Modus Ponens generalized

This page describes the design of a tactic replacing a predicate $P$ in either an hypothesis or a goal by $\btrue$ (respectively $\bfalse$ ) if $P$ (respectively $\lnot P$) is an hypothesis. That tactic is slightly different from the one requested : Feature Request #3137918

Objective

A sequent can be simplified by finding every occurrence of an hypothesis in the others hypothesis or goal, and replacing it by $\btrue$ or $\bfalse$ as explained below:

$\frac{P,\varphi(\btrue) \vdash G}{P,\varphi(P) \vdash G} \qquad \frac{P,\varphi(\bfalse) \vdash G}{\lnot P,\varphi(P) \vdash G} \qquad \frac{P \vdash \varphi(\btrue)}{P \vdash \varphi(P)} \qquad \frac{P \vdash \varphi(\bfalse)}{\lnot P \vdash \varphi(P)}$

Contrary to the Feature Request, the rule is applied on every occurrence of $P$ and not only on the one appearing in a disjunctive form. Also, not to make explode the analyzing-time of a sequent, only goal and visible (or selected, it is still in discussion) hypothesis can be re-written, using every hypothesis (global and local). The analyzing-time performance of a sequent can be problematic since every predicate of goal and visible (or selected) hypothesis are checked. But it is still worthy because it may simplify a lot, and so make proof obligation more legible to humans.

Legacy

There already exists a reasoner that does a part of that tactic : autoImpF (ImpE automatic). It looks for implicative hypothesis in a sequent. Among the predicates computed by breaking possible conjunctions in the left side of the implication, it checks if some appear in selected hypothesis. In that case, every predicates which are hypothesis (global or local) are removed from the left-sided-predicate of the implication. If it remains no predicates, then the right side of the implication is re-written as a list of conjunction, else implication is re-written without the removed predicates (possibly none).

Then this reasoner can be considered as "intelligent" because it finds itself the positions where apply, which infer a code hard to read. On the contrary, as reasoners are in the trust-base, they should be easily understandable.

Design Decision

The main drawback of this tactic is its execution time. Indeed, each (actually it is not the case but it helps to see how greedy this tactics can be) sub-predicate of each goal and visible (or selected) hypothesis is compared to the others hypothesis (global and local). Several strategies can be applied in order to reduce that execution-time :

• When a predicate match an hypothesis, its children must not be analyzed.

$\lnot P_1$

$P_1 \limp P_2$

$(P_1 \limp P_2) \limp P_3$

In that example, the sub-predicate $(P_1 \limp P_2)$ of the third predicate match to the second one. It has to be replaced by $\btrue$, and all its children have to be not analyzed (even if $P_1$ could have been replaced by $\bfalse$ because of the first predicate).

• Each hypothesis must be scanned only once to avoid accumulating predicate's running time. From that point of view, two decisions can be inferred :
  • A given predicate is compared once to every hypothesis. For instance, we do not try to find every occurrence of an hypothesis among the goal and the visible (or selected) hypothesis. In that case, each predicate would be scanned many times, accumulating execution time.
  • That tactic apply the rules specified here-above everywhere it is possible. Thus, from this set of hypothesis :
    • $\lnot P_1$
    • $P_1 \limp P_2$
    • $(P_1 \limp P_2) \limp P_3$

1. We will reach immediatly :
   • $\lnot P_1$
   • $\bfalse \limp P_2$
   • $\btrue \limp P_3$
2. Which will be re-written to :
   • $\lnot P_1$
   • $\btrue$
   • $P_3$

1. Instead of writing :
   • $\lnot P_1$
   • $\bfalse \limp P_2$
   • $(P_1 \limp P_2) \limp P_3$
2. Re-written :
   • $\lnot P_1$
   • $\btrue$
   • $(P_1 \limp P_2) \limp P_3$
3. And then :
   • $\lnot P_1$
   • $\btrue$
   • $(\bfalse \limp P_2) \limp P_3$
4. Re-written :
   • $\lnot P_1$
   • $\btrue$
   • $P_3$

• In the case of n-ary predicates, each possibility is not tested.

$P \lor Q$

$P \lor R \lor Q \lor S \lor T$

$\lnot P \land \lnot Q \limp R$

In that example, $P \lor Q$ could be replaced by $\btrue$ in the second predicate. In order not to add more calculation time, this kind of replacement is not computed.

That reasoner can be considered as "intelligent", similarly to autoImpF because it computes by its own where it has to apply the rules.

Implementation

In this part is explained how the previous choices have been implemented with Java.

1. First, every hypothesis of the sequent is stocked in a Set called hypSet. If the hypothesis is a negation (e.g. $\lnot P$) then its child ($P$) is actually the predicate stored in the set. Currently, if an hypothesis is in the form $\lnot\lnot P$, then $\lnot P$ is stored. Indeed, let's imagine that there is an hypothesis such as $\lnot...\lnot P$, it would be easy to store $P$ in the set. But it is hard ensure that there exists an hypothesis in the form $\lnot...\lnot P$ as we don't know in advance how many $\lnot$ there are in the predicate.
   In addition, predicate $\btrue$ or $\bfalse$ are not stored in the set. First, because i is fool to re-write such predicate. Moreover we can create infinite loop (drop of performance and the job performed is not worth). Indeed, let's imagine that we have the following sequent : $\bfalse, \lnot\btrue \vdash \lnot\btrue$. Here is how the tactic will re-write it :

   $\bfalse, \lnot\btrue \vdash \lnot\bfalse$

   $\bfalse, \lnot\btrue \vdash \lnot\btrue$

   $\bfalse, \lnot\btrue \vdash \lnot\bfalse$

   etc.

   • Then the goal is analyzed, i.e. each sub-predicate is compared to the predicates contained in hypSet (the set of hypotheses). If it matches, the predicate is stored in a Map (modifGoalMap) as well as its position, else its children are analyzed in the same way. Finally, the Map contains the sub-predicates which will be re-written (Map's key) mapped to the list of position where they will be replaced. Notice that the goal itself can be contained in the the Map modifGoalMap.
   • Visible (or selected) hypotheses are analyzed like the goal. To not compare to itself, it (or its child in case of a negation) is removed of the set hypSet. If the hypothesis can be re-written, then it is stored in a map whose the key is the hypothesis itself, and the value is the same as the Map describes for the goal. After analyzing, the hypothesis (or its child in case of a negation) is re-added to the set hypSet.
2. • Afterwards, goal is re-written using rewriteSubFormula given in Predicate. For each predicate in the Map modifGoalMap, we check if it is (or its negation) actually an hypothesis of the sequent. If so, we ensure that the it is equal to the predicate situated at the given position. Finally, if so, we replace it by $\btrue$ (or $\bfalse$). Then, we catch the re-written goal and the hypotheses needed to achieve it.
   • We proceed the same for hypothesis of the map modifHypMap. At the end, all IHypAction are stored in a List.
3. Finally, the proof rule is made. There are 4 cases :
   • the goal is re-written but not the hypotheses : an antecedent is created with no IHypAction, and the IProofRule is made with the needed hypotheses.
   • the goal and hypotheses are re-written : an antecedent is created with the IHypAction computed, and the IProofRule is made with the needed hypotheses.
   • hypotheses are re-written but not the goal : the IProofRule is made with the IHypAction computed.
   • nothing has been re-written : the tactic is no more applicable.

Currently, that implementation has few drawbacks.

• When there are multiple negations (e.g. $\lnot\lnot\lnot P$), that tactic does not work well (for better efficiency, all predicates should be put in the Negative Normal Form thanks to the NNFRewritesAutoTac) :
  • So, for the following predicates, no replacement will be proceeded :
    • $\lnot\lnot P_1$
    • $P_1$
  • whereas for those one, replacement can be proceeded on the second one :
    • $\lnot\lnot P_1$
    • $\lnot P_1$
  • re-written as following :
    • $\lnot\lnot P_1$
    • $\bfalse$
• Its time execution is enormous because of the work that tactic performed (intrinsic), and because of the test realized to ensure that all re-writing recorded in the maps are correct.