SEPARATING THE LOGICAL RULES IN DEEP INFERENCE
ANUPAM DAS
Abstract. We prove that the logical rules switch and medial can be separated in anySKS-derivation.
Notation 1. (ρ1, . . . , ρn) denotes the system with derivations of the form A
{ρ1}
...
{ρn}
B .
Lemma 2. There are derivations C(A)
SA
{s}
Aˆ
CA
{c↓}
A
and A
C0A {c↑}
Aˇ
S0A {s}
D(A)
, for every formula A, where
C(A) (resp. D(A)) is a formula in conjunctive normal form (resp. disjunctive normal form) logically equivalent toA.
Proof. We prove the existence of Φ, Ψ, whence the existence of Φ0, Ψ0 follows by duality. We define C(A) inductively as follows:
C(_
i
ai)≡_
i
ai C(B∧C)≡C(B)∧C(C) C(B∨(C∧D))≡C(B∨C)∧C(B∨D) Clearly, for any formula A, C(A) is a CNF formula logically equivalent to A. We define SA, CA inductively as follows, with the hypothesis IHthat A\∨B ≡Aˆ∨B.ˆ For the base case, we defineW[
iai≡W
iˆai≡W
iai.
C(W
iai)
≡ −−−−−−−−−
[ W
iai
≡ −−−−−−
W
iˆai
≡ −−−−−−
W
iai
C(B∧C)
≡ −−−−−−−−−−−−−−−−−−−−
C(B)
SB
{s}
Bˆ
CB
{c↓}
B
∧
C(C)
SC
{s}
Cˆ
CC
{c↓}
C
C(B∨(C∧D))
≡ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
C(B∨C)
SB∨C
{s}
B\∨C
IH−−−−−−−−−−−
Bˆ∨Cˆ
∧
C(B∨D)
SB∨D
{s}
B\∨D
IH−−−−−−−−−−−−
Bˆ∨Dˆ
2·s−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Bˆ∨Bˆ
c↓ −−−−−−−
Bˆ
CB {c↓}
B
∨
Cˆ
CC {c↓}
C
∧
Dˆ
CD {c↓}
D
We then simply define B\∧C ≡Bˆ∧C,ˆ [B∨\(C∧D)]≡Bˆ∨ Cˆ∧Dˆ
to finish the
construction.
Lemma 3. For every valid implication from a DNF to a CNF there is a derivation with the same premiss and conclusion in (c↑(0),aw↑,ai↑,ai↓,aw↓,ac↓).
Proof. Let W
i
V
jaij →V
s
W
tbst be a valid implication. Now for eachi we have that V
jaij →W
i
V
jaij is valid, for example by an application of weakening, and dually for each swe have thatV
s
W
tbst→W
tbstis valid. Therefore, by transivity of implication, for eachi, spair we have thatV
jaij →W
tbstis valid. We construct derivations Φi,s associated with these implications as follows:
Date: June 10, 2011.
1
2 ANUPAM DAS
(1) If there are a pair of dual atoms in
n
V
j=1
aij then we construct the following derivation:
n−2
V
j=1
aij aw↑ −−−
t
!
∧ a∧¯a
ai↑ −−−−−
f
=−−−−−−−−−−−−−−−−−−−−−−−−−−−
f
=−−−−−−−−−−−−−
"
v
W
t
f
aw↓ −−−
bst
#
(2) If there are a pair of dual atoms inW
tbstthen, dually to (1), we construct the appropriate derivation in (aw↑,ai↓,aw↓).
(3) If there is no pair of dual atoms in either the premiss or the conclusion, then it follows that {aij} ∩ {bst} 6=∅(otherwise the implication would not be valid), so letabe an atom in the intersection. We construct the following derivation:
n−1
V
j
aij
aw↑ −−−
t
!
∧a
=−−−−−−−−−−−−−−−−−−−−
a
=−−−−−−−−−−−−−−−−−−−
"
v−1
W
t
f
aw↓ −−−
bst
#
∨a We now combine these derivations as follows:
^
s
_
i
V
jaij
Φi,s
W
tbst
; ^
s
W
i
V
jaij W
iΦi,s
W
i
W
tbst
=−−−−−−−−−−
W
t
W
ibst {ac↓}
W
tbst
; W
i
V
jaij {c↑(0)}
V
s
W
i
V
jaij
(aw↑,ai↓,ai↑,aw↓,ac↓)
V
s
W
tbst
Theorem 4. For every valid implication there is a derivation with the same premiss and conclusion in (ac↑,aw↑,ai↓,m,s,m,ai↑,aw↓,ac↓).
Proof. For each valid implicationA→B we construct the following derivations:
A
(c↑,s)
D(A)
{c↑(0)}
D0(A)
(aw↑,ai↓,ai↑,aw↓)
C0(B)
{ac↓}
C(B)
(s,c↓)
B
;1
A
(ac↑,m,s)
D0(A)
(aw↑,ai↓,ai↑,aw↓)
C0(B)
(s,m,ac↓)
B
;2
A
(ac↑,aw↑,ai↓)
A0
(m,s,m)
B0
(ai↑,aw↓,ac↓)
B where D0(A) ≡V
s
W
i
V
jaij and C0(B)≡ V
s
W
t
W
ibst as in Lemma 3. The first derivation is obtained from the two lemmata. Transformation (1) is obtained by permuting the surface cocontractions above the switches, atomic contractions below the switches, and then reducing all contraction and cocontraction steps to atomic form, introducing medials and permuting as necessary. Finally transformation (3) is obtained by permuting all logical rules to the centre, as described in [Brunnler] and
SEPARATING THE LOGICAL RULES IN DEEP INFERENCE 3
collapsing the two switch sequences into one sequence, which gives us a derivation
in the required form.
Corollary 5. SKS\ {ai↓,ai↑}is complete over monotone implications.
Proof. If there is no negation in A or B, then there is no negation in D(A) and C(B). Therefore, in the proof of Lemma 3, each derivation from a conjunction of atoms to a disjunction of atoms falls under case (3), which does not introduce identities or cuts. Following this through Theorem 4 gives the required result.
Proposition 6 (Jerabek). MLKandKS∪ {ac↑,aw↑} are strongly equivalent.
Corollary 7. MLKis complete over monotone implications.
By restricting the above theorem to just proofs, we arrive at a similar decompo- sition for KS. While the above theorem gave us a medial-switch-medial sandwich, we can use this restriction to create another decomposition with a medial-switch- medial sandwich.
Lemma 8. Every tautology has aKS-proof of the form(ai↓,s,m,aw↓,ac↓).
Proof. For each tautology apply Theorem 4 with premisst. Since there are no atoms in the premiss, no coweakening or cut steps are introduced by the construction in Lemma 3. The surface cocontractions operate on a conjunction of t’s so they can
be replaced by =-steps.
Theorem 9. (ai↓,s,m,s,aw↓,ac↓,ai↑)is implicationally complete.
Proof. LetA→B be a valid implication, and let Φ be aKS-proof of ¯A∨B in the form of the above lemma. We construct the following derivations:
A
=−−−−−−−−−−−−−−−−−−−−−−−−
A∧ t
Φ
(ai↓,s,m,aw↓,ac↓)
A¯∨B
s−−−−−−−−−−−−−−−−−−−−−−−−
A∧A¯
i↑ −−−−−−
f ∨B
=−−−−−−−−−−−−−
B
;1
A
A∧Φ
(ai↓,s,m,aw↓,ac↓)
A∧A¯∨B
(s,ai↑)
B
;2
A
(ai↓,s,m,s,aw↓,ac↓,ai↑)
B
Transformation (1) is obtained by reducing the cut to atomic form, creating switches above all atomic cuts. Transformation 2 is obtained by permuting weakenings and
contractions beneath the newly created switches.
