Download Seminar on Decision Procedures Problem 1 The Simplex Method

Seminar on Decision Procedures
By Ruzica Piskac
Seminar on Decision Procedures
Ruzica Piskac
Problem 1
The Simplex Method
Apply the simplex method to decide whether the following formulas are satisfiable in
Q. You can use either the simplex method as described in the lecture textbook, or
you can use the table-based approach introduced in the lecture.
• x ≥ 1 ∧ 2x ≤ 1
• x + 2y ≥ 1 ∧ 2x + y ≥ 1 ∧ 2x + 2y ≤ 1
• x + 2y ≥ 1 ∧ 2x + y ≥ 1 ∧ x + y ≤ 1
Problem 2
Omega Test
Apply the Omega test to check the satisfiability of the following formulas:
• x + 3y + 5z = 9 ∧ 2x + 3y = 6 ∧ x + z ≤ 7
• 3x + 5y ≥ 15 ∧ x − 3y ≤ 0 ∧ 5y + x ≤ 11
Problem 3
Difference Logic
Consider the following formula x < y+5∧y ≤ 4∧x = z −1. First, convert the formula
in the standard form of difference logic formulas. Furthermore, check its satisfiability
by (both):
• using the conflict graph
• using the eager encoding to SAT. In the step 2, you should construct at least
one additional lemma.
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Seminar on Decision Procedures
Problem 4
By Ruzica Piskac
Boolean Algebra with Presburger Arithmetic
Apply the quantifier elimination technique to the following formulas. The resulting
formulas should be as simple as possible, containing only unquantified variables from
the original formula.
• ∀A.B ⊆ A
• ∃A.∃B.A ∪ B ⊆ C ∧ |A| ≥ |C| ∧ |B| ≥ 2
Using the quantifier elimination technique, check whether the following formulas are
satisfiable:
• ∀A.∃B.A ⊆ B ∧ |A| ≤ |B|
• ∃A.∀B.A ⊆ B ∧ |A| ≤ |B|
Problem 5
Quantifier-free Rational Arithmetic
1. Turn the following formulas into equisatisfiable formulas of the form ΦA ∧ Φ0 ,
where ΦA is a conjunction of linear equalities and Φ0 consists of elementary
atoms y ./ b, for ./ ∈ {<, ≤, =, 6=, ≥, >}.
• x ≤ 0 ∧ x + y > 2 ∨ (x + 2y − z = 0 ∨ x + y = 2)
• x < 2 ∨ x 6= 5
• x+y+z =0∨x+y+z >1
2. Use the algorithm presented in the lecture to find a satisfying assignment of the
following formula:
x + y = 1 ∧ x − y < −1 ∧ 2x + y < 1 .
Include all intermediary results (the tableau, the current assignment and bounds).
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