source: https://doi.org/10.7892/boris.26455 | downloaded: 10.8.2022
A Deep Inference System for the Modal Logic S5
Phiniki Stouppa
∗
March 1, 2006
Abstract
We present a cut-admissible system for the modal logic S5 in a for-
malism that makes explicit and intensive use of deep inference. Deep
inference is induced by the methods applied so far in conceptually pure
systems for this logic. The system enjoys systematicity and modularity,
two important properties that should be satisfied by modal systems.
Furthermore, it enjoys a simple and direct design: the rules are few
and the modal rules are in exact correspondence to the modal axioms.
Keywords modal logic S5, proof theory, deep inference, calculus of
structures, cut-admissibility.
1 Introduction
The failure of the sequent calculus to accommodate cut-admissible systems
for the important modal logic S5 (e.g. in Ohnishi and Matsumoto [21]) has
led to the development of a variety of new systems and calculi. A partial
solution to this problem has been presented in Shvarts [25] and Fitting [5],
where theorems of S5 are embedded into theorems of cut-free systems for
K45. These systems provide proof search procedures for S5, they are, how-
ever, systems of a weaker logic.
Complete solutions to the problem have been mainly obtained via two
techniques. The first one concerns the annotation of formulae with informa-
tion related to Kripke-frame semantics. This information is usually given
by means of labels or indices (e.g. in Kanger [16], Mints [18], Simpson [26],
Negri [20]). The second technique concerns the exhibition of formulae at
∗
Institut f¨ur Informatik und angewandte Mathematik, University of Bern. This work
has been completed while studying at the International Centre for Computational Logic,
Technische Universit¨at Dresden.
1
syntactic positions on which rules in usual Gentzen systems do not operate
(e.g. in Sato [24], Indrzejczak [11], Avron [1], Wansing [30]). These positions
may be obtained by adding new structural connectives to the usual sequents.
Systems built on the latter technique are conceptually pure, meaning that
their data structures correspond to modal formulae of S5. For this reas on,
this technique is in general preferable over the former.
A closer inspection on those systems shows that all of them overcome
the lack of cut-admissibility of the system for S5 in the sequent calculus,
following the same principle: the modal rules for S5 presented by Ohnishi
and Matsumoto [21]
1
α, 3Γ
1
, @∆
1
` 3Γ
2
, @∆
2
(3 `)
3α, 3Γ
1
, @∆
1
` 3Γ
2
, @∆
2
3Γ
1
, @∆
1
` 3Γ
2
, @∆
2
, α
(` @)
3Γ
1
, @∆
1
` 3Γ
2
, @∆
2
, @α
have the condition that all side formulae must be prefixed with a modality.
This restriction makes the cut-rule necessary in proofs of theorems like the-
orem B : p ⊃ @3p. The new systems allow derivations, the premise and
conclusion of which only partially match their corresponding in one of the
above rules, as the condition on side formulae is not satisfied. To illustrate
this, we present the cut-free pro of of axiom B in the hypersequent system
for S5 (see Avron [1]) (left) and its pro of with cut in the sequent system
(right):
p ` p
(` 3)
p ` 3p
(MS)
p ` | ` 3p
(` @)
p ` | ` @3p
(W
r
)
p ` @3p | ` @3p
(W
l
)
p ` @3p | p ` @3p
(C ext)
p ` @3p
(`⊃)
` p ⊃ @3p
p ` p
(` 3)
p ` 3p
3p ` 3p
(` @)
3p ` @3p
(cut)
p ` @3p
(`⊃)
` p ⊃ @3p
In the first proof, the partial matching of the (` @)-rule given above is
revealed by the derivation obtained when one removes the topmost and
lowest rule applications from it (i.e. the rules (` 3) and (`⊃)): each of the
sequents p ` 3p and p ` @3p matches the principal formula in the premise
and conclusion of the rule, respectively, the condition on side formulae is,
however, not satisfied.
1
We present their equivalent symmetric variants .
2
The admissibility of the cut rule in those systems relies strongly on such
partial matchings. Evidently, in the above example the premise and con-
clusion of the derivation match one of the premises and the conclusion of
the cut-rule (in the proof to the right), respectively. An overview of the
systems and details on how they allow such partial matchings can be found
in Stouppa [28].
Consequently, the systems provide mechanisms that allow, in certain
cases, deeper applications of the specific rules, so that the application affects
only a subsequent of a given sequent. Therefore, the formulation of cut-free
systems for S5 requires a technique to apply rules deeper on data structures.
Systems with rules that are applicable at any depth enjoy a form of deep
inference
2
. Thus, s uch systems allow nested structures of unbounded depth.
Among the existing systems for S5, only the system in display logic (Wansing
[30]) enjoys deep inference. The rest allow only nested structures of bounded
depth
3
. However, they have rules that are applicable at every syntactic
position. Such a rule is for instance the weakening, which in all systems (of
bounded depth) has two versions, one for each position.
Although deep inference is not necessary for just a cut-free formulation
for S5, its application allows for some proof theoretical advantages: both
the system in display logic and the system we are going to present next,
enjoy systematicity and modularity with respect to their modal rules. Sys-
tematicity refers to a clear technique for formulating the modal rules out
of the modal axioms, and modularity to that every axiom corresponds to
a finite number of rules. These properties have been introduced in Wans-
ing [30] and are among those that modal systems should satisfy, since they
are strongly related to the generality of a calculus. When one is concerned
with the latter, deep inference seems to be necessary as the only existing
conceptually pure system for B is the one in display logic (Wansing [30]) and
enjoys deep inference. On the other hand, rule applications in deep systems
can be combined in richer ways than in sequent systems and therefore, new
techniques related to proof search procedures are required.
In the following section we present a system for S5 in a deep inference
formalism, the c alculus of structures. The formalism accommodates, among
others, systems for classical logic (Br¨unnler [4]), as well as systems for dif-
ferent variants of linear logic (Straßburger [29], Guglielmi and Straßburger
2
Usually deep inference is applied on a calculus rather than on a particular system and
is used as a synonym to the calculus of structures. In this case though, it serves solely as
a property of a system.
3
The multiple sequent calculus allows for nested structures of unbounded depth; how-
ever, for simplification these have been dropped from the system for S5 (Indrzejczak [11]).
3
[8]) and several normal modal logics (Stewart and Stouppa [27]). The sys -
tem for S5 presented here, is a conservative extension of the system for S4
presented in Stewart and Stouppa [27], with rules that correspond to axiom
5. Apart from the properties described above, this system enjoys a simple
design with few rules, and a direct way of formalizing the modal rules out of
their axioms: for every axiom R ⊃ T ,
S{R}
S{T }
is a rule of the system. Also,
all rules can be restricted in such a way that their applications affect only a
bounded portion of the data structure. This property is called locality (see
Br¨unnler [4]) and induces a bounded computational cost for rule applica-
tions. The local system obtained remains simple, although the number of
rules is increased.
Cut-admissibility is presented in section 3 and has been obtained via
embedding of cut-free proofs from the hypersequent system for S5. The lat-
ter, as well as other hypersequent systems for non-classical logics, can also
be embedded into systems in the display logic, as the encoding of hyper-
sequents into display sequents in Wansing [31] suggests. Finally, section 4
summarizes our achievements and the possible research directions that can
strengthen the current results.
2 The System
The calculus of structures is a proof theoretical formalism introduced by
Guglielmi [7] that makes explicit and intensive use of deep inference: in this
formalism, all inference rules are granted with deep applicability. It is a
generalization of the one-sided sequent calculus, with formulae and sequents
being indistinguishable. Thus, all the connectives that appear in proofs are
logical ones and inference rules are defined only in terms of formulae. We
start with the syntactic presentation of modal formulae in the calculus of
structures:
Definition 1 Formulae in modal KS systems are built up as follows:
S ::= ff | tt | a | a | [S, S] | (S, S) | @ S | 3S,
where the units ff and tt stand for falsity and truth, the schematic letters
a, b, . . . and a, b, . . . for atoms and their complements, [S
1
, S
2
] and (S
1
, S
2
)
for disjunction and conjunction, and @S and 3S for the usual modal op-
erators. The formula context S{−} denotes a formula in which a positive
occurrence of a subformula is replaced by −, the hole, and the formula S{R}
is obtained by filling that hole with the formula R. Also, formulae of the form
4
[S
1
, [S
2
, . . . , [S
n−1
, S
n
] . . .]] are denoted by [S
1
, S
2
, . . . , S
n
] and those of the
form S{[S
1
, . . . , S
n
]} by S[S
1
, . . . , S
n
]. The analogous conventions are also
applied for conjunction.
Inference rules, rule applications and derivations are defined similarly to
those in the sequent calculus, with the distinction that now they range over
structures rather than sequents:
Definition 2 Modal structures are the classes of formulae obtained modulo
the equations of
• associativity: [R, [T, U]] = [[R, T ], U ] , (R, (T, U )) = ((R, T ), U)
• commutativity: [R, T ] = [T, R] , (R, T ) = (T, R)
• identity: R = [R, ff], R = (R, tt), tt = @tt , ff = 3ff
and the replacement theorem: If R and T are equivalent then so are S{R}
and S{T }, for any formula context S{−}.
Usually structures are denoted by one of their constitutive formulae and
so, a structure R denotes the class of formulae that are equivalent to formula
R (according to the above equations). Similarly, the structure R denotes the
class of formulae that are equivalent to the complement of R (in negation
normal form). Also, contrary to the sequent calculus, inference rules are now
deep and have precisely one premise. Thus, an inference rule is of the form
S{R}
S{T }
, and the tree-like notation on derivations is replaced by a linear one.
For instance, a derivation ∆ with the structures S
1
and S
2
as premise and
conclusion respectively, takes the form
S
1
k
k ∆
S
2
. Proofs are all derivations
with tt as a premise. As usual, derivations can be combined s equentially:
∆
1
; ∆
2
denotes the derivation obtained by extending ∆
1
with ∆
2
, provided
that the conclusion of ∆
1
and the premise of ∆
2
coincide. Moreover, given
a formula context S{−}, S{∆} denotes the derivation obtained from ∆ by
replacing every structure R in it with the structure S{R}.
Another characteristic of the calculus is that in every symmetric, non
cut-free system the dual rules are also rules of the system. The dual of a
rule is obtained by reversing and negating its premise and conclusion. This
symmetry is best esteemed in the case of the cut rule
S(R,
R)
i ↑
S{ff}
, which
5
