Homework 3 Solutions

# Homework 3 Solutions - 1 i S t = cfw s 1 s 3 a 1s0,s2,s4 T...

• Homework Help
• CountAtomAardvark3912
• 4
• 100% (1) 1 out of 1 people found this document helpful

This preview shows pages 1–3. Sign up to view the full content.

1) i. Sat( ϕ _1) = {s_0, s_1, s_2, s_3, s_4} TS ⊨ϕ ii. ϕ _2) = {s_4} ¬TS ⊨ϕ iii. ϕ ⊨ϕ 2) ϕ _1 = ∃◇∀ □c Subformulas: □c, c Sat(c) = {s_2, s_3, s_4} □c) = {s_2, s_3, s_4} ∃◇∀ □c) = {s_0, s_1, s_2, s_3, s_4} Therefore TS ⊨ϕ _1 ϕ _2 = (aU ∀◇ c) ∀◇ c, c ∀◇ c) = {s_0, s_1, s_2, s_3, s_4} ∀◇ c)) = {s_0, s_1, s_2, s_3, s_4} Or alternatively normalizing the formula for the algorithm: (aU¬ □¬c) Subformulas: c, ¬c, □¬c, ¬ □¬c Sat(¬c) = {s_0, s_1} □¬c) = {} Sat(¬ □¬c) = {s_0, s_1, s_2, s_3, s_4} And we get the same result. ⊨ϕ

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
3) (TS ⊨ ∃ ( ϕ Uψ)) = (TS' ⊨ ∃◇ ψ). Theorem Since TS has more or equal transitions than TS' and everything else the same, for any formula ϕ we have that (TS' ⊨ ϕ ) (TS ⊨ ϕ ) ­­­­­­­­­­­­­­­­ assump_0 I formalize the removal of outgoing transitions π:TS'∙ i∙ j∙j<i (π_j ¬(¬ ϕ∨ ψ) π_i=π_j) ­­­­ assump_1 Proving (TS ⊨ ∃ ( ϕ Uψ)) ⊨ ∃◇ ψ): TS ⊨ ∃◇ ψ def of = TS ⊨ ∃ (TUψ) base law
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

Homework 3 Solutions - 1 i S t = cfw s 1 s 3 a 1s0,s2,s4 T...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online