✎ aa:lab:11 [books]

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====== Lab 10: SAT ======


=== 0. Recap Conjunctive Normal ===


**CNF** is a form of boolean logic formulae that consists of a **conjunction** of one or more **clauses**. A **clause** is a **disjunction** of **literals**.

A good analogy for **conjunction** is **product** and for **disjunction** is **sum**. Thus, a formula in **CNF** can be seen as a product of sums.


Examples of **terminals**:

  * $math[X, Y, Z, a, b, X_{1}, X_{2}, ..., X_{n}]

Examples of **disjunctions**:

  * $math[X\vee Y\vee Z]
  * $math[a\vee b]
  * $math[X_{1}\vee X_{2}\vee X_{3}\vee ... \vee X_{n - 1}\vee X_{n}]

Examples of **conjunctions**:

  * $math[a\wedge b\wedge c]
  * $math[X\wedge  Z]

Examples of **CNF**:

  * $math[a\wedge (b\vee c)\wedge d]
  * $math[(X_{1}\vee X_{2})\wedge (Y_{1}\vee Y_{2})\wedge (Z_{1}\vee Z_{2})]
  * $math[(X\vee Y)\wedge (Y\vee Z)\wedge (Z\vee X)]

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** 1. Conjunctive Normal Form **


Rewrite the following formulae in Conjunctive Normal Form:

** 1.1. ** $math[\neg (a\leftrightarrow b)\wedge (b\vee c)]


** 1.2. ** $math[\neg ((a\vee c)\to (b \wedge \neg (a\vee c)))]

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** 2. Interpretations **

** 2.1. ** Write an interpretation which makes each of the previous formulae:
  * true
  * false

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** 3. Satisfiability and validity **

Write down a formula with 3 variables which is:
  * valid
  * satisfiable
  * invalid

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** 4. Binary Decision Diagrams **

Write binary decision diagrams for the formulae at 1.

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** 5. Encodings **


** 5.1. ** Express the following as ** CNF SAT ** formulae:

    * $math[a\oplus b]  (xor)

    * if c then t else p
    
    * exactly one of $math[{X_{1}, X_{2}, ..., X_{n}}] is true.


** 5.2. ** Encode the n-queens problem as a ** SAT ** formula. In the n-queens problem, we have a n x n board and we need to find a position for each n queens such that no queen is attacking another.


** 5.3. ** What is ** 3SAT **? Construct a transformation that takes a ** SAT ** formula and builds a ** 3SAT ** formula. Motivation: a lot of ** SAT ** solvers use as input ** 3SAT **.