Theorem proving is one of the oldest applications which require heuristics to prune the search space. One key observation about proof search is that some rules are invertible, that is, the premisses are derivable whenever the conclusion is derivable. We can apply invertible rules whenever possible without losing completeness. The proof search strategy that first applies all invertible rules is called inversion. An inversion calculus has been developed for classical, intuitionistic and linear logic. However, it has the following drawbacks:
Whenever a new logic comes along, its corresponding inversion calculus must be separately developed and proved to be correct. This is a time consuming and redundant process.
Inversion calculus processes connectives eagerly, often processing them even when it is not necessary.
To overcome these, we propose a general heuristic called nongshim which can be universally applied to a wide class of logics, often leading to inversion calculus.
2 A Universal Heuristic for Theorem Proving
Japaridze[1, 2, 3] have used an interesting heuristic in developing his proof theory, which is closely related to the Nongshim Cup game. The Nongshim Cup is a team competition for the two Go-playing countries such as China, Korea and Japan. It has an interesting tournament format: Two countries each starts with some number (typically five each) of players. The winner of each subsequent game stays in to play the next opponent from another country. In the end the team with remaining players is the winner of the tournament.
Let us assume Korea with korean players competes against China with chinese players . This is written as nongshim(Go,,). The following algorithm will clarify the Cup format.
set current game to nongshim(Go,,) % initialization
if Korea wins the current game of the form nongshim(Go,,), then update current game to nongshim(Go,,)
else update current game to nongshim(Go,,)
repeat step 2 until one team has no remaining player.
In this way, the decision of winning/losing can be achieved at the earliest possible time.
We are interested in applying the above idea to theorem proving. One instance of this heuristic – nongshim(literalization(F),M,O) – has been used to proof in first-order logic, where is a first-order logic formula, is a set of top-level occurrences of -formulas and is a set of top-level occurrences of -formulas. The process of literalization(F) transforms to its skeleton by removing and in it, as we shall see below. The resulting proof procedure turns out to be quite effective and performs better than the traditional inversion calculus.
In this paper, we apply this heuristic approach to propositional logic. That is, we apply
is a propositional formula,
is a set of top-level occurrences of -formulas and
is a set of top-level occurrences of -formulas.
That is, the resulting procedure, which we call LKg, is a - proof which captures - nature in proof search. It views
sequents as games between the machine and the environment,
proofs as a winning strategy of the machine, and
as the env’s resource and as the machine’s resource.
For propositional logic, it turns out that this heuristic does not derive a new calculus. Instead, it is able to derive an existing invertible sequent calculus from sequent calculus. This is meaningful, as it provides a deeper insight into why all the rules are invertible in propositional logic.
3 The logic LKg
The formulas are the standard classical propositional formulas, with the features that (a) are added, and (b) is only allowed to be applied to atomic formulas. Thus we assume that formulas are in negation normal form.
The deductive system LKg below axiomatizes the set of valid propositional formulas. LKg is a one-sided sequent calculus system, where a sequent is a multiset of formulas. Our presentation follows the one in .
First, we need to define some terminology.
A surface occurrence of a subformula is an occurrence that is not in the scope of any connectives ( and/or ).
A sequent is literal iff all of its formulas are so.
The literalization of a formula is the result of replacing in every surface occurrence of -subformulas by , and every surface occurrence of -subformulas by .
The literalization of a sequent is the propositional formula
A sequent is said to be stable iff its literalization is classically valid; otherwise it is unstable.
LKg has the four rules listed below where is a multiset of formulas and is a formula.
The deductive system LKg is shown below. Below, :stable means that must be stable. Similarly for :unstable. The Fail rule reads: an unstable sequent containing no surface occurrences of is not derivable.
A LKg-proof of a sequent is a sequence of sequents, with , such that, each follows by one of the rules of LKg from .
Below we describe some examples.
The formula is provable in LKg as follows:
3. from 1
4. from 2
5. from 3,4
4 The soundness and completeness of LKg
We now present the soundness and completeness of LKg.
If LKg terminates with success for , then is valid.
If LKg terminates with failure for , then is invalid.
Proof. Consider an arbitrary sequent .
Soundness: Induction on the length of derivatons.
Case 1: is derived from and by -rule. By the induction hypothesis, both and are valid, which implies that is valid.
Case 2: is derived from by -rule. By the induction hypothesis, is valid, which implies that is valid.
Case 3: is derived from by Succ.
In this case, we know that there is no surface occurrences of in and is classically valid. It is then easy to see that, reversing the literalization of (replacing by any formula of the form ) preserves validity. For example, if is , then is valid and is valid as well.
Completeness: Assume LKg terminates with failure.
We proceed by induction on the length of derivations.
If is stable, then there should be a LKg-unprovable sequent with the following condition.
Case 1: : has the form , and is either or . Suppose is LKg-unprovable. By the induction hypothesis, is invalid and is invalid as well. Similarly for .
Next, we consider the cases when is not stable. Then there are two cases to consider.
Case 2.1: Fail: In this case, there is no surface occurrence of and the algorithm terminates with failure. As is not stable, is not classically valid. If we reverse the propositionalization of by replacing by any formula with some surface occurrence of , we observe that invalidity is preserved. Therefore, is not valid.
Case 2.2: : In this case, has the form and is . In this case, is a LKg-unprovable sequent. By the induction hypothesis, is invalid. Therefore is not valid.
5 A simplified LKg
LKg in the previous section can be simplified by
observing the following:
A sequent is stable iff is either
This observation leads to a simplified version of
LKg, as shown below:
- if is
then return Yes.
- elsif is
then return and .
- elsif is
then return .
Of course, we can speed up the above procedure by processing in parallel.
- if is
then return Yes.
- elsif is
and and .
% total combinations above.
- elsif is
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