* *

*Jacques Pazelle nous introduit au système de Belnap
(référence : N.D. Belnap. A useful four-valued logic, pages 7-37. Modern
Uses of MultipleValued Logic. Reidel, 1977.), qui sont des travaux parmi
d'autres.*

*Comme introduction on peut consulter le passage du Stanford Encyclopedie
sur les logiques multivaluées : http://plato.stanford.edu/archives/fall2002/entries/logic-manyvalued/*

On y trouve entre autres une série de références d'ouvrages, ainsi qu'un
bref aperçu historique.

*Enfin j'attire l'attention sur cet article :*

**TI: The solution of logical equations**

AU: Lobanov,-I.

SO: Automatic-Documentation-and-Mathematical-Linguistics. 1998;
32(5): 16-27; Original: Nauchno-Tekhnicheskaya-Informatsiya,-Seriya-2.
1998; (9): 34-40

PB: Allerton Press

AB: The article considers the process of solving logical equations
and proposes a simple method that makes it possible to solve logical
equations with a large number of two-valued variables. Methods are developed
for minimization of functions in three- and four-valued logics for two-valued
arguments. A complete system of inverse functions is constructed by
means of the proposed methods. It is proved that a rigorous result may
be obtained only in four-valued logic.

**Références bibliographiques sur la logique
tétravalente, classées par thèmes.**

*- Sur les ordinateurs quantiques
:*

**Quantum computer with mixed states and
four-valued logic**

Tarasov,-Vasily-E. (Skobeltsyn Institute of Nuclear Physics, Moscow
State

University, Moscow (Russian Federation)). E-mail: tarasov@theory.sinp.msu.r

Journal-of-Physics.-A,-Mathematical-and-General (28 Jun 2002) v. 35(25)
p. 5207-5235

Country of input: International Atomic Energy Agency (IAEA); DOI: 10.1088/0305-4470/35/25/305
Available online at the Web site for the Journal of Physics. A, Mathematical
and General (ISSN 4361-6447) http://www.iop.org/

English

In this paper we discuss a model of quantum computer in which a state
is an operator of density matrix and gates are general quantum operations,
not necessarily unitary. A mixed state (operator of density matrix)
of n two-level quantum systems is considered as an element of 4 sup
n -dimensional operator Hilbert space (Liouville space). It allows us
to use a quantum computer model with four-valued logic. The gates of
this model are general superoperators which act on n-ququat state. Ququat
is a quantum state in a four-dimensional (operator) Hilbert space. Unitary
two-valued logic gates and quantum operations for an n-qubit open system
are considered as four-valued logic gates acting on n-ququats. We discuss
properties of quantum four-valued logic gates. In the paper we study
universality for quantum four-valued logic gates. (author) 33-035833
EBSCO Online

http://ejournals.ebsco.com/direct.asp?ArticleID=600H4KBR5KA355B65K44

InformationQuest http://www.eiq.com/usr_login.html?sici=0305-4470%28000000%2935%3A25%3C5207%3
AX%3E2.0.CO%3B2-X

Commander par : FORM@INIST

http://services.inist.fr/cgi-bin/manage?modele=fre/form01.htm&a=remplir&orig
ine=B&numero_document=[SP_MACRO]sp.rvalue field="SP$ISSN"[/SP_MACRO]&volume=[SP_MACRO]sp.rvalue

field="SP$VOLUME"[/SP_MACRO]&numero_fascicule=[SP_MACRO]sp.rvalue

field="SP$ISSUE"[/SP_MACRO]&annee_publication=[SP_MACRO]sp.rvalue

field="PY"[/SP_MACRO]&premiere_page=[SP_MACRO]sp.rvalue

field="SP$PAGE1"[/SP_MACRO]&type_document=PERIODIQUE""

**Some physical applications of nonstandard
logical systems. Quantum mechanics and possibility theory. Four-valued
logic with fuzzy truth values.**

Nekotorye fizicheskie primeneniya nestandartnykh sistem logiki.
Kvantovaya mekhanika i teoriya vozmozhnosti. 4-znachnaya logika s nechetkimi
istinnostnymi znacheniyami. Bogdanova,-N.B.; Kostenko,-B.F.

Joint Inst. for Nuclear Research, Dubna (USSR). Lab. of Computing Techniques
and Automation. 1989. 10 p.

Availability: INIS

JINR-R--5-89-403 (JINRR589403) Russian

A four-valued propositional calculus with two fuzzy truth values which
allows to describe physical phenomena with taking into account the Heisenberg
uncertainty relation is suggested. It is shown that this logic complies
with axioms and inference rules of the classical two-vaued logic Cl
sub 2 and is characterized by the equivalence classes algebra which
is isomorphous to the algebra of Cl sub 2. The meaning of the main logical
operations is discussed in detail. 12 refs.; 2 figs.; 2 tabs. 22-038902

Some physical applications of nonstandard logical systems. Quantum mechanics
and possibility theory. Four-valued logic with accidental truth values.

Nekotorye fizicheskie primeneniya nestandartnykh sistem logiki. Kvantovaya
mekhanika i teoriya vozmozhnosti. 4-znachnaya logika so sluchajnymi
istinnostnymi znacheniyami. Bogdanova,-N.B.; Kostenko,-B.F.

Joint Inst. for Nuclear Research, Dubna (USSR). Lab. of Computing Techniques
and Automation. 1989. 9 p. Availability: INIS

JINR-R--5-89-402 (JINRR589402) Russian

A four-valued propositional calculus with two accidental truth values
which corresponds to the empirical level of description of physical
phenomena with taking into account the Heisenberg uncertainty relation
is suggested. Some results of the traditional theory of the quantum
measurement are reformulated in a form of simple rules for successive
measurements. Arguments are brought showing orthodox or Copenhagen interpretation
of quantum mechanics which postulates the coexistence of two different
levels pf physical phenomena description (quantum and classical) may
be determined more accurately on the basis of imaginations of the physical
information transmission process. 12 refs.; 3 tabs.

*- en psychologie*

**TI: An inconsistency tolerant
model for belief representation and belief revision**

AU: Chopra,-S.; Parikh,-R.

ED: Dean,-T.

SO:

IJCAI-99.-Proceedings-of-the-Sixteenth-International-Joint-Conference-on-Artificial-Intelligence.
1999: 192-7 vol.1

PB: Morgan Kaufmann Publishers, San Francisco, CA, USA

AB: We propose a model for representing and revising belief structures,
which relies on a notion of partial language splitting and tolerates
some amount of inconsistency while retaining classical logic. The model
preserves an agent's ability to answer queries in a coherent way using
Belnap's four-valued logic. Axioms analogous to the AGM axioms hold
for this new model. The distinction between implicit and explicit beliefs
is represented and psychologically plausible, computationally tractable
procedures for query answering and belief base revision are obtained.

**TI: Relevance sensitive belief
structures**

AU: Chopra,-S.; Parikh,-R.

SO: Annals-of-Mathematics-and-Artificial-Intelligence. 2000; 28(1-4):

259-85

PB: Baltzer

AB: We propose a new relevance sensitive model for representing
and revising belief structures, which relies on a notion of partial
language splitting and tolerates some amount of inconsistency while
retaining classical logic. The model preserves an agent's ability to
answer queries in a coherent way using Belnap's four-valued logic. Axioms
analogous to the axioms of Alchourron, Gardenfors and Makinson (1985)
[AGM axioms] hold for this new model. The distinction between implicit
and explicit beliefs is represented and psychologically plausible, computationally
tractable procedures for query answering and belief base revision are
obtained.

PY: 2000

FTXT: SwetsWise

http://www.SwetsWise.com/link/access_db?issn=1012-2443&vol=28&iss=1-4&page=2
59&FT=1

**TI: A four-valued logic B(4) of
E(9) for modeling human communication**

AU: Rine,-D.; Alnakari,-R.

SO:

Proceedings-30th-IEEE-International-Symposium-on-Multiple-Valued-Logic-ISMVL

-2000. 2000: 285-90

PB: IEEE Comput. Soc, Los Alamitos, CA, USA

AB: In this paper a four-valued logic on B(4) called Hyawic Form
UniLogic is presented. Examples of this UniLogic are given which model
problems in the area of human interaction, dialog and communication,
i.e., human speech. The four values, or states, are isolated, conflicting,
coexisting and unifying, forming a square lattice of B(4), Post Algebra
of order 2 with 4 elements, inside a Post Algebra on E(9) of order 3
with 9=3/sup */3 elements. This logic has been used to represent human
speech problems and their solutions in college classrooms. This logic
is used in different universities, countries and languages including
France, Algeria, Egypt and USA. It is used for policy analysis and planning,
as well as for academic research in inter-communications analysis. Software
packages using this logic are available from the developer Dr. Raiek
Alnakari.

IB: 0769506925

PY: 2000

*- en électronique
et circuits intégrés*

**T1: Multiple-valued logic-in-memory
VLSI architecture based on floating-gate-MOS pass-transistor logic**

PA: HANYU-T; KAMEYAMA-M

SO: IEICE-Transactions-on-Electronics. 1999; v E82-C (9) : 1662-1668

IS: 0916-8524

PY: 1999

LA: English

AB: A new logic-in-memory VLSI architecture based on multiple-valued
floating-gate-MOS pass-transistor logic is proposed to solve the communication
bottleneck between memory and logic modules. Multiple-valued stored
data are represented by the threshold voltage of a floating-gate MOS
transistor, so that a single floating-gate MOS transistor is effectively
employed to merge multiple-valued threshold-literal and pass-switch
functions. As an application, a four-valued logic-in-memory VLSI
for high-speed pattern recognition is also presented. The proposed
VLSI detects a stored reference word with the minimum Manhattan distance
between a 16-bit input word and 16-bit stored reference words.
The effective chip area, the switching delay and the power dissipation
of a new four-valued full adder, which is a key component of the proposed
logic-in-memory VLSI, are reduced to about 33 percent, 67 percent and
24 percent, respectively, in comparison with those of the corresponding
binary CMOS implementation under a 0.5-<mu>m flash EEPROM technology.

**T1: Compact multiple-valued multiplexers
using negative differential resistance devices**

PA: CHAN-H-L; MOHAN-S; MAZUMDER-P; HADDAD-G-I

SO: IEEE-Journal-of-Solid-State-Circuits. 1996; 31 (8) : 1151-1156

IS: 0018-9200

PY: 1996

LA: English

AB: Quantum electronic devices with negative differential resistance
(NDR) characteristics have been used to design compact multiplexers.
These multiplexers may be used either as analog multiplexers where the
signal on a single select line selects one out of four analog inputs,
or as four-valued logic multiplexers where the select line and the input
lines represent one of four quantized signal values and the output line
corresponds to the selected input. Any four-valued logic function can
be implemented using only four-valued multiplexers (also known as T-gates),
and this T-gate uses just 13 devices (transistors) as compared to 44
devices in CMOS. The design of the T-gate was done using a combination
of resonant tunneling diodes (RTD's) and heterojunction bipolar transistors
(HBT's) with the folded I-V characteristic (NDR characteristic) of the
RTD's providing the compact logic implementation and the HBT's providing
the gain and isolation. The application of the same design principles
to the design of T-gates using other NDR devices such as resonant tunneling
hot electron transistors

(RHET's) and resonant tunneling bipolar transistors (RTBT's) is also
demonstrated.

**TI: Multiple-valued logic**

AU: Butler,-J.-T.

SO: IEEE-Potentials. April-May 1995; 14(2): 11-14

AB: The ultimate usefulness of a number system depends on its
implementation. Multiple-valued logic has been implemented in charge-coupled
devices (CCD). In this technology, logic values are encoded as charge.
For example, prototype four-valued logic devices have been implemented
at the University of Twente (Enschede, Holland). Hitachi has implemented
a 16-valued memory that stores the equivalent of 10/sup 6/ bits. CCD
is more compact than any other VLSI technology. Although it is slower
than CMOS (complementary metal oxide semiconductor), it is much faster
than the disk and has the potential of replacing the disk. The use of
multiple-valued logic in CCD increases its storage capacity significantly.
Multiple-valued logic has also been implemented in current-mode CMOS.

**TI: A four-valued logic and switch-level
differences**

AU: Mou-Hu

SO:Proceedings.-The-Twenty-Fourth-International-Symposium-on-Multiple-Valued-Lo
gic-Cat.-No.94CH3406-6. 1994: 362-7

PB: IEEE Comput. Soc. Press, Los Alamitos, CA, USA

AB: In this paper, the application of a four-valued logic to the
switch-level test generation is studied. A switch-level operator fault
model is proposed. Switch-level U difference and Z difference of a function
to a fault are defined. A method to derive switch-level differences
is given. Finally, a new switch-level test generation algorithm for
CMOS circuits is presented.

IB: 0818656506

PY: 1994

**TI: Four-valued logic system and
its application to describing the specification and design verification
of combinational circuits**

AU: Miura,-Y.; Li,-J.; Wang,-L.; Naito,-S.

SO: Computer-Systems-Science-and-Engineering. Jan. 1994; 9(1):
38-45

AB: This paper proposes a novel four-valued logic system and its
application to describing specification and verifying design. By the
logic system, we can describe the specification for incompletely specified
circuits easily and verify design by four-valued simulation. The logic
system consists of four basic events. The system includes two basic
events in the traditional two-valued logic system. First we show the
completeness of the logic system. Then we present a method for describing
specification, which is given as the set of correspondence from input
vector sets to output vector sets. We also present a design verification
method on the four-valued logic. The verification method is realized
using backward simulation by the four-value. It is shown that the four-valued
backward simulation can decrease the time complexity for verification.

PY: 1994

**TI: Pacific Rim Fault Tolerant
Computing (PRFTC) Conference**

SO: Computer-Systems-Science-and-Engineering. Jan. 1994;
9(1):

AB: The following topics were dealt with: fault tolerant computing
for safety critical applications in Japan; comparison-based fault diagnosis
in multiprocessor systems; algorithm for availability evaluation of
multidimensional voting; hyper-geometric distribution model for hardware
debugging process; WSI network design and its evaluation using yields;
four-valued logic system; self-checking systems hierarchy; fault tolerant
clock synchronisation algorithm; hardware designs verification and validation
via hardware Petri nets.

PY: 1994

*- en informatique
théorique et appliquée*

**T1: Hypothesis support for information
integration in four-valued logics.**

**Theoretical computer science : exploring new frontiers of theoretical
informatics : Sendai, 17-19 August 2000**

PA: LOYER-Y; SPYRATOS-N; STAMATE-D; VAN-LEEUWEN-Jan, ed;
WATANABE-Osamu,

ed; HAGIYA-Masami, ed; MOSSES-Peter-D, ed; ITO-Takayasu, ed

SO: Lecture-notes-in-computer-science. 2000; 1872 : 536-548

IS: 0302-9743

PY: 2000

LA: English

AB: We address the problem of integrating information coming from
different sources. The information consists of facts that a central
server collects and tries to combine using (a) a set of logical rules,
i.e. a logic program, and (b) a hypothesis representing the server's
own estimates. In such a setting incomplete information from a source
or contradictory information from different sources necessitate the
use of many-valued logics in which programs can be evaluated and hypotheses
can be tested. To carry out such activities we propose a formal framework
based on Belnap's four-valued logic. In this framework we work with
the class of programs defined by Fitting and we develop a theory for
information integration. We also establish an intuitively appealing
connection between our hypothesis testing mechanism on the one hand,
and the well-founded semantics and Kripke-Kleene semantics of Datalog
programs with negation, on the other hand.

**T1: Computing and comparing semantics
of programs in four-valued logics.**

MFCS '99 : mathematical foundations of computer science : Szklarska
Poreba, 6-10 September 1999

PA: LOYER-Y; SPYRATOS-N; STAMATE-D; KUTYOWSKI-Miroslalw, ed; PACHOLSKI-Leszek,
ed; WIERZBICKI-Tomasz, ed

SO: Lecture-notes-in-computer-science. 1999; 1672 : 59-69

IS: 0302-9743

PY: 1999

LA: English

AB: The different semantics that can be assigned to a logic program
correspond to different assumptions made concerning the atoms whose
logical values cannot be inferred from the rules. Thus, the well founded
semantics corresponds to the assumption that every such atom is false,
while the Kripke-Kleene semantics corresponds to the assumption that
every such atom is unknown. In this paper, we propose to unify and extend
this assumption-based approach by introducing parameterized semantics
for logic programs. The parameter holds the value that one assumes for
all atoms whose logical values cannot be inferred from the rules. We
work within Belnap's four-valued logic, and we consider the class of
logic programs defined by Fitting. Following Fitting's approach, we
define a simple operator that allows us to compute the parameterized
semantics, and to compare and combine semantics obtained for different
values of the parameter. The semantics proposed by Fitting corresponds
to the value false. We also show that our approach captures and extends
the usual semantics of conventional logic programs thereby unifying
their computation.

AN: 990504904

**T1: Towards a logical semantics
for pure Prolog. 6th European Symposium on Programming**

PA: BARBUTI-R; DE-FRANCESCO-N; MANCARELLA-P; SANTONE-A; NIELSON-Hanne-Riis,
ed

SO: Science-of-computer-programming. 1998; 32 (1-3) : 145-176

IS: 0167-6423

PY: 1998

LA: English

AB: The coincidence of the declarative and procedural interpretations
of logic programs does not apply to Prolog programs, due to the depth-first
left-to-right evaluation strategy of Prolog interpreters. We propose
a semantics for Prolog programs based on a four-valued logic. The semantics
is based on a new concept of completion analogous to Clark's and it
enjoys the nice properties of the declarative semantics of logic programming:
existence of the least Herbrand model, equivalence of the model-theoretic
and operational semantics.

**T1: Combining explicit negation and negation by failure via
Belnap's logic**

PA: RUET-P; FAGES-F

SO: Theoretical-computer-science. 1997; 171 (1-2) : 61-75

IS: 0304-3975

PY: 1997

LA: English

AB: Copyright (c) 1996 Elsevier Science B.V. All rights reserved.
This paper deals with logic programs containing two kinds of negation:
negation as failure and explicit negation. This allows two different
forms of reasoning in the presence of incomplete information. Such programs
have been introduced by Gelfond and Lifschitz and called extended programs.
We provide them with a logical semantics in the style of Kunen, based
on Belnap's four-valued logic, and an answer sets’ semantics
that is shown to be equivalent to that of Gelfond and Lifschitz. The
proofs rely on a translation into normal programs, and on a variant
of Fitting's extension of logic programming to bilattices.

**TI: Hypothesis support for information
integration in four-valued logics**

AU: Loyer,-Y.; Spyratos,-N.; Stamate,-D.

ED: van-Leeuwen,-J.; Watanabe,-O.; Hagiya,-M.; Mosses,-P.-D.;
Ito,-T.

SO:

Theoretical-Computer-Science.-Exploring-New-Frontiers-of-Theoretical-Informatics.-International-Conference-IFIP-TCS-2000.-Proceedings-Lecture-Notes-in-Computer-Science-Vol.1872.
2000: 536-48

PB: Springer-Verlag, Berlin, Germany

AB: We address the problem of integrating information coming from
different sources. The information consists of facts that a central
server collects and tries to combine using: 1) a set of logical rules,
i.e. a logic program, and 2) a hypothesis representing the server's
own estimates. In such a setting incomplete information from a source
or contradictory information from different sources necessitate the
use of many-valued logics in which programs can be evaluated and hypotheses
can be tested. To carry out such activities we propose a formal framework
based on Belnap's four-valued logic. In this framework we work with
the class of programs defined by fitting and we develop a theory for
information integration. We also establish an intuitively appealing
connection between our hypothesis testing mechanism on the one hand,
and the well-founded semantics and Kripke-Kleene semantics of Datalog
programs with negation, on the other hand.

**TI: Computing and comparing semantics
of programs in four-valued logics**

AU: Loyer,-Y.; Spyratos,-N.; Stamate,-D.

ED: Kutylowski,-M.; Pacholski,-L.; Wierzbicki,-T.

SO:

Mathematical-Foundations-of-Computer-Science-1999.-24th-International-Symposium,-MFCS'99.-Proceedings-Lecture-Notes-in-Computer-Science-Vol.1672.
1999: 59-69

PB: Springer-Verlag, Berlin, Germany

AB: The different semantics that can be assigned to a logic program
correspond to different assumptions made concerning the atoms whose
logical values cannot be inferred from the rules. Thus, the well founded
semantics corresponds to the assumption that every such atom is false,
while the Kripke-Kleene semantics corresponds to the assumption that
every such atom is unknown. In this paper, we propose to unify and extend
this assumption-based approach by introducing parameterized semantics
for logic programs. The parameter holds the value that one assumes for
all atoms whose logical values cannot be inferred from the rules. We
work within Belnap's four-valued logic, and we consider the class of
logic programs defined by Fitting. Following Fitting's approach, we
define a simple operator that allows us to compute the parameterized
semantics, and to compare and combine semantics obtained for different
values of the parameter. The semantics proposed by Fitting corresponds
to the value false. We also show that our approach captures and extends
the usual semantics of conventional logic programs thereby unifying
their computation.

IB: 3540664084

PY: 1999

**TI: DOLORES: a system for logic-based
retrieval of multimedia objects**

AU: Fuhr,-N.; Govert,-N.; Rolleke,-T.

ED: Bruce-Croft,-W.; Moffat,-A.; van-Rijsbergen,-C.-J.; Wilkinson,-R.;

Zobel,-J.

SO: Proceedings-of-the-21st-Annual-International-ACM-SIGIR-Conference-on-Research-and-Development-in-Information-Retrieval.
1998: 257-65

PB: ACM, New York, NY, USA

AB: We describe the design and implementation of a system for
logic-based multimedia retrieval. As high-level logic for retrieval
of hypermedia documents, we have developed a probabilistic object-oriented
logic (POOL) which supports aggregated objects, different kinds of propositions
(terms, classifications and attributes) and even rules as being contained
in objects. Based on a probabilistic four-valued logic, POOL uses an
implicit open world assumption, allows for closed world assumptions
and is able to deal with inconsistent knowledge. POOL programs and queries
are translated into probabilistic Datalog programs which can be interpreted
by the HySpirit inference engine. For storing the multimedia data, we
have developed a new basic IR system which yields physical data abstraction.
The overall architecture and the flexibility of each layer supports
logic-based methods for multimedia information retrieval.

IB: 1581130155

PY: 1998

**TI: Retrieval of complex objects
using a four-valued logic**

AU: Rolleke,-T.; Fuhr,-N.

SO: SIGIR-Forum. 1996; : 206-14

PB: ACM

AB: The aggregated structure of documents plays a key role in
full text, multimedia, and network information retrieval (IR). Considering
aggregation provides new querying facilities and improves retrieval
effectiveness. We present a knowledge representation for IR purposes
which pays special attention to this aggregated structure of objects.
In addition, further features of objects can be described. Thus, the
structure of full text documents, the heterogeneity and the spatial
and temporal relationships of objects typical for multimedia IR, and
meta information for network IR are representable within one integrated
framework. The model we propose allows for querying on the content of
documents (objects) as well as on other features. The query result may
contain objects having different types. Instead of retrieving only whole
documents, the retrieval process determines the least aggregated entities
that imply the query.

PY: 1996

*- en mathématiques et logique*

**T1: Logical inference of clauses
based on Petri net models**

PA: CHUANG-LIN; CHANSON-S-T

SO: International-journal-of-intelligent-systems. 1998; 13 (9)
: 821-840

IS: 0884-8173

PY: 1998

LA: English

AB: This paper extends the logical inference of Horn clauses in
Petri net models to cover a large class of non-Horn clauses. Based on
four-valued logic and the conflict transition concept, we show how the
Petri net model for this class of non-Horn clauses can be constructed.
The clause inference is solved by the T-invariant method or the fixpoint
of markings. Both forward and backward inference can be used in our
model. It is further shown that these techniques are efficient for the
common classes of monotonic reasoning.

AN: 980440135

FTXT: EBSCO Online

http://ejournals.ebsco.com/direct.asp?ArticleID=BV9BMNFA9EJRKNC19RK6

**T1: Motivation and demotivation
of a four-valued logic**

PA: FOX-J

SO: Notre-Dame-Journal-of-formal-Logic. 1990; 31 (1) : 76-80

IS: 0029-4527

PY: 1990

LA: English

AB: Belnap offers two arguments for the usefulness of four-valued
logic. I argue that one of them, which rests on interpreting valuations
as states of our information, when taken seriously collapses into an
argument for two-valued logic in which relevance is lost, and that the
other, resting on Scott's thesis, is not an argument for its usefulness

**TI: A first order, four-valued,
weakly paraconsistent logic and its relation with rough sets semantics**

AU: Tsoukias,-A.

SO: Foundations-of-Computing-and-Decision-Sciences. 2002; 27(2):
77-96

PB: Inst. Comput. Sci. Poznan Univ. Technol. Poland

AB: A first order four-valued logic, called DDT, is presented
in the paper as an extension of Belnap's logic using a weak negation
and establishing an appropriate semantic for the predicate calculus.
The logic uses a simple algebraic structure, that is the smallest non
trivial interlaced bilattice on the four truth values, thus resulting
in a boolean algebra on the set of truth values. The logic is a language
for reasoning under uncertainty, enabling to capture hesitation due
either to inconsistent or incomplete information, while keeping a clear
distinction between these epistemic states. The logic was originally
developed for preference modelling purposes (for which a brief account
is given in the paper). The paper demonstrates and discusses the equivalence
between the semantics of this logic and of rough sets semantics. On
this basis, this papers presents the possibility of inducing rules from
examples, that can be integrated in systems whose inference is expressed
in the above logic. Such an approach enhances the potentialities of
the use of rough sets in classification, reasoning and decision support.

**TI: Relative sets and rough sets**

AU: Mousavi,-A.; Jabedar-Maralani,-P.

SO: International-Journal-of-Applied-Mathematics-and-Computer-Science.
2001; 11(3): 637-53

PB: Tech. Univ. Zielona Gora; Lubusky Sci. Soc. Zielona Gora

AB: In this paper, by defining a pair of classical sets as a relative
set, an extension of the classical set algebra which is a counterpart
of Belnap's four-valued logic is achieved. Every relative set partitions
all objects into four distinct regions corresponding to four truth-values
of Belnap's logic. Like truth-values of Belnap's logic, relative sets
have two orderings; one is an order of inclusion and the other is an
order of knowledge or information. By defining a rough set as a pair
of definable sets, an integrated approach to relative sets and rough
sets is obtained. With this definition, we are able to define an approximation
of a rough set in an approximation space, and so we can obtain sequential
approximations of a set, which is a good model of communication among
agents.

**TI: Relative completeness with
respect to two unary functions**

AU: Pantovic,-J.; Tosic,-R.; Vojvodic,-G.

SO: Discrete-Applied-Mathematics. 15 Oct. 2001; 113(2-3): 337-42

PB: Elsevier

AB: One of the most important results is the fact that the number
of clones

is a continuum for k>or=3, while the corresponding set for k=2 is
countable.

This shows a sharp difference when we go from the binary to the ternary

case. This paper discusses the relative completeness with respect to
the

clone generated by two unary functions and show the sharp difference
when we

go from four-valued logic to k-valued logic for k>4, as well. The
number of

maximal clones over a finite set is finite and increases when k increases.

However, there are two relative maximal clones if k=3, 4 and there is
one

relative maximal clone if k>4.

**TI: The solution of logical equations**

AU: Lobanov,-I.

SO: Automatic-Documentation-and-Mathematical-Linguistics. 1998;
32(5): 16-27; Original: Nauchno-Tekhnicheskaya-Informatsiya,-Seriya-2.
1998; (9): 34-40

PB: Allerton Press

AB: The article considers the process of solving logical equations
and proposes a simple method that makes it possible to solve logical
equations with a large number of two-valued variables. Methods are developed
for minimization of functions in three- and four-valued logics for two-valued
arguments. A complete system of inverse functions is constructed by
means of the proposed methods. It is proved that a rigorous result may
be obtained only in four-valued logic.

**TI: Displaying the modal logic
of consistency**

AU: Wansing,-H.

SO: Journal-of-Symbolic-Logic. Dec. 1999; 64(4): 1573-90

PB: Assoc. Symbolic Logic

AB: It is shown that the constructive four-valued logic N4 can
be faithfully embedded into the modal logic S4. This embedding is used
to obtain complete, cut-free display sequent calculi for N4 and C4,
the modal logic of consistency over N4. C4 is a natural monotonic base
system for semantics-based non-monotonic reasoning.

PY: 1999

**TI: Comparison of logic operators
for use in multiple-valued sum-of-products expressions**

AU: Hozumi,-T.; Kakusho,-O.; Hata,-Y.

SO: IEICE-Transactions-on-Information-and-Systems. May 1999; E82-D(5):
933-9

PB: Inst. Electron. Inf. & Commun. Eng

AB: This paper shows the best operators for sum-of-products expressions.
We first describe conditions of functions for product and sum operations.
We examine all two-variable functions and select those that meet the
conditions and then evaluate the number of product terms needed in the
minimum sum-of-products expressions when each combination of selected
product and sum functions is used. As a result of this, we obtain three
product functions and nine sum functions in three-valued logic. We show
that each of three product functions can express the same functions
and MODSUM function is the most suitable for reduction of product terms.
Moreover, we show that similar results are obtained in four-valued logic.

PY: 1999

**TI: A four-valued fuzzy propositional
logic**

AU: Straccia,-U.

ED: Pollack,-M.-E.

SO:

IJCAI-97.-Proceedings-of-the-Fifteenth-International-Joint-Conference-on-Artificial-Intelligence.
1997: 128-33 vol.1

PB: Morgan Kaufmann Publishers, San Francisco, CA, USA

AB: It is generally accepted that knowledge based systems would
be smarter and more robust if they can manage inconsistent, incomplete
or imprecise knowledge. This paper is about a four-valued fuzzy propositional
logic, which is the result of the combination of a four-valued logic
and a fuzzy propositional logic. Besides the nice computational properties,
the logic enables us also to deal both with inconsistency and imprecise
predicates in a simple way.

IB: 1558604804

PY: 1997

**TI: Reed-Muller-Fourier versus
Galois field representations of four-valued logic functions**

AU: Stankovic,-R.-S.; Jankovic,-D.; Moraga,-C.

SO: Proceedings.-1998-28th-IEEE-International-Symposium-on-Multiple-Valued-Logic
-Cat.-No.98CB36138. 1998: 186-91

PB: IEEE Comput. Soc, Los Alamitos, CA, USA

AB: In this paper, we introduce Reed-Muller-Fourier representations
of four-valued functions over the ring of integers modulo 4 and compare
them with the Galois field representations over GF(4). It is reported
that on the average Reed-Muller-Fourier representations exhibit a lower
complexity than those based on Galois field polynomials.

IB: 0818683716

PY: 1998

**TI: A proof method for the six-valued
logic for incomplete information**

AU: Akama,-S.

SO:

Proceedings.-1997-27th-International-Symposium-on-Multiple-Valued-Logic-Cat.
-No.97CB36061. 1997: 223-6

PB: IEEE Comput. Soc. Press, Los Alamitos, CA, USA

AB: O. Garcia and M. Moussavi (1990) proposed a six-valued logic
to model incomplete information. Although their logic is promising,
it lacks a proof theory to serve as a basis for automated reasoning.
We describe a sequent style proof method for the logic and show that
the consequence relation in the logic is equivalent to the one by N.D.
Belnap's (1977) four-valued logic. We also suggest alternative formal
interpretations for the logic.

IB: 0818679107

PY: 1997

**TI: Combining explicit negation
and negation by failure via Belnap's logic**

AU: Ruet,-P.; Fages,-F.

SO: Theoretical-Computer-Science. 15 Jan. 1997; 171(1-2): 61-75

PB: Elsevier

AB: This paper deals with logic programs containing two kinds
of negation: negation as failure and explicit negation. This allows
two different forms of reasoning in the presence of incomplete information.
Such programs have been introduced by M. Gelfond and V. Lifschitz (1988)
and called extended programs. We provide them with a logical semantics
in the style of K. Kunen (1987), based on Belnap's four-valued logic,
and an answer set's semantics that is shown to be equivalent to that
of Gelfond and Lifschitz. The proofs rely on a translation into normal
programs, and on a variant of M. Fitting's extension of logic programming
to bilattices.

PY: 1997

FTXT: ScienceDirect (tm)

http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=SilverLinker&_ur

lversion=4&_method=citationSearch&_volkey=0304%2d3975%23171%2361&_version=1&

md5=e3d0de65b957cabbc9398cc10841e93f SwetsWise

**TI: Kleene's three valued logics
and their children**

AU: Fitting,-M.

SO: Fundamenta-Informaticae. March-May 1994; 20(1-3): 113-31

AB: Kleene's strong three-valued logic extends naturally to a
four-valued logic proposed by Belnap (1977). The author introduces a
guard connective into Belnap's logic and considers a few of its properties.
Then he shows that by using it four-valued analogs of Kleene's weak
three-valued logic, and the asymmetric logic of Lisp are also available.
He proposes an extension of these ideas to the family of distributive
bilattices. Finally he shows that for bilinear bilattices the extensions
do not produce any new equivalences.

PY: 1994

*- sur les applications en intelligence
artificielle*

**T1: Bilattices and reasoning in
artificial intelligence: Concepts and foundations**

PA: SIM-K-M

SO: Artificial-Intelligence-Review. 2001; 15 (3) : 219-240

IS: 0269-2821

PY: 2001

LA: English

AB: The past few decades have seen a resurgence of reasoning techniques
in artificial intelligence involving both classical and non-classical
logics. In his paper, "Multi-valued Logics: A Uniform Approach to Reasoning
in Artificial Intelligence", Ginsberg has shown that through the use
of bilattices, several reasoning techniques can be unified under a single
framework. A bilattice is a structure that can be viewed as a class
of truth values that can accommodate incomplete and inconsistent information
and in certain cases default information. In bilattice theory, knowledge
is ordered along two dimensions: truth/falsity and certainty/uncertainty.
By defining the corresponding bilattices as truth spaces. Ginsberg has
shown that the same theorem prover can be used to simulate reasoning
in first order logic, default logic, prioritized default logic and assumption
truth maintenance system. Although this is a significant contribution,
Ginsberg's paper was lengthy and involved. This paper summarizes some
of the essential concepts and foundations of bilattice theory. Furthermore,
it discusses the connections of bilattice theory and several other existing
multi-valued logics such as the various three-valued logics and Belnap's
four-valued logic. It is noted that the set of four truth values in
Belnap's logic form a lattice structure that is isomorphic to the simplest
bilattice. Subsequently, Fitting proposed a conflation operation that
can be used to select subsets of truth values from this and other bilattices.
This method of selecting sub-sets of truth values provides a means for
identifying sub-logic in a bilattice.

AN: 010231398

FTXT: Kluwer Academic Publishers http://kapis.www.wkap.nl/oasis.htm/239899

EBSCO Online

http://ejournals.ebsco.com/direct.asp?ArticleID=XGJY5TLDDW1UMLGU92W3
Kluwer Journals@Ovid

http://gateway.ovid.com/ovidweb.cgi?T=JS&PAGE=fulltext&D=kap&NEWS=n&DBC=n&AN
=239

**TI: FSimGEO: a test generation
method for path delay fault test using fault simulation and genetic
optimization**

AU: Sun-Yihe; Wu-Qifa

ED: Mukund,P.-R.; Chickanosky,J.; Krishnamurthy,R.-K.

SO:

Proceedings-14th-Annual-IEEE-International-ASIC/SOC-Conference-IEEE-Cat.-No.
01TH8558. 2001: 225-9

PB: IEEE, Piscataway, NJ, USA

AB: Presents FSimGEO, an efficient test generation method for
path delay fault test using fault simulation and genetic optimization.
A parallel-vector fault simulator is introduced first, which can simulate
several test vector pairs at the same time through introducing a four-valued
logic and the path status graph (PSG) of the circuit structure. After
this, a special genetic optimization algorithm is used to direct the
process of searching and to optimize test sets generated. We have given
the fitness function and genetic operators that affect the optimizing
efficiency of genetic algorithm (GA) in detail. Experimental results
have shown that the number of paths tested by FSimGEO is about 3.5 times
that by other methods.

*- Divers*

**TI: Nearness relations in environmental
space**

AU: Worboys,-M.-F.

SO: International-Journal-of-Geographical-Information-Science.
Oct.-Nov. 2001; 15(7): 633-51

PB: Taylor & Francis

AB: The notion of proximity is a fundamental component of any
comprehensive ontology of space. This paper presents an experiment with
human subjects concerning the vague spatial relation `near' in environmental
space. After the topic is introduced and relevant previous work surveyed,
the experiment is described. Three approaches to experimental analysis
are presented and discussed: nearness neighbourhoods as regions with
broad boundaries, fuzzy nearness and distance measures, and four-valued
logic. Issues discussed in further detail are the truth gap-truth glut
hypotheses regarding the psychology of vague predicates, and formal
properties of the three-valued nearness relation. The conclusion is
drawn that formal theories can usefully underpin and be applied to reasoning
with vague spatial notions, such as nearness.