Assignment 1

COMS E6998 Formal Verification of System Software
Fall 2018
Ronghui Gu

Submission

• Due 10/02/2018 at 6:10 pm
• Submit the hardcopy homework (printed or hand-written documents) in class
• Academic Honesty
  • The computer science department has very strict policies. Please read the department Academic Honesty page.
  • You MUST submit original homework written by you and you alone.
  • Do not look at anybody else’s answer. Do not show anybody your answer, or leave your answer where somebody else could see it.
  • Cases of non-original submission will receive a 0 and will be referred to the Judicial Committee.

Problem 1

Show “$(p → (q ∨ r)) , (q → s) , p \models ¬ r → s$” using semantic reasoning (you can use indirect proof).

Problem 2

Show “$(p → (q ∨ r)) , (q → s) , p \models ¬ r → s$” using syntactic reasoning.

Problem 3

Convert $((p \rightarrow q) ∨ (r ∧ s)) \rightarrow (¬q ∧ (p \rightarrow t))$ into CNF.

Problem 4

Check the satisfiability of

using DPLL. You should explain at each step which literal you split on, e.g., $s$-literal or $\neg s$-literal, and what the resulting clauses are.

Problem 5

Check the satisfiability of $(p∧q∧s → ⊥)∧(q∧s → p)∧(q → s)∧(⊤ → q)$.

Problem 6

Assume $\mathcal{F}:=\{O, S, +\}$ and $\mathcal{P}:=\{=, <,>,\leq, \geq,...\}$, prove

using (ref) and (=$e$) rules alone.