The New Sudoku Players' Forum

[eleven]

The strong links are there, if the puzzle has a solution or not.

Did you read a word I just said? Your statement is true if you're using mith's practical definition only. I was clearly speaking about the consequences of totuan's definition, which I consider the official one. Stick to that if you want to criticize what I wrote.

Yes, i read that, and from my answer you can see, that i do not share this definition, which i read for the first time in this thread, and is all but an official definition.
It may work fine for totuan, who obviously does not care about non solution puzzles. But for those it turns out to be irrelevant, while the really official definition is the one, our honored bot gave: at least one side has to be true (either obviously or by explanation).

Yeah, because your (like everyone else's) process looks like this:

1. Assume the puzzle has at least one solution.
2. Apply apparent strong links correctly, by mith's definition.
3. Arrive at a contradiction.
4. Conclude that that puzzle has no solution.

Point 1 is not needed (i just did it knowing that the puzzle has no solution), and it is not mith's definition, but the one, which is used by every AIC user for a decade.

What you're missing is a fifth step based on that conclusion:

5. Conclude that the apparent strong links weren't real, by totuan's definition.

The step is not needed, because i do not use totuan's definition. For me the strong links are real.

Personally I think we need both definitions: one globally (totuan's) and the other locally (mith's).

I disagree. The old definition is good enough for all purposes.
totuan's definition might be good enough for him in unique puzzles (and maybe multi solution puzzles).
But it may bring the honored bot to strange ideas:
Look at the solution and pick a number (presumable a backdoor). Then take any digit from the candidate grid (presumable one, which easily can be eliminated), and state, that there is a strong link between them. According to that definition it would be valid.

[Ajò Dimonios]

Hi Mith
Do you know the Eureka symbology?
In this symbology = indicates a strong inference and - indicates a weak inference
This is enough to read ¬A = A = B = ¬B => ¬A = ¬B.

[mith]

Question 6: Do you understand that "=" is not the boolean operator for equivalence in this context?

I think the answer to your question 6 is also the operator “NOT” is a Boolean operator while = is a mathematical operator it means equal to.

(emphasis mine)

"=" in this statement represents a strong link

Yes, I understand quite clearly. Just answer my questions, please.

[mith]

(Aside: It's worth pointing out the clever thing our troll bot did here in switching the negations around from the previous "theorem" he was using. Now you can't call into question that the conclusion is correct - ¬A = ¬B is really a valid strong link in the puzzle, for the propositions we are discussing - while the invalid A = B is hidden in the middle of the chain.

But questioning whether I know the symbology after a lengthy discussion of me using the symbology correctly - along with the symbology of propositional calculus and boolean operators in programming, and along with his occasional use of == for strong link - is next level trolling. If there were a chef's kiss emoji, I would be using it right now.)

[Ajò Dimonios]

Hi mith

Answer to question number 11. My answer is the following the definition of strong inference is redundant as I wrote previously.

I am fully convinced by the proof (A = B = C = D) → (A = D) that (A = D) is a strong inference. For this reason the strong inference definition: Definition 1: Let A, B be two assumptions. We say that A, B are thicklinked and note A == B when at least one of A, B is true. We say that A, B are thinlinked and note A - B when at most one of A, B is true should be generalized with: Definition 1: Let A, B be two assumptions. We say that A, B are thicklinked and note A == B when A is assumed false B turns out to be true and when B is assumed false A turns out to be True.
This uses the method of constructing a strong inference as a definition. This is certainly confirmed by experience, no one after writing an AIC that leads to certain eliminations checks after finding the solution that all the strong inferences of all the AICs written by the player are confirmed by the current definition of strong inference.

nswer to question 1: Yes, because ¬B and ¬A are both true.
Answer to question 2:No because as I wrote previously A = ¬A = ¬B = B => A = B. The strong inference arises from the fact that ¬B and ¬A are strongly linked because both are true.
Answer 3 :No as per previous demonstration A = ¬A = ¬B = B => A = B. The definition of strong inference is redundant. From its definition, as a premise, it is possible in certain cases to deduce the negation of the premise itself. This fact is clearly redundant but the logic is correct. In my opinion it means that the definition is not complete.
Answer to question 4: Yes

Paolo

[mith]

I'll simplify them in case it's a language barrier issue:

Question 7: A ≠ ¬A for any statement A. True or false?
Question 8: A = ¬A (strong link), A || ¬A (boolean operator), and A ∨ ¬A (logical disjunction) all mean the same thing. True or false?
Question 9: A = ¬A (strong link) and A == ¬A (boolean equivalence) do not mean the same thing. True or false?
Question 10: (A || B) && (B || C) && (C || D) => (A || D) for any statements A, B, C, D. True or false?
Question 11: A = B for any statements A and B. True or false?

[Ajò Dimonios]

NOT - simboli alternativi: x, ~, ¬, ! (in C, C++, C# e JavaScript)
AND - simboli alternativi: *, {\displaystyle \wedge }\wedge, &, && (in C, C++, C# e JavaScript), BUT (se usato insieme al NOT)
OR - simboli alternativi: +, {\displaystyle \vee }\vee, |, || (in C, C++, C# e JavaScript)
XOR - simboli alternativi: ⊕, {\displaystyle {\dot {\lor }}}{\dot {\lor }}, ∨, ^, EOR, orr
NAND - simbolo alternativo: ↑
NOR - simbolo alternativo: ↓
XNOR - simbolo alternativo: ≡, EQU

vero - simboli alternativi: true, 1, ON, SÌ (YES), alto
falso - simboli alternativi: false, 0, OFF, NO, basso

[SpAce]

A side note...

his occasional use of == for strong link

It's apparently taken from here, same as the terms "thicklink" and "thinlink".

On this forum we use the double == (and much more rarely --) to indicate derived links. For example, I could have written my recent UR solution like this:

(5)r1c1 == (9)r8c1 - (9=1)r8c2 - (1=5)r8c6 => -5 r1c6; stte

Or:

(5==9)r18c13 - (9=1)r8c2 - (1=5)r8c6 => -5 r1c6; stte

This would be bad form, because there's no native strong link between the first two nodes:

(5)r1c1 = (9)r8c1 - (9=1)r8c2 - (1=5)r8c6 => -5 r1c6; stte

If I see that as a reader, my first instinct is to suspect there's a typo or other mistake in the first link. The doubling tells me to look for another reason. That reason should also be explained, either externally or inline between the double symbols:

(5)r1c1 =UR= (9)r8c1 - (9=1)r8c2 - (1=5)r8c6 => -5 r1c6; stte

(5=UR=9)r18c13 - (9=1)r8c2 - (1=5)r8c6 => -5 r1c6; stte

[eleven]

Just a word, why i did not criticize totuan's definition:
It was clear to me, that he would not construct a link from the solution, but from an explanation derived from the candidates grid, which guarantees, that one side would be true in the solution (if one exists). In this sense it was correct, but sloppy formulated (due to language barriers).

[mith]

Question 7: A ≠ ¬A for any statement A. True or false?
Question 8: A = ¬A (strong link), A || ¬A (boolean operator), and A ∨ ¬A (logical disjunction) all mean the same thing. True or false?
Question 9: A = ¬A (strong link) and A == ¬A (boolean equivalence) do not mean the same thing. True or false?
Question 10: (A || B) && (B || C) && (C || D) => (A || D) for any statements A, B, C, D. True or false?
Question 11: A = B for any statements A and B. True or false?

[Ajò Dimonios]

question1 Ortho-cresol = 1 methil phenol true or false?
question2 p-cresol = 3 methil phenol true or false?
question3 m-cresol= 4 methil phenol true or false?
question 4 Catechol = 1,2-dihydroxybenzene or 2 hydroxy phenol true or false?

[mith]

Question 7: A ≠ ¬A for any statement A. True or false?
Question 8: A = ¬A (strong link), A || ¬A (boolean operator), and A ∨ ¬A (logical disjunction) all mean the same thing. True or false?
Question 9: A = ¬A (strong link) and A == ¬A (boolean equivalence) do not mean the same thing. True or false?
Question 10: (A || B) && (B || C) && (C || D) => (A || D) for any statements A, B, C, D. True or false?
Question 11: A = B for any statements A and B. True or false?

[mith]

question1 Ortho-cresol = 1 methil phenol true or false?
question2 p-cresol = 3 methil phenol true or false?
question3 m-cresol= 4 methil phenol true or false?
question 4 Catechol = 1,2-dihydroxybenzene or 2 hydroxy phenol true or false?

Ok, you got me, I can't resist.

For "=" as equality in this context, since there is no "strong link" that is relevant here.
1. False. o-cresol is 2-methylphenol.
2. False. p-cresol is 4-methylphenol.
3. False. m-cresol is 3-methylphenol
4. True. catechol is 1,2-dihydroxybenzene, 2-hydroxyphenol, 1,2-benzenediol...

I have a Chemistry degree, too. (But more importantly, I can read wikipedia.)

Your turn.

[SpAce]

Just a word, why i did not criticize totuan's definition:
It was clear to me, that he would not construct a link from the solution, but from an explanation derived from the candidates grid, which guarantees, that one side would be true in the solution (if one exists).

Are you perhaps suggesting the same was not obvious to everyone (except maybe the troll, but who cares)?

In this sense it was correct, but sloppy formulated (due to language barriers).

I don't think it was sloppily formulated at all. In fact, it's the only definition of a strong link I've seen that doesn't have any ambiguity. That's why I use it myself. In fact, I'm pretty sure it's exactly what David's strong link/inference definition implies, even if it has not been explicitly stated.

it is not mith's definition, but the one, which is used by every AIC user for a decade

Apparently neither totuan nor I are AIC users. (Edit. Well, that's an invalid conclusion, of course, because there's no weak link between the two definitions. Like I said, I use both.)

the really official definition is the one, our honored bot gave: at least one side has to be true (either obviously or by explanation).

Define 'true'. (Without that definition, the above definition doesn't mean anything. Talking about sloppily formulated.)

[mith]

(Paolo, It’s totally fine if you don’t know the answers. You can just say that.)