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ON SOME SYSTEMS OF MINIMAL MODAL LOGIC WITH HISTORY MECHANISM
Baghdasaryan A.
Russian-Armenian University, Yerevan, Armenia
ABSTRACT
Minimal logic being part of intuitionistic logic attracts a special interest being a constructive logic. Automated theorem proving problems in miminal logic have many applications in a variety of tasks, such as expert systems, software verification and synthesis problems and a number of other areas.
In this paper, both Hilbert style and Gentzen-style cut-free minimal modal logic systems are considered. In order to prevent loops, history mechanisms are added to Gentzen-style minimal modal logic system. Two systems of minimal fragment of modal 54 logic with history mechanisms are introduced and their equivalence is proved. Those systems may serve as a basis for automated theorem provers in minimal systems of modal logic.
Keywords: Minimal Logic, Modal Logic, Cut-Free Systems.
Introduction
In the system S5"Min, which was introduced in [1], four modal rules were added to the system GM [2]. As Ohnishi and Matsumoto showed [3] the cut rule can be eliminated from the sequent system of type 54, but the sequent calculus for 55 is not cut-free complete. When cut is removed from the calculus for 55 it results in a weaker system. For that reason, following [1] 54^„ minimal modal cut-free system of 54 type is formulated. Then, following [4] and [5], two types of history mechanisms will be added to those systems to develop new systems with loop detection.
System GMS4
Let H be the minimal propositional logic formulated in the Hilbert-style. The only rule of inference of H is modus ponens. The MML (minimal modal logic) of 54 type 54Min is obtained from H by adding the following three axioms:
1 Op 3 p
2 op 3 □□ p
3 O(p 3 q) 3 (op 3 oq)
and the rule of necessitation, i.e., from A infers
□A.
It can be easily showed that this is a minimal modal logic system of 54 type and with the principle of explosion (—A 3 (A 3 B)) it becomes intuitionistic IML of 54 type described in [6]. Moreover, adding the law of excluded middle (A V — A) leads the system to become 54. By taking the formulation of the sequent calculi, the corresponding sequential calculus for 54 form [3] would be modified. The system 54^n would be constructed by adding the following four modal rules to the GM-:
F^A
(Or)
A, Of^ OB
(Ol)
r^ OA v r OA, Or^ OB A,r ^ B Or ^ A
□A,r^ B (Ol) Or^ OA (Or) A,r ^ B Or ^ A
OA,r^ B (Ol) Or^ OA (Or) O r (OF) means the series of formulae, which is formed by prefixing O(O) in front of each formula of
r.
Theorem 1. The systems 54^n and S4Min are equivalent. That is, a sequence ^ a is deducible in 54^j„ if and only if a is deducible in S4Min.
To construct sequential systems for 54 minimal modal logic with history mechanisms, GMS4 system would be constructed on top of the propositional fragment of minimal sequential calculi introduced in [4] (GMp), which contains special notation of stoup (formula above the arrow) introduced by Girard in [7]. Modality rules in these systems would be the followings: r^A A, Or ^ OB
(Or)-oa- (0l)
r^ OA A,r^ B
da
r B
(□l)
oa
Or OB □ r^ A
□r ^ DA
(□r)
Theorem 2. The systems 54^„ and GM. equivalent. That is, a sequent ^5 is provable in 54^ if and only if ^5 is provable in CMS4.
S4 are
*
Min
Systems with history mechanism
The system SwMinS4 would be constructed by adding the following modal rules to the SwMin: r ^ A;H Or ^ A;H
(Or) ^^ ,, (or)
r ^ $A;H
A, ❖ r^ $B;e ❖a
❖ r ^B;H
□r ^ OA; H (❖L1),if A£ ❖r
❖a
❖ r 0B;H A,r ^ B;e
oa r B; H r^ B;H
(❖L2),if A £ ❖r
(□Li),if A£r
(□l2),if A£r
Oa
r B; H
To reduce the history and check the loops more easily context formulas can be dropped form history and only goal formulae need to be stored. In SwMinS4 system rules are such that the context cannot decrease; once a formula is in the context it will remain in the context of all sequents above it in the proof tree. For two sequents to be the same they obviously need to have the same context. We may empty the history every time the context is extended, since we will never get any of the sequents below the extended one again. Goal formulae are the only ones to be stored in the history. If we come across a goal already in the history, we have the same goal and the same context as another sequent, that is, a loop.
The well-known definitions of height and depth of the sequent in a proof tree will be used. The definition of the complexity of a formula would be extended for the cases of formulas containing O and 0 in this way: Definition 1. Let c(F) denote the complexity of formula F. If X is a propositional variable, then c (X) = 1. Also c(±) = 1. Then define inductively, c(A &B) = c(a) + c(B), c(AvB) = c(A) + c(B), c(A 3B) = c(A) + c(B) + 1, c(-A) = c(A) + 2, c(o.A) = c(A) + 2, c(0A) = c(A) + 2.
Theorem 3. The systems GMS4 and SwMinS4 without (*) restriction on rule (c) are equivalent. That is, a sequent r ^ S is provable in GMS4 if and only if r ^ S;e is provable in SwMinS4 (without (*)).
Proof. ^ direction, that is if r ^ S;e is provable in SwMinS4 then r ^ S is provable in GMS4.
Given the proof in SwMinS4, from each sequent of the proof tree the history has to be stripped. The valid GMS4 inference tree will be obtained except the cases of (OL2) and (0L2) rules. In these cases, implicit weakening rule needs to be applied:
r',A^B;H N p-AX^B , ,
-oa-(ol2) is transformed t^ ^-oa-(ol)
r',A — B; H r',A — B
Similarly, in the case of (0L2) rule. ^ direction, that is if r ^ S is provable in GMS4 then r ^ S;e is provable in SwMinS4. The proof is almost the same as for proof of the Theorem 2.1 in [4]. By the definition of a proof in GMS4 the proof tree is finite, that is, all branches of the tree terminate with
success at a finite height. If the next inference rule in the proof tree is (0R) (or (OR)): r^A
- (Or)
r^ OA ry then the next inference in the SwMinS4 is: r^A;H
r ^ OA; H (Or) The cases with side conditions are (OL) and (Ol). In these cases, the appropriate version of the rules, depending on the context would be applied. If the next inference rule in the proof tree is (OL):
(Ol)
r B
then the next inference in the SwMinS4 is: A,r ^ B;e r ^ B; H
—oa- (oli),if A £ r oa- OOaXif A £ r
r B;H r B;H
As the context is extended, the history can be emptied. The branch of proof tree in SwMinS4 for (Ol) rule can be constructed in similar way. If the next inference rule in the proof tree is (Ol): A, or^ OB
-oA- (°l)
Or OB
then the next inference in the SwMinS4 is: A, or^ OB; e
— (0L1), if A £ or
❖a
❖ r ❖B; H
❖ r^ ❖B; H
(❖L2),if A £ ❖r
❖a
❖ r ❖B; H
As the context is extended, the history can be emptied.
So, for the branches of the proof tree which contains the new four rules of the system GMS4 have their corresponding rules in the SwMinS4 system.
Theorem 4. The systems GMS4 with (*) restriction on rule (c) is equivalent to GMS4 without (*).
Proof. The only rules that cannot be done in the calculus with (*) that can be made in the calculus without (*) are ones with implication, negation, and conjunction as the goal. Rules with negation as a goal formula would be skipped as they are special case of implication. The proof would be constructed by induction on the depth of the proof and by the complexity of formulas like in the proof of the Theorem 2.2 in [4]. The formulas OA and ❖A selected for the stoup would be considered, as others are discussed in the proof of the Theorem 2.2 in [4].
1) The formula selected for the stoup is OA: X,r, OX^ADB X, r, OX ^ A&B
■ (Ol)and-ox-(Ol)
ox
r, OX
□x
r, OX A&B
r, OX ^ADB (cC) r, OX ^ A&B (c)
By induction on the depth of the proof, following proofs are valid in GMS4 with (*) restriction:
X,r, OX ^AdB and X, r, OX ^ A&B On the other hand, right inference rules could be applied to the above sequents: X,A,r, OX ^ B
(Ol)
and
□x
A, r, OX B A, r, OX ^B r,OX ^A^B
(c)
X, r, OX ^ A
□x
r, OX
r, OX ^ A
(□l)
X, r, OX ^ B
(□l)
□x
( , r, OX -> B ( .
(c) r, OX ^ B (c)
r, OX ^ A&B (&r)
Finally, induction on the complexity of the goals of the conclusions of the (c) rule would finish the proof.
2) The formula selected for the stoup is a OA. The proof can be done by
induction as in the previous part for OA selected as a stoup.
Theorem 5. The systems SwMinS4 with (*) restriction on rule (c) is equivalent to SwMinS4 without (*).
Proof. 1) If a sequent S is provable in SwMinS4 with (*) condition on rule (c), then it is obviously provable in SwMinS4 without (*). 2) Given a proof of r ^ A; H in SwMinS4 without (*), the history would be stripped away as in proof of Theorem 3. The proof of r ^ A sequent in CMS4 would be obtained. By Theorem 4 the proof of r ^ A sequent in CMS4 with (*) restriction also would be obtained. By the construction in the Theorem 3 we get a proof of r ^ A; e in SwMinS4 with (*).
The system ScMinS4 would be constructed by adding the following modal rules to the ScMin: r ^ A; (A, H) Or ^ A; (A, H)
( ) (Or) (Or)
r^ OA; H
A, Of^ OB; {OB}
oa
Or OB; H Or ^ OB; {OB, H}
oa
Or OB; H A,r ^ B; {B} oa r B; H B;{B, H}
Or ^ OA; H (OL1), if A g Or
(OL2), if A e Or, B g H
(oli), if agr
oa (ol2),if A e r, B g H
r B; H
Theorem 6. The systems CMS4 and ScMinS4 are equivalent. That is, a sequent T ^ 5 is provable in CMS4 if and only if T ^ 5; {5} is provable in ScMinS4.
The proof can be done in the similar way as for the systems with Swiss history mechanism.
Conclusion
In the theory of automated theorem proving systems of constructive logic are of a special interest due to an ability to extract rigorous information from the constructed proof. We have illustrated the use of the two history mechanisms for minimal modal logic. Two slightly different systems for minimal fragment of S4 modal logic (SwMinS4 and ScMinS4) are introduced. Both systems are based on the idea of adding context to the sequents. In one system, SwMinS4, the history is kept smaller, but ScMinS4 detects loops more quickly. The heart of the difference between the two systems is that in the SwMinS4 loop checking is done when a formula leaves the goal, whereas in the ScMinS4 it is done when it becomes the goal.
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