Download A Proof of Cut-Elimination Theorem for U Logic.

A Proof of Cut-Elimination Theorem for U
Logic.
Kiarash Rahmani
rahmani [email protected]
Department of Mathematical and Computer Sciences
Sharif University of Technology
September 2014
Abstract
In logical literature, the phrase basic logic refers to at least two different logical systems. The first one, basic propositional logic, BPL was introduced by A.
Visser in 1981. This logic is a subintuitionistic logic that can be obtained from
intuitionistic logic by weakening of modus ponens. The second logical system
with the name basic logic, is the system B that was introduced by G. Sambin in
1997. The goal of this logical system is to provide a common foundation for all
usual non-modal logics. The logic U introduced by M. Ardeshir and V. Vaezain
in 2012, provides a common base for these two logical systems such that, both of
them can be obtained as extensions of U logic.
Key words: Subintuitionistic Logics, Substructural Logics, Sambin’s Basic Logic,
Visser’s Basic Logic, Cut Elimination Theorem.
1
1
Introduction
Sambin’s basic logic, B: System B was introduced to offer a new perspective
and new tools to the search for unity in logic. The first version of this basic logic was
introduced by G. Sambin and G. Battilotti in 1997 [4]. A few years later, the second
version of basic logic was introduced as a logic that obeys three general properties:
reflection, symmetry and visibility [2]. The wish to unify all the usual logics, have been
met partly with introducing Sambin’s basic logic. It provides a common foundation
for classical, intuitionistic, non-modal linear, quantum-like and some other logics.
Sambins basic logic is a substructural (and also subintuitionistic) logic with no free
context and no structure.
Visser’s Basic Logic, BPL: Basic Propositional Logic, BPL, was introduced by
A. Visser in 1981 [5]. His aim, in contrast with Sambin’s, was to interpret implication
as formal provability. BPL can be obtained from intuitionistic propositional logic,
IPC by weakening of modus ponens. This weakening, allows construction of a larger
collection of logics, for example, formal provability logic, FPL from one side, and
IPC, from the other side. BPL, is related to modal logic K4 by Godel’s translation.
Visser’s propositional logic was extended to first order logic by W. Ruitenburg in
1998 [6].
U logic: M. Ardeshir and V. Vaezian introduced a new logic called U as a unifying
basis of basic logics of Sambin and Visser in 2012 [1]. They introduced some modifications and new axiomatizations for B and BPL to make them comparable in a
better way.
We briefly introduce each of the mentioned systems in following sections, and then in
chapter 5, we will give a proof of Cut-Elimination theorem for U logic.
2
Sambin’s Basic Logic
Sambin’s basic logic is a propositional logic with the language L = {⊗, , &, ⊕, →, ←
, 1, ⊥, 0, >} and characterized by the sequent calculus B with inference rules given in
the table 1, whereA, B, ... are formulae and Γ, Γ0 , ..., ∆, ∆0 , ... are finite lists of formulae
(i.e. sets in which repetitions are allowed and order is important).
Basic logic can be characterized as a logic that fulfils three general properties: reflection, symmetry and visibility.
&
Visibility: In any operational rule, the formula in the conclusion that contains the
introduced connective is called principal formula. The formulae in the assumptions that are the components of the principal formula are called secondary
formulae. Principal and secondary formulae together are called active formulae.
We say a rule satisfies visibility, if there is no context beside active formulae.
All the rules of basic logic satisfy visibility. A way to obtain extensions of B is
to relax visibility.
2
Table 1: Sambin’s Basic Logic (B)
A`A
Axioms:
Structural Rules:
Γ, A, B, Γ0 ` ∆
(eL)
0
Γ, B, A, Γ ` ∆
Γ ` ∆, A, B, ∆0
(eR)
Γ ` ∆, B, A, ∆0
Operational
Rules:
A ` ∆ B ` ∆0 ( L)
A B ` ∆, ∆0
`∆
(1L)
1`∆
( R)
Γ ` A Γ0 ` B
Γ, Γ0 ` A ⊗ B
Γ`
(⊥R)
Γ`⊥
(⊗R)
&
&
⊥ ` (⊥L)
Additives
(1R) ` 1
A ` ∆ B ` ∆ (⊕L)
A⊕B `∆
A`∆
B`∆
(&L)
A&B ` ∆
A&B ` ∆
0 ` ∆ (0L)
Implications
A`B
A←B`
Γ ` A, B
Γ`A B
&
(⊗L)
&
Multiplicatives
A, B ` ∆
A⊗B `∆
(&R)
(⊗R)
Γ`A Γ`B
Γ ` A&B
Γ`A
Γ`A⊕B
Γ`B
Γ`A⊕B
(>R) Γ ` >
(← L)
` A B ` ∆ (→ L)
A→B`∆
A`B C`D
(← U )
A←D`B←C
3
(→ R)
A`B
`A→B
Γ`A B`
Γ`A←B
A`B C`D
(→ U )
B→C`A→D
(← R)
Symmetry: The symmetry of B is a geometrical one, and rests on the symmetry
between the left side and the right side of a sequent. For any formula A, its
symmetric formula As is defined, and it is easily extended in a natural way, from
formulae to rules, derivations and logics themselves.
Reflection: Every new connective of B, is created by laying down the truth equation
it must satisfy. This equation, which is called, definitional equation, says which
conditions are sufficient to assert that connective and what to deduce from it.
For example, assume we wish the equation :
A ◦ B ` ∆ ⇐⇒ A, B ` ∆
to hold for any ∆. A direction of this equation is the rule:
A, B ` ∆
A◦B `∆
To define ◦ with no vicious circle, we must find a rule in which A ◦ B appears in
the conclusion, rather than in the premise. Hence, the following rule is derived
from the other direction of the definitional equation:
Γ`A
Γ0 ` B
0
ΓΓ ` A ◦ B
Detailed explanations of these characteristics can be found in [2].
3
Visser’s Basic Logic
Basic Propositional Logic, BPL, was invented by Albert Visser in 1981 [5]. He
wanted to interpret implication as formal provability. To protect his system against
the liar paradox, modus ponens is weakened. His axiomatization of BPL uses natural
deduction[3, p. 8]. The first sequent calculus for BPL, was given by M. Ardeshir
in 1995 [3]. Other sequent calculi then have also been given for B, but Ardeshir’s
system, GBPC, is more suitable for the main aim of introducing the U logic; Which
is finding a common base for BPL and B.
To make the two systems more comparable, Ardeshir and Vaezian in [1], introduced
a modified version of mentioned axiomatization, and called it GBPC*. They also
excluded connective ← from B and called the new system B’. They justified this action, by mentioning the main goal of Sambin’s Basic logic, (i.e. providing a common
foundation for all usual logics such that they can be obtained as extensions). According to their argument, although the new system B’, does not satisfy strong symmetry,
but its set of connectives is still sufficient for any sequent calculus.
4
Table 2: Visser’s Basic Logic (GBPC*)
A`A
Axioms:
`>
⊥`
Γ, A, B, Γ0 ` ∆
Structural Rules:
(eL)
0
Γ, B, A, Γ ` ∆
Γ, A, A ` ∆
Γ, A ` ∆
Γ`∆
Γ, A ` ∆
(eR)
(cL)
Γ ` ∆, A, B, ∆0
Γ ` ∆, B, A, ∆0
Γ ` ∆, A, A
Γ ` ∆, A
(cR)
(wL)
(wR)
Γ`∆
Γ ` ∆, A
Operational Rules:
Γ, A, B ` ∆
Γ, A ∧ B ` ∆
(∧L)
Γ, A ` ∆ Γ, B ` ∆
Γ, A ∨ B ` ∆
(∧R)
Γ ` ∆, A Γ ` ∆, B
Γ ` ∆, A ∧ B
(∨R)
(∨L)
(→ R)
(T r)
4
Γ ` ∆, A, B
Γ ` ∆, A ∨ B
Γ, A ` B
Γ`A→B
Γ`A→B Γ`B→C
Γ`A→C
(F ∧)
Γ`A→B Γ`A→C
Γ ` A → (B ∧ C)
(F ∨)
Γ`A→C Γ`B→C
Γ ` (A ∨ B) → C
U Logic
As we mentioned earlier, all the rules of basic logic of Sambin, satisfy visibility. One
way to obtain extensions of B is to relax visibility. For any sequent calculus X, the
version of X with context at the left side of operational rules is called XL. Symmetrically, XR is defined too.
Another way to obtain extensions of B is to add structural rules. We write XS for
the calculus obtained from X by adding the weakening and construction rules:
Γ, A, A ` ∆
Γ, A ` ∆
Γ ` ∆, A, A
Γ ` ∆, A
(cL)
Γ`∆
(wL)
Γ, A ` ∆
(cR)
Γ`∆
(wR)
Γ ` ∆, A
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Hence, the cube of basic logic extensions is obtained as follows:
Theorem. Neither GBPC* or B’ can be obtained as an extension of the other
system.
Proof. [1].
To find a direct relation between these two sub-intuitionistic logics, let T = {T r, F ∧, F ∨},
L indicates adding context beside active formula at the left side of sequents of operational rules, and S indicates the structural rules of weakening and contraction on the
both sides of sequents. Then it is straightforward that:
GBPC∗ = IPC − (→ L) + T
B0 = IPC(≡ B0 LS) − L − S
Note that, the rule (→ L) is equivalent to modes ponens, and does not hold in GBPC*.
Γ`A
B, Γ ` C
(→ L)
Γ, A → B ` C
Now we describe U, the system that unifies these two logics; The language of U is
L = {⊗, , &, ⊕, →, 1, ⊥, 0, >}, and the inference rules of U are given in the table 3.
&
Lemma 1. In U + (→ L)∗ ,
` A → B (→ L)*
A`B
the rules T ∗ {(T r)∗ , (F ∧)∗ , (F ∨)∗ } are admissible.
Proof. [1].
Since (→ L)∗ is implicit → reflection in Sambin’s terminology [2]; the following equality
holds:
B0 = U + (→ L)∗ + (→ U )
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Table 3: System (U)
A`A
Axioms:
Structural Rules:
Γ, A, B, Γ0 ` ∆
(eL)
0
Γ, B, A, Γ ` ∆
Γ ` ∆, A, B, ∆0
(eR)
Γ ` ∆, B, A, ∆0
Operational Rules:
(⊗L)
( R)
&
A ` ∆ B ` ∆0
A B ` ∆, ∆0
&
`∆
1`∆
&
A, B ` ∆
A⊗B `∆
( L)
(⊗R)
(1L)
Γ`
Γ`⊥
(1R) ` 1
A ` ∆ B ` ∆ (⊕L)
A⊕B `∆
A`∆
B`∆
(&L)
A&B ` ∆
A&B ` ∆
(&R)
(⊗R)
0 ` ∆ (0L)
Γ`A Γ`B
Γ ` A&B
Γ`A
Γ`A⊕B
(>R) Γ ` >
(→ R)
7
Γ ` A Γ0 ` B
Γ, Γ0 ` A ⊗ B
(⊥R)
⊥ ` (⊥L)
V Rules:
Γ ` A, B
Γ`A B
&
S Rules:
A`B
`A→B
(T r)*
`A→B `B→C
`A→C
(F ∧)*
`A→B `A→C
` A → (B&C)
(F ∨)*
`A→C `B→C
` (A ⊕ B) → C
Γ`B
Γ`A⊕B
It is proved in [1], that in system U augmented by structural rules of weakening
and contraction (on both sides), each pair of additive and multiplicative connectives
collapse in one connective. Thus, we can use the usual symbols ∧ and ∨ for conjunction
and disjunction, respectively.
Then It is proved the following equality holds:
GBPC∗ = U + L + S
Therefore, Ardeshir and Vaezian showed in [1], that both Sambin’s and Visser’s basic
logics can be obtained as extensions of logic U, as it is depicted in the following figure:
5
Proof of Cut-Elimination Theorem
Definition 1. Here, we call those operational rules of U logic that are inherited from
Sambin’s basic logic, S rules; And the other four rules, inherited from GBPC*, V
rules.
Proposition 1. If Γ ` ∆ is provable in U without any cut rule, then either Γ or ∆
contains at exactly one formula.
Proof. By immediate induction on cut free derivations:
- The sole axiom of U, A ` A, enjoys the property.
- All rules of system U preserve it.
-Because in S rules of U, always the principle formula is visible; That means
there is no context at the side of active formula (left side, for oL rules and right
side for oR rules) in all sequents.
-V rules are context free and their conclusions are always a single formula at
the right side of the ` .
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Remark. It is obvious that previous proof also shows consistency of U logic, Since `
is not derivable.
Proposition 2. The common formulation of Cut rule, which is usually called full
cut, is not valid in U.
C, Γ0 ` ∆0
Γ ` ∆, C
0
ΓΓ ` ∆∆0
(full cut)
Proof. Assuming full cut, the following derivation would be valid and possible, which
is (according to proposition 1) invalid.
B`B
E ` E ( L)
B`B
D ` D (⊗R)
B E ` B, E
B, D ` B ⊗ D
(f ullcut)
B E, D ` E, B ⊗ D
&
&
&
Definition 2. Two occurrences of the same formula, in two consecutive sequents,
are said to be linked, when they both are in the same place inside their containing
sequents.
Example. Any formula occurrence in the passive context of a sequent [as a premise
in a rule], is linked to the same formula, in the following conclusion sequent.
Example. A formula occurrence which is introduced by a rule, or by an axiom, is not
linked to any formula occurrence above it.
Definition 3. The history of a formula occurrence C in a derivation Π is the last
proof tree trunk, ΠC , which contains all the linked occurrences of C.
Intuitively, considering the history of formula C, amounts to climbing up Π, until C
is defined.
Lemma 2. (Substitution of a formula occurrence)
Let Π be a proof tree trunk in which the only rules applied, are oR rules of system U,
or the V rules. Consequently, all sequents in Π, contain passive or empty context at
the left side.
If Π has conclusion Γ ` ∆, and C ∈ Γ, then the result of substitution of any list
of formulas Σ for C , in Π ; Π[Σ/C] is again a proof tree trunk with the following
conclusion: Γ[Σ/C] ` ∆.
According to visibility, All rules of U are independent of the contexts; Therefor, when
applying a rule, any formula other than the active formula, could be replaced without
affecting the active side of the conclusion.
A dual statement holds, when C is in the right side of the sequent.
Proposition 3. (Cut Elimination Theorem)
The following formulations of cut rule is admissible in U.
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Π1
Π2
..
.
..
.
Γ, C ` ∆
Σ`C
(Cut L)
Γ, Σ ` ∆
Π1
Π2
..
.
..
.
Σ ` C, Σ
C`∆
(Cut R)
Σ ` ∆, Λ
Proof. We describe the case of Cut L. By symmetry Cut R is eliminated in the same
manner.
Case 1. One of the premises is the axiom A ` A.
Π1
Π2
..
.
..
.
A`A
Γ, A ` ∆
(Cut L)
Γ, A ` ∆
is replaced by:
Π2
..
.
Γ, A ` ∆
and
Π2
Π1
..
.
..
.
Γ ` ∆, A
A`A
(Cut L)
Γ ` ∆, A
is replaced by:
Π2
..
.
Γ ` ∆, A
Case 2. The cut formula is active in both premises.
In this case, cut formula is visible in both premises. That means, there is no context
at its side in sequents. That is because, it is principle in both sequents, and because
U logic obeys visibility.
Thus, the main problem reduces to eliminating the following formulation of cut rule:
10
Π1
Π2
..
.
..
.
C ` ∆ (Cut*)
Γ`C
Γ, A ` ∆
This is done by a simple induction on the level of the cut (i.e. complexity of the cut
formula).
1. C ≡ A ⊗ B
Γ`A
Γ0 ` B (⊗L)
0
Γ, Γ ` A ⊗ B
A, B ` ∆
(⊗R)
A⊗B `∆
(cut*)
Γ, Γ0 ` ∆
is replaced by:
Γ`A
Γ0 ` B
A, B ` ∆
(cut*)
Γ, B ` ∆
(cut*)
Γ, Γ0 ` ∆
2. C ≡ A B
&
A`∆
B ` ∆0
A B ` ∆, ∆0
&
Γ ` A, B
Γ`A B
(Cut*)
&
Γ ` ∆, ∆0
is replaced by:
Γ ` A, B
A`∆
(Cut*)
Γ ` B, ∆
Γ ` ∆, ∆
0
3. C ≡ 1
Π
..
.
`1
`∆
1`∆
`∆
is replaced by:
Π
..
.
`∆
11
B ` ∆0
(Cut*)
4. C ≡ ⊥
Π
..
.
Γ`
Γ`⊥
Γ`
⊥`
is replaced by:
Π
..
.
Γ`
5. C ≡ A ⊕ B
Π1
Π2
Π3
..
.
..
.
..
.
A`∆
B`∆
A⊕B `∆
Γ`∆
Γ`A
Γ`A⊕B
is replaced by:
Π1
Π2
..
.
..
.
Γ`A
A`∆
Γ`∆
6. C ≡ A&B
Π1
Π2
Π3
..
.
..
.
..
.
Γ`A
A`∆
Γ`B
Γ ` A&B
A&B ` ∆
Γ`∆
is replaced by:
Π1
Π3
..
.
..
.
Γ`A
A`∆
Γ`∆
Remark. The remaining rules (namely; (→ R), (T r)∗, (F ∧)∗, (f ∨)∗) never appear in the
premises of cut rules in this case. Because connective ”→”, never appears in the left
side of sequents according to rules of system U.
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Case 3. The cut formula is principle in the right premise of cut rule, and is not
principle in the left premise.
In this case, because C is active in the right premise, according to visibility of U,
it must be visible with no context. Therefor, this case, reduces to the problem in
previous case.
Case 4. The cut formula is principle in the left premise and is not principle in the
right premise.
Consider the history of C in right premise’s derivation. That is, we climb up the
tree, on all branches, until we reach the rules that have introduced C. By visibility,
the conclusion on these rules must be of the form: C ` ∆i
..
.
..
.
C ` ∆i
..
.
C ` ∆1
..
.
C ` ∆3
.
Σ`C
..
.
..
..
.
..
.
C ` ∆2
..
.
..
.
. . ..
..
..
.
..
.
.
.
. . ..
..
..
.
..
.
..
..
.
..
Π1
..
.
..
.
..
.
Γ, C ` ∆
Γ, Σ ` ∆
By replacing C by Σ in the history of C, It would turn into a valid proof tree with
conclusion Γ, Σ ` ∆, as required, and Σ ` ∆1 , Σ ` ∆2 , ...., Σ ` ∆n as new leaves. (Lemma
1)
Now, we only need to find proof for each leaf of the new derivation tree, in order to
construct a complete proof tree for Γ, Σ ` ∆. It is easily done, by using a cut* rule(case
2), and the conclusion of Π1 :
Π1
..
.
..
.
..
.
Σ`C
C ` ∆i
(Cut*)
Σ ` ∆i
Therefor, the first tree, is replaced by:
Π1
Π1
..
.
..
.
..
.
..
.
Σ`C
C ` ∆i
(Cut*)
Σ ` ∆i
..
.
..
.
Σ`C
C ` ∆1
(Cut*)
Σ ` ∆1
. . ..
..
..
.
..
.
Γ, Σ ` ∆
13
..
.
..
.
Σ`C
C ` ∆2
Σ ` ∆2
..
.
.
.
..
.
..
..
..
.
Π1
..
.
Case 5. The cut formula is not principle in either premises.
This case is similar to the previous case; Here, again we need to make a derivation tree for Γ, Σ ` ∆, by replacing Σ for C in the history of C in Π2 .
After substituting Σ for C, Π2 would turn into a proof tree with conclusion Γ, Σ ` ∆,
and leaves of the form: Σ ` ∆i .
Now to construct a proof tree for each Σ ` ∆i , consider, the history of C in Π1 :
..
.
..
.
..
.
Σ1 ` C
..
Σ2 ` C
..
.
..
.
Σj ` C
.
. . ..
..
..
.
..
.
. . ..
..
..
.
..
.
.
..
.
..
.
Σ3 ` C
..
.
..
..
.
..
.
..
.
Σ`C
Γ, C ` ∆
Σ, Γ ` ∆
Then we substitute each occurrence of C in ΠC
1 by ∆i :
..
.
..
.
..
.
Σ1 ` ∆i
Σ 2 ` ∆i
..
.
..
.
Σ j ` ∆i
.
. . ..
..
..
.
..
.
. . ..
..
..
.
..
.
.
..
.
..
..
.
Σ 3 ` ∆i
..
.
..
..
.
..
.
..
.
Σ ` ∆i
Now, we only need to construct a proof tree for each leaf of this new tree; Which are
of the form: Σj ` ∆i
It is easily done by the following cut*:
..
.
..
.
Σj ` C
..
.
..
.
C ` ∆i
Σ j ` ∆i
(Cut*)
In which, the right premise is from Π2 and the left premise is from Π1 .
Therefor, we proved the Cut Elimination theorem for logic U. This proof is similar
to the proof explained in [2] for B. Here, we showed it works for U as well.
14
6
Acknowledgments
My best regards to Professor Mohammad Ardeshir, who introduced me to logic.
I would also like to thank Dr. Mojtaba Mojtahedi and Mohsen Shahriari, for their
helpful comments and companionship.
References
[1] M. Ardeshir, V. Vaezian, A unification of the basic logics of Sambin and Visser,
Logic Journal of the IGPL, Vol. 20, Issue 6, 2012.
[2] G. Sambin and G. Battilotti and C. Faggian, Basic Logic: reflection, symmetry,
visibility , The Journal of Symbolic Logic, 65, 979-1013, 2000.
[3] M. Ardeshir, Aspects of Basic Logic. PhD Thesis, Department of Mathematics,
Statistics and Computer Science, Marquette University, 1995.
[4] G. Sambin and G. Battilotti., Basic Logic and the cube of its extensions. In logic
in Florence’ 95. A. Cantini, E. Casari and P. Minari, eds. kluwer, 1997.
[5] A. Visser, A propositional logic with explicit fixed points. Studia Logica, 40, 155175, 1981.
[6] W. Ruitenburg, Basic Predicate Calculus. Notre Dame Journal of Formal Logic,
39, 18-46, 1998.
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