟐 𝒊𝒇 𝒙 > 𝒄
(5)
For example, the following is a display of the cumulative density function of a triangular random variable
with a minimum value of 2, a maximum of 8, and with a peak at 4.
3
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
Random Number Generation of Triangular Random Variables:
The CDF expression given by formula (5) can be used to generate random values according to a specific
triangular distribution. In this method, first a standard uniform random value 𝒓 is created. This value is
then used as a cumulative probability and replaces 𝑭(𝒙) in formula (5). The formula is then solved for
the random variable 𝒙. The following rule describes this random number generation:
𝒙 = {
𝒂 + √𝒓 (𝒃 − 𝒂)(𝒄 − 𝒂) 𝒊𝒇 𝒓 ≤
𝒄−𝒂
𝒃−𝒂
𝒃 − √(𝟏 − 𝒓)(𝒃 − 𝒂)(𝒃 − 𝒄) 𝒊𝒇 𝒓 >
𝒄−𝒂
𝒃−𝒂
(6)
Example:
In this example, we will simulate ten million triangular random values in R. We will then compare the
numerical characteristics of this randomly generated set with the expected values.
1. Specify the specification of the triangular distribution:
> a<-2
>
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CP 2011 Poster
- 3. A complete equi-propagator for a constraint can be implemented using binary decision diagrams (BDDs) and can be evaluate in polynomial time.
- 4. When C(X1,…,Xk) a constraint about (fixed) “k” integers with n bits each, the BDD representing it is of size O(n^k).
- 5. Global constraints (such as allDiff) implemented using Ad-Hoc rules.
- 6. There is a strong connection between Simplify and Encoding because Simplify done on the “encoding bits” and might change the encoding accordingly.
- 8. Ben-Gurion University Solver (http://www.cs.bgu.ac.il/~benr/nonograms/)BEE is faster than the dedicated solvers on the hard puzzles. Selected human puzzles 5,000 random 30x30 puzzles