A Top-down Approach to Combining Logics

Christoph Benzm¨uller

Dept. of Mathematics and Computer Science, Freie Universit¨at Berlin, Berlin, Germany

Keywords:

Combination of Logics, Context, Expressive Ontologies, Multi-agent Systems, Higher-order Logic, Semantic

Embedding, Proof Automation.

Abstract:

The mechanization and automation of combination of logics, expressive ontologies and notions of context are

prominent current challenge problems. I propose to approach these challenge topics from the perspective of

classical higher-order logic. From this perspective these topics are closely related and a common, uniform

solution appears in reach.

1 MOTIVATION

The mechanization and automation of (A) combina-

tion of logics, (B) context and (C) expressive on-

tologies are prominent current challenge problems.

Their solution is of signiﬁcant interest to computer

scientists, artiﬁcial intelligence researchers, computa-

tional linguists and philosophers. Application areas

include, for example, logic-based knowledge repre-

sentation and reasoning, multi-agent systems, the se-

mantic web, and computational social choice.

I propose to approach these challenge topics from

an higher-order logic (HOL) perspective. From this

perspective these topics are closely related and a uni-

form solution appears in reach. Moreover, off-the-

shelf higher-order automated theorem provers (HOL-

ATPs) can be readily employed for the automation of

reasoning with respect to these challenge topics.

(A) Combining Logics. Researchers from various

disciplines have developed and studied a wide range

of classical and non-classical logics (Woods and Gab-

bay, 2004). These developments were often targeted

at desirable properties that previous logics did not

adequately address. Examples include intuitionistic

logics, deontic logics, epistemic logics, many-valued

logics, relevant logic, linear logic, etc.

Philosophers and logicians have recently devel-

oped an increased interest in combining logics, and

in general theories of logical systems; new schools

and new journals have emerged. The interest is in

tools and techniques for modeling and analyzing inte-

grations, extensions, embeddings, and translations of

logics, and for studying commonalities and deviations

of logics, e.g., with respect to fundamental theorems

or computational properties.

Computer scientists and artiﬁcial intelligence re-

searchers in contrast have a strong practical interest

in combinations of logics and in developing effective

reasoning systems for them. This is because complex

and real-world reasoning tasks often require reason-

ing about beliefs, obligations, actions and change and

a host of other tasks that cannot be adequately mod-

eled in simpler logics.

However, building automated reasoning systems

that support combinations of logics is a very demand-

ing endeavor. One option is to develop a speciﬁc sys-

tem for each particular logic combination in question.

Doing this for all relevant and interesting combina-

tions is hardly feasible. In fact, there is a strong dis-

crepancy between the number of combined reasoning

systems that have been sketched on paper, and the

number of (non-trivial) combined reasoning systems

that have actually been implemented.

A second option is to develop ﬂexible, plug-and-

play frameworks for various logics and their combi-

nations. Notable developments in this direction in-

clude: Logic Workbench (LWB), LoTREC, Tableaux

Workbench (TWB), FaCT, ileanCoP, and the transla-

tion based MSPASS system.

System websites: LWB: http://www.lwb.unibe.ch/,

LoTREC: http://www.irit.fr/Lotrec/,

TWB: http://twb.rsise.anu.edu.au/,

FaCT: http://www.cs.man.ac.uk/∼horrocks/FaCT/,

ileanCoP: http://www.leancop.de/ileancop/,

MSPASS: http://www.cs.man.ac.uk/∼schmidt/mspass/.

Systems overview: http://www.cs.man.ac.uk/∼schmidt/

tools/

346

Benzmüller C..

A Top-down Approach to Combining Logics.

DOI: 10.5220/0004324803460351

In Proceedings of the 5th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2013), pages 346-351

ISBN: 978-989-8565-39-6

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

However, these systems are mainly restricted

to comparably inexpressive propositional logics and

in particular their support for ﬂexible logic com-

binations is still very limited. For example, none

of these systems supports quantiﬁed modal logics

(QMMLs) or quantiﬁed conditional logics (QCLs)

and their combinations. Only for ﬁrst-order quantiﬁed

monomodal logics some specialist provers do exist,

such as MleanCoP, MleanTAP, MleanSeP, GQML,

and f2p+MSPASS

, but none of these systems cur-

rently supports logic combinations.

(B) Context. The study of notions of context has a

long history in philosophy, linguistics, and artiﬁcial

intelligence. In artiﬁcial intelligence, a main motiva-

tion has been to resolve the problem of generality of

computer programs as identiﬁed by McCarthy (Mc-

Carthy, 1987). The generality aspect of context scru-

tinizes ﬂexible combinations (nestings) of contexts in

combination with rich context descriptions.

Giunchiglia (Giunchiglia, 1993) additionally em-

phasizes the locality aspect and the need for struc-

tured representations of knowledge. The locality as-

pect is particularly important for large knowledge

bases; the challenge is to effectively identify and ac-

cess information that is relevant within a given rea-

soning context.

Different approaches to formalizing and mecha-

nizing context have been proposed in the last decades;

many of these are outlined in overview articles (Ak-

man and Surav, 1996; de Paiva, 2003; Seraﬁni

and Bouquet, 2004) or in special issues of journals

(Lehmann et al., 2012).

McCarthy (McCarthy, 1993) has pioneered

the modeling of contexts as ﬁrst class objects (in

ﬁrst-order logic) and he introduced the predicate

ist. For example, in his approach the expression

ist(context of(“Ben’s Knowledge”),likes(Sue,Bill))

encodes that proposition Sue likes Bill is true in the

context of Ben’s knowledge. A main motivation

of McCarthy’s approach actually is to avoid modal

logics (here for the modeling of Ben’s knowledge).

His line of research has been followed by a number

of researchers, including, for example, Guha (who

has put contexts into Cyc), Buvac and Mason (Buvac

et al., 1995; Guha, 1991). Also Giunchiglia and

Seraﬁni (Giunchiglia and Seraﬁni, 1994) avoid modal

logics and propose the use of so called multilanguage

systems. They show various equivalence results to

common modal logics, but they also discuss several

properties of multilanguage systems not supported in

modal logics.

See http://www.cs.uni-potsdam.de/ti/iltp/qmltp/

systems.html for more information on these systems.

All of the above approaches avoid a higher-order

perspective on context. My position is complemen-

tary and I argue that a solid higher-order perspec-

tive on context can be very valuable for various rea-

sons. On the theory side the twist between formalisms

based on modal logic and formalisms based on ﬁrst-

order logic seems to dissolve, since both modal logics

(and other non-classical logics) and ﬁrst-order logics

are just natural fragments of HOL. Moreover, modal

(and other) contexts can be elegantly combined and

nested in HOL, so that a ﬂexible solution to Mc-

Carthy’s generality problem appears feasible.

Giunchiglia’s locality aspect can also be ad-

dressed. The means for this is relevance ﬁlter-

ing (Meng and Paulson, 2009; Pease et al., 2010;

Hoder and Voronkov, 2011).

such as the Suggested Upper Merged Ontology

SUMO (Pease, 2011) or CYC (Ramachandran et al.,

2005) contain a small but signiﬁcant number of

higher-order representations (Benzm¨uller and Pease,

2012). They are particularly employed for modeling

contexts, including temporal, epistemic, or doxastic

contexts. In SUMO, for example, a statement like

(loves Bill Mary) can be restricted to the year 2009

by wrapping it (at subterm level) into respective con-

text information: (holdsDuring (YearFn 2009) (loves

Bill Mary)). Similarly, the statement can be put into

an epistemic or doxastic context: (knows/believes Ben

(loves Bill Mary)). Moreover, contexts can be ﬂexibly

combined and the embedded formulas may be com-

plex: (believes Bill (knows Ben (forall (?X) ((woman

?X) => (loves Bill ?X)))). The close relation to Mc-

Carthy’s approach is obvious. A crucial requirement

for challenge (C) thus is to support ﬂexible context

reasoning in combination with other ﬁrst-order and

even higher-order reasoning aspects, and in combina-

tion with relevance ﬁltering in large knowledge bases.

The proposed Solution. The challenges (A), (B)

and (C) are addressed from a fresh, analytical per-

spective. The starting point is HOL, that is, classi-

cal higher-order logic (Church, 1940). Instead of syn-

thesizing new logic combinations from source logics

bottom-up as typically done in other approaches, the

approach works top-down: HOL is decomposed into

its embedded logic fragments (respectively their com-

positions). Recent work has shown that these HOL

fragments comprise prominent non-classical logics

such as propositional (normal) modal logics, intu-

itionistic logic, security logics, conditional logics and

logics for time and space (Benzm¨uller and Paulson,

2008, 2010, 2012; Benzm¨uller, 2011; Benzm¨uller and

ATop-downApproachtoCombiningLogics

347

Genovese, 2011; Benzm¨uller at al., 2012). These

fragments also comprise ﬁrst-order and even higher-

order extensions of non-classical logics, for which

only little practical automation support has been avail-

able so far. Most importantly, however, combinations

of embedded logics can be elegantly achieved in this

approach.

The HOL approach bridges between the Tarski

view of logics (for ’metalogic’ HOL) and the Kripke

view (for the embedded source logics) and exploits

the fact that well known translations of logics, such

as the relational translation (Ohlbach, 1991), can be

easily formalized in HOL. This way HOL-ATP sys-

tems can be uniformly applied to reason within and

also about embedded logics and their combinations.

2 HOL AND ITS AUTOMATION

Classical higher-order logic HOL (Andrews, 2002;

Church, 1940) is built on top of the simply typed λ-

calculus. The set T of simple types is usually freely

generated from a set of basic types {o, ι} (where o

is the type of Booleans and ι is the type of individu-

als) using the function type constructor . Instead of

{o, ι} a set of base types {o, ι, µ} is used, providing an

additional base type µ (the type of possible worlds).

The simple type theory language HOL is deﬁned

by (α, β, o ∈ T ):

s,t ::= p

| X

| (λX

)

αβ

| (s

αβ

)

| (¬

oo

)

∨

ooo

)

| (s

ααo

)

| (Π

(αo)o

αo

)

denotes typed constants and X

typed variables

(distinct from p

). Complex typed terms are con-

structed via abstraction and application. The logi-

cal connectives of choice are ¬

oo

, ∨

ooo

, =

ααo

and Π

(αo)o

(for each type α). From these connec-

tives, other logical connectives can be deﬁned in the

usual way. Often binder notation ∀X

s is used for

(αo)o

(λX

The semantics of HOL is well understood and

thoroughly documented in the literature (Andrews,

1972; Henkin, 1950; Benzm¨uller et al., 2004).

When choosing Henkin semantics (Andrews,

1972; Henkin, 1950) then G¨odel’s incompleteness re-

sults are circumvented and a framework is achieved

that obeys important theoretical properties such as re-

cursive axiomatizability, compactness, and countable

models. Theoretically the expressive power of HOL

with Henkin semantics corresponds to that of multi-

sorted ﬁrst-order logic enriched by inﬁnitely many

comprehension axioms. In some applications, how-

ever, HOL is better suited in practice both for elegant

modeling and for proof automation — this is a main

hypothesis of my work.

Automation of HOL has been pioneered by the

work of Andrews on resolution in type theory (An-

drews, 1971), by Huet’s pre-uniﬁcation algorithm

(Huet, 1975) and his constrained resolution calcu-

lus (Huet, 1973), and by Jensen and Pietroswski’s

(Pietrzykowski and Jensen, 1972) work. More re-

cently extensionality and equality reasoning in HOL

has been studied (Brown, 2007; Benzm¨uller et al.,

2004; Benzm¨uller, 2002; Benzm¨uller, 1999). The

TPS system

which is based on a higher-order mat-

ing calculus, is a pioneering ATP system for HOL.

The automation of HOL has recently made strong

progress. This has been fostered by the recent exten-

sion of the successful TPTP infrastructure for ﬁrst-

order logic (Sutcliffe, 2009) to higher-order logic,

called TPTP THF0 (Sutcliffe and Benzm¨uller, 2010;

Benzm¨uller et al., 2008).

Meanwhile several higher-order provers and

model ﬁnders accept the THF0 language as input.

These systems are available online via the Syste-

mOnTPTP tool (Sutcliffe, 2007), through which they

can be easily employed avoiding local installations.

These THF0 compliant systems currently include four

HOL-ATPs (TPS, LEO-II, Isabelle, and Satallax) and

three HOL (counter-)model ﬁnders (Refute, Nitpick,

and Satallax).

The progress in automating HOL is measurable in

terms of the improvement rates achieved in the yearly

THF0 CASC competitions

: In 2010 the winner

LEO-II performed 56% better than the 2009 cham-

pion TPS, the 2011 winner Satallax was 21% better

than the 2010 champion LEO-II, and in 2012 Isabelle-

HOL was 35% better than 2011 winner Satallax.

3 THE PROPOSED APPROACH

To illustrate the approach the embedding of QMML

in HOL is sketched next. The idea is simple: QMML

formulas are lifted in HOL to predicates over possi-

ble worlds, that is, HOL terms of type µ  o, where

µ is the reserved base type denoting the set of possi-

ble worlds. Modal operators such as ¬, ∨, 2 , and

http://gtps.math.cmu.edu/tps.html

The system websites are:

TPS: http://gtps.math.cmu.edu/tps.html,

LEO-II: http://www.leoprover.org,

Satallax: http://www.ps.uni-saarland.de/∼cebrown/

satallax/,

Isabelle: http://isabelle.in.tum.de/.

Refute/Nitpick: http://www4.in.tum.de/∼blanchet/

nitpick.html.

SystemOnTPTP: http://www.cs.miami.edu/∼tptp/cgi-bin/

SystemOnTPTP

http://www.cs.miami.edu/∼tptp/CASC/

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348

even quantiﬁcation over individuals ∀

ind

and propo-

sitions ∀

prop

can then be elegantly deﬁned as abbre-

viations of proper HOL terms, for example, ∨ =

λφ

µo

λψ

µo

λW

φW ∨ ψW and 2 = λR

ιιo

λφ

ιo

λW

∀V

¬RW V ∨ φV.

Similarly, the notion of validity of QMML for-

mulas can be explicitly deﬁned as an abbreviation in

HOL: valid = λφ

µo

∀W

φW. Thus, QMML proof

problems can be formulated in HOL in its original

syntax, for example, valid2

∃

prop

µo

Using

rewriting or deﬁnition expanding, such proof prob-

lems are reduced to corresponding statements con-

taining only the basic connectives of HOL; no ex-

ternal, error-prone transformation mechanism (as e.g.

needed in the ﬁrst-order based MSPASS approach) is

required for this. For the trivial example formula this

leads to ∀W

∀Y

¬rW Y ∨ ¬∀P

µo

¬(PY). This

formula is obviously valid in Henkin semantics (put

P = λY

⊤), and it can be effectively proved in a frac-

tion of a second by the above HOL provers.

Speciﬁc modal logics can now be easily axiom-

atized, for example, to model 2

as an S5 modal-

ity the axioms (M) valid∀

prop

φ 2

φ ⊃ φ and (5)

valid∀

prop

φ 3

φ ⊃ 2

φ are postulated. Al-

ternatively, corresponding semantic properties like

(reﬂexiver and (serialr) could be stated.

Similar semantic embeddings already exist e.g.

for intuitionistic logics, for access control logics, for

logics for spatial reasoning, for propositional con-

ditional logics and for quantiﬁed conditional log-

ics (Benzm¨uller and Paulson, 2008, 2010, 2012;

Benzm¨uller, 2011; Benzm¨uller and Genovese, 2011;

Benzm¨uller at al., 2012): in each case the logical con-

nectives of the source logic are equated to speciﬁc λ-

terms in HOL. All these logics are thus natural frag-

ments of HOL. The embedding of conditional logics

is particularly interesting since its selection function

semantics (Stalnaker, 1968), which is what we have

studied, can be seen as a higher-order extension of

Kripke semantics for modal logics and cannot be nat-

urally embedded into ﬁrst-order logic.

Figure 1 illustrates how logic combinations can be

achieved in the HOL approach (Benzm¨uller, 2011).

A small epistemic puzzle is presented, and Baldoni’s

formalization in QMML (Baldoni, 1998) is adapted

to the HOL framework.

The formalization employs a 4-dimensional quan-

Note how the deﬁniens of 2 abstracts over accessibil-

ity relations R. Via function application to concrete acces-

sibility relations r multiple 2

operators are obtained.

stands for the application of 2 to relation r.

In the HOL approach it is even possible to effectively

prove the correspondence of semantic properties and their

corresponding axioms (Benzm¨uller, 2011).

tiﬁed modal logic, combining the four epistemic

modalities 2

, 2

, and 2

fool

. The acces-

sibility relations associated with these box operators

are a, b, c, and fool. They are all of type µ  µ  o,

hence, they all range over the same world type µ. In

this sense the particular notion of a logic combination

employed here is related to that of a fusion (Thoma-

son, 1984). In order to model the example as a prod-

uct (Segerberg, 1973) of four logics, different world

types µ

, . . . , µ

could be introduced, and the modal

connectives could be copied for each of those. More-

over, axioms could be postulated to model the desired

product properties.

An important observation concerns the bridge

rules in Figure 1 (axioms 6, 7 and 8); they express mu-

tual relations in the scenario between the local epis-

temic contexts of the agents a, b, c and fool. Such

bridge rules, which are often crucial in the modeling

of logic combinations, can be directly expressed as

axioms in the HOL approach. In traditional bottom-

up approaches to combining logic, however, this is

often not possible due the lack of expressiveness and

axiom schemata or new calculus rules are needed in-

stead. This clearly poses a challenge, in particular, for

ﬂexible proof automation.

In ongoing work I also study semantic embed-

dings of the OWL2-full ontology language and the

Dolce ontology in HOL. The embedding is easy and

straightforward in both cases. This also implies that

the HOL approach is potentially suited as a ﬂexible

reasoning framework for semantic web applications

that require combinations of OWL2-full with other,

more expressive logics. Note that hardly any imple-

mented provers exist to date for such kind of logic

combinations.

4 SUMMARY

I have outlined a plug-and-play environment for rea-

soning within and about combinations of logics. This

HOL based approach even supports combinations of

very expressive logics (such as QMMLs and QCLs)

that are hardly supported to date. At the same time the

approach enables the integration of existing special-

ist reasoners, if available, for single embedded logics

(or logic combinations) and it is capable of cooper-

ating with them. This unique combination has the

potential to signiﬁcantly advance the state-of-the art

for the combinations of logics challenge in practice.

This challenge is timely and relevant for various ar-

eas, including artiﬁcial intelligence, formal methods,

computer security, and the semantic web.

From a philosophical perspective a framework

ATop-downApproachtoCombiningLogics

349

(Wise Men Puzzle) Once upon a time, a king wanted to ﬁnd the wisest out of his three wisest men. He ar-

ranged them in a circle and told them that he would put a white or a black spot on their foreheads and that one

of the three spots would certainly be white. The three wise men could see and hear each other but, of course,

they could not see their faces reﬂected anywhere. The king, then, asked to each of them to ﬁnd out the color

of his own spot. After a while, the wisest correctly answered that his spot was white. (How could he do this?)

We introduce epistemic modalities 2

, 2

, and 2

, for the three wise men, and 2

fool

, for encoding common

knowledge. The entire problem encoding consists now of the following axioms for X,Y, Z ∈ {a, b, c} and

X 6= Y 6= Z:

valid 2

fool

((ws a) ∨ (ws b) ∨ (ws c)) (1)

valid 2

fool

((ws X) ⊃ 2

(ws X)) (2)

valid 2

fool

(¬(ws X) ⊃ 2

¬(ws X)) (3)

valid ∀

prop

φ 2

fool

φ ⊃ φ (4)

valid ∀

prop

φ 2

fool

φ ⊃ 2

fool

φ (5)

valid ∀

prop

φ 2

fool

φ ⊃ 2

φ (6)

valid ∀

prop

φ ¬2

φ ⊃ 2

¬2

φ (7)

valid ∀

prop

φ 2

φ ⊃ 2

φ (8)

valid ¬ 2

(ws a) (9)

valid ¬ 2

(ws b) (10)

Axiom (1) says that a, b, or c must have a white spot and that this information is known to everybody. Axioms

(2) and (3) express that it is generally known that if someone has a white spot (or not) then the others see

and hence know this. Common knowledge 2

fool

is axiomatized as an S4 modality in axioms (4) and (5). For

, 2

, and 2

it is sufﬁcient to consider K modalities. The relation between those and common knowledge

fool

modality) is axiomatized in inclusion axioms (6). Axioms (7) and (8) encode that whenever a wise man

does (not) know something the others know that he does (not) know this. Axioms (9) and (10) say that a and

b do not know whether they have a white spot. Finally, the conjecture valid 2

(ws c) states that that c knows

he has a white spot. To solve the puzzle we thus want to prove (1), . .. , (10) |= valid 2

(ws c); the fastest

HOL-ATPs can solve this problems in a fraction of a second. For more details see (Benzm¨uller, 2011).

Figure 1: The wise men puzzle in the HOL based approach; adapting Baldoni’s formalization in QMML (Baldoni, 1998).

with some universal logic characteristics is provided.

This framework not only supports the reasoning

within logic combinations but also about their meta-

logical properties. To the best of my knowledge no

other such framework exists to date.

Another aim of my work is to attack the long-

standing preconception that HOL-ATP is not feasi-

ble in practice, and that it is not suited for applica-

tions. This preconception has had a strong inﬂuence

on the development of logic based approaches in vari-

ous research areas since the beginning of the last cen-

tury: HOL-ATP has usually been avoided due to its

allowedly quite unfavorable worst case complexity.

However, worst case complexity alone not necessar-

ily provides a good basis for judging about a partic-

ular approaches competitiveness and effectiveness in

speciﬁc application domains. In fact, modern HOL-

ATPs such as Satallax, LEO-II and Isabelle do inte-

grate specialist reasoners such as state of the art SAT

solvers or ﬁrst-order ATPs, and unsurprisingly they

can be quite competitive for these logic fragments. At

the same time they offer support for the automation of

more expressive formalizations whenever needed.

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