Download Implicative Formulae in the Vroofs as Computations” Analogy

Implicative Formulae in the
Vroofs as Computations” Analogy
Andrea Asperti, Gian Luigi Ferrari, Roberto Gorrieri
Universiti di Pisa, Dipartimento di Informatica
Corso Italia 40, I - 56125 Pisa Italy
asperti (giangi, gorrieri [email protected],it
ABSTRACT
1. INTRODUCTION
In [As871 a correspondence between the subset of Linear
Petri Nets pet621 are the first and maybe the most popular
Logic [Gi86] involving the conjunctive tensor product only
formalism
and PlacelTransition Petri Nets IRei
is established. In
activities. A Petri Net is defined as a set of places which
this correspondence, formulae are regarded as distributed
can contain tokens, a set of transitions and a flow relation
states and provable seqnents are computations in the net.
connecting places and transitions. A place in the net is
Developing this i&a, Marti-Oliet and Meseguer [MaM89]
interpreted as a resource type, a token in the place as an
have suggested that all the other computations of Linear
instance of the resource associated to the place, and a
Logic, which do not have an immediate correspondence
transition as an activity which consumes resources and
with Petri Nets, should be regarded as “gedanken” or
produces other resources in accordance with the flow
idealized processes, providing a richer language for the
relation.
specification and the study of properties of distributed
the introduction of linear implication allows us to observe
role of conjunction (see also [GG89, MaM89]). Linear
the net at a lower, more decentralized level of atomicity,
Logic is a logical calculus where a particular care is devoted
where the preemption of each resource neededfor thefiring
to the introduction and deletion of hypothesis during the
of a transition is represented as a separate move. We give a
deductive process, with the effect that formulae look like
conservative theorem relating computations at different
resources subject to a limited use. The main idea of the
levels of abstraction. The categorical semantics establishes
relation between tensor theories and Petri Nets is that an
a tight correspondence among Petri nets, monoidal closed
atomic formula A can be thought of as a token in a place
named A, and a tensor formula AI@ ...@A. as a marking
categories and tensor theories, reminiscent of the well
languages, Cartesian
i.e. a distribution of tokens on the places Al,. . .,A,. The
closed categories and intuitionistic logic [LS86]. The
identification of computations in the categorical model
suggests the generalisation
proof of a sequent of the form Al ,..., A, 1 B is then a
computation from the marking Al@ ...@A. to B. A
of the notion of
specific net is described by a set of external axioms
process [DMM89] at the lower level of atomicity.
Research partially supported by Joint Collaboration
Contract ST2J-0374~C(EDB) of EEC and by Esprit Basic
Research Actions, project 3011 CEDISYS.
Machinery.
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tensor theory) and the dynamic behaviour of the net is
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theory in the language of Linear Logic [Gi86], involving
only the connective of tensor product @, which plays the
fundamental connective of linear implication. We prove that
naturally
and modelling
In [As871 it is proved that a net N can be regarded as a
computations. In this paper we apply this program to the
known relation among functional
for specifying
59
described by the inference rules of the propositional tensor
proof theoretic setting of the implicative fragment of Linear
fragment. It is easily proved that a net computation
Logic.
corresponds to a proof in the tensor fragment, and
The main result we prove about the introduction of linear
conversely every proof has an associated computation on
the net.
implication is that it is conservative with respect to the
class of concluded derivations. If a sequent r !- A is
The expressive power of full Linear Logic is far beyond the
provable in a tensor theory T with the implicative fragment
expressive power of Petri Nets. In this respect, [MaM89]
suggests that all the computations of Linear Logic which
of the logic and A does not contain the connective of
implication, then there exists a derivation of r I- A within
do not have an immediate correspondence with Petri Nets
the pure tensor fragment. Moreover the two derivations
should be regarded as gedanken or idealized
processes,
have the same (categorical) semantics, that is they describes
providing a richer language for the specification and the
the same process. This result nicely relates the two different
study of net properties. They also develop a few examples
levels of atomicity
involving the two additive connectives of conjunction and
behaviour of the net.
disjunction, relating them lo situations of internal and
from which we can observe the
cxtemal nondcterminism (see also [GG89] on this subject).
The paper is organized as follows. Section 2 introduces the
tensor fragment, while its categorical semantics is
In this paper we pursue their program, studying the
investigated
fundamental ccnnective of linear implication. We suggest
to look at an implicative formula (e.g. B + C) as a sort of
correspondence between tensor theories, Petri nets, and
non-concluded distributed state of a given computation,
extended with linear implication and introduces the notions
where some resources (B) are needed to properly terminate
of concluded
with the production of other resources (C). We prove that
semantics in monoidal closed categories naturally suggests
this approach enables one to observe the system at a lower,
the right generalization of the notion of process [DMM89].
more decentralized level of atomicity: indeed we can
Finally, in Section 6 the relation between the different
observe, as a separate move, the preemption of every
levels of atomicity is established as a corollary of a weaker
resource needed for the firing of a transition. In other words,
version of Gentzen’s Hauptsatz.
in Section
3. Section
4 settles
the
monoidal categories. Section 5 studies the fragment
computation
and deadlock.
Moreover the
if a place A is a pre-condition of more than one transition,
then each token in A can autonomously decide in which
transition to be involved, reaching an intermediate state
2, THE TENSOR
where it can only wait for the tokens taking part to the
FRAGMENT
OF
LINEAR
LOGIC
selected transition, This behavior, not allowed in classical
Linear Logic [Gi86] is essentially a Gentzen Calculus of
Petri Net Theory, represents a sort of more decentralized
Sequents (see [Fr65]) without weakening and contraction
description of distributed computations. However, a serious
rules, and with a duplication
problem arises: the possibility of deadlock. For instance,
suppose to have two transitions t: A,B I- C and t’: A$ t D;
from the state A@B it can happen that t preempts the token
of the usual logical
connectives of conjunction and disjunction, naturally
suggested by the lack of such structural rules. This means
that neither uselesspremises can be freely added during the
in A and t’ the token in B, yielding the deadlock state
(B+C)@(A-+D).
Deadlock situations and strategies for
inference, nor different occurrences of the same formula in
the premises can be identified: each hypothesis is “used”
controlling the decentralized execution of net specifications
once and only once. In this sense, logical formulae loose
can be formally understood and profitably studied in the
their abstract, platonistic countenance of truth values, or
60
types, gaining the more concrete nature of “resources” or
“states”. Moreover, any step of logical deduction modifics
the state of its premises. This is very appealing for
computer science applications, and in particulai for
concurrent computations since it puts emphasis on
dynamics.
A derivafion D is a tree of sequcntssatisfying the following
conditions:
l
l
The Tensor Fragment is the part of linear logic in which
the only logical connective is the conjunctive
multiplicative tensor 8. This subcalculushas (formally) an
the topmost sequents of D are logical axioms
every sequentin D except the lowest one (the root) is an
upper sequentof an inference whose lower sequent is also
in 0.
The root sequentis also called$finulsequent.
intuitionistic nature, since only one formula can appear in ,
the right hand side of a sequent. The alphabet of the
fragment is given by atomic propositions and by the tensor
8. A formula is either an atomic proposition or the product
A8B of two formulae. An intuitionistic sequent has the
syntactic structure l? I- B where r is a finite (possibly
A tensor theory T is a set of sequents,playing the role of
extra logical axioms. If T is a tensor theory, a T-derivation
D is a derivation where the topmost sequentscan be in T. A
marked tensor theory Th is a pair c A, T 1, where A is a
formula and T is a tensor theory. When considering marked
theories, we are interested in all the derivations of the
theory having A as 1.h.s.of their final sequents.
empty) list of formulae, and B is another formula. The
inference rules of the calculus formalize the process of
construction of complex proofs by meansof simpler ones.
3. CATEGORICAL
Gw
Al-A
The categorical semantics of full Linear Logic has been
mostly developedby Lafont [La881 and Seely [S&7], with
contributions of several other people ([DP89], [MaM89],
see also [AL901 for an updated account). In particular, a
model for the tensor fragment is a Symmetric Mono&l
Category, whose relevance in Concurrency Theory has
beenemphasizedin [DMM89].
I-t-A
@xc,1)
r’l-A
r’ permutation of r
l-lt A
A,rzl-B
(cut)
hr2t
( A is called cutfomdu
SEMANTICS
Definition
B
3.1 (Symmetric Monoidul Categories)
A Symmetric Monoidal Category is a category C with a
functor Q: CxC+C (called tensor product), and an object
)
1 such that:
1) assoc: X@(YYZ))n(X@Y)@Z
2) ins: X s 1 Q X
3) exch: X 62~
Y z Y @X
l-, A, B t C
ml)
T,A@Bt-C
are natural isomorphisms,called structural isomorphisms.
Moreover theseisomorphismsmust satisfy the well-known
MacLane-Kelly coherence equations [ML7 11. When the
structural isomorphisms are actually identities, i.e. l)-3)
aboveare equations, the monoidal categoryis called strict.
61
Definition
the two frameworks, prefixing each sequentin a derivation
with its associatedterm in the category.
3.2 (The Categorical Inrerpreration)
Given a tensor theory T and a symmetricmonoidal category
C, the categorical interpretation I of T over C associates
every atomic formula A in T with an object I(A) of the
category.The interpretation I is then inductively extendedto
ali the formulas by I(A@B) = I(A)@I(B), and to sequences
I-= A 1,..., A, by I(T) = I(Al)@...@‘I(A,). Every provable
4. TENSOR
idA:A
t-A
f exch : B@A t C
formula A. The deductive engine of the tensor fragment
provides the tools to represent the dynamic behaviour: the
(@,r ) rule describesthe parallel execution of computations
l
g:A@Ct-D
(cut>
and the cut rule their sequentialcomposition.
g 0 f@idC: BBC I- D
f:Al-B
NETS
whose individual componentsare A and B. In other words,
an atomic formula can be understood as a f&en in a place,
and any formula as a distribution of tokens in the net
places. A sequent r l- A describes a computation starting
from the state F and ending in the state representedby the
@xc)
FBI-A
AND PETRI
The analogy between formulae and states (proofs and
computations),and its possible application to the theory of
Petri Nets was pointed out by the first author [As87], and
has been recently revisited and extended in [MaM89] and
[GG89]. Any formula of a tensor fragment can be
interpreted as representingthe distributed state of a system,
describing its individual components; for instance the
formula AQDB,where A and B are atomic, representsa state
sequentr 1 A is interpretedas a morphism f: I(IT)+I(A). In
particular I fixes the interpretation for the extra-logical
axioms, while every logical axiom A f- A is interpreted as
the identity morphism of I(A). The interpretation of the
provable sequentsis inductively defined on the derivation
tree, according to the following rules.
ow
THEORIES
Now we introduce the classical definition of Petri Nets. By
Petri Nets we meanPlace/TransitionPetri Nets llXei85].
g:CkD
fag: A@C I- B@D
Definition
4.1 (Place Transition Petri Nets)
A Petri Net N is a quadruple (S, T, F, MO) where:
Notice that there is no interpretation for (B, I ) rule since
the comma in the left hand side of the sequent is already
identified with 8.
l
l
l
Fixed a theory T, there exists an obvious monoidal category
C(T) freely generatedby T. For many respects,it is easier
to work in this free category, than in the original logic
framework. The most important advantage is that it is
simpler to handle arrows of the category than derivationsof
the logic. Another drawback of the logic formalization,
which disappears at the categorical level, is the annoying
syntactic distinction between the comma in the 1.h.s.of a
sequent and the connective of tensor product. In the
following, we shall often make a blend use of notations in
S is a non empty set of places,
T is a set of transitions, S A T = 0,
F is a multiset relation over (SxT) u (TxS) called the
causal dependencyrelation,
. MO is a non-empty multiset of places, called the initial
marking
The causal dependencyrelation (also calledflow relation)
can be interpreted as a function from (S x T) u (‘T x S) to
the set N of natural numbers. Also markings can be
interpreted as functions from S to N. For simplicity, a
marking M is usually represented as {nlsl, . . . . npsp)
62
where the natural number ni z 0 indicatesthe number of the
occurrences(tokens) of the place si in M.
fragment provides the tools to represent the dynamic
behaviour.
For any transition t its preset is defined as the multiset over
S given by pre(t)(s) = F(s,t). Similarly, the postset
post(t)(s) = F(t,s). The dynamic behaviour of P/T Nets is
defined by the token game. Each firing of a transition
removesits presetand producesits postset.
Theorem 4.6
Given a net N, Mn is reachable from MO if and only if
@MOi- A is a provable sequentin the tensor theory Th(N)
Definition 4.2 (The Token Game)
Let N be a net, and let M be a marking. A transition t iienabled at M if pre(t)(s)l M(s). If t is enabled at M, then
the firing of t (M -t+M’) transforms M in M’: M’(s) =
M(s) - preW(s) + PosttW.
Definition 4.3 (Firing Sequence and Reachable Marking)
Let N be a net, and let MO be the initial marking of N. A
firing sequence is a sequence(MO, tI, M 1, . . . , tn, Mn) such
that MO -tl+MI . . . tn+Mn. Mn is called reachable from
MO-
andp(A)=Mn.
The proof (straightforward) of the last theorem settles a
consuuctive correspondencebetween firing sequencesand
derivations.
Interpreting the functor Q as parallel composition and as
l
sequential composition, the equational nature of the
categorical semantics, developed in Section 3, imposes
some relevant identifications among computations. In
addition to the basic categoricalequations:
= f (g h)
idqf = f = f-id
tf
l
g)
o
h
l
l
associativity of composition
identity is an idle computation
we have:
Let An indicate the tensor n-power i.e the formula
ABA@. . .@A where the atomic formula A occurs n times.
(f @ g) (f @ g’) = (f f) Q (g g3
Let A be a formula, p(A) denotesthe multiset of the atomic
formulae occurring in A; this operation is obviously
extendedto sequencesof formulae via multiset union.
This law describes one of the basic property about
concurrency: the parallel composition of two given
independent computations f 0 f and g 0 g’ has the same
l
l
l
functuriality of 63
effect of the computation whose steps are the parallel
composition f@g and f@ g’. As an interesting instance
Definition 4.4 (Marking-formulae)
Given a marking M = (nlsl, . . . . npsp] the associated
(marking-)formula is @M = s lnl@. . . @sp”P.
of this law, consider the following example. Given two
computations tl: A I- B, t2: C 1 D, we have:
Definition 4.5 (Petri Net Theories)
Given a Petri Net N = (S, T, F, MO) the marked tensor
theory associated with N is the pair Th(N) = 40M0, Th,
(id@tI); (t@idB) = ‘1 @ 12 = (@idA) ; (idD@tI)
that is the well-known property that the concurrent
execution of two independent transitions tl and 12, is
where Th is the set of the extra logical axioms, defined in
the alphabet S of atomic formulae, as follows: for any t E
T, t: r I- A is an axiom in Th where pre(t) = p( r) and
equivalentto their execution in any order.
If the symmetric monoidal category C is strict, we have the
law:
post(t) = N A )+
The formula @MOis called initial formula of ( @MO,Th ).
f@g=g@f
A tensor theory provides the logic counterpartof Petri Nets:
in fact a net is uniquely characterizedby the preset and the
postsetof its transitions. The deductiveengine of the tensor
commutativity of @
which states the intuitive fact lhat parallel composition is
commutaative.These interesting laws were first pointed out
by Meseguer and Montanari [MM881 which gave a
63
characterization of Petri Nets as monoidal categories. This
approach to an algebraic description of Petri Nets as
monoids has been further investigated in [DMM89], where
the above outlined notion of computations suggestedby the
equation of monoidal categories (called commutarive
processes in [DMM89]) is compared with the well-known
notion of process [GR83]. They prove that commutative
processesare the least abstract model which is more abstract
In the categorical semantics, the connective of linear
implication + is interpreted as the right adjoint to the
tensor product @, that is as a bifunctor +: CxC+C
such
that there exists a natural isomorphism
(1) A : C[c@a,b] %C[c,a+b].
Mono&l
Categories where the tensor product @ has a right
adjoint are called Monoidol Closed.
than both tiring sequences and processes(see aIso [BD87]).
Property 5.2 (Monoidal closed categories)
Remark From a proof-theoretic perspective, it would be
The existence of the adjunction in (1) is equivalently
interesting to single out which is, in a class of equivalent
characterizedas follows:
proofs, the more reduced, i.e. the proof with only the
for all objects a, b, c in C :
an object a+b
a morphism evala,b: (a-+b)@a -+ b
strictly necessary cuts. Since the cut rule corresponds to
l
sequential composition, Cuntcr and Gchlot [GG89] point
l
out that
this requirement can bc restated in concurrency
(evaluation map, i.e. modus ponens)
theory as follows: among all the equivalent computations,
which is the maximally concurrent? In the categorical
l
approach this problem can be solved in a natural way by
appropriately “orienting”
above (with
such that for all morphisms f: &a-b,
the semantic equations given
some care to the associativity
for every object c an operation
I\c : C[c@a,b]-+C[c,a+bl
h: c -+ (a-+b), the
following equations hold:
of the
I3 eV&,b 0 &O@iQ
=f
0 h@ida) = h
q) &$Wt&,b
composition). The simplicity of this solution is due to the
fact that it is much simpler to work with terms than with
derivation trees. This result, based on a suggestion of
Definition 5.3 (The Categorical Interpretation )
Montanari, will be the object of a forthcoming paper.
The categorical interpretation
of linear implication
is
defined as follows:
5. DECENTRALIZED
LINEAR
COMPUTATIONS
f:C@A 1- B
AND
IMPLICATION
(-+ r>
A&:
In this section, we enrich the tensor fragment with the
introduction
of the linear implication
connective. The
f:DtA
Linear Logic rules for implication are:
Definition
C t A+B
g:B@El-C
(+,I)
g o (eVdA,BO
5.1 (Linear Implicarion)
(idA+B@‘f))@idE: (A+B)@D@E l- C
A formula without + represents a state where all the
r,AkB
activated transitions, if any, have been terminated; a
6-h 0
formula containing the linear implication represents an
Tl-A+B
intermediate, non-concluded state of a computation. The
l-1 F-A
B,I’2k
C
introduction of the connective of implication allows us to
(+,l)
have a finer vision of the computations. Each sequent of a
A-B,
Fl, I-2 J- C
tensor theory T represents an atomic move; the implication
64
rules allow us to break it down into a computation of
simpler moves.
Definition 5.4 (Concluded and Open Sequents)
A sequent IT I- A is concluded iff A does not contain any
occurrence of the implication
informally
outlined
above is further stressed by the categorical semantics. In
fact the P-equation
eVdg,C
o (I\A(f)@idg) = f: A@B /- C
connective; it is open
expressesthe fact that a computation f: AQ9Bl- C regarded
otherwise
Example
The property of linear implication
as an atomic step, is equivalent to a sequence of simpler
5.5
Consider the transition t: A, B I- C, and suppose to be in
the state A. In the pure multiplicative
fragment, no
computation was possible other than the idle move Al-A .
consecufive steps, namely the preemption of a by f, and
the conclusion of the open computation AA(f) when b
becomesavailable.
On the contrary, in the implicative fragment, we have also
the computation A(t): Al-B-C. where t locally gets the
Example 5.6
Suppose to have two transitions
only available resource A, reaching the non concluded state
B-Z. When B becomes available, from B+C by modus
and to start with an initial marking [email protected] is a possible
ponens we obtain C. This is described by the following
f: C@A t- B, g: D I- A,
computation:
evalA$ 0 (AC(f)@& : COD t- B
derivation D:
Bt-B
A,Bt-C
Al-B4
Looking for C and A, f finds only C and preempts it,
yielding the open state A+B. At the same time g produces
A from D. Now the open computation AC(f) can be
cl-c
B-C, Bl- C (modus ponens)
A,Bl-C
Intuitively,
f o id@g: C@D l- B
this computation is equivalent to the direct
firing of t, that is to the trivial derivation D’ = A, B l- C.
The derivation D’ and D can be naively represented in terms
of the two nets as illustrated below according to their wellknown graphical representation.
0
A
B
concluded with the production of B. Another possible
strategy for the execution off and g is the following:
A
B
which can be read as follows: first g is fired, while C stays
idle; then the pair C@A evolves into B by f. The previous
computations are identified by the equational theory :
evdA,B o (k(f)@& = evdA,B o ($$@idA o id&$)
by the functoriality of @
= (eValA,B 0 hC(f)@idA) 0 id&0g by associativily
=foidC@g
by P
The introduction of the linear implication increases the
Y
C
potential non determinism of the system; as a consequence
some computations have to be properly understood as
unsuccessful. A typical example is provided by the two
transitions t: A,B t C and t’: A,B I- D, in the initial state
A@B, when t preempts the token in A and t’ the token in
B, yielding the state (B--+C)@(A-+D). In this case both the
transitions cannot terminate. This situation is known under
the name of deadlock.
avoid the occurrence of deadlock situations, namely by
means of a metaaleveldeduction system which provides a
guideline to the application of the rules in the derivation.
This topics will be further investigated elsewhere.
Definition 5.7 (Conclusive Sequent and Deadlock)
Given a tensor theory T, a sequentf’ I- A is T-conclusive iff
there exist a concludedT-derivation AI-A’. A sequentr I- A
which is not (Tj-conclusive is called a (T-1 deadlock.
Example
Example 5.8 (The Dining Philosophers).
5.9
Consider the transitions f: A t- B, g : C I- D , h : D I- E.
The computations:
f@(h 0 g): A@C I- B@E
eval 0 A(f@h)@g: A@C I- B@E
are semanticallyequivalent,as
evalo A(f@h)@g= eval 0 A(f@h)@id0 id@g=
The Dining Philosophers can be represented by the
following theory T (we consider the case of three
philosophers tl, t2, t3 and three forks Cl, C2, C3 ):
t1: Cl, c2 t C1@C2,
t2: C2, C3 t C@C3,
t3: c3, Cl t c3@21
= f@h 0 iddag = (f 0 id)@(h 0 g) = f@(h 0 g)
The second computation can be read as follows. From the
state ABC, the transition f could fire; instead, the decision
to execute a parallel transition f@h is taken, and A is
t3
t1
c3
c2
preempted waiting for the availability of D. This is
describedin the computation h(f@h) leading from A to the
open state D+(BQE). At the sametime g producesD from
C, and the whole computation can terminate with the
parallel firing off and h. The interesting fact is that in
evalo A(f@h)@g: ABC t B@E,
the production of E seemsto depend from D (consider the
subcomputationh(f@h): A I- D-+(B@E) ). This dependence
does not exist in the computation f@(h 0 g): ABC I- B@E.
The following are examplesof computations:
(a) (idC,@tz) 0 (tl@idC3) : Cl,C2.C3 I- C1@C2@C3,
(b) idCl@A(tl 0 exch)@A(tx):
Cl,C2,C3
We are now able to observea richer class of dependencies.
Thesedependencesare temporal dependenciessince they are
originated by the execution strategy; they can be eliminated
in every equivalent computation where the transitions are
executedas atomic moves(seeCorollary 6.4).
t CI~(Cl~(Cl~C2))~(CI~(C3~CI)),
(c) ~01)@Nt2)@w3)
:
The sequent(a) is concluded: it representsthe computation
where the first and second philosopher have eaten the one
after the other. The sequent (b) is conclusive: here the first
and third philosopher have respectively taken at the same
time the fork at their left and their right; now they are
competing for the fork Cl. Finally (c) is a deadlock: all the
6.
CONCLUDED
COMPUTATIONS:
OPEN CUT ELIMINATION
THE
THEOREM,
In this section we relate computationsat a different level of
atomicity. We prove that any concluded computation is
equivalent to a computation where every transition is
executedas an atomic move. This follows as a corollary of
a weak version of Gentzen’s Hauptsatz. Note that we need
to prove anew this result since a consequence of the
philosophershave taken the fork at their right.
As a remark, notice that a distributed implementation of a
Petri net specification can be achievedby the definition of
strategies of deduction in the implicative fragment which
66
introduction of “external”
axioms, is the failure of the
application of (+, r) and (-+, 1) can be in D.
general Gentzen’s Hauptsatz.
Definition
6.1 (open formulae)
A formula is an open formula, if it contains a subformula
whose main connective is the implication. A open cut (ocut) is a cut whose cut formula is open. A o-cuf free
derivation is a derivation which contains no open cuts.
Theorem 6.2, (The o-cut Elimination
Acknowledgements
The authors would like to thank P. Degano, G. Longo, S.
Martini, J. Meseguer and U. Montanari for the stimulating
discussions on the topics of this paper.
Theorem)
Let T be a tensor theory, and let D be a T-derivation of
the sequent I- t- A in the implicative fragment. Then there
exists a o-cut free derivation
appear in I- I- A, no one has been introduced, and no
IS of the same sequent.
REFERENCES
[AL901
Asperti A,, Longo G., Categories, Types and
Structures. Book, to appear MIT Press, 1990.
[As871
Asperti A., A Logic for Concurrency,
Technical Report, Dipartimento di Informatica,
Univ. Pisa, 1987.
lBD871
Best E., Devillers
Moreover the final sequents associated with D and D’
have the same categorical semantics
The previous theorem is easily obtained from the following
lemma, by induction on the number of the o-cut rules
occurring in the derivation D. The proof of the lemma can
Concurrent
Theoretical
be found in the appendix.
Lemma 6.3
[DP87]
De Paiva V., Z’he Dialectica Categories, in
AMS Conference on Categories in Computer
Science and Logic, (Gray-Scedrov Eds.)
Boulder, 1987.
[DMM89]
Degano P., Montanari U., and Meseguer J.,
Let T be a set of axioms which only contain atomic
formulas, and let D be a T-derivation of the sequent r l- A
which contains only one o-cut rule occurring as the last
inference. Then there exists a o-cut fret derivation D’ of the
R., Sequential
and
Behaviour in Petri Net Theory,
Computer Science, 55, 1987.
Axiomatizing
Processes,
same sequent. Moreover the final sequents associated with
Net
Computations
and
in Proc. Logics in Computer
Science ‘89, AsiIomar, 1989.
D and D’ have the same categorical semantics.
Corollary
6.4
Let T be a tensor theory, and let D be a T-derivation of the
concluded sequent r t A in the implicative fragment. Then
[GG89]
Gunter C., Gehiot V., Nets us Tensor
Theories,
in Proc. 10th International
Conference on Application and Theory of Petri
Nets, Bonn, 1989.
[G i861
Girard J. Y., Linear Logic,
Computer Science, SO, 1986
[GR83]
Goltz U., Reisig W., The Non Sequential
Behaviour of Petri Nets, Information
and
Computation, 51, 1983.
[La881
Lafont Y., The Linear
there exists a T-derivation D’ of the same sequent r I- A in
the tensor fragment. Moreover the final sequents of D and
D’ have the same categorical semantics,
Proof By the o-cut elimination theorem we know that
there exists a o-cut free T-derivation D of I- l- A. However,
in a o-cut free derivation every open formula which has
been introduced at same stage of the derivation will
definitively appear as a subformula in some formula of the
Theoretical
final sequent (subformula property). Since no such formula
67
Theoretical
Abstract Machine,
Computer Science, 59, 1988.
[LS861
[ML711
Lambek J., Scott P.J., Introduction lo Higher
Order Categorical
Logic, Cambridge
Univertsity Press, 1986.
Mac Lane S., Categories for lhe Working
Marhematicians, Springer-Verlag, 1971.
FIaM891 Marti-Olitet N., MeseguerJ., From Petri Nets
to Linear Logic, to appear in Proc. Third
Conferenceon Category Theory and Computer
Science,Manchester,1989.
W4881
Meseguer J., Montanari U., Petri Nets are
Monoids: A New Algebraic Foundation for Net
Theory, In Proc Logics in Computer Science
‘88, 1988 (full
version to appear in
Information and Computation).
Ipeal
Petri C.A., Kommunicalion mit Automaten,
Schriften des Institutes fur Instrumentelle
Mathematik, Bonn 1962.
I3651
Prawitz D., Natural Deduction: A ProofTheoretic Study, Almqvist and Wiksell,
Stockholm, 1965.
IRei
Reisig W., Petri Nets: An Introduction,
Springer Verlag, 1985.
[Se871
Seely R., Linear
Logic, *-Autonomous
Categories and Cofree Coalgebras, in AMS
Conferenceon Categoriesin ComputerScience
and Logic, (Gray-ScedrovEds.) Boulder, 1987.
APPENDIX:
Proof of Lemma 6.3
We define two scales for measuring the complexity of a
proof: the rank and the grade.
A path in a derivation D from a leaf to the root is called a
thread. Given a threadT and an occurrenceof a formula A in
the final sequentof T, we call rank of A in T the number of
consecutivesequentsin T, counting upward, which contain
the sameoccurrence of the formula A. The rank of every
thread is at least 1. Given a derivation D with only one ocut as last inference,we call a thread a left (right) thread if it
contains the left (right) upper sequent of the cut. The left
(right) rank of a derivation D is the maximum among the
ranks of the left (right) threads in D. The rank of a
derivation D is the sum of its left and right ranks. The rank
of a derivation is at least 2.
The grade of a formula A (denoted by g(A) ) is the number
of logical symbols contained in A. The grade of a cut is the
grade of the cut formula. When a derivation D has only one
cut as the last inference, we define the grade of D (denoted
by g(D) ) to be the grade of this cut.
We prove the lemmaby double induction on the grade g and
rank r of the derivation D. The proof is subdivided in two
main cases,namely r-2 and ~2.
Case 1: r = 2
We distinguish casesaccording to the forms of the proofs of
the upper scquentsSI and S2 of the cut rule.
1.1) SI or S2 is an axiom. This case is not possible, since
by hypothesis we only consider axioms with atomic
formulae.
1.2) SI or S2 is the lower sequent of a cut rule.
Impossible, since r=2.
1.3) both S l and S2 are the Iower sequents of logicat
inferences. Since the Ieft and right rank of D is 1, the cut
formulae on each side arc the principal formulas of the
logical inferences.
68
We USCinduction on the grade. reducing the proof to
This proof contains at most an o-cut as its last infcrcncc.
another proof with cuts of lcsscr grade. We distinguish two
Furthcrmorc the grade of the cut formula is less than
g(A@B). By induction hypothesis, we have then a o-cut free
derivation D” of r, rt, B I- C. Then we have :
casesaccording to the outermost logical symbol of A.
(+) the derivation D has the structure:
r, A I- B
l-l I-A
l? I- A-B
BJ2
ry-B
I-C
r,rl,
r,rl,r2
A+B, l-l, l-2 I-C
l-c
Again this proof contains at most one o-cut as its last
r,rl,r21-c
inference and the grade of the cut formula is less than
g(A@B). By induction hypothesis, we have then a rcquestcd
o-cut free derivation D” of r, rl, B I- C.
where by assumption the proofs ending with r, A I- B ,
rl I- A or r2, B I- C do not contain any o-cut. Then
consider the following derivation, and note that both the
Case 2: r > 2
cuts have lesser grade than the original one:
rl I- A
B I-C
The induction hypothesis is that we can eliminate the cut
l-, A I- B
from every derivation D’ which contains only one cut as the
last inference, and which satisfies either g(D’)<g(D), or
r, rll-
B
l3, B I- C
r2,r,rl
g@‘)=g(D) and rank(D’)aankfD).
There are two main cases, namely: rankr(D)>l; rankl(D)>l
I- c
(and rank,(D) = 1).
2.1) rat&r(D) > 1
exchanges
r, rl,r2
we distinguish several subcases according to the logical
inference whose lower sequent is S2.
I- c
2.1.1) the sequent S2 is the lower sequent of a cut rule. The
(63) The derivation has the structure:
derivation D has three possible structures:
l-11-A
I-21-B
1-, A, B I- C
2.1.1.1)
l-1, I-2 I- AQDB
r~,A,l-2l-C
l-, A@B I- C
r I- A
r,rl,r2t-c
Al, rl, r, r2. 4 I-B
where by assumption the proofs ending with rl I- A , r2 IB and l7, A, B I- C do not contain any o-cut.
Where by hypothesis, C cannot be a open formula.
Consider the derivation:
Consider the derivation
rll- A
l-, A, B I- C
r,q,~
At, C, A2 I- B
r I- A
l-c
rl, A, r2 I- C
rl,r,r2
69
I- c
the grade of D’ is the same of D, namely g(A). Moreover
the two derivations have the sameleft rank, while rankr(D’)
= rankr(D)-1, thus we can apply the induction hypothesis,
obtaining a o-cut free derivation of rl, r, r2 I- C. Then
has the same grade of D, namely g(A).Moreover the two
derivations have the same left rank, while rankr(D’) =
rankf(D) - 1, thus we can apply the induction hypothesis,
obtaining a o-cut free derivation of 61, r, A2 I- B . With
an exchangerule we than obtain the requestedproof of rI ,
r, r2 I- B.
with a cut of C we obtain:
rl, r, r2 I- c
Al,C, A2l-B
2.1.3) the sequent S2 is the lower sequent of a logical
inference J. In this case the principal formula of the
inferencecannot be the cut formula. We distinguish several
casesaccording to the J rule:
Al, rl, r, r2,62 I-B
2.1.1.2)
rl I-C
A~,A,A~,CJQI-B
2.1.3.1) J = (-+, r)
I- I- A
AlAA2J-l,A3I-B
The derivation D has the structure:
rl, A, r2, B I- c
Al, r, 62, rl. A3 I- B
ri- A
analogousto 2.1.1.l)
2.1.1.3j
rl i-c
rl, A, r2 I- B-C
rl, r. r2 i- ~4
Al, C, A2, A, -43 I- B
Considerthe derivation D’:
r I- A
A1,rl,Ap%A3I-B
Al> rl,
TI-A
A21 r, A3 1-B
rl, r, r2, B i- c
analogous to 2.1.1.1)
D’ has the samegrade of D, namely g(A). Moreover the two
derivations have the same left rank, while rank,(D’)=
rankf(D)-1, thus we can apply the induction hypothesis,
obtaining a o-cut free derivation of rl, r, r2, B I- C. With
2.1.2) the sequent S2 is the lower sequentof an exchange
rule. The derivation has the structure:
Al, A, A2 I- B
m-A
rl, A, r2, B I- c
an application of (-+. r) we than obtain the requestedproof
0f rI, r, r2 i- ~4.
rl, A, r2 I- B
2.1.3.2) J = (+, 1) . Similarly
rl, r, r2 I- B
2.1.3.3) J = (8, r )
where Al, A2 is a permutation of rl, r2 .
The derivation D has two possible structures,namely:
The derivation D’
l- I- A
rl,A,r2i-B
A I-C
Al, A, A2 I- B
ri-A
l-l, A, l-2, A I- B@C
A1,r,A21-B
rl,r,r2,AI-B@c
70
A I-B
I- I- A
l-1, A, r2 I- C
A , I-1, A, r2 I- BW
A, rl, r, 5 I- B@C
We treat only the first case,the other one being completely
analogous.
Considerthe derivation D
Ti-A
Q,A,r2l-B
rl, r. r2 I-B
D’ has the samegrade of D, namely g(A). Moreover the two
derivations have the same left rank, while rank,(r)‘)=
rankr(D)-1, thus we can apply the induction hypothesis,
obtaining a o-cut free derivation of l-1, r, r2 I- B. With au
application of (8~.r ) we than obtain:
rl, r, r2 i-B
lYl,l-,1-2,A
A I-C
I-B&E
2.1.3.4) J = (a, 1 ). Similarly.
2.2) rankl(D) > 1 (and rankr(D) = 1)
This caseis proved in the sameway as 2.1 above.
This completesthe proof of Lemma 6.3 and henceof the ocut-elimination theorem.
71