Acyclicity and Coherence in Multiplicative Exponential Linear Logic.

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Acyclicity and Coherence in Multiplicative
Exponential Linear Logic
(Extended abstract with appendix)
Michele Pagani?
Dipartimento di Filosofia – Universit` a Roma Tre
Via Ostiense 231 – 00124 – Roma
pagani@uniroma3.it
Abstract. We give a geometric condition that characterizes MELL
proof structures whose interpretation is a clique in non-uniform coherent
spaces: visible acyclicity.
We define the visible paths and we prove that the proof structures which
have no visible cycles are exactly those whose interpretation is a clique.
It turns out that visible acyclicity has also nice computational properties,
especially it is stable under cut reduction.
1Introduction
Proof nets are a graph-theoretical presentation of linear logic proofs, that gives
a more geometric account of logic and computation. Indeed, proof nets are in
a wider set of graphs, that of proof structures. Specifically, proof nets are those
proof structures which correspond to logically correct proofs, i.e. sequent calculus
proofs.
A striking feature of the theory of proof nets is the characterization by ge-
ometric conditions of such a logical correctness. In multiplicative linear logic
(MLL) Danos and Regnier give (see [1]) a very simple correctness condition,
which consists in associating with any proof structure a set of subgraphs, called
switchings, and then in characterizing the proof nets as those structures whose
switchings are connected and acyclic.
Later Fleury and Retor´ e relax in [2] Danos-Regnier’s condition, proving that
acyclicity alone characterizes those structures which correspond to the proofs of
MLL enlarged with the mix rule (Figure 4). More precisely, the authors define
the feasible paths as those paths which are ”feasible” in the switchings, then they
prove that a proof structure is associated with a proof of MLL plus mix if and
only if all its feasible paths are acyclic.
Proof structures are worthy since cut reduction is defined straight on them,
not only on proof nets. We can thus consider a concrete denotational semantics
for proof structures, as for example the relational semantics. Here the key notion
is that of experiment, introduced by Girard in [3]. Experiments allow to associate
?This work has been partially supported by MIUR national project FOLLIA.

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with any proof structure (not only proof net) π a set of points ?π? invariant under
the reduction of the cuts in π.
In [4] Bucciarelli and Ehrhard provide a notion of coherence between the
points of the relational semantics, so introducing non-uniform coherent spaces
and non-uniform cliques. Such spaces are called non-uniform since the web of
their exponentials does not depend on the coherence relation, as it does instead
in case of usual (uniform) coherent spaces (see [3]).1
Hence we have a geometrical notion – feasible acyclicity – dealing with logical
correctness, and a semantical one – clique – defined in the framework of coherent
spaces. Such notions are deeply related in MLL: from [3] it is known that for any
proof structure π, if π is feasible acyclic then its interpretation ?π? is a clique;
conversely Retor´ e proves in [5] that for any cut-free π, if ?π? is a clique then π
is feasible acyclic.
Such results tighten the link between coherent spaces and multiplicative proof
nets: as a corollary we derive in [6] the full-completeness of coherent spaces for
MLL with mix.
What happens to this correspondence in presence of the exponentials, i.e. in
multiplicative exponential linear logic (MELL)?
The notion of feasible path can be easily extended to MELL (Definition
6). In this framework feasible acyclicity characterizes the proof structures which
correspond to the proofs of MELL sequent calculus (Figure 3) enlarged with the
rules of mix and daimon (Figure 4). However the link between feasible acyclicity
and coherent spaces fails: there are proof structures which are associated with
cliques even if they have feasible cycles (for example Figure 5).
The main novelty of our paper is to find out a geometrical condition on
MELL proof structures, which recovers the missed link with coherent spaces.
In Definition 7 we introduce the visible paths, which are an improvement of the
feasible paths in presence of exponentials. Then we prove in Theorems 9 and 10:
– for any MELL proof structure π, if π is visible acyclic, then ?π? is a non-
uniform clique;
– for any MELL cut-free proof structure π, if ?π? is a non-uniform clique then
π is visible acyclic.
Finally, it turns out that visible acyclicity has also nice computational proper-
ties, especially it is stable under cut reduction (Theorem 18), moreover it assures
confluence and strong normalization.
Acknowledgments. I would like to thank Lorenzo Tortora de Falco for his invalu-
able support and Paolo Di Giamberardino, whose M´ emoire de D.E.A. has been
the starting point of this work.
1Actually the difference between uniform and non-uniform spaces has a relevance only
in presence of exponentials: in MLL we can speak simply of coherent spaces and
cliques.

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2Proof structures
The formulas of MELL are defined by the following grammar:
F ::= X | X⊥| F?F | F ⊗ F | ?F | !F
As usual we set (A?B)⊥= A⊥⊗ B⊥, (A ⊗ B)⊥= A⊥?B⊥, (?F)⊥=!F⊥
and (!F)⊥=?F⊥. We denote by capital Greek letters Σ,Π,... the multisets of
formulas. We write A1?...?An−1?Anfor A1?(...?(An−1?An)...), where
? is ? or ⊗.
Proof structures are directed graphs with boxes – highlighted subgraphs –
and pending edges – edges without target. Edges are labeled by MELL formulas
and nodes (called links) are labeled by MELL rules. Links are defined together
with both an arity (the number of incident edges, called the premises of the link)
and a coarity (the number of emergent edges, called the conclusions of the link).
The links of MELL are ax, cut, ⊗, ?, !, ?d, ?c, ?w, as defined in Figure 1. The
orientation of the edges will be always from top to bottom, so that we may omit
the arrows of the edges.
ax
AA⊥
cut
A⊥
A
⊗
A
B
A ⊗ B
?
B
A?B
A
!
!A
A
?d
?A
A
?A
?A
?c
?A
?w
?A
Fig.1. MELL links.
Definition 1 (Proof structure, [3]). A proof structure π is a directed graph
whose nodes are MELL links and such that:
1. every edge is conclusion of exactly one link and premise of at most one link.
The edges which are not premise of any link are the conclusions of the proof
structure;
2. with every link ! o is associated a unique subgraph of π, denoted by πo,
satisfying condition 1 and such that one conclusion of πois the premise of o
and all further conclusions of πoare labeled by ?-formulas. πois called the
exponential box of o (or simply the box of o) and it is represented by a dash
frame. The conclusion of o is called the principal door of πo, the conclusions
of πolabeled by ?-formulas are called the auxiliary doors of πo;

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3. two exponential boxes are either disjoint or included one in the other.
The depth of a link in π is the number of boxes in which it is contained. The
depth of an edge a is 0 in case a is a conclusion of π, otherwise it is the depth
of the link of which a is premise.
The depth of π is the maximum depth of its links, the size of π is the number
of its links, the cosize of π is the number of its links ?c.
A link l is terminal when either l is a ! and all the doors of πlare conclusions
of π, or l is not a ! and all the conclusions of l are conclusions of π.
Proof structures are denoted by Greek letters: π,σ,..., edges by initial Latin
letters: a,b,c... and links by middle-position Latin letters: l,m,n,o.... We write
a : A if a is an edge labeled by the formula A.
A proof structure without cuts is called cut-free. The cut reduction rules
are graph rewriting rules which modify a proof structure π, obtaining a proof
structure π?with the same conclusions as π. We do not give here the cut reduction
rules, which are completely standard (see [3]). However we remark that at the
level of proof structures there exist cuts, called deadlocks, which are irreducible.
These are the cuts between the two premises of an axiom, and the cuts between
two doors of an exponential box (see Figure 2).
cut
ax
AA⊥
!
cut
!A
...
?Γ
?A⊥
πo
Fig.2. Examples of deadlocks.
We denote by π ?β π?whenever π?is the result of the reduction of a cut
in π. As always, →βis the reflexive and transitive closure of ?βand =βis the
symmetrical closure of →β.
3Non-uniform coherent spaces
We recall that a multiset v is a set of elements in which repetitions can occur.
We denote multisets by square brackets, for example [a,a,b] is the multiset
containing twice a and once b. The plus symbol + denotes the disjoint union of
multisets, whose neutral element is the empty multiset ∅. If n is a number and v
a multiset, we denote by nv the multiset v + ... + v
????
n times
. The support of v, denoted
by Supp(v), is the set of elements of v, for example Supp([a,a,b]) = {a,b}.

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If C is a set, by M(C) (resp. Mfin(C)) we mean the set of all multisets (resp.
finite multisets) of C.
Definition 2 (Non-uniform coherent space, [4]). A non-uniform coherent
space X is a triple (|X|,?
?,≡), where |X| is a set, called the web of X, while
and ≡ are two binary symmetric relations on |X|, such that for every x,y ∈ X,
x ≡ y implies x?
?y.
of X.
A clique of X is a subset C of |X| such that for every x,y ∈ C, x?
?
?
?
?(resp. ≡) is called the coherence (resp. the neutrality)
?y.
Remark the difference from Girard’s (uniform) coherent spaces: we do not
require the relation
?
?to be also reflexive.
We will write x?
?y [X] and x ≡ y [X] if we want to state explicitly which
coherent space
?
?and ≡ refer to. We introduce the following notation, well-
known in the framework of coherent spaces:
strict coherence: x?y[X], if x?
incoherence: x?
?y [X], if not x?y [X];
strict incoherence: x?y [X], if x?
?y[X] and x ?≡ y [X];
?y [X] and x ?≡ y[X].
Notice that we may define a non-uniform coherent space by specifying its
web and two well chosen relations among ≡,
Let X be a non-uniform coherent space, the non-uniform coherent model on
X (nuCohX) associates with formulas non-uniform coherent spaces, by induction
on the formulas, as follows:
?
?,
?,
?
?,
?.
– with X it associates X;
– with A⊥it associates the following A⊥: |A⊥| = |A|, the neutrality and
coherence of A⊥are as follows:
• a ≡ a??A⊥?iff a ≡ a?[A],
• x?
?y[A];
– with A⊗B it associates the following A⊗B: |A⊗B| = |A|×|B|, the neutrality
and coherence of A ⊗ B are as follows:
• < a,b >≡< a?,b?> [A ⊗ B] iff a ≡ a?[A] and b ≡ b?[B],
• < a,b >?
– with !A it associates the following !A: |!A| = Mfin(|A|), the strict incoher-
ence and neutrality of !A are as follows:
• v?u[!A] iff ∃a ∈ v and ∃a?∈ u, s.t. a?a?[A],
• v ≡ u[!A] iff not v?u[!A] and there is an enumeration of v (resp. of u)
v = [a1,...,an] (resp. u = [a?
?y?A⊥?iff x?
?< a?,b?> [A ⊗ B] iff a?
?a?[A] and b?
?b?[B];
1,...,a?
n]), s.t. for each i ≤ n, ai≡ a?
i[A].
Of course the space A?B is defined by (A⊥⊗ B⊥)⊥as well as ?A is defined
by (!A⊥)⊥.
Remark that !A may have elements strictly incoherent with themselves, i.e.
?
?is not reflexive. For example, suppose a,b are two elements in A such that