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Comp280 Hw02

Module by: Ian Barland, John Greiner

Summary: 2004.spring Comp280 Hw02, Rice University.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

All (non-practice, non-extra-credit) problems due 2004.Feb.03 (Tue) 17:00.

Note:

The final exercise is deferred until hw03.

Aside:

Rice Students: If you like, you can play WaterWorld on OwlNet, in /home/comp280/bin/waterworld. To run Waterworld on your home computer, download (from owlnet) the directory /home/scheme/plt/205/collects/waterworld/, and (from drscheme) add it as a teachpack.

Reasoning with Inference Rules

For proofs on this homework, Remember that each step must be justified by one of: "premise", a listed inference rule (.ps) (a previous line number and, if ambiguous, substitutions for the inference rule's metavariables), or a WaterWorld domain axiom.

Exercise 1

N.B.:

(Carried over from original hw01 assignment.)

(2 pts.) Fill in the blank reasons in the following proof that ∨∨ commutes, that is, χ ∨ υ ⊢ υ ∨ χχ ∨ υ ⊢ υ ∨ χ.

Table 1
1χ ∨ υχ ∨ υ  premise
2subproof:χ ⊢ υ ∨ χχ ⊢ υ ∨ χ  
2.a  χχ premise for subproof
2.b  υ ∨ χυ ∨ χ ∨∨Intro
3subproof:υ ⊢ υ ∨ χυ ⊢ υ ∨ χ  
3.a  υυ premise for subproof
3.b  υ ∨ χυ ∨ χ ____________
4υ ∨ χυ ∨ χ  ____________

Exercise 2

(5pts.) Show that τ ∧ χτ → υχ → ω ⊢ υ ∧ ωτ ∧ χτ → υχ → ω ⊢ υ ∧ ω. (It may take around 8 steps.)

Exercise 3

(4pts) Using the inference rule RAA, prove ¬χ ⊢ ¬χ ∧ υ¬χ ⊢ ¬χ ∧ υ.

Exercise 4

(6pts.) Show that ¬W-safe ∨ ¬Y-unsafe ⊢ W-unsafe ∨ Y-safe¬W-safe ∨ ¬Y-unsafe ⊢ W-unsafe ∨ Y-safe.

hint:

Use the ∨∨Elim inference rule to get the final result.

Exercise 5

(5pts) For the following WaterWorld proof of X-has-1 ⊢ W-unsafe ∨ Y-unsafeX-has-1 ⊢ W-unsafe ∨ Y-unsafe, provide the missing steps and/or reasons:

Table 2
1X-has-1X-has-1  ____________
2____________  WaterWorld domain axiom
3____________  →→Elim
4subproof:W-safe ∧ Y-unsafe ⊢ W-unsafe ∨ Y-unsafeW-safe ∧ Y-unsafe ⊢ W-unsafe ∨ Y-unsafe  
4.a  W-safe ∧ Y-unsafeW-safe ∧ Y-unsafe premise for subproof
4.b  Y-unsafeY-unsafe____________
4.c  W-unsafe ∨ Y-unsafeW-unsafe ∨ Y-unsafe ____________
5subproof:¬W-safe ∧ Y-unsafe ⊢ W-unsafe ∨ Y-unsafe¬W-safe ∧ Y-unsafe ⊢ W-unsafe ∨ Y-unsafe  
5.a  ¬W-safe ∧ Y-unsafe¬W-safe ∧ Y-unsafe premise for subproof
5.b  W-unsafe ∧ Y-safeW-unsafe ∧ Y-safe CaseElim (left), where φφ=____________, and ψψ=____________.
5.c  ____________ ____________
5.d  W-unsafe ∨ Y-unsafeW-unsafe ∨ Y-unsafe ____________
6W-safe ∧ Y-unsafe ∨ ¬W-safe ∧ Y-unsafeW-safe ∧ Y-unsafe ∨ ¬W-safe ∧ Y-unsafe From class, φ ∨ ¬φφ ∨ ¬φ, with φ____________φ____________
7W-unsafe ∨ Y-unsafeW-unsafe ∨ Y-unsafe  ____________

Exercise 6

[practice; 0pts]

Figure 1: A sample WaterWorld board
Figure 1 (hwA-1.png)

Given the above figure, and using any of the immediately obvious facts as premises, prove that location P is safe by using our proof system and the WaterWorld domain axioms.

While this proof is longer (nearly 30 steps), it's not too bad. To make life easier, you may also use the following:

  • ¬φ ⊢ ¬φ ∧ ψ¬φ ⊢ ¬φ ∧ ψ
  • Q-has-1 → P-safe ∧ R-safe ∨ P-safe ∧ W-safe ∨ R-safe ∧ W-safeQ-has-1 → P-safe ∧ R-safe ∨ P-safe ∧ W-safe ∨ R-safe ∧ W-safe.
  • Not only is Y-safeY-safe an allowable premise (reading off the given board), but so is ¬Y-unsafe¬Y-unsafe.

Solution

Table 3
1Q-has-1Q-has-1  premise
2X-has-1X-has-1  premise
3¬Y-unsafe¬Y-unsafe  premise
4X-has-1 → Y-unsafe ∨ W-unsafeX-has-1 → Y-unsafe ∨ W-unsafe th'm: previous problem (and, →→Intro)
5W-unsafe ∨ Y-unsafeW-unsafe ∨ Y-unsafe  →→Elim, by lines 2,4
6Y-unsafe ∨ W-unsafeY-unsafe ∨ W-unsafe  ∨∨ commutes; line 5
7W-unsafeW-unsafe  CaseElim, by lines 3,6
8subproof:¬¬P-safe ∧ W-safe ⊢ false¬¬P-safe ∧ W-safe ⊢   
8.a  ¬¬P-safe ∧ W-safe¬¬P-safe ∧ W-safe premise for subproof
8.b  P-safe ∧ W-safeP-safe ∧ W-safe ¬¬Elim, by line 8.1
8.c  W-safeW-safe ∧∧Elim
8.d  W-safe → ¬W-unsafeW-safe → ¬W-unsafe WaterWorld domain axiom
8.e  ¬W-unsafe¬W-unsafe →→Elim, by lines 8.2,8.3
8.f  false falseIntro, by lines 7,8.4
9¬P-safe ∧ W-safe¬P-safe ∧ W-safe  RAA, by line 8
10subproof:¬¬R-safe ∧ W-safe ⊢ false¬¬R-safe ∧ W-safe ⊢   
10.a  ¬¬R-safe ∧ W-safe¬¬R-safe ∧ W-safe premise for subproof
10.b  R-safe ∧ W-safeR-safe ∧ W-safe ¬¬Elim, by line 10.1
10.c  W-safeW-safe ∧∧Elim
10.d  W-safe → ¬W-unsafeW-safe → ¬W-unsafe WaterWorld domain axiom
10.e  ¬W-unsafe¬W-unsafe →→Elim, by lines 10.2,10.3
10.f  false falseIntro, by lines 7,10.4
11¬R-safe ∧ W-safe¬R-safe ∧ W-safe  RAA, by line 10
12Q-has-1 → P-safe ∧ R-safe ∨ P-safe ∧ W-safe ∨ R-safe ∧ W-safeQ-has-1 → P-safe ∧ R-safe ∨ P-safe ∧ W-safe ∨ R-safe ∧ W-safe  Allowed by problem statement
13P-safe ∧ R-safe ∨ P-safe ∧ W-safe ∨ R-safe ∧ W-safeP-safe ∧ R-safe ∨ P-safe ∧ W-safe ∨ R-safe ∧ W-safe  →→Elim, by lines 1,12
14R-safe ∧ W-safe ∨ P-safe ∧ R-safe ∨ P-safe ∧ W-safeR-safe ∧ W-safe ∨ P-safe ∧ R-safe ∨ P-safe ∧ W-safe  ∨∨ commutes (ex.1), by line 13
15P-safe ∧ R-safe ∨ P-safe ∧ W-safeP-safe ∧ R-safe ∨ P-safe ∧ W-safe  CaseElim, by lines 9,14
16P-safe ∧ W-safe ∨ P-safe ∧ R-safeP-safe ∧ W-safe ∨ P-safe ∧ R-safe  Theorem: ∨∨ commutes (ex.1), line 15.
17P-safe ∧ R-safeP-safe ∧ R-safe  CaseElim, by lines 11,16.
18P-safeP-safe ∧∧Elim -- Whew!

The subproofs could easily have been pulled out into lemmas (making the proof clearer, exactly as subroutines make programs clearer, even if only called once). Observe how the second subproof had to repeat lines identical from the first -- it would have been incorrect to cite those earlier subproof lines, because the previous subproof had been completed.

note:
Interestingly, we didn't need to use R-safeR-safe as a premise. (In fact, we nearly proved that ¬R-safe¬R-safe would have been inconsistent with the other premises.)

Exercise 7

(6 pts)

  • State where on a board pirates could be positioned, so that: P-has-1 ∧ U-has-1 ∧ W-has-1P-has-1 ∧ U-has-1 ∧ W-has-1, but X-safeX-safe.
  • Compare this with a previous theorem (example 11) B-has-1 ∧ G-has-1 ∧ J-has-1 → K-unsafeB-has-1 ∧ G-has-1 ∧ J-has-1 → K-unsafe, the same idea shifted down a couple of rows. Suppose we try to translate this proof so as to conclude ¬X-safe¬X-safe (clearly untrue, by the above). What is the first step of the proof which doesn't hold when B,G,J,K are replaced with P,U,W,X, respectively? (Just give a line number; no explanation needed. Your answer will be of the form "Lemma A line 1" or "main proof line 2".)
  • If you replace such steps with the closest true formulas you can ("translating" instead of "transliterating", if you will), 1 where is the first line of the proof which fails and cannot be patched? (Just give a line number; no explanation needed. Your answer will be of the form "Lemma A line 1" or "main proof line 222".)

Exercise 8

(Extra Credit 3 pts) Describe a board and an accompanying (simple) WFF where you know the formula is true, yet it's not possible to prove it via the WaterWorld domain axioms without mentioning propositions about the total-pirate-count totCount-is-1totCount-is-1, totCount-is-2totCount-is-2, …. (Indeed, you may have noticed that list of domain axioms makes only a paltry attempt at including domain axioms involving totCount-is-5totCount-is-5. There are quite a few omitted subcases as it is, even without considering other total-pirate-counts.) As always, a simpler example is better.

Suppose we don't patch up the WaterWorld domain axioms with numerous axioms for each of the total-pirate-count cases. What does your example above say about the soundness and the completeness of our WaterWorld domain axioms? (Recall that the pertinent problem when playing waterworld is not that you sometimes have to guess, but rather knowing whether a guess is the only option.)

Relations

Exercise 9

(5 pts) Are each of the following formulas true for all interpretations ("valid")? (Remember that the relation-names are just names in the formula; don't assume the name has to have any bearing on their interpretation.)

  • For arbitrary aa,bb in the domain, atLeastAsWiseAsab ∨ atLeastAsWiseAsbaatLeastAsWiseAsab ∨ atLeastAsWiseAsba
  • For arbitrary aa in the domain, primea → odda → primeaprimea → odda → primea
  • For arbitrary aa,bb in the domain, betterThanab → ¬betterThanbabetterThanab → ¬betterThanba
For each, if it is true or false under all interpretations, prove that. (For these small examples, a truth table will probably be easier than using boolean algebra or an inference-system proof.) Otherwise, give and interpretation in which it is true, and one in which it is false.

Hint:

As always, look at trivial and small test cases first. Here, try domains with zero, one, or two elements, and small relations.

Quantifiers

Exercise 10

(2 pts.) Rosen, Section 1.3, pg.40, #6.

Exercise 11

(3 pts.) Rosen, Section 1.3, pg.40, #10.

Exercise 12

(3 pts.) Rosen, Section 1.3, pg.41, #16.

Exercise 13

Deferred:

This exercise will be moved to hw03 -- Thurs's lecture didn't get far enough!

(4 pts.) Rosen, Section 1.3, pg.44, #58.

Footnotes

  1. "transliterate" meaning a word-for-word substitution, while "translation" tries to preserve meanings/idioms. So, the German "Übung macht den Meister" transliterates to "Drill makes the master", but translates to "practice makes perfect".

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