TIERED LOGIC FOR AGENTS

Rosalito Perez Cruz and John Newsome Crossley

∗

Faculty of Information Technology, Monash University, Australia

Keywords:

Agents, Logics for agent systems, Ontologies and agent systems.

Abstract:

We introduce a new kind of logic for agents in different localities, which works in tiers or layers.

At the base are local worlds with their own logic. Above them is a global logic that takes statements from the

local worlds and combines them. This allows communications between the different localities.

We give a basic example using ﬁrst order logic as the local logic and propositional calculus at the global level.

As a more sophisticated example we use the algebraic speciﬁcation language CASL and take the locations as

speciﬁcationsm. Moreover we then permit the combination of such speciﬁcations according to the architectural

speciﬁcations of CASL.

Although we only consider two layers in the present paper, we see no reason why the approach should not be

extended to any ﬁnite number of tiers. We prove soundness and completeness proofs for our logics.

1 INTRODUCTION

It is well established that the work of agents in a

multi-agent system is enhanced by the presence of on-

tologies. For an ontology to be useful, people will

have to agree to its terms and usage in the spirit of

sharing. However, human nature ensures that people

will not agree nor use something like an ontologycon-

sistently. Thus the idea of arriving at a global ontol-

ogy for a domain of application appears to be wishful

thinking. So it seems more appropriate to conceive of

pockets of communities sharing their ontologies and

coping with any differences. It is more realistic to

think of communities adopting a number of ontolo-

gies, each created within their local community.

We shall adopt an approach which contextualizes

the logics that support these ontologies, and thereby

point a way for agent systems to deal with heteroge-

nous ontologies. We shall describe two logics:

a ﬁrst

order logic of localities, Tiered FOL, which we use

as a basis, then we extend this technique to a language

Tiered CASL, where the localities are architectural

speciﬁcations in the Common Algebraic Speciﬁca-

tion Language, CASL, see (CASL, 2001; Bidoit and

Mosses, 2004). We prove completeness results for

both these logics.

∗

The authors would like to thank four referees for their

(solely) constructive comments and suggestions for reduc-

ing the length of this paper. A full account of technicalities

may be found in (Cruz and Crossley, 2008).

We use natural deduction systems throughout.

In the ﬁeld of AI and, by association, Logic, there

are two major styles of embedding localities

in a log-

ical system. The ﬁrst is in the Propositional Logic

of Context (PLC) of Buvac-Mason (Buvaˇc et al.,

1995) and their extension of this to FOL. The second

is the Local Models Sematics/MultiContext Systems

(LMS/MCS) of (Giunchiglia and Ghidini, 2000; Ghi-

dini and Seraﬁni, 1998). By no means do we imply

that these are the only two possible styles: there are

others such as in (Akman and Surav, 1996).

One example of an LMS approach in the ﬁeld of

Description Logic (DL) is that taken by Borgida and

Seraﬁn, who describe a Distributed DL in (Borgida

and Seraﬁni, 2003). A major problem has been the

transfer of knowledge between localities. Bridge

rules (see Section 2) were introduced in (Ghidini and

Seraﬁni, 1998), but the form of the rules was very

limited and only allowed the (partial) identiﬁcation of

one concept as a subset of another in a different lo-

cality. The idea is to align ontologies (or knowledge

bases) by expressing the connections between them.

The intent is that the logical system should allow the

relationship of concepts to be stated in the said ontolo-

gies, for example subsumption of concepts between

ontologies. To do this, Borgida and Seraﬁni extend

the usual DL formulation, taking their cue from the

Distributed First Order Logic (DFOL) of (Ghidini and

Seraﬁni, 1998). In their formulation a DL statement is

We use ”locality” rather than ”context” because the lat-

ter is so ambiguous.

369

Perez Cruz R. and Newsome Crossley J. (2009).

TIERED LOGIC FOR AGENTS.

In Proceedings of the International Conference on Agents and Artiﬁcial Intelligence, pages 369-376

DOI: 10.5220/0001120803690376

 SciTePress

preceded by a label that stands for the ontology. Then

they state bridge rules, which relate a concept in one

ontology to another one in a different ontology (see

(Borgida and Seraﬁni, 2003)). Thus they have seman-

tic mappings in the system.

Seraﬁni/Borgida/Ghidini take their technique

from Giunchiglia’s LMS which they call the

compose-and-conquer way of dealing with differ-

ences of languages in contexts. PLC uses a divide-

and-conquer technique and since we take our cue

from PLC, Tiered Logic is a divide-and-conquertech-

nique, though the terminology may not be entirely ap-

propriate as there are similarities to both.

In relation to Gabbay’s Fibring of Logics, see e.g.

(Gabbay and Nossum, 1997), we can easily see that

the way we chose the global model has strong afﬁni-

ties in ﬁbring. This can be seen in how we deﬁne what

models X

, and note that fib evaluates to m

itself in

our system(see Section 2).

In global (natural language) discourse one often

sees or hears statements in a foreign language used

in the middle of something in the local language, for

example in a television broadcast where the spoken

foreign language is accompanied by subtitles. Refer-

ences may then neeed to be changed or at least clari-

ﬁed. Consider the following two assertions:

“Le pr´esident `a dit qu’il n’y a aucune arme de

destruction de masse en Irak.”

“The President said that there are weapons of

mass destruction in Iraq.”

Here the references are to the same country, however

the reference to the president refers, in the ﬁrst case,

to the French one, and in the second, to the US Pres-

ident.

There is no contradiction between the quota-

tions, but there is between the two men.

In the media there would be an indication of the

locality, i.e. country. Thus we might have found in the

USA: “The President of France said that there are no

weapons of mass destruction in Iraq,” and in France:

“Aux

Etats Unis, le Pr´esident a dit qu’il y a armes de

destruction de masse en Irak”. Finally, in a third coun-

try: “In the USA, the President said there are weapons

of mass destruction in Iraq, but in France, its Presi-

dent said there are no weapons of mass destruction.”

Semantically we understand these utterances because

we tag each utterance with its context or, as we shall

say, “locality”, in these cases, France and the USA,

respectively. Then we interpret them in that locality.

“The President said that there are no weapons of mass

destruction in Iraq.”

The reference to weapons of mass destruction was

more problematic because we did not know whether there

were any in Iraq!

For agents in localities we again have the problem

of them communicating across different languages.

This paper is an attempt to provide a basic method

of formalizing such situations.

We give the ﬁrst presentation of what we call

“Tiered Logic”, which allows the inclusion of pow-

erful bridge rules. In our logic, statements made in a

local language are tagged with that local locality and

then become “atomic” statements or basic proposi-

tions in a higher tier of what we call the global logic.

With bridge rules any statement in one locality can

have consequences in another. So information can be

conveyed, or even translated, from one locality to an-

other.

We provide soundness and completeness proofs

for two varieties of our underlying idea of tiered logic.

For simplicity we assume that all our localities have

the same underlying logic, but different languages.

This restriction is not essential but a completely gen-

eral approach would be notationally horrendous. The

complications in our presentation come from the in-

teractions between the tiers: when a sentence from

one locality is used in a different locality, one has to

refer back to the ﬁrst locality in order to determine the

semantics.

Additionally we use Saˇsa Buvaˇc’s, see e.g. (Buvaˇc

et al., 1995) notion of ﬂatness (see Section 2). This

entails that once a statement has been made (and its

semantics determined for its own locality) then the

truth or falsehood of the statement is unaffected by

reporting it in another locality. Thus in the example

above, a US newspaper reporting what had been said

in the USA might include the statement that it had

been reported in France that the (US) President had

said there were weapons of mass destruction in Iraq.

The semantics here would only depend on what was

said in the US, not what was reported in France (as-

suming that the media tell the truth).

2 TIERED FOL

First we consider the informal semantics. We have a

number of localities, think of France, the USA, etc.,

each with its own local theory. In our ﬁrst example

we simply use ﬁrst order logic at each locality. These

comprise Tier 0. At each locality we have a traditional

model of the local theory, that is to say, a ﬁrst order

model. We collect these together to form a model for

the global (tier 1) language. The underlying seman-

tics at tier 1 is the standard semantics of propositional

calculus except that traditional propositional letters

are replaced by what we call ”basic” global formulae.

However, we also have interaction between the

ICAART 2009 - International Conference on Agents and Artificial Intelligence

370

Γ ⊢

(Exit)

Γ ⊢

(Enter)

Γ ⊢

(A → B)

Γ ⊢

→ B

)

(K)

Γ ⊢

(¬A)

→ ¬(A

)

(D)

Γ ⊢

¬(A

) → (¬A)

(T)

Γ ⊢

)

↔ A

(Flat)

Γ ⊢

A ↔ A

(Flat-0)

Figure 1: The transfer rules. Note that A and B must be

local formulae of l for the (Exit) and (Entry) rules. (Of

course this includes global formulae).

global scene and the localities. So we have to specify

how the semantics (the models) interact between tier

0 and tier 1. From an intuitive point of view the inter-

action is relatively simple and reﬂects our earlier in-

formal example. Intuitively: a formula is interpreted

in its local locality, so that a tier 0 formula is inter-

preted in a traditional ﬁrst order logic model (in tier

0 at a locality, l, say). On the other hand a global, or

tier 1, formula is interpreted using the values from the

tier 0 model (or models) according to the usual rules

for propositional calculus. When we go back down

from tier 1 to tier 0, the semantic value is unchanged.

(This depends on the fact that our formulae at tier 1

have no free variables and are therefore true or false.)

The formal deﬁnitions follow the usual pattern.

Syntax. Because of going up and down between tiers

the syntax looks a little complicated, however the ac-

tual formulae should be easily readable. We let L be

a set of localities. At each locality l ∈ L we have a

ﬁrst order logic with a language L

as usual. These

generate the strictly local formulae, which we denote

by ϕ, ψ, etc. Going up to the global level (tier 1)

we deﬁne the basic global formulae as strictly local

sentences tagged by their locality, e.g. ϕ

. These are

combined as in an ordinary propositional calculus and

we denote global formulae by Φ,Ψ, etc. But now we

can take these back down to the local level, where

they interact with formulae already there (including

strictly local formulae). We then take the inductive

closure in the usual way, to get the set of local formu-

lae at that locality.

Thus local formulae and global formulae are in-

ductively deﬁned using a pair of interacting inductive

deﬁnitions. Notice that although global formulae are

local formulae (for any locality) the reverse is deﬁ-

nitely not the case. For example, a strictly local for-

mula of locality l is not a global formula.

Examples. We assume that the language of local-

ity l has only the predicate letter P, and that the local-

ity k has only the predicate letters P

and P

Strictly local formulae: ∀xP(x) in the locality l;

(x) → P

(x)) in the locality k; and ∃yP

(y) in the

locality k.

Global formulae: ∀xP(x)

, ((∀xP(x))

→

(∃xP

(x))

), (∃yP

(y))

. Notice that the localities

are superscripts in the global formulae; Each global

example is either a superscripted local sentence or a

propositional combination of such sentences.

Local formulae for the locality k: (∀xP(x))

(x) → P

(x)), and ∃y((∀xP(x))

→ P

(y)). The

ﬁrst formula, (∀xP(x))

, is local (even in the locality

k) because it is a global formula; the second is local

in k because it is a strictly local formula of k; and the

third is local in k, because it is a ﬁrst order logic com-

bination of a strictly local (and therefore also local)

formula, P

(y), of k and a global (therefore also local)

formula, (∀xP(x))

Our axiom system is designed from reﬂecting on

the semantics. The (strictly) local syntax is simply

ﬁrst order logic in the language L

for tier 0 and

propositional calculus for tier 1. In addition to these

we havethe rules in Figure 1 which are essentially due

to (Buvaˇc et al., 1995). We read Γ ⊢

A as “Γ globally

proves A”and Γ ⊢

A as “Γ proves A in the locality l”.

The (Exit) and (Enter) rules allow us to move up

and down between the tiers, provided we appropri-

ately tag or untag the formula. The rules (K), (D) and

(T),

when used together with the (Exit) and (Enter)

rules, ensure that the propositional connectives com-

mute with moving between the tiers.

The rule (Flat), see (Buvaˇc et al., 1995), ensures

that once a statement has been made in one locality its

truth-value is unchanged when it is taken into another

locality. (Flat-0), which is our addition to the ideas of

Buvaˇc ensures consistency between local and global

versions of a statement. cf. footnote 7 above.

Remark 1

. If Ξ is the strictly local theory in the lo-

ality l, then we deﬁne the lifting of Ξ to the global

tier to be Ξ

= {ϕ

: Ξ ⊢

ϕ}.

Lemma 1

. 1. If Φ is a global formula, then Γ ⊢

(Φ ↔ Φ

) f

or any locality l.

2. Ξ ⊢

ϕ is equivalent to Ξ

⊢

Lemma 2

. If Φ and all formulae in Γ are global for-

ulae, and Γ ⊢

Φ, then Γ ⊢

Φ.

The proofs of these and all other results may be

found in (Cruz and Crossley, 2008).

We have retained Buvaˇc’s arrangement and labels.

If we did not have the (Exit) and (Enter) rules we would

be able to have, say, the apparent inconsistency of having

¬A at the local level and yet A

at the global level.

TIERED LOGIC FOR AGENTS

371

Theorem 1 (CNF for Global Formulae). Every

global formula is provably equivalent to a conjunc-

tion of disjunctions of basic global formulae.

Proof. First show that every global formulais globally

provably equivalent to a propositional combination of

basic global formulae, and then, as usual, put this into

conjunctive normal form. 

Formal Semantics. We ﬁrst deﬁne a strictly local

model for a locality l as being a model in the usual

ﬁrst order logic sense, and we denote such models as

. These are the tier 0 models. Then a model for the

global system, or tier 1, model is a set of such models:

M = {m

: l ∈ L}.

In order to deﬁne global satisfaction we need si-

multaneously to deﬁne local satisfaction, so we have

a double inductive deﬁnition. The reader should be

warned that the formal deﬁnitions, which may be

found in (Cruz and Crossley, 2008) look much more

forbidding than they are in practice. He or she should

refer back to our motivating section 2, and here we

shall only give an intuitive picture.

Given a basic global sentence ϕ

(which means

ϕ is a strictly local sentence of locality l), then ϕ

(globally) true in M = {m

: l ∈ L}, written M |=

if, and only if, m

ϕ. In this case we also say ϕ

locally satisﬁed at l, and we write this as M |=

If Φ is a global sentence, then we use the usual

rules of propositional calculus to compute its truth

value. This also covers local satisfaction.

This only partly deﬁnes global satisfaction, for it

only deﬁnes it for propositional combinations of basic

global sentences.

Remark 2 (Overlap Requirements)

. It is possible to

ave overlaps in the languages at the different local-

ities. Then we impose the requirement that if two

atomic sentences from different localities, are syntac-

tically identical, then they are semantically identical

also. This will then carry over to more complicated

formulae in the usual way.

It remains to deﬁne local satisfaction for local formu-

lae that are not global formulae. Such formulae may

contain free variables from a particular locality. We

simply do this in the obvious way, except that, be-

cause global formulae are sentences and have no free

variables, we can simply use the truth values of any

global sentences contained in such a formula.

Thus a local sentence A is locally satisﬁed in l if,

and only if, m

A. We also use the locutions “A is

(strictly locally) true in m

(at l)”, and “m

is a model

of (the sentence) A”.

There will be no ambiguity, because strictly local satis-

faction is not deﬁned for such formulae.

To determine global satisfaction of a global for-

mula put the formula into conjunctive normal form

by Theorem 1, then determine the truth value of eac h

basic global sub-formula ϕ

by determining the local

truth value of ϕ in l. Finally compute the global truth

value from these truth values.

Consistency and Soundness. There are many vari-

eties of consistency: strictly local, global and local.

Happily, because of our rule system they are all es-

sentially equivalent. For example, we say that a set of

global formulae Γ is globally consistent if Γ 6⊢

⊥and

that a set, Γ

, of formulae local in l is locally consis-

tent in l if Γ

6⊢

⊥. It then follows that if Σ is a set of

strictly local formulae then Σ is strictly locally con-

sistent if, and only if, it is locally consistent; if Σ is

a locally consistent set of local formulae in a locality

l, then Σ

= {A

: A ∈ Σ} is globally consistent; and

that if Σ is a set of global formulae, then Σ is glob-

ally consistent if, and only if it is locally consistent at

some locality l if, and only if, it is locally consistent

for every locality.

We deﬁne soundness in the obvious way: A rule

Γ,A, B ⊢

C is sound if whenever Γ,A and B are satis-

ﬁed (globally, or locally at l) then so isC, respectively.

Theorem 2

. 1. The axioms and rules for Tiered FOL

re both globally sound, and locally sound for any

locality l.

2. The rules and axioms for Tiered FOL are con-

sistent. 

Bridge Rules. Bridge rules are global formulae in-

volving local formulae from different localities. The

original rules are given by (Ghidini and Seraﬁni,

1998) and also used in description logics (Borgida and

Seraﬁni, 2003).

In description logic, suppose we have concepts, C

and D, in localities k and l, respectively, then, our ver-

sion of the rules in (Borgida and Seraﬁni, 2003) would

mean we would write C

⊑ D

which corresponds to

the informal sentence ∀x(C

(x) → D

(x)). However,

we cannot model this directly in our system.

Never-

theless we can certainly imitate the intent of Borgida

and Seraﬁni by adding rules of the form: For all con-

stants c common to localities k and l

Γ ⊢

(c)

Γ ⊢

(c)

However, our system admits very powerful rules.

For example, we can have rules that depend on not

just one locality inﬂuencing another, but more than

For an implementation of our scheme using description

logic see the ﬁrst author’s forthcoming thesis (Cruz, 2008).

ICAART 2009 - International Conference on Agents and Artificial Intelligence

372

one. We can have bridge axioms of the form ϕ

∧

→ χ

or bridge rules of the form

Γ ⊢

or with even more premises. Further examples of

bridge rules involving quantiﬁcation are: ∀xP(x)

→

∃xQ(x)

, and (∀xP

(x) → ∀yP

(y))

→ ((∃zQ

(z) →

∀wQ

(w) ∧ ∃vQ

(v))

Completeness and Decidability. In order to prove

the completeness of our system under the tier scheme,

we follow the technique of Leon Henkin (Henkin,

1949). Given a set, Γ, of consistent global formulae,

we extend this to a maximal consistent set, Γ

∞

, and

show this has a model.

The main difference from

the classical scheme is that we make maximal consis-

tent sets of sentences both at the global level, Γ

∞

, and

at each locality.

Now consider the strictly local sentences in (Γ

∞

)

These include the atomic (strictly) local sentences and

it is just these that are used, in the standard Henkin

way, to build a local model, m

. Then we collect these

into M = {m

: l ∈ L} as a global model for Γ

∞

The only unusual part is to show that Γ

∞

is closed

under the (Enter) and (Exit) rules. Suppose A is a

local sentence of l not in (Γ

∞

)

. Then we cannot have

in Γ

∞

. Hence ¬(A

) is in Γ

∞

, and by rule (D), (¬A)

is in in Γ

∞

and ﬁnally by (Enter), A is in (Γ

∞

)

, which

is a contradiction.

Theorem 3 (Completeness)

. The system of rules and

xioms for Tiered FOL is complete (both locally and

globally).

For decidablility we restrict ourselves to systems in

which the ﬁrst order logics in every locality are de-

cidable and there is only a ﬁnite number of localities

in our system.

Theorem 4

. If 1. the global system has only a ﬁnite

umber of localities and the strictly local theories at

each locality are decidable, and 2. there is a ﬁnite

number of bridge rules, them the global system is de-

cidable.

Proof. To decide whether

Γ ⊢

{Ξ

: l is a locality} → Φ

The restriction to global formulae is merely for conve-

nience. (Replace local formulae ∆ in a locality l by the set

of global formulae {ϕ

: ϕ ∈ ∆} and use the rules (Enter)

and (Exit).

The proof is as usual except that we have to ensure con-

sistency across localities. This is ensured by the model com-

monality requirement, see Remark 2 above.

express the sentence as a propositional combination

of basic global sentences.

Now use the truth values

of these basic global sentences to compute the value

of the sentence.

3 CASL

In the previous part of the paper there was no direct

interaction between localities except in the presence

of bridge rules, or overlapping languages (cf. Re-

mark 2). There are other possibilities dealing with

structured localities (Gabbay and Nossum, 1997).

Here we consider algebraic speciﬁcations, where new

speciﬁcations are built from old ones, as the local-

ities. From an ontology point of view, there is a

strong reason to use CASL typed languages as ontol-

ogy languages, primarily because the operations pro-

vided by CASL ﬂow over to the operations one might

want to do to ontologies, e.g. translate one to an-

other ( with operation), combine them ( and opera-

tion), hide some parts ( hide operation), extend them

( then operation).

Each locality l will now be a speciﬁcation de-

scribed in a language such as CASL (CASL, 2001;

Bidoit and Mosses, 2004). There is no necessity for

these speciﬁcations to be ﬁnite but in practice we

would expect them to be so.

CASL stands for “Common Algebraic Speciﬁ-

cation Language”, see (CASL, 2001; Bidoit and

Mosses, 2004). It was designed by the Common

Framework Initiative (CoFI) for algebraic speciﬁca-

tion and development. It is a tool for specifying the

modular and functional requirements of software, and

has ﬁrst order logic as its base language and as such it

may be used for for tier 0. A good overview of CASL

from an applied logic standpoint may be found in (Po-

ernomo et al., 2005) but we give a very brief review

of CASL here. We note that the constructions we use

are architectural speciﬁcations, this is to ensure the

uniformity of constructions and to avoid clashes of

notations.

CASL builds other speciﬁcations from basic spec-

iﬁcations. A basic speciﬁcation is an ordinary ﬁrst or-

der many-sorted logic of the form S

= < Σ, Ax >,

where Σ =< S, TF, P > is the signature which com-

prises sorts, functions and predicates, Ax is a set of

axiom formulae whose members come from the set

of well formed formulae of SP (WFF(SP)). Mod-

els for CASL speciﬁcations are ordinary many-sorted

models for ﬁrst order logic. Such a model M, is a Σ-

structure comprising non-empty carrier sets s

for all

See Remark 1 and Corollary 1 re the deﬁnition of Ξ

Thanks to Peter Mosses for clariﬁcation on this point.

TIERED LOGIC FOR AGENTS

373

s ∈ S, a function f

from w

→ s

for each f ∈ TF

w,s

a relation P

⊆ s

× ... × s

for each P ∈ P

with

w = s

...s

as the set of all Σ − models. We also de-

note the set of models SP by Mod(SP).

CASL Algebraic Operations. CASL provides al-

gebraic operations for building speciﬁcations. One

starts with basic speciﬁcations and then uses the op-

erations of translation, union, extension and hiding,

which we brieﬂy describe below. We use the archi-

tectural speciﬁcations of CASL so that we preserve

the categorical structuring of the set of speciﬁcations.

In practice this means that we have no problems of

clashes of names.

When one views a CASL speciﬁcation as a de-

scription of a theory i.e. a locality or ontology

(L¨uttich and Mossakowski, 2004), then we readily

have ontology operations at our ﬁnger tips. The oper-

ations that may be performed on CASL speciﬁcations

are deﬁned by speciﬁcation expressions in CASL lit-

erature.

Structured speciﬁcations are ways of combining

basic speciﬁcations. Fuller details of all our con-

structions may be found in the CASL Manual (CASL,

2001) or (Poernomo et al., 2005).

Translation is simply the renaming of constants,

predicates and functions in a speciﬁcation. Formally

a translation is the inductive closure of a symbol map-

ping ρ, which maps the symbols of SP to another

speciﬁcation, preserving sorts, etc..

This is written

in CASL as SP with ρ.

In CASL the union of two speciﬁcations (possibly

with some amalgamation) is achieved in such a way

that the union speciﬁcation is a conservative exten-

sion of the two given speciﬁcations

and, moreover,

the models of the union are always such that they have

reducts that are models of the originally given speciﬁ-

cations, see e.g. (Poernomo et al., 2005) or (Cengarle,

1994). Formally we proceed as follows.

Formally, the amalgamated union of two spec-

iﬁcations, written SP 1 and SP 2 is deﬁned as the

pushout in the following diagram.

SP 1

SP 2

inr

SP 1 and SP 2

inl

If symbols are in SP but not in the domain of ρ we make

the convention that they are left unchanged. However, we

also insist that this is done in such a way that there is no

clash of names.

I.e. no new sentences in the language of either speciﬁ-

cation are provable from the theory of the union speciﬁca-

tion.

Extensions are deﬁned in a very similar way to

unions except that we can extend by a partial speciﬁ-

cation. The extension of SP by SP EXT is denoted as

SP then SP EXT For examples, see (Poernomo et al.,

2005).

Hiding may perhaps be regarded as an opposite

of taking extensions. Given a SP and a symbol list

SL the operation SP hide SL cuts down the signature

SP hide SL to SP/SL). The models of SP hide SL are

Mod(SP hide SL) = {m|

: m ∈ Mod(SP)} where σ is

the injection from Σ to sig (SP), see (Poernomo et al.,

2005).

4 THE TIERED CASL SYSTEM

Syntax. We use architectural speciﬁcations as local-

ities and we recall that a speciﬁcation has a language

inside it and this we designate as the “local language”.

We then follow the same model as before (see Sec-

tion 2). In Tiered CASL, the strictly local formulae

are simply ﬁrst order formulae in the syntax of the

locality SP. Basic global formulae are strictly local

sentences annotated by superscripts that are speciﬁ-

cation [names]. Thus a strictly local sentence, ϕ, is

lifted to the global level as a basic global sentence

. Local formulae in a speciﬁcation (locality) SP

are the inductive closure of the strictly local formulae

and the global formulae.

Examples: We assume that the language of local-

ity SP 1 has only the predicate letter P, that the local-

ity SP 2 has only the predicate letters P

and P

, and

that locality SP 3 has only the predicate letter Q.

Strictly local formulae: ∀x : s•P(x) in the locality

SP 1 and ∀x : s• P(x) in the locality SP 1 and SP 2;

∀x : s• (P

(x) → P

(x)) in the locality SP 2; and ∃y :

s• P

(y) in the locality SP 1.

Global formulae: (∀x : s• P(x))

SP 1

((∀x : s• P(x))

SP 1 and SP 2

→ (∃x : s• P

(x))

SP 2

(∃y : s• P

(y))

SP 2

Local formulae for the locality SP 2:

(∀x : s • P(x))

SP 1

, ∀x : s • (P

(x) → P

(x)), ∃y : s •

((∀x : s • P(x))

SP 1

→ P

(y)), (∀x : s • P(x))

SP 1

→

(∀x : s• Q(x))

SP 3

and [(∀x : s• P(x))

SP 1

]

SP 3

The ﬁrst formula, (∀x : s• P(x))

SP 1

is local (even

in the locality SP 2) because it is a global formula;

the second is local in SP 2 because it is a strictly lo-

cal formula of SP 2; and the third is local in SP 2,

because it is a ﬁrst order logic combination of a

strictly local (and therefore also local) formula, P

(y),

of SP 2, and a global (therefore also local) formula,

(∀x : s• P(x))

SP 1

. The fourth is a mixture of global

ICAART 2009 - International Conference on Agents and Artificial Intelligence

374

Γ ⊢

′

ρ(Γ) ⊢

ρ(A)

′

with ρ

(trans)

If SL is any symbol list

Γ ⊢

′

Γ ⊢

′

hide SL

(hide)

provided sig {A∪ SP} does not contain SL.

Γ ⊢

SP 1

Γ ⊢

inl(A)

SP 1 & SP 2

(union

)

Γ ⊢

SP 2

Γ ⊢

inr(A)

SP 1 & SP 2

(union

)

Γ ⊢

SP 1

Γ ⊢

inl(A)

SP 1 then SP EXT

(ext

)

Γ ⊢

SP EXT

Γ ⊢

inr(A)

SP 1 then SP EXT

(ext

)

Figure 2: The structural rules involving speciﬁcations.

Γ ⊢

Γ, SP ⊢

(Enter)

Γ, SP ⊢

Γ ⊢

(Exit)

provided A is a local SP formula and

Γ is a set of global formulae.

Figure 3: The locality changing rules in Tiered CASL:

going from global to local and vice versa.

formulas from SP 1 and SP 3. The last one is a local

formula for it is derived from a global formula.

Bridge rules: (∀x : s • P(x))

SP 1

→ (∀x :

s • Q(x))

SP 3

, (P(a))

SP 1

→ (Q(b))

SP 3

, (∃x : s •

P(x))

SP 1

↔ (∃x : s• Q(x))

SP 3

We deﬁne derivations as before using the same

schemata, but add rules for structured speciﬁcations.

I.e., the rules of the global system Tiered CASL are

given by ﬁrst order logic at the local level and propo-

sitional calculus at the global level with transfer rules

as in Figure 1, and structural rules as in 2.

Consistency, strictly local, global and local is de-

ﬁned exactly as above in Section 2, and as before we

assume that all of the basic speciﬁcations, SP, in our

system are consistent.

The categorical nature of the construction of the non-

basic speciﬁcations guarantees that all of the speciﬁcations

constructed are consistent (provided the basic ones are!).

Semantics. Again we deﬁne the semantics of our sys-

tem, strictly local, global and local, exactly as in Sec-

tion 2, except that the models we are now considering

are many-sorted. Global models M will now be sets

of models m

such that SP i

s a speciﬁcation in our

system. However, because of the structural rules of

Figure 2, such a global model M must also include

models for all the speciﬁcations constructed from the

basic speciﬁcations using translation, union, exten-

sions and hiding.

The soundness of Tiered CASL is proved as be-

fore except that we now have also to consider the

structural rules. Since the other rules are treated in

the usual way, we only need consider the structural

rules and we take (union

) as an example.

Assume M |=

, then the local model m

SP 1

in M is such that m

SP 1

A. Let m

SP 2

be any model

of SP 2. Then the amalgamated union of m

SP 1

and

SP 2

is a model of inl(A). Since this is true for all

such pairs of models we have Γ |=

inl(A)

SP 1 & SP 2

The other cases are similar.

The initial idea of the completeness proof was in-

spired by that in Section 2. However, because changes

in basic speciﬁcations cause changes in any structural

speciﬁcation constructed from them, we have to mod-

ify our strategy.

First recall that localities (i.e. speciﬁcations) may

be built from other localities, so when we add wit-

nesses to each basic speciﬁcation, SP to get a new ba-

sic speciﬁcation SP+, this expands the speciﬁcation

at that locality in a trivial way, but it carrries over to

constructed speciﬁcations, so that for SP 1 and SP 2

we now have SP

1+ and SP 2+, to which we add new

constants to obtain (SP 1+ and SP 2+)+. Similarly

for speciﬁcations using the other operations of Sec-

tion 3: extension, hiding and translation.

When we construct the model the cases for the ba-

sic sets of rules proceed as before. We give just one

example for the structural rules.

(union

) Assume that A

SP 1+

∈ Γ

∞

. We now test

if inl(A)

(SP 1+ & SP 2+)+

∈ Γ

∞

. Suppose not, then we

have ¬(inl(A)

(SP 1+ & SP 2+)+

) ∈ Γ

∞

by maximality.

Thereforeinl(¬A)

(SP 1+ & SP 2+)+

∈ Γ

∞

since negation

commutes with the locality (by rules (D), (T) and (K))

and with inl by the deﬁnition of inl. But then by (hide)

inl(¬A)

inl(SP 1+)

∈ Γ

∞

and (¬A)

SP 1+

∈ Γ

∞

by (trans)

using the map (inl)

−1

. Finally using (D) and (T) once

more ¬(A

SP 1+)

∈ Γ

∞

which is a contradiction. 

Now, for each speciﬁcation SP+ we construct a lo-

cal model m

SP+

as in Section 2, and the global model

M = {m

SP+

: SP is a speciﬁcation}.

Theorem 5 (Completeness of Tiered CASL). The

system of rules and axioms for Tiered CASL is

complete (both locally and globally), i.e. if, for

TIERED LOGIC FOR AGENTS

375

every global model M and every global sentence Φ

we have M |=

Φ ↔ ⊢

Φ, and similarly for local

sentences for each speciﬁcation.

5 FUTURE WORK

We have described a scheme that provides for global

communication between agents in different localities,

possibly with different logics, but certainly with dif-

ferent languages. In doing so we have allowed one

locality to inﬂuence another by bridge rules. The new

range of rules is much more complex than those in

e.g. (Ghidini and Seraﬁni, 1998) and (Borgida and

Seraﬁni, 2003), since two (or more) localities may af-

fect what happens in another locality.

We have proved completeness and consistency re-

sults for a basic system and also for a system, Tiered

CASL, which allows the localities to be structured

speciﬁcations in CASL.

For a practical implementation of our scheme we

have built software where the local logic is PROLOG

and the global logic is propositional calculus.

There remains one general area that particularly

requires further investigation. How do we do quantiﬁ-

cation at the global level? (Buvaˇc et al., 1995) devel-

oped quantiﬁcation over localities and we see no dif-

ﬁculty in extending our work in that direction. How-

ever we would like to imitate Borgida’s C

⊑ D

di-

rectly , but it does not seem to make sense to write

∀x(C(x)

→ D(x)

) since some elements in locality k

may not be in locality l. So we remain like the ancient

Chinese mathematician, Liu Hui, see p. 74 of (Li Yan

and Du Shiran, 1987), “... not daring to guess, [we]

wait for a capable person to solve it.”

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