Proof Search and Counter-model Construction
for Bi-intuitionistic Propositional Logic with
Labelled Sequents
Lu´ıs Pinto
1
and Tarmo Uustalu
2
1
Centro de Matem´atica, Universidade do Minho,
Campus de Gualtar, P-4710-057 Braga, Portugal, luis@math.uminho.pt
2
Institute of Cybernetics at Tallinn University of Technology,
Akadeemia tee 21, EE-12618 Tallinn, Estonia, tarmo@cs.ioc.ee
Abstract. Bi-intuitionistic logic is a conservative extension of intu-
itionistic logic with a connective dual to implication, called exclusion.
We present a sound and complete cut-free labelled sequent calculus for
bi-intuitionistic propositional logic, BiInt, following S. Negri’s general
metho d for devising sequent calculi for normal modal logics. Although it
arises as a natural formalization of the Kripke semantics, it is does not
directly support proof search. To describe a proof search procedure, we
develop a more algorithmic version that also allows for counter-model
extraction from a failed proof attempt.
1 Introduction
Bi-intuitionistic logic (also known as Heyting-Brouwer logic, subtractive logic) is
an extension of intuitionistic logic with a connective dual to implication, called
exclusion (coimplication, subtraction), a symmetrization of intuitionistic logic. It
first got the attention of C. Rauszer [14–16], who studied its algebraic and Kripke
semantics, alongside adequate Hilbert-style systems and sequent calculi. More
recently, it has been of interest to ÃLukowski [9], Restall [17], Crolard [2] and Gor´e
with colleagues [6, 1, 7, 8]. Part of the motivation is the expected computational
significance of the logic: one would expect proof systems working as languages
for programming with values and continuations in a symmetric way.
A particularity of bi-intuitionistic logic is that it admits simple sequent calculi
obtained from the standard ones for intuitionistic logic essentially by dualizing
the rule for implication. Although several authors have stated or “proved” that
these calculi enjoy cut elimination (most notably Rauszer [15] for her sequent
calculus), they are in fact incomplete without cut and thus not directly suitable
for backward (i.e., root-first) proof search. The reasons of the failure are similar
to those for the modal logic S5 (S4 + symmetry) and the future-past tense
logic KtS4 (S4 + modalities for the converse of the accessibility relation). A
closer analysis suggests that finding remedies that are satisfactory, both from
the structural proof theory and automated theorem proving points of view, is
challenging and provides insights into the subtleties of the logic.
In this paper we propose one solution to the problem. We describe a cut-free
labelled sequent calculus for bi-intuitionistic propositional logic, BiInt, where
the labels are interpreted as worlds in Kripke structures. Exploiting the fact
that BiInt admits a translation to the future-past tense logic KtS4, we obtain
it by the general method of S. Negri [12] for devising sequent calculi for normal
modal logics. Then, to formulate a search procedure and obtain a termination
argument we fine-tune it for the constructive logic situation with monotonicity of
truth. This approach is in line with S. Negri’s method where frame conditions are
uniformly transformed into inference rules, but termination of proof search of the
resulting sequent calculus must be obtained on a case-by-case basis. Interestingly,
bi-intuitionistic logic turns out to be a rather delicate case.
Cut-free sequent calculi for BiInt have also been proposed by Gor´e and col-
leagues. Gor´e’s first formulation [6] was in the display logic format, inspired
by a general method for devising display systems for normal modal logics. The
next formulation by Postniece and Gor´e [1, 7] achieves cut-freedom by combining
refutation with proof (passing failure information from premise to premise) to
be able to glue counter-models together without the risk of violating the mono-
tonicity condition of interpretations. The new nested sequent calculus by Gor´e,
Postniece and Tiu [8] is a refinement of the display logic version and basically
allows reasoning in a local world of a Kripke structure with references to facts
about its neighbouring worlds captured in the nested structure.
The paper is organized as follows. In Sect. 2, we introduce BiInt with its
Kripke semantics and the translation to KtS4. We also show its Dragalin-style
sequent calculus and why cut elimination fails. In Sect. 3, we introduce a labelled
sequent calculus for BiInt designed according to S. Negri’s recipe. In Sect. 4,
we refine this declarative system into a more algorithmic version, show that it
is sound and its rules also preserve falsifiability. In the next section (Sect. 5) we
define a proof search procedure for the calculus and show that it terminates. In
Sect. 6 we put the pieces together to conclude completeness. In the final section
we sum up and outline some directions for further enquiry.
2 Bi-intuitionistic propositional logic, Dragalin-style
sequent calculus and failure of cut elimination
We start by defining the logic BiInt. The language extends that of intuitionistic
propositional logic, Int, by one connective, exclusion, thus the formulae are given
by the grammar:
A, B := p | > | ⊥ | A ∧ B | A ∨ B | A ⊃ B | A B
where p ranges over a denumerable set of propositional variables which give
us atoms; the formula A B is the exclusion of B from A. We do not take
negations as primitive, but in addition to the intuitionistic (or strong) negation,
we have dual-intuitionistic (or weak) negation, definable by ¬A := A ⊃ ⊥ and
v A := > A.
2
The Kripke semantics defines truth relative to worlds in Kripke structures
that are the same as for Int. A Kripke structure is a triple K = (W, ≤, I) where
W is a non-empty set whose elements we think of as worlds, ≤ is a preorder
(reflexive-transitive binary relation) on W (the accessibility relation) and I—the
interpretation—is an assignment of sets of propositional variables to the worlds,
which is monotone w.r.t. ≤, i.e., whenever w ≤ w
0
, we have I(w) ⊆ I(w
0
).
Truth in Kripke structures is defined as for Int, but covers also exclusion,
interpreted dually to implication as possibility in the past:
– w |= p iff p ∈ I(w);
– w |= > always; w |= ⊥ never;
– w |= A ∧ B iff w |= A and w |= B; w |= A ∨ B iff w |= A or w |= B;
– w |= B ⊃ A iff, for any w
0
≥ w, w
0
6|= B or w
0
|= A;
– w |= A B iff, for some w
0
≤ w, w
0
|= A and w
0
6|= B.
A formula is called valid if it is true in all worlds of all structures. It is easy to see
that monotonicity extends from atoms to all formulae thanks to the universal
and existential semantics of implication and exclusion.
It is also a basic observation that the G¨odel translation of Int into the modal
logic S4 extends to a translation into the future-past tense logic KtS4 (cf. [9]).
As the semantics of KtS4 does not enforce monotonicity of interpretations,
atoms must be translated as future necessities or past possibilities (these are
always monotone): p
#
= ¤p (or ¨p); >
#
= >; ⊥
#
= ⊥; (A ∧ B)
#
= A
#
∧ B
#
;
(A ∨ B)
#
= A
#
∨ B
#
; (B ⊃ A)
#
= ¤(B
#
⊃ A
#
); (A B)
#
= ¨(A
#
B
#
).
A sequent calculus for BiInt is most easily obtained from Dragalin’s sequent
calculus for Int (as has been done by Restall [17] and Crolard [2]; Rauszer’s
[15] original sequent calculus was different). In Dragalin’s system sequents are
multiple-conclusion, but the implication-right rule is constrained. The extension
imposes a dual constraint on the exclusion-left rule. The sequents are pairs Γ ` ∆
where Γ, ∆ (the antecedent and succedent) are finite multisets of formulae (we
omit braces and denote union by comma as usual). Such a sequent is taken to
be valid if, for any Kripke structure K and world w, some formula in Γ is false
or some formula in ∆ is true. The inference rules are displayed in Fig. 1.
Note that the context ∆ is missing in the premise of the ⊃R rule and dually in
the premise of L we do not have the context Γ . The rules ⊃L and R involve
some contraction. This is necessary because we have chosen not to include a
general contraction rule.
This calculus is sound and complete w.r.t. the above-defined notion of validity
(completeness can be shown going through the algebraic semantics in terms of
Heyting-Brouwer algebras [14]). However it is incomplete without cut, as shown
by Pinto and Uustalu in 2003 (private email message from T. Uustalu to R. Gor´e,
13 Sept. 2004, quoted in [1]). It suffices to consider the obviously valid sequent
p ` q, r ⊃ ((p q) ∧ r). The only possible last inference in a proof could be
?
p, r ` (p q) ∧ r
p ` q, r ⊃ ((p q) ∧ r)
⊃R
3
initial rule and cut:
Γ, A ` A, ∆
hyp
Γ ` A, ∆ Γ, A ` ∆
Γ ` ∆
cut
logical rules:
Γ ` ∆
Γ, > ` ∆
>L
Γ ` >, ∆
>R
Γ, A, B ` ∆
Γ, A ∧ B ` ∆
∧L
Γ ` A, ∆ Γ ` B, ∆
Γ ` A ∧ B, ∆
∧R
Γ, ⊥ ` ∆
⊥L
Γ ` ∆
Γ ` ⊥, ∆
⊥R
Γ, A ` ∆ Γ, B ` ∆
Γ, A ∨ B ` ∆
∨L
Γ ` A, B, ∆
Γ ` A ∨ B, ∆
∨R
Γ, B ⊃ A ` B, ∆ Γ, A ` ∆
Γ, B ⊃ A ` ∆
⊃L
Γ, B ` A
Γ ` B ⊃ A, ∆
⊃R
A ` B, ∆
Γ, A B ` ∆
L
Γ ` A, ∆ Γ, B ` A B, ∆
Γ ` A B, ∆
R
Fig. 1. Dragalin-style sequent calculus for BiInt
but the premise is invalid as the succedent formula q has been lost. With cut,
the sequent is proved as follows:
p ` q, p, . . .
hyp
p, q ` q, p q, . . .
hyp
p ` q, p q, . . .
R
p, p q, r ` p q
hyp
p, p q, r ` r
hyp
p, p q, r ` (p q) ∧ r
∧R
p, p q ` q, r ⊃ ((p q) ∧ r)
⊃R
p ` q, r ⊃ ((p q) ∧ r)
cut
Cut elimination fails as we cannot permute the cut on the exclusion p q up
past the ⊃R inference for which the cut formula is a side formula. This is one
type of cuts that cannot be eliminated, there are altogether 3 such types [11].
This situation is similar to the naive sequent calculus for S5 where the sequent
p ` ¤♦p cannot be proved without cut, but can be proved by applying cut to
the sequents p ` ♦p and ♦p ` ¤♦p that are provable without cut.
3 L: a labelled sequent calculus
We now proceed to a labelled sequent calculus for bi-intuitionistic logic that we
call L. This calculus turns out to be complete without a cut rule. Essentially it is
a formalization of the first-order theory of the Kripke semantics in such a fashion
that the extralogical axioms corresponding to the reflexivity-transitivity condi-
tion on frames and monotonicity condition on interpretations do not necessitate
cut. Our design follows the method of S. Negri [12].
We proceed from a denumerable set of labels. A labelled formula is a pair
x : A where x is a label and A a formula. The intended meaning is truth of the
formula at a particular world.
Sequents are triples Γ `
G
∆ where Γ and ∆ are finite multisets of labelled
formulae, and G is a finite binary relation on labels called the graph. Graphs
4
preorder rules:
Γ `
G∪{(x,x)}
∆
Γ `
G
∆
refl
xGy yGz Γ `
G∪{(x,z)}
∆
Γ `
G
∆
trans
initial rule and monotonicity rules:
Γ, x : A `
G
x : A, ∆
hyp
xGy Γ, x : A, y : A `
G
∆
Γ, x : A `
G
∆
monL
yGx Γ `
G
y : A, x : A, ∆
Γ `
G
x : A, ∆
monR
logical rules:
Γ `
G
∆
Γ, x : > `
G
∆
>L
Γ `
G
x : >, ∆
>R
Γ, x : A, x : B `
G
∆
Γ, x : A ∧ B `
G
∆
∧L
Γ `
G
x : A, ∆ Γ `
G
x : B, ∆
Γ `
G
x : A ∧ B, ∆
∧R
Γ, x : ⊥ `
G
∆
⊥L
Γ `
G
∆
Γ `
G
x : ⊥, ∆
⊥R
Γ, x : A `
G
∆ Γ, x : B `
G
∆
Γ, x : A ∨ B `
G
∆
∨L
Γ `
G
x : A, x : B, ∆
Γ `
G
x : A ∨ B, ∆
∨R
xGy Γ `
G
y : B, ∆ Γ, y : A `
G
∆
Γ, x : B ⊃ A `
G
∆
⊃L
y /∈ G, Γ, ∆ Γ, y : B `
G∪{(x,y)}
y : A, ∆
Γ `
G
x : B ⊃ A, ∆
⊃R
y /∈ G, Γ, ∆ Γ, y : A `
G∪{(y,x)}
y : B, ∆
Γ, x : A B `
G
∆
L
yGx Γ `
G
y : A, ∆ Γ, y : B `
G
∆
Γ `
G
x : A B, ∆
R
Fig. 2. Labelled sequent calculus L
are a means to keep track of label dependencies and thus induce an accessibility
relation on worlds.
The inference rules are presented in Fig. 2. Some of them have provisos, that
we also write as rule premises. We let xGy abbreviate (x, y) ∈ G. Following
usual sequent calculus terminology, at a given rule, we call the explicit labelled
formula in the conclusion the labelled formula introduced by the rule or the main
labelled formula of the rule and the explicit labelled formulae in the premises the
side labelled formulae.
The interesting logical rules are those for implication and exclusion which
are dual. Notice the freshness condition on the label y in the rules ⊃R and L,
guaranteeing their soundness. We call label y the eigenlabel of the rule and x
the parent of y. Note also the presence of the monotonicity rules accounting for
propagation of truth (resp. falsity) to future (resp. past) worlds and preorder
rules which account for reflexivity and transitivity of accessibility.
The counter-example to cut elimination for the Dragalin-style sequent calcu-
lus is proved in L as follows:
x : p, y : r `
(x,y)
x : q, x : p
hyp
x : p, y : r, x : q `
(x,y)
x : q
hyp
x : p, y : r `
(x,y)
x : q, y : p q
R
x : p, y : r `
(x,y)
x : q, y : r
hyp
x : p, y : r `
(x,y)
x : q, y : (p q) ∧ r
∧R
x : p `
∅
x : q, x : r ⊃ ((p q) ∧ r)
⊃R
Notice the downward information propagation in the R inference to an already
existing label.
In a L-derivation the names of the eigenlabels can be changed (to new names
not occurring in the derivation) without changing the end sequent. This property
5
