A Proposal for a New Foundation for
Proof Theory
Alessio Guglielmi
University of Bath
11 March 2016
This talk is available at http://cs.bath.ac.uk/ag/t/PNFPT.pdf
Deep inference web site: http://alessio.guglielmi.name/res/cos/
1 / 46
Outline
Proof semantics – When are two given proofs the same?
Deep inference – Free composition of proofs
Deep inference for classical logic
Cut elimination
The noncommutative linear logic BV
Atomic ows – Locality yields topology
Normalisation with atomic ows – Topology is enough to normalise
2 / 46
When are two given proofs the same?
Remove bureaucracy; compare shapes. Proofs → proof nets:
Proof Nets and the Identity of Proofs
11
id
id
!
! a⊥ , a
"
!
!
id
id
! a, a⊥
! a⊥ , a ! a, a⊥
a⊥
exch
"
id
"(a ! a), a⊥
!
a⊥ , a
! a⊥ "(a ! a), a⊥ ! a⊥ , a
! a⊥ "(a ! a), a, a⊥ ! a⊥
! a⊥ "(a ! a), a "(a⊥ ! a⊥ )
exch
id
!
↓
"
! a⊥ , a
"
id
! a, a⊥
!
!
!
! a⊥ , a
! a, a⊥ ! a⊥ , a
a, a, a⊥
! a⊥
! a, a "(a⊥ ! a⊥ )
! a⊥ , a ! a, a "(a⊥ ! a⊥ )
! a⊥ "(a ! a), a "(a⊥ ! a⊥ )
id
!
!
"
exch
↓
"
a, a⊥
id
! a⊥ , a
!
"
!
!
id
! a,, a⊥
! a⊥ , a ! a,, a⊥
a⊥
!
a⊥ , a
! a⊥ "(a
"( ! a),
), a⊥ ! a⊥ , a
exch
"
id
"( ! a),
"(a
), a⊥
! a⊥ "(a
"( ! a),
), a,, a⊥ ! a⊥
! a⊥ "(a
"( ! a),
), a "(a
"( ⊥ ! a⊥ )
!
! a,, a⊥
exch
id
!
↓
"
! a⊥ , a
"
id
!
a,, a,, a⊥
! a⊥
! a,, a "(a
"( ⊥ ! a⊥ )
! a⊥ , a ! a,, a "(a
"( ⊥ ! a⊥ )
! a⊥ "(a
"( ! a),
), a "(a
"( ⊥ ! a⊥ )
!
! a⊥ , a
"
!
id
! a,, a⊥
! a⊥ , a ! a,, a⊥
id
! a⊥ "(a
"( ! a),
), a⊥
exch
"
! a⊥ , a
! a⊥ "(a
"( ! a),
), a⊥ ! a⊥ , a
! a⊥ "(a
"( ! a),
), a,, a⊥ ! a⊥
! a⊥ "(a
"( ! a),
), a "(a
"( ⊥ ! a⊥ )
↓
a⊥
a
"
a
!
a⊥
a
"
!
! a,, a⊥
exch
id
!
"
! a⊥ , a
"
a⊥
!
id
id
!
!
"
exch
"
a,, a⊥
! a,, a,, a⊥ ! a⊥
! a,, a "(a
"( ⊥ ! a⊥ )
! a⊥ , a ! a,, a "(a
"( ⊥ ! a⊥ )
! a⊥ "(a
"( ! a),
), a "(a
"( ⊥ ! a⊥ )
↓
a⊥
a
"
a
!
a⊥
a
"
a⊥
!
id
! a⊥ , a
exch
!
! a,, a⊥
! a⊥ , a ! a,, a⊥
! a⊥ , a⊥ , a ! a
a⊥ , a⊥
"( ! a))
"(a
! a,, a⊥ ! a⊥ , a⊥ "(a
"( ! a))
! a "(
"(a⊥ ! a⊥ ), a⊥ "(a
"( ! a))
! a⊥ "(a
"( ! a),
), a "(a
"( ⊥ ! a⊥ )
↓
!
! a⊥ , a
! a,, a⊥ ! a⊥ , a
"(a ! a)
↓
!
↓
id
a⊥ , a⊥
! a, a⊥ ! a⊥ , a⊥ "(a ! a)
id
id
! a⊥ , a⊥ , a ! a
! a⊥ "(a ! a), a "(a⊥ ! a⊥ )
! a⊥ , a
! a,, a⊥ ! a⊥ , a
!
! a, a⊥
! a⊥ , a ! a, a⊥
! a "(a⊥ ! a⊥ ), a⊥ "(a ! a)
id
id
id
! a⊥ , a
exch
! a⊥ , a
id
!
"
! a,, a⊥
"
exch
id
! a,, a⊥
! a⊥ , a ! a,, a⊥
exch
! a⊥ , a⊥ , a ! a
! a⊥ , a⊥ "(a
"( ! a))
! a,, a⊥ ! a⊥ , a⊥ "(a
"( ! a))
! a "(a
"( ⊥ ! a⊥ ), a⊥ "(a
"( ! a))
! a⊥ "(a
"( ! a),
), a "(a
"( ⊥ ! a⊥ )
↓
a⊥
a
"
a
!
Figure 2: From sequent calculus to proof nets via coherence graphs
a⊥
a
"
a⊥
!
Figure taken from [20]
Remove too much → no more a proof system: proof nets are not a
proof system.
2.3.4 Exercise Reduce in (6) the leftmost instance of id to atomic version. And draw the proof net according
to the method in Figure 1. What does change compared to the net in (9)?
For dealing with cuts (without forgetting them!), we can prevent the flow-graph from flowing through the
cut, i.e., by keeping the information that there is a cut. What is meant by this is shown in Figure 4.
2.3.5
Exercise Compare the net obtained in Figure 4 with your result of Exercise 2.3.4.
Proof semantics – When are two given proofs the same?
3 / 46
Proof Systems
I
Proof system = algorithm checking proofs in polytime.
I
Theorem (Cook and Reckhow):
∃ super proof system
iff
NP = co-NP
where
super = with polysize proofs over each proved tautology
Proof semantics – When are two given proofs the same?
4 / 46
A1 β1 , . . . , Ah
Compressing
proofs
↔
↔β
↔ β ) ∧ · · · ∧ (A ↔ β ) .
h , . . . , (A1
1
h
h
ON THE PROOF COMPLEXITY OF DEEP INFERENCE
Corollary
Systems
xFrege
and xSKSg
are
p-equivalent.
Known
proof
theoretic
How can 5.6.
we make
proofs
smaller?
Structural
rules mechanisms:
Log
Proof
theory.
1.
Re-use
the
same
sub-proof:
cut
rule.
We now move to the
substitution rule.
A
A
A ∧ Ā
w↑ is a Frege system
c↑
2. Re-use
theAsame
sub-proof:
dagness,
or
cocontraction:
.
i↑ Frege

Definition
5.7.
substitution
(proof
)
system
augmented

∧A

t
A
f


A


cointeraction
coweakening
cocontraction
. We
denote equivalent
by sFrege
the
proof system
where a proof
the3.substitution
rule sub
Substitution:
In Frege,
to (4).



Aσ or cut


derivation with noSKSg
premisses, conclusion αk , and shape
4. Tseitin extension:
p ↔ A (where
p is a fresh atom). Optimal?

t
A∨ A
A

αf ≡
αi1i↓≡ nd


w↓ i h
c↓
s

5. Higher orders (including
2
order
propositional).

"#
$
!
"#
$
!

A
A ,
(A
 , α σA,∨αĀ , . . . , α
α1 , . . . , αi
j1 1
i1 +1
i h −1 , α j h σ h , αi h +1 , . . . , αk

1 −1

 interaction
weakening
contraction
where
all the
conclusions
ofor
instances
αi1 , . . . , αih are
singled out, α
1– 4 (and
a bit
also 5) have
asubstitution
lot
to do with
proof composition
– our
identity
{α
,
.
.
.
,
α
},
.
.
.
,
α
∈
{α
,
.
.
.
,
α
},
and
the
rest
of
the
proof
is
as
in Frege.
main
tool.
1
i1 −1
jh
1
i h −1
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).
Logica
and sFrege are p-equivalent. Atomic structural rules

a ∧ āthe same substitution
a
We can extend SKSg with
rule as afor Frege. The rule is used
ai↑
aw
↑instances are
acinterleaved
↑


other
proper
rules
of
system
SKSg
,
so
its
with =-rule instan
t
a ∧a
Proof semantics – When 
are two given proofsf the same?
5 / 46

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.
Proof semantics – When are two given proofs the same?
6 / 46
We want to design Formalism B such that it ex-
GM G McCusk
Towards proof
systems
closest
to semantics
presses
proof systems
with the following
characPB P Bruscoli
teristics:
RA research a
Proof systems (proof complexity)
•
They have minimal complexity relative
Normalisationto
andall
analyticity The main goal o
(proof theory)
known proof systems (efficiency
Deep).inference
Formalism B as
Formalism B
•Frege The
proofsCalculus
theyof represent
are closer
to seWe need to ass
Truth
Gentzen
Open
Atomic
Girard
tables
systems
formalisms structures
flows
proof nets
mantics
than thosededuction
of existing proof
systems
Syntax
Semantics
in the
broadest
late 1800s
∼1900
1935
2001
2010
2008
1987
(naturality).
avoid idiosyncra
Atproof
thesystems
corewith
of normalisation
Formalism
there and
is athat
substitution
The Mathe
Figure 1: Converging towards
andBanalyticity
are as close as possibletion.
to natural
semantic structures; the horizontal
linefor
carries
formalisms
of increasing
abstraction. low complexity provides a pers
notion
atomic
flows
that ensures
I Open deduction:
inference
– fullbureaunormalisation
and theestablished
removal of adeep
certain
kind of proof
ory and normal
the notion
of proof system
give The
us a natural
logics
thanthen
Gentzen’s
[15, 2].
theory.
cracy.
set oftarget
atomic flows
must
be closed
perience in sem
for designing formalisms. Our goal is the answer to: Atomic flows (2008) They prove that normalisasubstitution
and subjected
to+ certain
equa- (possibly
languages [6, 24
I `Formalism under
B':
getting
the
power
of
Frege
substitution
tion is an independent phenomenon from synQuestion Which formalism describes the proof systions.
For
example,
this
is
what
should
happen
when
theory
tax
and
suggests
further
abstractions
for-as an or
tems
with the most
abstract
proofs? by incorporating substitution, guided by for
optimal
proof
system)
the
malisms.
Atomic
flows, contrary totry
proof
nets,
the
atomic
flow
on
the
right
inside
the
parentheses
as
a foundat
The question
is vague of
but atomic
it is useful toows:
understand
geometry
are
purely
geometric
objects
[16,
19].
is substituted
into
the atomic flow on the left:
this project and our previous
work on deep
inferThere are two t
Open deduction (2010) Inspired by atomic flows, it
removes a specific and pervasive kind
Effof buEfficienc
reaucracy while generalising and preserving
Thisfortheme will
=
all the properties
of the. more syntactic
malisms [18].16
absolute and re
In deep inference we can express important
logics logic pr
classical
Proof semantics itself is not set in stone, it is an acfor the
algebras that cannot
The
picture
describes part
of averification
proof inofa process
dag-like
atomic
flows, w
tive field of research, very
close
to the semantics
Proof semantics – When are two given proofs the same?
be expressed in Gentzen theory [3, 17, 33]. So, our 7 / 46
ence. Our goal is to design a formalism of efficient
and natural proof systems, and�the idea
�is to pro�
gressively refine the syntax, while being guided by →
semantics (naturality) and while making sure that
the complexity of proofs decreases (efficiency).
What is deep inference?
It's the free composition of proofs via the same connectives as formulae.
If
C
A
Φ=
B
and
Ψ=
D
are two proofs with, respectively, premisses A and C and conclusions B
and D, then
(Φ ∧ Ψ) =
(A ∧ C)
(B ∧ D)
and
[Φ ∨ Ψ] =
[A ∨ C]
[B ∨ D]
are valid proofs with, respectively, premisses (A ∧ C) and [A ∨ C], and
conclusions (B ∧ D) and [B ∨ D].
Deep inference – Free composition of proofs
8 / 46
Why deep inference?
I
To recover a De Morgan premiss-conclusion symmetry that is lost
in Gentzen [2].
I
To obtain new notions of normalisation in addition to cut
elimination [11, 10].
I
To shorten analytic proofs by exponential factors compared to
Gentzen [6, 8].
I
To obtain quasipolynomial-time normalisation for propositional
logic [7].
I
To express logics that cannot be expressed in Gentzen [22, 3].
I
To make the proof theory of a vast range of logics regular and
modular [3].
I
To get proof systems whose inference rules are local, which is
usually impossible in Gentzen [19].
Deep inference – Free composition of proofs
9 / 46
Why deep inference? (cont.)
I
To inspire a new generation of proof nets and semantics of proofs
[21].
I
To investigate the nature of cut elimination [10, 12].
I
To type optimal versions of the λ-calculus that are not typeable in
Gentzen [13, 14].
I
To model process algebras [5, 16, 17, 18].
I
To model quantum causal evolution [1] …
I
… and much more.
Deep inference – Free composition of proofs
10 / 46
Why deep inference? (cont.)
Several formalisms can be designed in deep inference: Calculus of
Structures (CoS), Nested Sequents, Open Deduction, Formalism B, …
CoS and open deduction are equivalent under any reasonable point of
view, so we adopt open deduction. (CoS is convenient for certain
technical aspects.)
Nested sequents is not full deep inference.
Formalism B is still in development.
Deep inference – Free composition of proofs
11 / 46
As an example, I will show the standard deep inference system for
tional logic. System SKS is a proof system defined by the following
inference rules (where a and ā are dual atoms)
Deep inference system SKS for classical logic
i#
t
w#
a ā
_
I
Atomic/structural rules:
identity
f
c#
a
weakening
a_a
a
contraction
,
i"
a ^ ā
w"
f
cut
a
c"
t
coweakening
a
a^a
cocontraction
and by the following two logical inference rules:
I
Linear/logical rules:
s
A ^ [B _ C]
m
(A ^ B) _ C
(A ^ B) _ (C ^ D)
[A _ C] ^ [B _ D]
switch
I
I
.
medial
A cut-free derivation is a derivation where i" is not used, i.e., a d
Plus an `=' linearSKS
rule
(associativity, commutativity, units).
\ {i"}. In addition to these rules, there is a rule
Negation on atoms only.
=
The cut is atomic.
C
D
,
such that C and D are opposite sides in one of the following equati
SKS is complete for propositional logic. See [4].
Deep inference for classical logic
A_B =B_A
A _ f = A 12 / 46
Example
[A _ B] _ C = A _ [B _ C]
t_t=t
(A ^ B) ^ C = A ^ (B ^ C)
f ^f =f
.
We do not always show the instances of rule =, and when we do show them, we
gather several contiguous instances into one.
For example, this is a valid derivation:
[a _ b] ^ a
([a _ b] ^ a) ^ ([a _ b] ^ a)
c"
=
m
a
a^a
_ c"
b
b^b
[a _ b] ^ [a _ b]
^ c"
a
a^a
.
This derivation illustrates a general principle in deep inference: structural rules
on generic formulae (in this case a cocontraction) can be replaced by corresponding structural rules on atoms (in this case c").
3
Structural rules on generic formulae can be replaced by structural rules
on atoms.
Proof-Theoretical Properties
Locality and linearity are foundational concepts for deep inference, in the same
spirit as they are for linear logic. Going for locality and linearity basically means
going for complexity bounded by a constant. This last idea introduces geometry
Deep inference for classical logic
13 / 46
Example with quanti
ers
ALESSIO GUGLIELMI
i#
c#
2
9x8y 4
w#
f
p(x)
_
3
t
2
p y 5 _ 9x8y 4p (x) _
h
i
9x8y p (x) _ p y
w#
f
p y
3
5
This is more natural than in Gentzen because there is no waste in the
proof.
Deep inference for classical logic
14 / 46
∨t
∨ tLocality
=
ac↑
=
a ∧ ā
(a
∧ a) ∧ ā
(a ∧ ā) for locality,
a ∧allows
Deep inference
ai↑
a∧f
m
=
([a ∨ b ] ∧ [a ∨ b ]) ∧ (a ∧ a)
([a ∨ b ] ∧ a) ∧ ([a ∨ b ] ∧ a)
i.e.,
+
*


t
inference
∧ ā ∨can be checked
 a steps
 in constant time (so, they are small).


ā ∨ a
s



∧ ā 
ā ∨ ā
a
b



∨
ā]
a
a
a∧


∧
∨
∧
 E.g., 
a a b ∧b
atomic ācocontraction: 
m
a∧a
a∧a

[a ∨ b ] ∧ [a ∨ b ]
f
∨ t
=

a ∧ ā
a ∧ Gundersen, Heijltjes and Parigot obtained a typed
Thanks to locality
f
λ-calculus that achieves fully lazy sharing [13].
In Gentzen:
E 2.
I
no locality for (co)contraction (counterexample in [2]),
I
no local reduction of cut into atomic form.
Examples
offorderivations
in CoS and Formalism A notation,
Deep inference
classical logic
15 / 46
Reduction of cut to atomic form
Apply repeatedly—and locally:
i↑
[A ∨ B] ∧ (Ā ∧ B̄)
f
s
=
s
i↑
[A ∨ B] ∧ B̄
A ∨ (B ∧ B̄)
A ∧ Ā
f
∨
i↑
∧ Ā
B ∧ B̄
f
Proof complexity does not increase!
Deep inference for classical logic
16 / 46
t for the same impact that hypersequents have on the branching fac
proof
search space,
we can
obtain
much
Analyticity
costs
much
less
(1)smaller proofs than in Ge
ems, thanks to the better proof representation in deep inference. I will
example here, by reasoning on the first three Statman tautologies (see
tautologies:
formalStatman
definitions):
S1 = (a ^ b) _ ā _ b̄ ,
⇥
⇤
⇥
⇤
S2 = (c ^ d) _ c̄ _ d¯ ^ a ^ c̄ _ d¯ ^ b _ ā _ b̄ ,
⇥
⇤
⇥
⇤
S3 = (e⇥ ^ f ) _⇤ ⇥ē _ f¯⇤ ^ c ^ ⇥ē _ f¯ ⇤^ d⇥ _ ⇤
ē _ f¯ ^ c̄ _ d¯ ^ a ^ ē _ f¯ ^ c̄ _ d¯ ^ b _ ā _ b̄
.
and so on…
It is well known, and the reader will have no difficulty in seeing it, tha
In the cut-free
sequent
exponentially.with n. The stru
of cut-free
sequent
proofscalculus
of Snproofs
growsgrow
exponentially
on is that the external connectives in formulae force repeated duplic
he context. Let us see what happens if we could just access conne
mediately below the external ones.
Deep inference for classical logic
17 / 46
Analyticity costs much less (2)
deduction
proof of
S1 :
For S1Open
we have
a trivial
cut-free
proof in SKS:
t
i#
i#,s
=
s
(a ^ b) _ ā _ b̄
s
t
a _ ā
^ i#
[a _ ā] ^ b
(a ^ b) _ ā
t
b _ b̄
_
.
b̄
For S2 we can obtain:
t
i#,s
Deep inference for classical logic
18 / 46
Analyticity costs much less (3)
or S2 Open
we can
obtain:proof of S2 :
deduction
t
i#,s
t
t
i#,s
(c ^ d) _ c̄ _ d¯
^
a^
i#,s
(c ^ d) _ c̄ _ d¯
^
b
s
(c ^ d) _ (c ^ d)
c#,m
c^d
_
⇥
_
ā _ b̄
⇤
⇥
⇤
c̄ _ d¯ ^ a ^ c̄ _ d¯ ^ b
Deephow
inference the
for classical
logic
19 / 46
we see
external
atoms c and d are ‘brought inside’ the tauto
Analyticity costs much less (4)
Open deduction proof of S3 :
t
i#,s
t
t
i#,s
(c ^ d) _ c̄ _ d¯
^
a^
i#,s
(c ^ d) _ c̄ _ d¯
^
b
s
(c ^ d) _ (c ^ d)
c#,m
t
t
i#,s
(e ^ f ) _ ē _ f¯
^
c^
i#,s
(e ^ f ) _ ē _ f¯
^
d
_
0
B
@
t
i#,s
(e ^ f ) _ ē _ f¯
^
⇥
⇤
c̄ _ d¯ ^ a ^
t
i#,s
(e ^ f ) _ ē _ f¯
^
⇥
1
⇤ C
c̄ _ d¯ ^ bA
_
ā _ b̄
s
(e ^ f ) _ (e ^ f ) _ (e ^ f ) _ (e ^ f )
c#,m
e^f
_
⇥
⇤
⇥
⇤
ē _ f¯ ^ c ^ ē _ f¯ ^ d
_
⇥
⇤ ⇥
⇤
⇥
⇤ ⇥
⇤
ē _ f¯ ^ c̄ _ d¯ ^ a ^ ē _ f¯ ^ c̄ _ d¯ ^ b
In open deduction analytic Statman proofs grow polynomially.
Deep inference for classical logic
20 / 46
ism in deep inference [Gug07b], and in particular for its proof system SKS, introduced
by Brünnler in [Brü04] and then extensively studied [Brü03a, Brü03b, Brü06a, Brü06d,
BG04, BT01].
Our contributions fit in the following picture.
Deep inference and proof complexity (size)
cut-free
op. ded.
open
op. ded.
op. ded. +
extension
2
Brünnler
’04
1×
Frege
×
Gentzen
op. ded. +
substitution
4
open
CookReckhow ’74
Statman ’78
cut-free
Gentzen
!
3
5
!
Krajíček-Pudlák ’89
Frege +
extension
Frege +
substitution
Cook-Reckhow ’79
The notation
indicates that formalism " polynomially simulates formalism
"
#simulates'.
−→
= `polynomially
# ; the notation " × # indicates that it is known that this does not happen.
The left side of the picture represents, in part, the following. Analytic Gentzen sysOpen
tems, i.e.,deduction:
Gentzen proof systems without the cut rule, can only prove certain formulae,
which we call ‘Statman tautologies’, with proofs that grow exponentially in the size of
I
in
theOncut-free
case,
thanks
towith
deep
inference,
hasStatman
an exponential
the formulae.
the contrary,
Gentzen
systems
the cut
rule can prove
tautologies
by
polynomially
growing
proofs.
So,
Gentzen
systems
p-simulate
analytic
speed-up over the cut-free sequent calculus (e.g., over Statman
tautologies)—1, see [6];
has as small proofs as the best formalisms—2, 3, 4, 5, see [6];
thanks to dagness, has quasipolynomial cut elimination (instead of
1
exponential) [7, 15].
Cut free deep inference outperforms the sequent calculus.
Date: March 15, 2010.
This research was partially supported by EPSRC grant EP/E042805/1 Complexity and Non-determinism in
DeepI
Inference.
$
c ACM, 2009. This is the authors’ version of the work. It is posted here by permission of ACM for your
personal use. Not for redistribution. The definitive version was published in ACM Transactions on ComputaI
tional Logic 10 (2:14) 2009, pp. 1–34, http://doi.acm.org/10.1145/1462179.1462186 .
I
Deep inference for classical logic
21 / 46
Deep inference 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
[8]. 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 nite, not
necessarily to make it ef cient.
Deep inference for classical logic
22 / 46
Cut elimination
by `experiments'
(for logics with
contraction)
Experiment
f!
over
a ^ ā a proof:ā
"
t^f
!
a
a ^ ā
"
t^f
a
f _a
#
ā f_ !
a
!
a
f _a
#
ā _ 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?
Cut elimination
23 / 46
Normalisation in the linear fragment: Splitting
Theorem (Splitting) For every proof
t
K{A ∧ B}
there are proofs
KA ∨ KB ∨ { }
t
t
K{ }
KA ∨ A
KB ∨ B
Similar theorems hold for every logics we tried so far (including logics
that for Gentzen theory are hopeless).
Cut elimination
24 / 46
Splitting for an atomic cut
t
Therefore for every cut-free proof
K{a ∧ ā}
there are cut-free proofs
K0 {ā} ∨ K00 {a} ∨ { }
t
t
K{ }
K0 {ā} ∨ a
K00 {a} ∨ ā
i↓
therefore we can build
t
ā
a
K0 {ā}
∨
K00 {a}
K{f }
therefore a cut at the is admissible.
Cut elimination
25 / 46
Ingredients for a Kleene algebra
Two things are necessary:
1. A sequentiality operator: a.b;
2. The Kleene star: a? = {, a, a.a, a.a.a, . . . }.
This stuff is the basis of many process algebras, e.g., CCS.
Surprise(?): sequentiality cannot be captured in an analytic Gentzen
system. It requires deep inference [9, 22].
The moral reason is that sequentiality and the Kleene's star are self-dual
and noncommutative:
a.b | a.b = a.b | ā.b̄ → ◦ .
Technically, Tiu's counterexample applies (see next slides and [22, 23]).
The noncommutative linear logic BV
26 / 46
System BV
BV = MLL + self-dual noncommutative operator [9, 22]:
I
Equations:
A B = Ā O B̄
A O B = Ā B̄
AB=BA
AOB=BOA
A / B = Ā / B̄
A (B C) = (A B) C
A / hB / Ci = hA / Bi / C
A O [B O C] = [A O B] O C
A◦=A/◦=◦/A=AO◦=A
i↑
I
a ā
◦
Rules:
i↓
q↑
◦
a O ā
The noncommutative linear logic BV
s
A [B O C]
(A B) O C
q↓
hA / Bi hC / Di
(A C) / (B D)
[A O C] / [B O D]
hA / Bi O hC / Di
27 / 46
a
ā
b
Tiu's counterexample: ·
·
BV is not expressible in Gentzen
·
All the dual atoms in this structure can be made into communication by identifying two time points
Graphical
representation
a proof
BV: the structure in the following way:
between
two structures
connected byof
a par.
We canin
“prove”
·
·
c
·
c
c̄
·
b
·
·
c̄
b̄
b
·
ā
a
·
·
=⇒
·
·
·
·
b̄
·
a
·
=⇒
·
·
c
c̄
=⇒
·
·
c
c̄
b
b̄
b
b̄
ā
a
ā
·
·
·
·
a
·
·
·
·
ā
·
·
c
·
c̄
·
=⇒
b
c
b̄
·
a
·
=⇒
b
ā
·
c̄
=⇒
b̄
c
c̄
=⇒
◦
·
·
·
The noncommutative linear logic BV
¯
28 / 46
Tiu's counterexample:
BV is not expressible in Gentzen (cont.)
We can build a growing fractal of growing depth; the next step is:
6
Alwen F. Tiu
·
·
·
·
c2
·
·
c1
b̄1
·
a1
·
c̄
b
·
a2
c̄1
·
a
b1
·
·
c̄
b̄2
b̄
ā2
b2
·
·
ā1
·
·
ā
·
c̄2
·
·
·
Fig. 2 Graphical representation of the structure S1
…and each of its cut-free proofs has to start deeper inside.
b̄ and c̄) by nesting them inside other S structures:
Thereforea and
BVb (orcannot
be captured
by shallow
inference!
·
·
0
The noncommutative linear logic BV
·
·
·
·
29 / 46
Splitting for BV
Theorem (Splitting) For every proof
◦
K{A B}
there are proofs
KA O KB O { }
◦
◦
K{ }
KA O A
KB O B
and for every proof
◦
K{A / B}
there are proofs
hKA / KB i O { }
◦
◦
K{ }
KA O A
KB O B
Splitting recovers Gentzen's notion of analyticity without imposing it on
the meta-level of the formalism.
The noncommutative linear logic BV
30 / 46
=
f t
a∧f
Atomic t ows – Locality
yields
topology
+
*


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
CoS and
Below the
proofs,
their (atomic)
owsin [10]
areFormalism
shown: A notation,
and associated atomic flows.
I only structural information is retained in ows;
I logical information is lost;
the right flow cannot:
I ow size is polynomially related to derivation size;
I composition of proofs naturally correspond to composition of
ψ
ψ!
φ
ows.
,
and
.
Atomic ows – Locality yields topology
a
a∧a
31 / 46
Flow reductions:
(co)weakening (1)
NORMALISATION CONTROL IN DEEP INFERENCE VIA ATOMIC FLOWS
→
→
→
→
15
→
→
→
Each ow reduction
→ corresponds to a correct proof
→ reduction.
Atomic ows – Locality yields topology
32 / 46
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
t
−
$
Π!! $
"
#
a ε ∨ ā
$
Φ$
!
→ a
ζ
"
ε
t
$
Ψ$
ξ t∨
becomes
$
Φ{a ε /t} $
→
−
$
Π!! $
$
ā
;
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
ε
a
ξ s
∨t
(f ∧ t)measures.
I we get natural, $syntax-independent induction
Φ$
$
becomes
;
→
! ε"
Φ{a ε /t} $
a
Atomic ows – Locality yields topology
33 / 46
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:
Atomic ows – Locality yields topology
+
34 / 46
proof of Theorem 27, respectively as Case (4), (1), (2) and (3). If we apply the full set of
weakening
and coweakening
until possible, starting from a proof in cut-free
Generalising
thereductions
cut-free
The equalityform
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.
Normalisation with atomic ows – Topology is enough to normalise
35 / 46
” (Definition 2.9) is
the same atomic types, as indicated in (5), and let
flow
be the
How do we break paths?
With
path breaker [11]:
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
Normalisation with atomic ows – Topology is enough to normalise
36 / 46
rem 4.2 and Theorem 7.3.
, respectively. We can now build
We can do the same on derivations,
of 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
That we call
. We also have
and
and
.
[1] K. Brünnler. Deep Inference and Symmetry for Class
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 with atomic ows – Topology is enough to normalise
37 / 46
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
φ2
φ2
φ2
φ2
φ2
φ2
φ2
φ1
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
1
φ1
φ1
φ1
φ1
φ1
φ1
φ1
φ1
φ1
φ1
φ1
φ1
φ2
φ1
φ1
φ1
Normalisation with atomic ows – Topology is enough to normalise
38 / 46
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 correNote local cocontraction
(=appearing
better
not
available
in
sponding to the atom occurrences
in thesharing,
conclusion α of Π.
We then
have that,
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
Normalisation with atomic ows – Topology is enough to normalise
�
0
�
n
k
39 / 46
Conclusions
I
Composition in Gentzen is too 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.
Conclusions
40 / 46
References
[1]
Richard F. Blute, Alessio Guglielmi, Ivan T. Ivanov, Prakash Panangaden & Lutz Stra burger
(2014): A Logical Basis for Quantum Evolution and Entanglement.
In Claudia Casadio, Bob Coecke, Michael Moortgat & Philip Scott, editors: Categories and
Types in Logic, Language, and Physics, Lecture Notes in Computer Science 8222, Springer-Verlag,
pp. 90–107, doi:10.1007/978-3-642-54789-8_6.
Available at http://cs.bath.ac.uk/ag/p/LBQEE.pdf.
[2]
Kai Brünnler (2004): Deep Inference and Symmetry in Classical Proofs.
Logos Verlag, Berlin.
Available at http://cs.bath.ac.uk/ag/kai/phd.pdf.
[3]
Kai Brünnler (2010): Nested Sequents.
Available at http://arxiv.org/pdf/1004.1845.
Habilitation Thesis.
[4]
Kai Brünnler & Alwen Fernanto Tiu (2001): A Local System for Classical Logic.
In R. Nieuwenhuis & Andrei Voronkov, editors: Logic for Programming, Arti cial Intelligence, and
Reasoning (LPAR), Lecture Notes in Computer Science 2250, Springer-Verlag, pp. 347–361,
doi:10.1007/3-540-45653-8_24.
Available at http://cs.bath.ac.uk/ag/kai/lcl-lpar.pdf.
[5]
Paola Bruscoli (2002): A Purely Logical Account of Sequentiality in Proof Search.
In Peter J. Stuckey, editor: Logic Programming, 18th International Conference (ICLP), Lecture Notes
in Computer Science 2401, Springer-Verlag, pp. 302–316, doi:10.1007/3-540-45619-8_21.
Available at http://cs.bath.ac.uk/pb/bvl/bvl.pdf.
Conclusions
41 / 46
References (cont.)
[6]
Paola Bruscoli & Alessio Guglielmi (2009): On the Proof Complexity of Deep Inference.
ACM Transactions on Computational Logic 10(2), pp. 14:1–34, doi:10.1145/1462179.1462186.
Available at http://cs.bath.ac.uk/ag/p/PrComplDI.pdf.
[7]
Paola Bruscoli, Alessio Guglielmi, Tom Gundersen & Michel Parigot (2010): A Quasipolynomial
Cut-Elimination Procedure in Deep Inference Via Atomic Flows and Threshold Formulae.
In Edmund M. Clarke & Andrei Voronkov, editors: Logic for Programming, Arti cial Intelligence,
and Reasoning (LPAR-16), Lecture Notes in Computer Science 6355, Springer-Verlag, pp. 136–153,
doi:10.1007/978-3-642-17511-4_9.
Available at http://cs.bath.ac.uk/ag/p/QPNDI.pdf.
[8]
Anupam Das (2011): On the Proof Complexity of Cut-Free Bounded Deep Inference.
In Kai Brünnler & George Metcalfe, editors: Tableaux 2011, Lecture Notes in Arti cial
Intelligence 6793, Springer-Verlag, pp. 134–148, doi:10.1007/978-3-642-22119-4_12.
Available at http:
//www.anupamdas.com/items/PrCompII/ProofComplexityBoundedDI.pdf.
[9]
Alessio Guglielmi (2007): A System of Interaction and Structure.
ACM Transactions on Computational Logic 8(1), pp. 1:1–64, doi:10.1145/1182613.1182614.
Available at http://cs.bath.ac.uk/ag/p/SystIntStr.pdf.
[10] Alessio Guglielmi & Tom Gundersen (2008): Normalisation Control in Deep Inference Via
Atomic Flows.
Logical Methods in Computer Science 4(1), pp. 9:1–36, doi:10.2168/LMCS-4(1:9)2008.
Available at http://arxiv.org/pdf/0709.1205.pdf.
Conclusions
42 / 46
References (cont.)
[11] Alessio Guglielmi, Tom Gundersen & Lutz Stra burger (2010): Breaking Paths in Atomic Flows
for Classical Logic.
In Jean-Pierre Jouannaud, editor: 25th Annual IEEE Symposium on Logic in Computer Science
(LICS), IEEE, pp. 284–293, doi:10.1109/LICS.2010.12.
Available at http://www.lix.polytechnique.fr/~lutz/papers/AFII.pdf.
[12] Tom Gundersen (2009): A General View of Normalisation Through Atomic Flows.
Ph.D. thesis, University of Bath.
Available at
http://tel.archives-ouvertes.fr/docs/00/50/92/41/PDF/thesis.pdf.
[13] Tom Gundersen, Willem Heijltjes & Michel Parigot (2013): Atomic Lambda Calculus: A Typed
Lambda-Calculus with Explicit Sharing.
In Orna Kupferman, editor: 28th Annual IEEE Symposium on Logic in Computer Science (LICS),
IEEE, pp. 311–320, doi:10.1109/LICS.2013.37.
Available at http://opus.bath.ac.uk/34527/1/AL.pdf.
Conclusions
43 / 46
References (cont.)
[14] Tom Gundersen, Willem Heijltjes & Michel Parigot (2013): A Proof of Strong Normalisation of
the Typed Atomic Lambda-Calculus.
In Ken McMillan, Aart Middeldorp & Andrei Voronkov, editors: Logic for Programming, Arti cial
Intelligence, and Reasoning (LPAR-19), Lecture Notes in Computer Science 8312, Springer-Verlag,
pp. 340–354, doi:10.1007/978-3-642-45221-5_24.
Available at http://www.cs.bath.ac.uk/~wbh22/pdf/
strong-normalisation-atomic-lambda-gundersen-heijltjes-parigot-2013.
pdf.
[15] Emil Jerábek (2009): Proof Complexity of the Cut-Free Calculus of Structures.
Journal of Logic and Computation 19(2), pp. 323–339, doi:10.1093/logcom/exn054.
Available at http://www.math.cas.cz/~jerabek/papers/cos.pdf.
[16] Ozan Kahramanogullar (2005): Towards Planning as Concurrency.
In M.H. Hamza, editor: Arti cial Intelligence and Applications (AIA), ACTA Press, pp. 197–202.
Available at http://www.wv.inf.tu-dresden.de/~guglielm/ok/aia05.pdf.
[17] Luca Roversi (2011): Linear Lambda Calculus and Deep Inference.
In Luke Ong, editor: Typed Lambda Calculi and Applications, Lecture Notes in Computer Science
6690, Springer-Verlag, pp. 184–197, doi:10.1007/978-3-642-21691-6_16.
Available at http://www.di.unito.it/~rover/RESEARCH/PUBLICATIONS/
2011-TLCA/Roversi2011TLCA.pdf.
Conclusions
44 / 46
References (cont.)
[18] Luca Roversi (2014): A Deep Inference System with a Self-Dual Binder Which Is Complete for
Linear Lambda Calculus.
Journal of Logic and Computation, doi:10.1093/logcom/exu033.
Available at http://www.di.unito.it/~rover/RESEARCH/PUBLICATIONS/
2014-JLC/Roversi2014JLC.pdf.
To appear.
[19] Lutz Stra burger (2003): Linear Logic and Noncommutativity in the Calculus of Structures.
Ph.D. thesis, Technische Universität Dresden.
Available at
http://www.lix.polytechnique.fr/~lutz/papers/dissvonlutz.pdf.
[20] Lutz Stra burger (2006): Proof Nets and the Identity of Proofs.
Technical Report 6013, INRIA.
Available at http://hal.inria.fr/docs/00/11/43/20/PDF/RR-6013.pdf.
[21] Lutz Stra burger (2011): From Deep Inference to Proof Nets Via Cut Elimination.
Journal of Logic and Computation 21(4), pp. 589–624.
Available at http://www.lix.polytechnique.fr/~lutz/papers/deepnet.pdf.
[22] Alwen Tiu (2006): A System of Interaction and Structure II: The Need for Deep Inference.
Logical Methods in Computer Science 2(2), pp. 4:1–24, doi:10.2168/LMCS-2(2:4)2006.
Available at http://arxiv.org/pdf/cs.LO/0512036.pdf.
Conclusions
45 / 46
References (cont.)
[23] Alwen Fernanto Tiu (2001): Properties of a Logical System in the Calculus of Structures.
Technical Report WV-01-06, Technische Universität Dresden.
Available at http://users.cecs.anu.edu.au/~tiu/thesisc.pdf.
Conclusions
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