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:
2
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).
2
We use natural deduction systems throughout.
In the field of AI and, by association, Logic, there
are two major styles of embedding localities
3
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
3
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
Copyright
c
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 Gabbays 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
k
, and note that fib evaluates to m
k
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.
4
“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.
5
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.
4
“The President said that there are no weapons of mass
destruction in Iraq.
5
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
l
A
Γ
γ
A
l
(Exit)
Γ
γ
A
l
Γ
l
A
(Enter)
Γ
γ
(A B)
l
Γ
γ
(A
l
B
l
)
(K)
Γ
γ
(¬A)
l
¬(A
l
)
(D)
Γ
γ
¬(A
l
) (¬A)
l
(T)
Γ
γ
(A
l
)
k
A
l
(Flat)
Γ
l
A A
l
(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
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
l
. 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
1
and P
2
.
Strictly local formulae: xP(x) in the locality l;
(P
1
(x) P
2
(x)) in the locality k; and yP
2
(y) in the
locality k.
Global formulae: xP(x)
l
, ((xP(x))
l
(xP
1
(x))
k
), (yP
2
(y))
k
. 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))
l
,
(P
1
(x) P
2
(x)), and y((xP(x))
l
P
2
(y)). The
first formula, (xP(x))
l
, 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
2
(y), of k and a global (therefore also local)
formula, (xP(x))
l
.
Our axiom system is designed from reflecting on
the semantics. The (strictly) local syntax is simply
first order logic in the language L
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
γ
A as Γ globally
proves A”and Γ
l
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),
6
when used together with the (Exit) and (Enter)
rules, ensure that the propositional connectives com-
mute with moving between the tiers.
7
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-
c
ality l, then we define the lifting of Ξ to the global
tier to be Ξ
l
= {ϕ
l
: Ξ
l
ϕ}.
Lemma 1
. 1. If Φ is a global formula, then Γ
γ
(Φ Φ
l
) f
or any locality l.
2. Ξ
l
ϕ is equivalent to Ξ
l
γ
ϕ
l
.
Lemma 2
. If Φ and all formulae in Γ are global for-
m
ulae, and Γ
l
Φ, then Γ
γ
Φ.
The proofs of these and all other results may be
found in (Cruz and Crossley, 2008).
6
We have retained Buvaˇc’s arrangement and labels.
7
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
l
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
m
l
. 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 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
l
(which means
ϕ is a strictly local sentence of locality l), then ϕ
l
is
(globally) true in M = {m
l
: l L}, written M |=
γ
ϕ
l
,
if, and only if, m
l
|=
l
ϕ. In this case we also say ϕ
l
is
locally satisfied at l, and we write this as M |=
l
ϕ
l
.
8
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
h
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
l
|=
l
A. We also use the locutions “A is
(strictly locally) true in m
l
(at l)”, and “m
l
is a model
of (the sentence) A”.
8
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
l
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⊢
γ
and
that a set, Γ
l
, of formulae local in l is locally consis-
tent in l if Γ
l
6⊢
l
. 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
l
= {A
l
: 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
x
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
a
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
k
D
l
which corresponds to
the informal sentence x(C
k
(x) D
l
(x)). However,
we cannot model this directly in our system.
9
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
γ
D
l
(c)
Γ
γ
C
k
(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
9
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
k
ψ
l
χ
m
or bridge rules of the form
Γ
γ
ϕ
l
Γ
γ
ψ
k
Γ
γ
χ
m
or with even more premises. Further examples of
bridge rules involving quantification are: xP(x)
k
xQ(x)
l
, and (xP
1
(x) yP
2
(y))
k
((zQ
1
(z)
wQ
2
(w) vQ
3
(v))
l
.
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.
10
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.
11
Now consider the strictly local sentences in (Γ
)
l
.
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
l
. Then we collect these
into M = {m
l
: 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
)
l
. Then we cannot have
A
l
in Γ
. Hence ¬(A
l
) is in Γ
, and by rule (D), (¬A)
l
is in in Γ
and finally by (Enter), A is in (Γ
)
l
, which
is a contradiction.
Theorem 3 (Completeness)
. The system of rules and
a
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
n
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
γ
^
{Ξ
l
: l is a locality} Φ
10
The restriction to global formulae is merely for conve-
nience. (Replace local formulae in a locality l by the set
of global formulae {ϕ
l
: ϕ } and use the rules (Enter)
and (Exit).
11
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.
12
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.
13
CASL builds other specifications from basic spec-
ifications. A basic specification is an ordinary first or-
der many-sorted logic of the form S
P
= < 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
M
for all
12
See Remark 1 and Corollary 1 re the definition of Ξ
l
.
13
Thanks to Peter Mosses for clarification on this point.
TIERED LOGIC FOR AGENTS
373
s S, a function f
M
from w
M
s
M
for each f TF
w,s
,
a relation P
M
s
M
1
× ... × s
M
n
for each P P
w
with
w = s
1
...s
n
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..
14
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
15
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
i
1
-
SP 1
SP 2
i
2
?
inr
-
SP 1 and SP 2
inl
?
14
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.
15
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|
σ
: 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
ϕ
SP
. 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
1
and P
2
, and
that locality SP 3 has only the predicate letter Q.
Strictly local formulae: x : sP(x) in the locality
SP 1 and x : s P(x) in the locality SP 1 and SP 2;
x : s (P
1
(x) P
2
(x)) in the locality SP 2; and y :
s P
2
(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
1
(x))
SP 2
),
(y : s P
2
(y))
SP 2
.
Local formulae for the locality SP 2:
(x : s P(x))
SP 1
, x : s (P
1
(x) P
2
(x)), y : s
((x : s P(x))
SP 1
P
2
(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
2
(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
γ
A
SP
ρ(Γ)
γ
ρ(A)
SP
with ρ
(trans)
If SL is any symbol list
Γ
γ
A
SP
Γ
γ
A
SP
hide SL
(hide)
provided sig {A SP} does not contain SL.
Γ
γ
A
SP 1
Γ
γ
inl(A)
SP 1 & SP 2
(union
1
)
Γ
γ
A
SP 2
Γ
γ
inr(A)
SP 1 & SP 2
(union
2
)
Γ
γ
A
SP 1
Γ
γ
inl(A)
SP 1 then SP EXT
(ext
1
)
Γ
γ
A
SP EXT
Γ
γ
inr(A)
SP 1 then SP EXT
(ext
2
)
Figure 2: The structural rules involving specifications.
Γ
γ
A
SP
Γ, SP
λ
A
(Enter)
Γ, SP
λ
A
Γ
γ
A
SP
(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.
16
16
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
SP
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
1
) as an example.
Assume M |=
γ
A
SP
1
, then the local model m
SP 1
in M is such that m
SP 1
|=
λ
A. Let m
SP 2
be any model
of SP 2. Then the amalgamated union of m
SP 1
and
m
SP 2
is a model of inl(A). Since this is true for all
such pairs of models we have Γ |=
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
1
) 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 |=
γ
Φ
γ
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.
17
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
k
D
l
di-
rectly , but it does not seem to make sense to write
x(C(x)
k
D(x)
l
) 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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