Computational Logic

Instructions:

1. For your solution use the template file that was posted on the course news, and follow the instructions in it.

In particular: (a) Include at the top of the first page: full name, student number, and email address. (b) Assignments have to be created with Latex, and submitted in pdf format. (c) Every problem solution MUST include the problem statement. The source file for this assignment is provided.

Latex has to be used as such, not as you would use a text editor, such as Notepad. In particular, formulas have to be written using Latex’s mathe- matical features, and then compiled.

2. Assignments are individual, no groups.

3. Submit by email to the instructor, with “Assignment ”Number”, CompLog” in the subject. Include your last name in the file name! For example, in the subject: “Assig. 1 CompLog”. The file name: “[url removed, login to view]”.

Only a single pdf file will be accepted as submission. No tar or zip files (or anything like that), please. Keep your Latex source files in case you are requested to show them.

4. Explain your solution very carefully, but still be succinct with your answers. No unnecessary verbose arguments, please. Go to the point.

Make explicit all your assumptions.

5. Not following the instructions above or the solution template file

will make you lose points.

1.

includes only Chile, Peru, Argentina and Bolivia (cf. Fig. 1) cannot be painted with 2 colors if adjacent countries must have different colors. To simplify, assume that the colors are blue and green.

More precisely:

(a) Write in propositional logic, outside Prover9, the knowledge basis.

(b) Describethemethodologyyouwilluse(allthisbeforegoingintoProver9).

(c) Do the proof with Prover9, including the text file for/by Prover9 as an appendix (better use the verbatim environment in Latex). It has to contain all the formulas, the run, and the final clean proof.

Prove using PROPOSITIONAL logic and Prover9 that the map that

Figure 1: Map of South America

(d) Discuss to what extent your methodology is general and declarative (as opposed to a hack for this very particular problem). In particular, would your program be essentially different it the question were about coloring with 3 colors?

2. There are two sealed boxes, of gold and silver. It is known that only one of them contains the diamond. Each of the boxes has a label on top, saying:

• Label on Gold Box: “The diamond is not here”.

• Label on Silver Box: “Exactly one of the labels is true”.

It is also known that at most one of the labels is true. Determine using PROPOSITIONAL logic and Prover9 where the diamond is. More specif- ically,

(a) Write a propositional knowledge basis describing the above situation. Explain.

(b) Describe how to use Prover9 to deduce in which box is the diamond. Include the file and run as in problem 1.

(c) Explain what Prover9 did.

2

3. An equivalence relation (ER) on a set can be seen as a structure ⟨A, R⟩, where

A is a non-empty set, and R ⊆ A × A is a binary relation with the following properties:

1. For every a ∈ A : (a, a) ∈ R (reflexivity)

2. For every a,b ∈ A: (a,b) ∈ R ⇒ (b,a) ∈ R (symmetry)

3. Forevery a,b,c∈A: (a,b)∈R and (b,c)∈R ⇒ (a,c)∈R (tran- sitivity)

(a) Show a concrete finite ER on a finite set, including the proof that it is an ER.

(b) Prove, as usual, from 1. - 3. the following theorem of the theory of ERs: (no Prover9 here)

“Forevery a,b,c∈A: (a,c)∈R and (b,c)∈R ⇒ (a,b)∈R”

(c) Show by means of a finite example (structure) that: 1. & 2. ̸⇒ 3. That is, a counterexample that proves the independence of 3. from 1. and 2. (d) Show with a finite example (structure) that: 1. & 2. ̸⇒ ¬3.

That is, a counterexample that proves the consistency of 3. with 1. and 2. (¬[url removed, login to view]:“Therearea,b,c∈Awith: (a,b)∈R,(b,c)∈R, and (a,c)∈/R”)

C Programming Computer Graphics Computer Support R Programming Language

Project ID: #9579787

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