by Implication is one of the advanced techniques. It works with Booleans and requires the establishment of hypotheses. It produces a simplification of the number of Booleans. It is an extremely powerful instruction and is widely used in puzzles marked “often difficult.”
The implication is not easy to understand. To help, we can translate it with the formula “if… then.”
If it rains, then there are clouds.
This sentence is perfect for understanding – keeping it in mind helps avoid many mistakes.
Actilud deals with the following case:
If a ⇒ b and b ⇒ a, then a⇔b and we can replace one with the other.
To perform an implication, we choose a boolean a and observe what happens when we set it to true. If we find a boolean b that becomes true as a result of a, we reset a to its initial state and observe what happens when we set b to true . If a becomes true in turn, we have demonstrated the equivalence and we can replace one with the other. On the site we can memorize a configuration, this allows us to make hypotheses and go back. It is advisable to start with the most frequent booleans.
But be careful!
A common mistake is to believe that if a ⇒ b and ¬a ⇒ ¬b, then a ⇔ b. This is false! As false as saying “if it is not raining, then there are no clouds.” Applied to Booleans, this translates as follows: if a true => b true and if a false => b false, we are not allowed to conclude that a ⇔ b and we should not replace one with the other, even if we really want to, because the behaviors of the two Booleans seem to agree.
Note: the symbol ¬ is the negation: b ⇒ ¬a means b implies ” not a” ; if b is true, then “¬a “ is true, therefore a is false.
The truth table of implication is as follows; it is not intuitive. The implication a ⇒ b is false in only one case, when a is true and b is false. This means that a true premise must not imply a false conclusion; but, if the premise is false, it does not matter what the conclusion is: the implication is always true.
| a | b | a ⇒ b |
|---|---|---|
| TRUE | TRUE | TRUE |
| TRUE | FALSE | FALSE |
| FALSE | TRUE | TRUE |
| FALSE | FALSE | TRUE |
Let’s analyze the common error. When we set a to true, we select the first two rows of our table. Then, we prove that a ⇒ b is true (by setting a to true, we see that b becomes true in turn); we see in the table that we must retain the first row: a true, a ⇒ b true, therefore b is true.
Let’s see what happens with ¬a ⇒ ¬b. When we set a to false, we select the last two rows of our table. Then, when we prove that ¬a ⇒ ¬b (by setting a to false we see that b becomes false in turn), the implication is true, so we are always on the last two rows. And we see in this case, that the implication is true regardless of the state of b, true or false. So we have not proven anything.
Inconsistencies
This is a common case that is often appreciated by players, because it allows them to move forward very quickly in the resolution! Sometimes an implication produces an inconsistency. There are several types:
- rows with only false boxes
- rows with multiple true signs
- Boolean that cannot take any value: neither true nor false.
- alternative that cannot be resolved
In this case, it is sufficient to put the inverse value in the boolean: if hypothesis a is true results in an inconsistency, setting a to false ; conversely, if hypothesis a is false results in an inconsistency, setting a to true.
Inconsistency is to integrams what reasoning by contradiction is to mathematics. If a logical argument leads to an absurdity, the premises have been shown to be false.
Boolean true… but false
This is a special case of inconsistency, which is handled by Actilud:
If a ⇒ b and b ⇒ ¬a then a = false
To work the corollary, after having demonstrated that a => b , we must therefore demonstrate that b => ¬a . By setting the boolean b to true, we discover that the boolean a ends up becoming false. If this is the case, we can replace all the booleans a in the grids with false signs .
Invariance
If, whatever the value of a boolean a , a boolean b is always true or always false , then we can assign this value to the boolean b .
Since the February 2025 update, we have incorporated into implication this case of Boolean-specific invariance, when a Boolean acts on another Boolean. The case where a Boolean modification produces constant empty boxes is still handled by invariance.
How is implication carried out on Actilud?
To keep the puzzles from being too difficult, the solver limits his deductions.
To perform the implication, the solver goes through all the Booleans, starting with the most frequent. It performs the hypotheses by searching by level (see below).
Only puzzles marked as often difficult are likely to use implication or invariance. In basic puzzles (jogging, fashion show, land yachting, orchard) the solver adds the square configuration; therefore, there is not systematically implication or invariance. Moreover, the difficulty may not appear because it depends on the order in which the propositions are treated.
Since the February 2025 update, the solver always favors solutions that require the least depth of analysis. They are thus -sometimes- more understandable and make the puzzles more motivating. In addition, the solver takes inconsistencies into account. To help you a little, the difficulty level of the implication and the invariance is indicated by the Advisor, when you use the solver in “step by step” mode. The level corresponds to the number of successful operations that had to be done to obtain a result. Level 1 is therefore the easiest: the implication only required a single operation. Level 7 is the most difficult.
The operations performed are: intersections, complement, repercussion, fusion; the complement being executed multiple times. Intersection also includes consistency.