Proof composition mechanisms and
their geometric interpretation
(Somewhat IDEOLOGICAL talk)
Alessio Guglielmi
University of Bath
5 November 2013
This talk is available at http://cs.bath.ac.uk/ag/t/PCMTGI.pdf
Deep inference web site: http://alessio.guglielmi.name/res/cos/
Outline
Problem: Compressing proofs and proof (search) spaces
Solution: Two proof composition mechanisms beyond Gentzen
Open deduction: composition by connectives and inference
Open deduction and complexity: smaller proofs than in Gentzen, locality
Atomic ows: locality brings geometry
Cut elimination by experiments: Gentzen's structure is too rigid
Normalisation with atomic ows: geometry is enough to normalise
Composition by substitution: more geometry, more ef ciency, more
naturality
1
1
h
Compressing proofs
h
1
1
h
h
ON THE PROOF COMPLEXITY OF DEEP INFERENCE
Corollary
Systems
xFrege
and xSKSg
areStructural
p-equivalent.
Known
mechanisms:
How can 5.6.
we make
proofs
smaller?
rules
Lo
1. Re-use
the to
same
sub-proof:
cut
rule.
Proof
theory.
We
now move
the substitution
rule.

A
A
A ∧ Ā
1
w↑ is a Frege system
c↑ augmented
2.
Re-use
the
same
sub-proof:
dagness
, system
or cocontraction:
.
i↑Frege

Definition 5.7. A substitution
(proof
)


t
A∧ A
f


A


coweakening
cocontraction
. cointeraction
We
denote equivalent
by sFrege
the
proof system
where a proof
the3.substitution
rule sub 
In Frege,
to (4).
Substitution:



Aσ

or cut

derivation with no SKSg
premisses, conclusion αk , and shape
4. Tseitin extension:
p ↔ A (where
p is a fresh atom). Optimal?∨

t
A A
A

α f≡
αi1 i≡


↓2nd order propositional).
w↓ i h
c↓
s

5. Higher orders (including

"#
$
!
"#
$
!

(

, . . . , αih −1 , α jhAσ h , αih +1 , . . . , αk A ,
α1 , . . . , αi1 −1
, α j1 σ1A, α∨iĀ


1 +1

 interaction
weakening
contraction
where
all see
the that
conclusions
substitution
instances
αi1do
, . .with
. , αih are singled out, α
We will
1– 4 (andofaor
bitidentity
also 5) have
a lot to
{α
αi1 −1 }, . . . , α jhthis
∈ {α
, . . . ,main
αih −1 },
and the rest of the proof is as in Frege.
proof
is1our
tool.
1 , . . . ,composition:
FIGURE 2. Systems SKSg and KSg.
We rely on the following result.
Our main objective is providing for small spaces of canonical proofs
Theorem
5.8. (Cook-Reckhow
and Krajíček-Pudlák,
[CR79, KP89])
Systems xF
(= eliminating
bureaucracy = getting
good proof semantics).
Atomic structural rules
Logic
and sFrege are p-equivalent.
1
Dagness is a terrible
 word. It∧ is a combination of everything evil in this world. For
a the
ā same substitution
a
a Frege. The rule is used
We can extend SKSg with
rule as for
example lies, smoke,
gall, ugly
darkness,aw
greedy,
venereal ac
disease,
war, death and so
ai↑ teeth,
↑instances
↑


other
proper
rules
of
system
SKSg
,
so
its
are
interleaved
∧ a “My with

f
t
a
on. The word dagness
can
be
compared
to
the
devil
but
it
is
much
worse.
biggest=-rule
fear instan




is dagness”. The
Dictionary.

Definition
5.9.Urban
An cointeraction
sSKSg
proof is coweakening
a proof of SKSgcocontraction
where, in addition to the infer
Compressing proof (search) spaces
How can we make proof (search) spaces smaller? This also has a lot to
do with proof composition:
By allowing for more composed proofs we get:
I
More proofs in the proof (search) space. This might be bad.
I
Small (search) subspaces of canonical proofs. This is good.
How can we get better composition
mechanisms?
Less is more. Let's make better use of what we have already.
Given two proofs φ : A ⇒ B and ψ : C ⇒ D:
1. For a logical connective ? we have:
φ ? ψ : (A ? C) ⇒ (B ? D) .
Proofs are composed by the connectives of the formula language.
2. For an inference rule B/C we have:
φ/ψ : A ⇒ D .
Proofs are composed by inference rules.
3. For an atom a we have:
φ{a
ψ} : A{a
C, ā
D̄} ⇒ B{a
Proofs are composed by substitution.
D, ā
C̄} .
Two new formalisms
Open deduction: composition by (1) connectives and (2) inference rules.
I
I
I
I
I
I
It exists [7] and it can be taken as a de nition for deep inference.
It generalises2 the sequent calculus by removing its restrictions.
Sequents ≈ `depth-1 inference without full inference composition'.
Hypersequents ≈ `depth-2 inference without full inference
composition'.
It compresses proofs by cut, dagness and depth itself (new,
exponential speed-up).
I'll show the main ideas and some results.
`Formalism B': open deduction + (3) substitution.
I It almost exists (work with Bruscoli, Gundersen and Parigot).
I It further compresses proofs by substitution (conjectured further
superpolynomial speed-up, it is equivalent to Frege + substitution).
I I'll show some ideas and what I think we can get.
2
It does not extend it. `Extension' is another terrible word much worse than the
devil, Agata.
notion of substitution to derivations in the natural way, but this requires
The issue is clarified in Section 3 (see, in particular, Notations 3 and 5 and Pro
The issue is clarified in Section 3 (see, in particular, Notations 3 and 5 and
System SKS is a CoS proof system, defined by the following structural infe
System SKS is a CoS proof system, defined by the following structural i
t t
f f
a ∨ aa ∨ a
Open-deduction system SKS
ai↓ ai↓ −−−−−−
I
Atomic rules:
a ∨ aā ∨ ā
identity
identity
a ∧ aā ∧ ā
ai↑ ai↑ −−−−−−
f f
cutcut
aw↓aw↓ −−−
a a
weakening
weakening
ac↓ ac↓ −−−−−−
a a
contraction
contraction
, ,
a a
t t
a ∧ aa ∧ a
coweakening
cocontraction
coweakening
cocontraction
a a
aw↑aw↑ −−−
ac↑ ac↑ −−−−−−
following
logical
inference
rules:
andand
by by
thethe
following
twotwo
logical
inference
rules:
I
I
I
The
SKS
∧ B) ∨ (C ∧ D)
(Aβ)
∧
∨
∧ δ)
(α
−−−−−−−−−(γ
−−−−−−−
−−−−−−−−
m−
∨ C ] ∧ [B ∨ D] .
Linear rules:
[A
[α ∨ γ ] ∧ [β ∨ δ] .
medial
medial
C
In addition to these rules, there is a rule γ= −−−, such that C and D are oppos
In addition to these rules, there is a rule = ,Dsuch that γ and δ are opposite
of
the following
equations: commutativity,
Plus an `='
linear
rule (associativity,
units).
δ
of the following equations:
∨B =B ∨A
A
A∨ f = A
Negation on atoms only.
∨α
∨f =α
α ∨Aβ∧=
β
α
∧
B =B A
A∧ t = A
(1)
.
∧β=β∧α
∧t=α
α
α
∨
∨
∨
∨
[A B] C = A [B C ]
t ∨ t = t.
(1)
∨ β] ∨ γ = α ∨ [β ∨ γ ]
cut is atomic.
[α (A
t ∨ tf =
t
∧ B) ∧ C = A ∧ (B ∧ C )
∧f =f
(α ∧ β) ∧ γ = α ∧ (β ∧ γ )
f ∧f =f
We do not always show the instances of rule =, and when we do show th
is complete for propositional logic. See [2].
instances
into one.
We consider
the =we
rule
implicitl
Weseveral
do notcontiguous
always show
the instances
of rule
=, and when
doasshow
them
s
∧ [B ∨ C ]
∨ γ]
αs∧A
−−[β
−−−−−−−−
−−−−−−
∧ B) ∨ C
(A
(α ∧ β) ∨ γ
switch
switch
m
=
[f ∨ a] ∧ ā

=
(ā ∧ [a ∨ t]) ∨ t
s

Examples
of
open deductionac↑ a ∧ ā
[(ā ∧ a) ∨ t] ∨ t

a

=
(a ∧ a) ∧ ā

=
(a ∧ ā)b ∨ t
∧ ā 
a
ai↑ ∨
a ∧ (a ∧ ā)

a
a
ai↑

tb
I
∧ a fb∨ ∧
∧
a

=
a∧f
m
a∧a
a∧a
t ∨ b]
[a ∨ b ] ∧ [a
+
*

∧ ā
t
 a ∧ ā ∨
t

ā ∨ a
s
a ∨ ā

m
ā ∨ ā

 a∧
[a ∨ t] ∧ [t ∨ ā]
a

s


ā ∨
I

[a ∨ t] ∧ ā
a∧a
s

f
 a ∧ ā
∨ t
=


∨t
a ∧ ā
f
a∧
+
m
=
∧





ā 



[(a ∧ a) ∨
([a ∨ b ] ∧
([a ∨ b ] ∧
a
m
[a ∨ b ] ∧ [
f
ns in CoS and Formalism A notation,
Proofs are composed by the same operators as formulae (horizontally)
and by inference rules (vertically).
Top-down symmetry: so inference steps can be made atomic
(the medial rule, m, is illegal in Gentzen).
∨
a ∧a b
BG04, BT01].
contributions fit inand
the following
picture.
OpenOurdeduction
proof
complexity (size)
open
cut-free
op. ded.
op. ded.
op. ded. +
extension
2
1×
Brünnler
’04
Frege
cut-free
Gentzen
×
Gentzen
op. ded. +
substitution
4
open
CookReckhow ’74
Statman ’78
!
3
5
!
Krajíček-Pudlák ’89
Frege +
extension
Frege +
substitution
Cook-Reckhow ’79
−→
`polynomially
The=notation
" simulates'.
# indicates that formalism " polynomially simulates formalism
# ; the notation " × # indicates that it is known that this does not happen.
Thededuction:
left side of the picture represents, in part, the following. Analytic Gentzen sysOpen
tems, i.e., Gentzen proof systems without the cut rule, can only prove certain formulae,
I
the
thanks with
to deep
inference,
an exponential
whichinwe
callcut-free
‘Statmancase,
tautologies’,
proofs
that growhas
exponentially
in the size of
speed-upOn
over
cut-free
sequent
calculus
(e.g.cut
, over
the formulae.
the the
contrary,
Gentzen
systems
with the
rule Statman
can prove Statman
tautologies
by polynomially
growing proofs. So, Gentzen systems p-simulate analytic
tautologies)—1,
see [3];
IDate:
has
as small
March
15, 2010.proofs
as the best formalisms—2, 3, 4, 5, see [3];
This research was partially supported by EPSRC grant EP/E042805/1 Complexity and Non-determinism in
I Inference.
thanks
Deep
to dagness, has quasipolynomial cut elimination (instead of
$
c exponential)
ACM, 2009. This is
the10].
authors’ version of the work. It is posted here by permission of ACM for your
[4,
personal use. Not for redistribution. The definitive version was published in ACM Transactions on Computa-
Open deduction and proof search complexity
Unconstrained bottom-up formula-driven proof search has horrendous
complexity due to deep inference, because every connective can make
the search tree branch.
However:
1. Das proved that in the presence of distributivity, a depth 2 proof
system polynomially simulates any unbounded depth proof system
[5]. This means that a very moderate increase of nondeterminism
buys exponentially smaller proofs.
2. Focusing techniques should be facilitated by the more liberal proof
composition.
3. In particular it should be possible to con ne the search inside small
sub-spaces of canonical proofs.
4. THE SEQUENT CALCULUS WAS DESIGNED TO MAKE PROOF
SEARCH FINITE, NOT NECESSARILY TO MAKE IT EFFICIENT.
5. SUGGESTION: To exploit (1) design sort of a hypersequent calculus
(so, depth 2 and no more) with full inference composition (=
top-down symmetry). Start from Statman tautologies.
∨t
Locality
Deep
ac↑
=
a ∧ ā
=
(a ∧ a) ∧ ā
(a ∧ ā) for
a ∧allows
inference
ai↑
a∧f
([a ∨ b ] ∧ [a ∨ b ]) ∧ (a ∧ a)
([a ∨ b ] ∧ a) ∧ ([a ∨ b ] ∧ a)
locality,
i.e.,
+ checked
* can be
 steps
 in constant time
inference
t
∧ small).

ā ∨
(so, 
theyaare


ā ∨ a
s



∧ ā 
ā ∨ ā
a
b

 a∧

∨
ā]
a
a


∧
∨
∧
 E.g., 
a a b ∧b
atomic ācocontraction: 
∧
m
a a
a∧a

[a ∨ b ] ∧ [a ∨ b ]
f
∨ t
=

a ∧ ā
Thanks to locality
a ∧ Gundersen, Heijltjes and Parigot obtained a typed
f
λ-calculus that achieves
fully lazy sharing [9].
In Gentzen:
I no locality for (co)contraction (counterexample in [1]),
I no local reduction of cut into atomic form.
THIS IS BECAUSE AT THE TIME OF GENTZEN COMPLEXITY AND
SEMANTICS
WERE NOT
A CONCERN:
GENTZEN
ONLY WANTED A
E 2. Examples
of derivations
in CoS
and Formalism
A notation,
WELL-ORDER.
ociated atomic flows.
=
f t
a∧f
Atomic t ows: locality
brings
geometry
+
*


t
 a ∧ ā ∨

ā ∨ a
s

∨
ā ā

 a∧
a

ā ∨

a∧a
t
m
s
a ∨ ā
[a ∨ t] ∧ [t ∨ ā]
[a ∨ t] ∧ ā
s
 a ∧ ā

∨t
∨


t

=
f
f
a∧
a ∧ ā
∧




ā 



a
m
∨
b
a ∧a b ∧ b
[a ∨ b ] ∧ [a ∨ b ]
∧
f
FIGURE
2. Examples
of derivations
CoSare
and shown:
Formalism A notation,
Below the
proofs,
their (atomic)
owsin [6]
and associated atomic flows.
I
only structural information is retained in ows;
I
logical information is lost;
the right flow cannot:
I
I
ow size is polynomially related to derivation size;
composition
of proofs
naturally
correspond to composition of
ψ
ψ!
φ
,
and
.
ows.
a
a∧a
Flow reductions: (co)weakening (1)
NORMALISATION CONTROL IN DEEP INFERENCE VIA ATOMIC FLOWS
→
→
→
→
→
→
→
Each ow reduction
→ corresponds to a correct proof
→ reduction.
15
substitute over them. We repeatedly examine each coweakening instance aw↑
Flow reductions: (co)weakening (2)
a
t
in Π! , fo
some edge ε of φ, and we perform one transformation out of the following exhaustiv
speciζ {es}:that
E.g.
, for some Π!!→
list of
cases,
, Φ, Ψ, ξ { } and
(1)
→
−
$
Π!! $
!
ξ
−
$
Π!! $
"
t
#
a ε ∨ ā
$
Φ$
!
→ aε
ζ
t
$
Ψ$
"
ξ
becomes
t∨
$
Φ{a /t} $
→
−
$
Π!! $
$
ā
ε
;
ζ { t}
$
Ψ$
α
α
(2)
f
−
$
" reductions instead than
We can operate on! ow
Π!! on derivations:
! $
"
f
f ∧ [ t ∨ t]
I much easier,ξ
aε
ξ s
∨t
(f ∧ t)measures.
I we get natural, $syntax-independent induction
Φ$
$
ε
becomes
;
→
! ε"
Φ{a /t} $
a
Flow reductions: (co)contraction
→
→
→
→
→
Figure 2: Atomic-flow reduction rules.
We
to use the conserve
reductions in
2 as and
ruleslength
for rewriting
inside generic
I would
Theselikereductions
theFigure
number
of paths.
atomic flows. To do so, in general, we should have matching upper and lower edges in
I Open
problem:indoes
cocontraction
superpolynomial
the flows
that participate
the reduction,
and theyield
reductions
in the figure clearly do so.
However, compression?
we also have to pay attention to polarities, not to disrupt atomic flows. In fact,
consider the following example.
Example 4.2. The ‘reduction’ on the left, when used inside a larger atomic flow, might
create a situation as on the right:
+
Cut
Experiment
elimination over a proof:
a ā
"
by
t f
`experiments'
^
^
f!
f!
a
a ^ āā
!a "
t^f
!
f _a
#
ā _ a
a
ā
a
many identities
A1
↵
We do:
↵1
!
B
A2n
↵2n
···
B
B
all assignments
W
‘experiments’
many contractions
B
proof with n cuts
I
I
I
cut-free proof
Simple, exponential cut elimination;
2n experiments, where n is the number of atoms;
fairly syntax independent method.
The secret of success is in the proof composition mechanism.
WHY IS THIS IMPOSSIBLE IN THE SEQUENT CALCULUS?
f _a
#
ā _ a
proof of Theorem 27, respectively as Case (4), (1), (2) and (3). If we apply the full set of
weakening
and coweakening
until form
possible, starting from a proof in cut-free
Generalising
thereductions
cut-free
The equality follows by induction on the number of ve
form, we obtain a proof of the same formula and whose flow has shape
edges to cross, For ai↑ we proceed dually.
Definition 2.9. An atomic flow is weakly stream
.
(resp., streamlined and strongly streamlined) if it ca
represented as the flow on the left (resp., in the middle
on consisting
the right): of the reductions in Figure 6 is confluent.
Note that the graph rewriting system
I
Normalised proof:
8. FINAL COMMENTS
System
aSKS is not aderivation:
minimal complete system for propositional logic, because the
I Normalised
atomic cocontraction rule ac↑ is admissible (via ac↓, ai↑ and s). Removing ac↑ from
aSKS yields system KS. A natural question is whether quasipolynomial normalisation
holds for KS as well. We do not know, and all indications and intuition point to an
essential role being played by cocontraction in keeping the complexity low. Analysing
Figure 5I shows
how cocontraction
provides
for a typical ‘dag-like’ speed-up over the
The symmetric
form is called
streamlined.
corresponding ‘tree-like’ expansion consisting in generating some sort of Gentzen tree.
I Cut elimination is a corollary of streamlining.
However,
we are aware that in the past this kind of intuition has been fallacious.
I
We just need to break the paths between identities and cuts, and
(co)weakenings do the rest.
” (Definition 2.9) is
the same atomic types, as indicated in (5), and let
flow
How do we break
paths?
be the
With
path breaker [8]:
ed atomic
flowthe
has
Figure
3.
, and this normalLocal rewrite rules
nition 5.1. Let be an atomic flow of the shape
mination here since
at the crucial point
ition 4.1, we have
preserves the prop→
n the normal form
is no generator ,
.
(5)
.
(6)
e the wires of the selected
and
generators carry
ame atomic types, as indicated in (5), and let be the
ns
esent a method for
ly streamlined
call identity
a path breaker
withleft,
respect
, and
Even if one.
there is aThen
path we
between
and cutofon the
theretois none
eration on
to proofs
in
write
.
the right.
way to break paths
rem 4.2 and Theorem 7.3.
That we
call derivations,
and , respectively. We can of
now build
We can do the same
on
course
References
with flow is as shown
and be
whose
in given.
(8). By
pplying (9) we get the derivation
Lemma 7.6. The relation
can be lifted to
Proof. Let
with
applying (9) we have a derivation
,
, from which we can obtain a derivation
with
. We also have
and
.
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Proofs. PhD thesis, Technische Universität Dresden, 20
[2] K. Brünnler and A. F. Tiu. A local system for classical lo
In R. Nieuwenhuis and A. Voronkov, editors, LPAR 20
volume 2250 of LNAI, pages 347–361. Springer, 2001.
[3] P. Bruscoli and A. Guglielmi. On the proof complexit
deep inference. ACM Transactions on Computational Lo
10(2):1–34, 2009. Article 14.
,
,
[4] P. Bruscoli, A. Guglielmi, T. Gundersen, and M. Pari
A quasipolynomial cut-elimination procedure in deep in
ence via atomic flows and threshold formulae. submit
2010.
[5] S. R. Buss. The undecidability of -provability. Annal
Pure and Applied Logic, 53:72–102, 1991.
[6] V. Danos and L. Regnier. The structure of multiplicati
Annals of Mathematical Logic, 28:181–203, 1989.
[7] J.-Y. Girard. Linear logic. Theoretical Computer Scie
50:1–102, 1987.
J.-Y.inGirard.
whose atomic flow is as [8]
shown
(6). Proof Theory and Logical Complexity, Vol
I, volume 1 of Studies in Proof Theory. Bibliopolis, ediz
scienze,
Theorem 7.7. The relation di filosofia
can be elifted
to 1987.
.
and
be given. By
→
I
We can compose this as many times as there are paths between
,
identities and
cut.
I
10
We obtain a family of normalisers that
only depends on n.
I
there exists a weakly-streamlined
The construction is exponential.
I
Finding something
Proof. Immediate from Lemmas 7.5 and 7.6.
Proof Theorem 7.1. For every
-derivation
-derivation
by Theorem 5.7 and Theorem 7.7; for every weakly-derivation without ows.
there exists a
like streamlined
this is unthinkable
strongly streamlined
-derivation
by Theorem 4.2 and Theorem 7.3.
NORMALISATION CONTROL IN DEEP INFERENCE VIA ATOMIC FLOWS II
Example for 2 cuts
where φ1 is the atomic flow associated with a, φ2 is the atomic flow associated with b and
a and b are the only non-weakly-streamlined atoms in Φ, then the atomic flow associated
with Norm2 (a, b , Core(Φ)) is
φ1
→,
φ2
ψ
φ2
φ1
φ1
1 is the atomic flow associated with a, φ2 is the atomic flow associated with b and
are the only non-weakly-streamlined atoms in Φ, then the atomic flow associated
rm2 (a, b , Core(Φ)) is
φ1
φ1
φ1
φ1
φ1
φ1
φ2
φ2
φ2
φ2
φ2
φ1
φ1
φ1
φ1
φ2
φ1
φ1
φ1
φ2
φ1
φ1
φ1
φ2
24
PAOLA BRUSCOLI, ALESSIO GUGLIELMI, TOM GUNDERSEN, AND MICHEL PARIGOT
Quasipolynomial
cut elimination
by
threshold functions
φ!0
φ
..
.
θ1
θk
φ!k
ψk
φ
..
.
YT
TTYJVi.,
θk+1
θn
φ!n
φ
··· ···
α
I
I
Atomic flow of a proof in cut-free form.
Only n + 1 copies ofF the5.proof
are stitched together.
where ψ is the union of flows φ , . . . , φ , and where we denote by α the edges corresponding to the atom occurrences
in thesharing,
conclusion α of Π.
We then
have that,
Note local cocontraction
(=appearing
better
not
available
in
for 0 < k < n, the flow of Φ is φ , as in Figure 5, where ψ is the flow of the derivation
Gentzen). Ψ . The flows of Φ and Φ are, respectively, φ and φ .
IGURE
1
k
k
0
n
�
k
n
�
0
�
n
k
•
They have minimal complexity relative to all
The main goa
known
proof systems
(efficiency).
`Formalism B':
allowing
substitution
Formalism B a
•
The proofs they represent are closer to seWe need to a
mantics than those of existing proof systems
in the broade
(naturality).Proof systems (proof complexity)
avoid idiosync
Normalisation and analyticity
At the core of Formalism B there is a substitution
(proof theory)
tion. The Mat
Deep inference
B
notion for atomic flows thatFormalism
ensures
low complexity provides a pe
Truth
Frege
Gentzen
Calculus of Open
Atomic
Girard
and the
removal
ofstructures
a certain
kind of proofflows
bureauorySemantics
and norm
tables
systems
formalisms
deduction
proof nets
Syntax
late 1800s
1900
2001
2010
2008
1987
cracy. ∼The
set1935
of atomic
flows
must then be
closed
perience in se
under substitution and subjected to certain equalanguages [6,
Figure 1: Converging towards proof systems with normalisation and analyticity and that are as close as possible to natural
Forline
example,
thisofisincreasing
what abstraction.
should happen when theory as an o
semantic structures;tions.
the horizontal
carries formalisms
Achieving the power of Frege + substitution (possibly optimal proof
the atomic flow on the right inside the parentheses
try as a found
system)
incorporating
substitution,
guided
geometry
the notion ofby
proof
system give us a natural
target
logics by
thanthe
Gentzen’s
[15, 2]. of ows:
is substituted into the atomic flow on the left:
There are two
for designing formalisms. Our goal is the answer to: Atomic flows (2008) They prove that normalisa-
tion is an independent phenomenon from synEff Efficien
tax and suggests further abstractions for formalisms.
Atomic
flows,
contrary
to
proof theme
nets,
This
w
→
=
.
The question is vague but it is useful to understand
are purely geometric objects [16, 19].
this project and our previous work on deep inferabsolute
and
Open deduction (2010) Inspired by atomic flows, it
ence. Our goal is to design a formalism of efficient
classical
removes a specific and pervasive kind
of bu- logic
and natural proof systems, and the idea is to proreaucracy while generalising and preserving
picture
describes
atomic flows,
gressively refine The
the syntax,
while being
guided bypart of a proof in a dag-like
all the properties of the more syntactic forsemantics (naturality)
andsystem
while making
that
proof
( sure
represents
dag
sharing)
a) depth of in
malisms
[18].16 where a
the complexity of proofs decreases (efficiency).
contraction ( ) occurs. The
ideainference
is thatwethe
In deep
can geoexpress important logics
Question Which formalism describes the proof sys-
� proofs?
�
tems with the most abstract
�
F IGURE 3. Example of flow substitutions.
Example of ow substitution
!
←
"

















−
"
"
,
−
,
−
"
,
−
F IGURE 4. Example of flow substitution.

















The ows represent proofs. The bigger the set on the right, the more
bureaucracy is captured by the substitution, the smaller the set of
canonical proofs is.
the subflows of kind ,
and
are replaced according to the obvious
dual
rules; of
theshapes,
paths ofall
β are
respected
the introduction
#co$contraction,
Note the
variety
of which
areby
equivalent.
This isoffar
more
identity and cut nodes, as per Conventions 7 and 9.
exible than permutation of rules and similar Gentzen mechanisms.
,
Gundersen's substitution trick
Substituting a proof φ inside an identity or cut stands for a set of proofs
with as many elements as ways to break φ.
This iterated mechanism alone generates one canonical form for an
exponentially big class of proofs.
first building block is connected components and so flows, which are then decorated
with extra relations between edges, that in turn form derivations and formulae. In
summary, we do not extarct flows from derivations, rather we build derivations on
top of flows.
Consider the following two synchronal open deduction derivations:
Lifting ow substitutions to proofs
t
i↓ −−−−−−−−−−−−−−−−
a
φ!
∨ ā
c↑ −−−−−−− ∨ ā
∧
a a
! −−−−−−−−−−−−−−−−−−−−−−−−−−−−
ā ∨ ā
c↓ −−−−−−−
ā
#a ∧ a$ ∨
w↑ −−−
and
ψ!
f
b ∨ −−−
b
.
−−−−−−−−
b
t
We want to define a denotation for the formal substitution φ | a ← ψ. One element in
the set of denotations of φ | a ← ψ is
t
i↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
b∨f
−−−−−−−−−−−−"
−−−−−−−−−−−−−−−−−−
c↑ !
#
$
∨
f
b̄ ∧ t
∧ %b ∨ f&
b ∨ −−−
∨
b̄
−−−−−−
b̄ ∧ b̄
b
−−−−−−−−−−−−−,
−−−
! −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+

b ∨ b
−−−−−−
b
∧

f
b ∨ −−−
b
−−−−−−−−
b
∨
#
b̄ ∧ t
$
b̄
b̄ ∧ −−−
t
∨
c↓ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
b̄ ∧ t
w↑ −−−−−−
t
.
Conclusions
I
We are interested in proof composition (so in the rst and second
order propositional proof theory).
I
Composition in Gentzen is rigid (it was designed for consistency
proofs, not much else).
I
Deep inference composition is free and yields local proof systems.
I
Locality = linearity + atomicity, so we are doing an extreme form
of linear logic.
I
Because of locality we obtain a sort of geometric control over
proofs.
I
So we obtain an ef cient and natural formalism for proofs, where
more proof theory can be done with lower complexity.
I
We are obtaining interesting notions of proof semantics.
This talk is available at http://cs.bath.ac.uk/ag/t/PCMTGI.pdf
Deep inference web site: http://alessio.guglielmi.name/res/cos/
References
[1]
K. Brünnler.
Deep Inference and Symmetry in Classical Proofs.
Logos Verlag, Berlin, 2004.
http://www.iam.unibe.ch/~kai/Papers/phd.pdf.
[2]
K. Brünnler and A. F. Tiu.
A local system for classical logic.
In R. Nieuwenhuis and A. Voronkov, editors, Logic for Programming, Arti cial Intelligence, and Reasoning (LPAR), volume 2250 of Lecture Notes in Computer Science, pages 347–361.
Springer-Verlag, 2001.
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ACM Transactions on Computational Logic, 10(2):14:1–34, 2009.
http://cs.bath.ac.uk/ag/p/PrComplDI.pdf.
[4]
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A quasipolynomial cut-elimination procedure in deep inference via atomic ows and threshold formulae.
In E. M. Clarke and A. Voronkov, editors, Logic for Programming, Arti cial Intelligence, and Reasoning (LPAR-16), volume 6355 of Lecture Notes in Computer Science, pages 136–153.
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[5]
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http://www.anupamdas.com/items/PrCompII/ProofComplexityBoundedDI.pdf.
[6]
A. Guglielmi and T. Gundersen.
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http://www.lmcs- online.org/ojs/viewarticle.php?id=341.
[7]
A. Guglielmi, T. Gundersen, and M. Parigot.
A proof calculus which reduces syntactic bureaucracy.
In C. Lynch, editor, 21st International Conference on Rewriting Techniques and Applications, volume 6 of Leibniz International Proceedings in Informatics (LIPIcs), pages 135–150. Schloss
Dagstuhl–Leibniz-Zentrum für Informatik, 2010.
http://drops.dagstuhl.de/opus/volltexte/2010/2649.
[8]
A. Guglielmi, T. Gundersen, and L. Stra burger.
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In J.-P. Jouannaud, editor, 25th Annual IEEE Symposium on Logic in Computer Science (LICS), pages 284–293. IEEE, 2010.
http://www.lix.polytechnique.fr/~lutz/papers/AFII.pdf.
[9]
T. Gundersen, W. Heijltjes, and M. Parigot.
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In 28th Annual IEEE Symposium on Logic in Computer Science (LICS). IEEE, 2013.
In press.
[10]
E. Jerábek.
Proof complexity of the cut-free calculus of structures.
Journal of Logic and Computation, 19(2):323–339, 2009.
http://www.math.cas.cz/~jerabek/papers/cos.pdf.