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2 Translating Formulæ from B to PFOL

# 2 Translating Formulæ from B to PFOL

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208

P. Halmagrand

θ(E) =

|

|

|

|

|

T

t

match E with

xT

E1 , E 2

E1 × E2

P(E)

f P(T ) (. . .)

Δ(x)

Pair(θ(E1 ), θ(E2 ))

Set(Pair(θ(E1 ), θ(E2 )))

Set(θ(E))

Set( T t )

= match T with

if id ∈ Ω then return Ω(id)

else Ω := Ω, (id, αid ) return αid

if id ∈ Ω then return Ω(id)

| id when f lag = gl →

else T := T , kid :: 0 ; Ω := Ω, (id, kid ) return kid

| T1 × T2

→ Pair( T1 t , T2 t )

| P(T )

→ Set( T t )

| id when f lag = ax →

P

f

=

|

|

|

|

|

match P with

⊥|

P1 ∧ P 2

P1 ⇒ P 2

¬P

∀xT · P

| E1 = E2

| E1 ∈ E2

⊥|

P1 f ∧ P 2 f

P1 f ⇒ P 2 f

¬ P f

∀x : T t . P f and Δ := Δ, x : T t

∀x1 : T1 t .∀x2 : T2 t . P f

and Δ := Δ, x1 : T1 t , x2 : T2 t

→ E1 e =θ(E1 ) E2 e

→ E1 e ∈θ(E1 ) E2 e

= match E with

| xT

| E1 , E 2

→x

→ ( E1 e , E2 e )θ(E1 ),θ(E2 )

| ∀(xT1 1 , xT2 2 ) · P

E

e

θ(E1 ) = Set(τ1 )

θ(E2 ) = Set(τ2 )

| P(E)

→ Pτ ( E e ) where θ(E) = Set(τ )

| f P(T ) (E1 , . . . , En ) →

if f : Πα1 . . . αm . τ1 → . . . → τn → τ ∈ T

then T := T , f : Sig(f P(T ) (E1 , . . . , En ))

⎨ θ(E1 ) = τ1 (τ1 , . . . , τm )

···

return f (τ1 , . . . , τm ; E1 e , . . . , En e ) where

θ(En ) = τn (τ1 , . . . , τm )

| E 1 × E2

→ E1

e

×τ1 ,τ2 E2

e

where

Fig. 4. Translation from B to PFOL

Finally, the translation of the goal (we unfold the ⊆ deﬁnition, see Sect. 2.1) is:

∀s : Set(k1 ), t : Set(k2 ), a : Set(Pair(k1 , k2 )), b : Set(Pair(k1 , k2 )).

(a, b) ∈ P(s × t) × P(s × t) ⇒ f (a, b) ∈ P(s × t)

Soundly Proving B Method Formulæ Using Typed Sequent Calculus

5

209

Translating LLproof Proofs into B Proofs

-1

In Fig. 5, we present the reverse translation, denoted ϕ , to translate

monomorphic PFOL formulæ into B formulæ. This reverse translation is simpler than the one presented in Sect. 4.2 because we do not need to translate

types, annotations for bound variables and function symbols not being necessary anymore.

ϕ

-1

f

e

-1

e

=

|

|

|

|

|

|

|

match ϕ with

⊥|

ϕ1 ∧ ϕ2

ϕ1 ⇒ ϕ2

¬ϕ

∀x : τ. ϕ

e1 =τ e2

e1 ∈τ e2

⊥|

-1

ϕ1 -1

f ∧ ϕ2 f

-1

ϕ1 f ⇒ ϕ2 -1

f

¬ ϕ -1

f

∀x · ϕ -1

f

-1

e1 -1

e = e2 e

-1

-1

e1 e ∈ e2 e

=

|

|

|

|

|

match E with

x

(e1 , e2 )τ1 ,τ2

e1 ×τ1 ,τ2 e2

Pτ (e)

f (τ1 , . . . , τm ; e1 , . . . , en )

x

-1

e1 -1

e , e2 e

-1

e1 e × e2 -1

e

P( e -1

e )

f ( e1 -1

e , . . . , en

-1

e )

Fig. 5. Translation from PFOL to B

Theorem 1. For a set of B formulæ Γ and a B goal P , if there exists a LLproof

proof of the sequent Γ , ¬P LL ⊥, then there exists a set Γ of monomorphic

-1

, ¬P B ⊥.

instances of Γ , and a B proof of the sequent Γ

Proof. We present a sketch of the proof.

-1

1. We show that if P is a B goal, then we have P

⇔ P.

2. Given a proof Π of the sequent Γ , ¬P LL ⊥, there exists a proof ΠKleene

of the sequent, starting with all applications of ∀type rules on polymorphic

formulæ, thanks to the permutation of inference rules in sequent calculus [15].

3. We take the subproof Πmono of ΠKleene , where we removed all the ∀type nodes

and the remaining polymorphic formulæ.

4. The set Γ of monomorphic instances of Γ is made of the root node formulæ

of Πmono , except ¬P .

5. We extend the reverse translation to LLproof sequents,

-1

-1

-1

-1

→ P1 , . . . , Pn

, and to LLproof proof

P1 , . . . , Pn LL Q

B Q

nodes in Figs. 6 and 7.

-1

-1

, ¬P B ⊥.

6. Πmono is a B proof of the sequent Γ

210

P. Halmagrand

Axiom

P

-1

P

BR3

¬P

P, ¬P

¬P

-1

R5

-1

BR3

=

t =τ t

-1

R10

¬(t =τ t)

¬(t =τ t)

-1

¬(t =τ t)

R5

-1

BR3

Sym

t =τ u

t =τ u

-1

t =τ t

BR3

-1

¬P

¬P

-1

BR3

¬¬P

¬¬P

¬(t =τ t)

¬(t =τ t)

t =τ u, ¬(u =τ t)

¬¬

R10

P

R5

¬¬P

P ∧Q

P ∧Q

P ∧Q

-1

P ∧Q

R2

-1

P

BR3

P ⇒ Q, ¬P

P ⇒Q

-1

P

Q

¬⊥

-1

-1

BR4

-1

BR3

-1

R2

-1

P ∧ Q, P, Q

-1

-1

BR4

BR4

BR6

R5

-1

-1

R5

R9

¬¬P, P

P ∧ Q, P

P ∧Q

P ∧Q

BR3

BR3

-1

-1

P ∧Q

-1

-1

-1

¬¬P

¬(t =τ t)

P ⇒Q

P ⇒Q

Q

P ⇒Q

-1

-1

P ⇒Q

BR3

MP

P ⇒ Q, Q

-1

-1

BR4

Fig. 6. Translations of LLproof Rules into B Proof System (part 1)

We give in Figs. 6 and 7 the translations for each LLproof proof node. Each

node can be translated to a B derivation where all PFOL sequents are translated

into B sequents, leading to a B proof tree. To lighten the presentation, we omit to

indicate the context Γ and some useless formulæ (removable by applying BR2) on

the left-hand side of sequents, and we use for LL . For instance, the translation

of the LLproof Axiom rule should be:

Γ, P, ¬P, ¬⊥

LL

P

-1

BR3

Γ, P, ¬P

Γ, P, ¬P, ¬⊥

LL

-1

LL

¬P

-1

BR3

R5

Example 5. The proof of the running example is too big to be presented here.

Instead, we present the proof translation for the following B formula, given s:

∀x · (x ∈ s ⇒ x ∈ s)

Soundly Proving B Method Formulỉ Using Typed Sequent Calculus

¬∧

¬(P ∧ Q), ¬P

-1

¬(P ∧ Q)

¬⊥

BR6

-1

R5

-1

P

¬(P ∧ Q), ¬Q

¬(P ∧ Q)

¬(P ∧ Q)

where Π :=

¬⇒

¬(P ∧ Q)

¬(P ⇒ Q), P, ¬Q

-1

¬(P ⇒ Q), P

¬(P ⇒ Q)

¬(P ∧ Q)

¬⊥

Q

P ∧Q

¬∀x : τ.P (x), ¬P (c)

¬∀x : τ.P (x)

¬(P ⇒ Q)

¬∀x : τ.P (x)

-1

-1

¬⊥

∀x : τ.P (x)

-1

∀x : τ. P (x)

∀x : τ. P (x)

∀x : τ. P (x)

P (t)

-1

R5

¬(P ⇒ Q)

-1

BR3

R5

BR6

R7

¬∀x : τ.P (x)

¬∀x : τ.P (x)

-1

BR3

R5

-1

BR3

R8

-1

Π

-1

-1

¬∀x : τ.P (x)

R1

R5

-1

P (c)

R5

-1

R5

¬(P ⇒ Q)

¬∀

Q

BR6

-1

BR6

-1

R3

-1

¬⊥

-1

¬(P ∧ Q)

BR3

-1

-1

P ⇒Q

-1

211

∀x : τ. P (x), P (t)

∀x : τ. P (x)

-1

-1

BR4

Subst

P (t), ¬(t =τ u)

P (t)

-1

¬⊥

t =τ u

-1

P (t)

-1

BR6

R5

P (u)

P (t)

P (t)

-1

-1

P (t)

BR3

R9

P (t), P (u)

-1

Fig. 7. Translation of LLproof Rules into B Proof System (part 2)

The latter leads to the PFOL formula, where k is a constant:

∀s : Set(k). ∀x : k. x ∈ s ⇒ x ∈ s

The LLproof proof is:

Ax

cx ∈k cs , cx ∈k cs LL ⊥

¬⇒

¬(cx ∈k cs ⇒ cx ∈k cs ) LL ⊥

¬∀

¬∀x : k. x ∈k cs ⇒ x ∈k cs LL ⊥

¬∀

¬∀s : Set(k). ∀x : k. x ∈k s ⇒ x ∈k s LL ⊥

-1

BR4

212

P. Halmagrand

We obtain the B proof (we removed the universal quantiﬁcation over the given

set s, the ﬁrst R5 node in the translation of ¬∀, some useless formulỉ on the

left-hand side of sequents and used for B , c for cx and s for cs ):

c∈s

BR3

BR3

c∈s

c∈s c∈s

R5

c ∈ s, c ∈ s ⊥

c∈s c∈s

R3

c∈s⇒c∈s

¬⊥

BR6

R5

¬(c ∈ s ⇒ c ∈ s)

¬(c ∈ s ⇒ c ∈ s)

6

¬(c ∈ s ⇒ c ∈ s)

c∈s⇒c∈s

∀x · (x ∈ s ⇒ x ∈ s)

BR3

R5

¬⊥

BR6

R5

R7

Conclusion

Automated theorem provers are in general made of thousands lines of code, using

elaborate decision procedures and speciﬁc heuristics. The conﬁdence in such

tools may therefore be questioned. The correctness of Zenon proofs is already

guaranteed by the checking of proof certiﬁcates by an external proof checker.

But to prove B proof obligations, Zenon relies on two external tools, bpo2why

and Why3, to translate proof obligations into its input format, which raises the

question whether the proof found still corresponds to a proof of the original

statement.

In this paper, we have formalized a diﬀerent and direct translation from the B

Method to a polymorphic ﬁrst-order logic. The main purpose of this work is not

to replace bpo2why, but to validate the use of Zenon to prove B proof obligations.

One of the most challenging part of this translation deals with the encoding of

the B notion of types. Our solution to make the axioms polymorphic allows us

to beneﬁt from the ﬂexibility of polymorphism. Furthermore, we showed that

this translation is sound and gave a procedure to translate Zenon proofs in the

B proof system.

As future work, we want to prove the soundness and completeness of the

deduction modulo theory [11] extension of the proof system LLproof with regard

to those of LLproof, in particular in the case of the B Method.

References

1. Abrial, J.R.: The B-Book: Assigning Programs to Meanings. Cambridge University

Press, Cambridge (1996)

2. Blanchette, J.C., Bă

ohme, S., Popescu, A., Smallbone, N.: Encoding monomorphic and polymorphic types. In: Piterman, N., Smolka, S.A. (eds.) TACAS

2013. LNCS, vol. 7795, pp. 493–507. Springer, Heidelberg (2013). doi:10.1007/

978-3-642-36742-7 34

3. Bobot, F., Filliˆ

of provers. In: International Workshop on Intermediate Verification Languages

(Boogie) (2011)

Soundly Proving B Method Formulæ Using Typed Sequent Calculus

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4. Bodeveix, J.-P., Filali, M.: Type synthesis in B and the translation of B to PVS.

In: Bert, D., Bowen, J.P., Henson, M.C., Robinson, K. (eds.) ZB 2002. LNCS, vol.

2272, pp. 350–369. Springer, Heidelberg (2002). doi:10.1007/3-540-45648-1 18

5. Boespflug, M., Carbonneaux, Q., Hermant, O.: The λΠ-calculus modulo as a universal proof language. In: Proof Exchange for Theorem Proving (PxTP) (2012)

6. Bonichon, R., Delahaye, D., Doligez, D.: Zenon: an extensible automated theorem

prover producing checkable proofs. In: Dershowitz, N., Voronkov, A. (eds.) LPAR

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1007/978-3-540-75560-9 13

7. Bury, G., Delahaye, D., Doligez, D., Halmagrand, P., Hermant, O.: Automated

deduction in the B set theory using typed proof search and deduction modulo.

In: LPAR 20 : 20th International Conference on Logic for Programming, Artificial

Intelligence and Reasoning, Suva, Fiji (2015)

8. Cauderlier, R., Halmagrand, P.: Checking Zenon modulo proofs in Dedukti.

In: Fourth Workshop on Proof eXchange for Theorem Proving (PxTP), Berlin,

Germany (2015)

9. Delahaye, D., Doligez, D., Gilbert, F., Halmagrand, P., Hermant, O.: Zenon modulo: when achilles outruns the tortoise using deduction modulo. In: McMillan, K.,

Middeldorp, A., Voronkov, A. (eds.) LPAR 2013. LNCS, vol. 8312, pp. 274–290.

Springer, Heidelberg (2013). doi:10.1007/978-3-642-45221-5 20

10. Delahaye, D., Dubois, C., March´e, C., Mentr´e, D.: The Bware project: building a

proof platform for the automated verification of B proof obligations. In: Ameur,

Y.A., Schewe, K.-S. (eds.) Abstract State Machines, Alloy, B, VDM, and Z (ABZ).

LNCS, vol. 8477, pp. 290–293. Springer, Heidelberg (2014)

11. Dowek, G., Hardin, T., Kirchner, C.: Theorem proving Modulo. J. Autom.

Reasoning (JAR) 31, 33–72 (2003)

12. Dowek, G., Miquel, A.: Cut elimination for zermelo set theory. Archive for Mathematical Logic. Springer, Heidelberg (2007, submitted)

13. Jacquel, M., Berkani, K., Delahaye, D., Dubois, C.: Verifying B proof rules using

deep embedding and automated theorem proving. Softw. Eng. Formal Methods

7041, 253–268 (2011)

´ Dubois, C.: Why would you trust B ? In: Dershowitz, N., Voronkov,

14. Jaeger, E.,

A. (eds.) LPAR 2007. LNCS (LNAI), vol. 4790, pp. 288–302. Springer, Heidelberg

(2007). doi:10.1007/978-3-540-75560-9 22

15. Kleene, S.C.: Permutability of inferences in Gentzens calculi LK and LJ. In:

Bulletin Of The American Mathematical Society, vol. 57, pp. 485–485. Amer Mathematical Soc, Providence (1951)

16. Mentr´e, D., March´e, C., Filliˆ

atre, J.-C., Asuka, M.: Discharging proof obligations from Atelier B using multiple automated provers. In: Derrick, J., Fitzgerald,

J., Gnesi, S., Khurshid, S., Leuschel, M., Reeves, S., Riccobene, E. (eds.) ABZ

2012. LNCS, vol. 7316, pp. 238–251. Springer, Heidelberg (2012). doi:10.1007/

978-3-642-30885-7 17

17. Schmalz, M.: Formalizing the logic of event-B. Ph.D. thesis, Diss., Eidgenă

ossische

Technische Hochschule ETH Ză

urich, Nr. 20516, 2012 (2012)

18. ClearSy: Atelier B 4.1 (2013). http://www.atelierb.eu/

Deriving Inverse Operators for Modal Logic

Michell Guzm´

an1(B) , Salim Perchy1 , Camilo Rueda3 , and Frank D. Valencia2,3

1

2

´

Inria-LIX, Ecole

Polytechnique de Paris, Palaiseau, France

michell.guzman@inria.fr

´

CNRS-LIX, Ecole

Polytechnique de Paris, Palaiseau, France

3

Pontificia Universidad Javeriana de Cali, Cali, Colombia

Abstract. Spatial constraint systems are algebraic structures from concurrent constraint programming to specify spatial and epistemic behavior in multi-agent systems. We shall use spatial constraint systems to

give an abstract characterization of the notion of normality in modal

logic and to derive right inverse/reverse operators for modal languages.

In particular, we shall identify the weakest condition for the existence

of right inverses and show that the abstract notion of normality corresponds to the preservation of finite suprema. We shall apply our results

to existing modal languages such as the weakest normal modal logic,

Hennessy-Milner logic, and linear-time temporal logic. We shall discuss

our results in the context of modal concepts such as bisimilarity and

inconsistency invariance.

Keywords: Modal logic · Inverse operators

Modal algebra · Bisimulation

1

·

Constraint systems

·

Introduction

Constraint systems (cs’s) provide the basic domains and operations for the

semantic foundations of several declarative models and process calculi from concurrent constraint programming (ccp) [3,8,9,11,15,18,23,25]. In these calculi,

processes can be thought of as both concurrent computational entities and logic

speciﬁcations (e.g., process composition can be seen as parallel execution and

conjunction). All ccp process calculi are parametric in a cs that speciﬁes partial

information upon which programs (processes) may act.

A cs is often represented as a complete algebraic lattice (Con, ). The elements of Con, the constraints, represent partial information and we shall think

of them as being assertions. The intended meaning of c d is that d speciﬁes

at least as much information as c (i.e., d entails c). The join operation , the

This work has been partially supported by the ANR project 12IS02001 PACE,

the Colciencias project 125171250031 CLASSIC, and Labex DigiCosme (project

ANR-11-LABEX-0045-DIGICOSME) operated by ANR as part of the program

“Investissement d’Avenir” Idex Paris-Saclay (ANR-11-IDEX-0003-02).

c Springer International Publishing AG 2016

A. Sampaio and F. Wang (Eds.): ICTAC 2016, LNCS 9965, pp. 214–232, 2016.

DOI: 10.1007/978-3-319-46750-4 13

Deriving Inverse Operators for Modal Logic

215

bottom true and the top false of the lattice (Con, ) correspond to conjunction, the empty information and the join of all information, respectively. The

ccp operations and their logical counterparts typically have a corresponding elementary construct or operation on the elements of the constraint system. In particular, parallel composition and conjunction correspond to the join operation,

and existential quantiﬁcation and local variables correspond to a cylindriﬁcation

operation on the set of constraints [25].

Similarly, the notion of computational space and the epistemic notion of belief

in the sccp process calculi [15] correspond to a family of functions [·]i : Con →

Con on the elements of the constraint system Con that preserve ﬁnite suprema.

These functions are called space functions. A cs equipped with space functions is

called a spatial constraint system (scs). From a computational point of view the

assertion (constraint) [c]i speciﬁes that c resides within the space of agent i. From

an epistemic point of view, the assertion [c]i speciﬁes that agent i considers c to

be true (i.e. that in the world of agent i the assertion c is true). Both intuitions

convey the idea of c being local to agent i.

The Extrusion Problem. Given a space function [·]i , the extrusion problem consists in ﬁnding/constructing a right inverse of [·]i , called extrusion function, satisfying some basic requirements (e.g., preservation of ﬁnite suprema). By right

inverse of [·]i we mean a function ↑i : Con → Con such that [↑i c]i = c. From a

computational point of view, the intended meaning of [↑i c]i = c is that within a

space context [·]i , ↑i c extrudes c from agent i’s space. From an epistemic point of

view, we can use [↑i c]i to express utterances by agent i, i.e., to specify that agent

i wishes to say c to the outside world. One can then think of extrusion/utterance

as the right inverse of space/belief.

Modal logics [21] extend classical logic to include operators expressing modalities. Depending on the intended meaning of the modalities, a particular modal

logic can be used to reason about space, knowledge, belief or time, among others. Some modal logics have been extended with inverse modalities to specify,

for example, past tense assertions in temporal logic [24], utterances in epistemic

logic [13], and backward moves in modal logic for concurrency [19], among others.

Although the notion of spatial constraint system is intended to give an algebraic

account of spatial and epistemic assertions, we shall show that it is suﬃciently

robust to give an algebraic account of more general modal assertions.

Contributions. We shall study the extrusion problem for a meaningful family

of scs’s that can be used as semantic structures for modal logics. These scs’s

are called Kripke spatial constraint systems because its elements are Kripke

structures. We shall show that the extrusion functions of Kripke scs’s, i.e. the

right inverses of the space functions, correspond to right inverse modalities in

modal logic. We shall derive a complete characterization for the existence of right

inverses of space functions: The weakest restriction on the elements of Kripke

scs’s that guarantees the existence of right inverses. We shall also give an algebraic characterization of the modal logic notion of normality as maps that preserve ﬁnite suprema. We then give a complete characterization and derivations of

216

M. Guzm´

an et al.

extrusion functions that are normal (and thus they correspond to normal inverse

modalities). Finally, we use the above-mentioned contributions to the problem

of whether a given modal language can be extended with right inverse operators. We discuss the implications of our results for speciﬁc modal languages

and modal concepts such the minimal modal logic Kn [10], Hennessy-Milner

logic [14], a modal logic of linear-time [20], and bisimulation.

2

Background: Spatial Constraint Systems

In this section we recall the notion of basic constraint system [3] and the more

recent notion of spatial constraint system [15]. We presuppose basic knowledge

of order theory and modal logic [1,2,10,21].

The concurrent constraint programming model of computation [25] is parametric in a constraint system (cs) specifying the structure and interdependencies

of the partial information that computational agents can ask of and post in a

shared store. This information is represented as assertions traditionally referred

to as constraints.

Constraint systems can be formalized as complete algebraic lattices [3]1 . The

elements of the lattice, the constraints, represent (partial) information. A constraint c can be viewed as an assertion (or a proposition). The lattice order is

meant to capture entailment of information: c d, alternatively written d c,

means that the assertion d represents as much information as c. Thus we may

think of c d as saying that d entails c or that c can be derived from d. The

least upper bound (lub) operator represents join of information; c

d, the

least element in the underlying lattice above c and d. Thus c

d can be seen

as an assertion stating that both c and d hold. The top element represents the

lub of all, possibly inconsistent, information, hence it is referred to as false. The

bottom element true represents the empty information.

Definition 1 (Constraint Systems [3]). A constraint system (cs) C is a

complete algebraic lattice (Con, ). The elements of Con are called constraints.

The symbols , true and false will be used to denote the least upper bound (lub)

operation, the bottom, and the top element of C, respectively.

We shall use the following notions and notations from order theory.

Notation 1 (Lattices). Let C be a partially ordered set (poset) (Con, ).

We shall use S to denote the least upper bound (lub) (or supremum or join)

of the elements in S, and S is the greatest lower bound (glb) ( inﬁmum or

meet) of the elements in S. We say that C is a complete lattice iff each subset

of Con has a supremum and an inﬁmum in Con. A non-empty set S ⊆ Con

is directed iff every ﬁnite subset of S has an upper bound in S. Also c ∈Con

is compact iff for any directed subset D of Con, c

D implies c

d for

1

An alternative syntactic characterization of cs, akin to Scott information systems, is

given in [25].

Deriving Inverse Operators for Modal Logic

217

some d ∈ D. A complete lattice C is said to be algebraic iff for each c ∈ Con,

the set of compact elements below it forms a directed set and the lub of this

directed set is c. A self-map on Con is a function f :Con → Con. Let (Con,

) be a complete lattice. The self-map f on Con preserves the supremum of

a set S ⊆ Con iff f ( S) = {f (c) | c ∈ S}. The preservation of the inﬁmum of a set is deﬁned analogously. We say f preserves ﬁnite/inﬁnite suprema

iﬀ it preserves the supremum of arbitrary ﬁnite/inﬁnite sets. Preservation of

ﬁnite/inﬁnite inﬁma is deﬁned similarly.

Spatial Constraint Systems. The authors of [15] extended the notion of cs to

account for distributed and multi-agent scenarios where agents have their own

space for local information and for performing their computations.

Intuitively, each agent i has a space function [·]i from constraints to constraints. Recall that constraints can be viewed as assertions. We can then think

of [c]i as an assertion stating that c is a piece of information residing within a

space attributed to agent i. An alternative epistemic logic interpretation of [c]i

is an assertion stating that agent i believes c or that c holds within the space of

agent i (but it may not hold elsewhere). Both interpretations convey the idea that

c is local to agent i. Similarly, [[c]j ]i is a hierarchical spatial speciﬁcation stating

that c holds within the local space the agent i attributes to agent j. Nesting of

spaces can be of any depth. We can think of a constraint of the form [c]i [d]j as

an assertion specifying that c and d hold within two parallel/neighboring spaces

that belong to agents i and j, respectively. From a computational/ concurrency

point of view, we think of as parallel composition. As mentioned before, from

a logic point of view the join of information corresponds to conjunction.

Definition 2 (Spatial Constraint System [15]). An n-agent spatial constraint system (n-scs) C is a cs (Con, ) equipped with n self-maps [·]1 , . . . , [·]n

over its set of constraints Con such that: (S.1) [true]i = true, and (S.2) [c

d]i =

[d]i for each c, d ∈ Con.

[c]i

Axiom S.1 requires space functions to be strict maps (i.e. bottom preserving).

Intuitively, it states that having an empty local space amounts to nothing. Axiom

S.2 states that the information in a given space can be distributed. Notice that

requiring S.1 and S.2 is equivalent to requiring that each [·]i preserves ﬁnite

suprema. Also S.2 implies that each [·]i is monotonic: I.e., if c d then [c]i [d]i .

Extrusion and utterance. We can also equip each agent i with an extrusion

function ↑i : Con → Con. Intuitively, within a space context [·]i , the assertion ↑i c

speciﬁes that c must be posted outside of (or extruded from) agent i’s space. This

is captured by requiring the extrusion axiom [ ↑i c ]i = c. In other words, we view

extrusion/utterance as the right inverse of space/belief (and thus space/belief as

the left inverse of extrusion/utterance).

Definition 3 (Extrusion). Given an n-scs (Con, , [·]1 , . . . , [·]n ), we say that

↑i is extrusion function for the space [·]i iﬀ ↑i is a right inverse of [·]i , i.e., iﬀ

[ ↑i c ]i = c.

218

M. Guzm´

an et al.

From the above deﬁnitions it follows that [c ↑i d]i = [c]i d. From a spatial

point of view, agent i extrudes d from its local space. From an epistemic view

this can be seen as an agent i that believes c and utters d to the outside world.

If d is inconsistent with c, i.e., c d = false, we can see the utterance as an

intentional lie by agent i: The agent i utters an assertion inconsistent with their

own beliefs.

The Extrusion/Right Inverse Problem. A legitimate question is: Given

space [·]i can we derive an extrusion function ↑i for it? From set theory we

know that there is an extrusion function (i.e., a right inverse) ↑i for [·]i iff [·]i

is surjective. Recall that the pre-image of y ∈ Y under f : X → Y is the set

f −1 (y) = {x ∈ X | y = f (x)}. Thus the extrusion ↑i can be deﬁned as a

function, called choice function, that maps each element c to some element from

the pre-image of c under [·]i .

The existence of the above-mentioned choice function assumes the Axiom

of Choice. The next proposition from [13] gives some constructive extrusion

functions. It also identiﬁes a distinctive property of space functions for which a

right inverse exists.

Proposition 1. Let [·]i be a space function of scs. Then

1. If [false]i = false then [·]i does not have any right inverse.

2. If [·]i is surjective and preserves arbitrary suprema then ↑i : c →

right inverse of [·]i and preserve arbitrary inﬁma.

3. If [·]i is surjective and preserves arbitrary inﬁma then ↑i : c →

right inverse of [·]i and preserve arbitrary suprema.

−1

[c]i

−1

[c]i

is a

is a

We have presented spatial constraint systems as algebraic structures for spatial

and epistemic behaviour as that was their intended meaning. Nevertheless, we

shall see that they can also provide an algebraic structure to reason about Kripke

models with applications to modal logics.

In Sect. 4 we shall study the existence, constructions and properties of right

inverses for a meaningful family of scs’s; the Kripke scs’s. The importance of

such a study is the connections we shall establish between right inverses and

reverse modalities which are present in temporal, epistemic and other modal

logics. Property (1) in Proposition 1 can be used as a test for the non-existence

of a right-inverse. The space functions of Kripke scs’s preserve arbitrary suprema,

thus Property (2) will be useful. They do not preserve in general arbitrary (or

even ﬁnite) inﬁma so we will not apply Property (3).

It is worth to point out that the derived extrusion ↑i in Property (3), preserves

arbitrary suprema, this implies ↑i is normal in a sense we shall make precise next.

Normal self-maps give an abstract characterization of normal modal operators, a

fundamental concept in modal logic. We will be therefore interested in deriving

normal inverses.

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