PROOF COMPLEXITY
ANUPAM DAS
Abstract. We prove that, from the point of view of proof complexity, dagness is equivalent to the presence of cocontraction. We present some applications of our technique to derive both new and existing results.
1. Introduction
Dagness and cocontraction are both mechanisms for sharing information, and can both be used to compress proofs. In the Gentzen formalism it is known that, in the absence of cut, dag-like systems exhibit an exponential speedup over tree- like systems, witnessed in proofs of the Statman tautologies. Cocontraction in deep inference systems can be considered as a generalization of dagness since it allows information to be shared at any depth. So while the two notions coincide for Gentzen systems, it is believed that cocontraction in deep inference allows for a further speedup over dagness.
In this note we utilise one half of the depth-change trick to show that, from the point of view of proof complexity, the two notions can be considered equivalent, even in deep inference systems, by showing that cocontraction can always be restricted to surface applications. The transformation given induces a natural normalisation procedure to eliminate cocontractions in a proof, and we present some applications of this procedure where there is only a polynomial increase in size, yielding results on relative efficiency of proof systems and size of proofs of some variants of the pigeonhole principle.
2. Polynomial Equivalence of Dagness and Cocontraction We offer a definition of dagness for sequential derivations (i.e. derivations in the calculus of structures), which extends to open deduction derivations in general, in the natural way.
Definition 1 (Dag-like deep inference). For a CoS systemS, we define “dag-like S” to be the system S ∪ {dag}, where dag is the inference rule t
dag−−
A with the additional proviso thatAappears above the dagstep in the derivation.
Proposition 2. Dag-likeKSgis strongly polynomially equivalent toKSg∪ {c↑(0)}.
Proof. Apply the following transformations inductively to derivations. The first takes a dag-like KSg derivation to a tree-like KSg∪ {c↑(0)} derivation, and the
Date: June 13, 2011.
1
second vice versa.
A
Φ
B
Ψ
ξ ( t
dag−−
B )
;
A
Φ
B
c↑(0)−−−−−−−−−
B
Ψ
ξ{t}
∧B
ss==========
ξ{B}
A
Φ
B
c↑(0)−−−−−−−
B∧B
;
A
Φ
B
=−−−−−−−−−−−
B∧ t
dag−−
B
where ssstands for superswitch, as introduced in [Brunnler].
Lemma 3 (Depth-change for cocontraction). In the presence of contraction and switch, cocontraction steps can be pushed to the surface with only polynomial in- crease in size of derivation.
Proof. Apply the following transformations inductively to derivations:
A
Φ
ξ (
C∨ D
c↑ −−−−−−−
D∧D )
Ψ
B
;
A
Φ
ξ
C∨D
c↑ −−−−−−−−−−−−−−−−−−−−
[C∨D]∧[C∨D]
2·s−−−−−−−−−−−−−−−−−−−−−−
C∨C
c↓ −−−−−−−
C ∨(D∧D)
Ψ
B
A
Φ
ξ (
C∧ D
c↑ −−−−−−−
D∧D )
Ψ
B
;
A
Φ
ξ
C∧D
c↑ −−−−−−−−−−−−−−−−−−−−−
(C∧D)∧(C∧D)
=−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
C∧
C
=−−−−−−−−−−
f
w↓ −−
t ∨C
s−−−−−−−−−−−−−−−−−
C∧t
=−−−−−
C ∨C
c↓ −−−−−−−−−−−−−
C
∧(D∧D)
Ψ
B
Theorem 4. Dag-like KSg is polynomially equivalent toKSg+.
Proof. Immediate from Proposition 2 and Lemma 3.
Corollary 5 (Cocontraction-elimination for KSg+). First push all cocontraction instances in a proof to the surface, then apply the following transformation induc- tively to surface cocontraction steps, starting with the topmost.
− Φ
KSg
A
c↑ −−−−−−
A∧A
;
− Φ
KSg
A
∧
− Φ
KSg
A
Clearly, the transformation can blow up a proof exponentially, however it is often the case that eliminating cocontractions in this manner does not blow up a proof, and this can often allow us to show that certain complexity properties ofKSg+also hold for KSg. We give examples in the following sections.
3. Relative Efficiency of Deep Inference, Gentzen and Truth Tables Bruscoli and Guglielmi have already proved that cut-free Gentzen systems can- not polynomially simulate cut-free deep inference systems, by way of the Statman tautologies. We offer a new proof here, via truth tables, that is nonconstructive, making use of the cocontraction elimination in the previous section.
Proposition 6 (D’Agostino). Cut-free Gentzen systems cannot polynomially sim- ulate truth tables.
Lemma 7. A truth table proof for a formulaAhas size|A| ·2#A, where#Ais the number of distinct propositional variables in A.
Proof. A truth table for Ais a matrix whose height is the number of assignments onAand whose width is the number of propositional variables in A.
Theorem 8. KSg+ polynomially simulates truth tables.
Proof. Letτ be a tautology. For each partial assignmentA, defined on just those atoms appearing in τ, we construct a derivation ΦA(τ) by structural induction on τ:
ΦA(a)≡a ΦA(A∧B)≡ΦA(A)∧ΦA(B) ΦA(A∨B)≡
ΦA(B)
=−−−−−−−−−−
f
w↓ −−
A∨B In the last case, when there is a disjunction, choose a disjunctB such thatAB.
It is clear that each ΦA(τ) has conclusion τ and premiss a conjunction of literals;
moreover this conjunction of literals is satisfied by A. Let φA be the conjunction of literals that is satisfied by A such that each atom or its dual appears exactly
once. Then it follows that there is a derivation of the form:
φA
{ac↑}
φ0A
{aw↑}
P r(ΦA(τ))
. We now
construct a proof Ψ ofW
AφAby induction on the number of distinct atoms inτ:
Base case: t
ai↓ −−−−−
a∨¯a Inductive step:
Ψ−
{ai↓,ac↑,s}
W
A
φA
=−−−−−−−−−−−−−−−−−−−−−−−−
t
ai↓ −−−−−
a∨¯a∧
φA
ac↑==========
φA ∧φA
s=====================================
W
A(a∧φA)∨W
A(¯a∧φA)
Putting this all together we obtain the following proof:
Ψ−
{ai↓,ac↑,s}
W
A
φA
{ac↑,aw↑}
P r(ΦA)
ΦA {w↓}
τ
c↓=======================
τ
. Fi-
nally we eliminate all coweakenings in the usual way to obtain the desired result.
Corollary 9. KSg polynomially simulates truth tables.
Proof. In the above proof, permute all atomic identities to the top, followed by all atomic cocontractions, which will have as conclusion a formula of the formV
i[(ai∧
. . . ai)∨(¯ai∧. . .¯ai)]. Now apply Corollary 5 to each conjunct separately, resulting in only a quadratic blowup of the entire formula. Eliminate all coweakenings in the
usual way to obtain the desired result.
Corollary 10. Cut-free Gentzen systems cannot polynomially simulate cut-free deep inference systems.
Proof. Immediate from Proposition 6 and Corollary 9.
Remark 11. It should be noted that D’Agostino’s separation is only quasipolyno- mial, and so the separation in the corollary above is also only quasipolynomial, while the proof using the Statman tautologies gives an explicit exponential separation.
4. Polynomial Size Proofs of Some Pigeonhole Principles inKSg Jerabek has shown that there are polynomial-size proofs of the functional and onto variants of the pigeonhole principle in KSg+. We show that there is only non-essential use of cocontraction in these proofs and so they can be eliminated.
Definition 12 (Functional pigeonhole principle).
PHPn+1n ≡ _
i<n+1
^
j<n
¯ aij∨ _
j<n
^
i<n+1
¯
aij∨ _
j<n,i<i0<n+1
(aij∧ai0j)
Proposition 13 (Jerabek). Every SKSg-proof of a formula Acan be transformed in polynomial time into a KSg-proof of A∨W
a∈A(a∧¯a).
Corollary 14 (Jerabek). There are polynomial-size proofs in KSg+ of the func- tional pigeonhole principle.
Proof. Buss has constructed polynomial-size Frege proofs of PHPn+1n ; transform these polynomially into SKSg-proofs and apply the proposition above to obtain polynomial-size KSg-proofs ofPHPn+1n ∨W
a∈PHPn+1n (a∧a). Now construct¯ KSg+- derivations with premiss a∧¯a and conclusion PHPn+1n for each a ∈ PHPn+1n as follows (for simplicity we construct the derivation for a ≡a00, whence the others follow by symmetry):
a00∧a¯00
i↓=====================================
a00∧¯a00∧V
j6=0¯a0j∨W
j6=0a0j s=====================================
V
ja¯0j∨a00∧W
j6=0a0j c↑=====================================
V
ja¯0j∨(V
j6=0a00∧W
j6=0a0j)
s=====================================
V
j¯a0j∨W
j6=0(a00∧a0j)
w↓=============================
PHPn+1n
Finally just apply contraction to obtain the desired result.
Corollary 15. There are polynomial-size proofs inKSgof the functional pigeonhole principle.
Proof outline. In the derivation above apply the cocontraction elimination algo- rithm from Corollary 5 to obtain a derivation inKSgwith premissV
j6=0(a00∧¯a00) and same conclusion. An application of = gives V
j6=0a00∧V
j6=0¯a00, so now re- construct the proof above with V
j6=0a00 substituted for a00 andV
j6=0¯a00 for ¯a00, applying weakening to the whole formula if required. If a00 (or ¯a00) was created by an identity step then, once this procedure has been done for all atoms, push
all would-be identity steps to the top and apply cocontraction elimination to each conjunct, giving only a quadratic blowup in proof size.
