BRAIN. Broad Research in Artificial Intelligence and Neuroscience,
ISSN 2067-3957, Volume 1, July 2010,
Special Issue on Complexity in Sciences and Artificial Intelligence,
Eds. B. Iantovics, D. Rǎdoiu, M. Mǎruşteri and M. Dehmer

ÃLUKASIEWICZ-MOISIL MANY-VALUED LOGIC ALGEBRA
OF HIGHLY-COMPLEX SYSTEMS

Ion C. Baianu, George Georgescu,
James F. Glazebrook and Ronald Brown

Abstract.
The fundamentals of ÃLukasiewicz-Moisil logic algebras and their applica-

tions to complex genetic network dynamics and highly complex systems are
presented in the context of a categorical ontology theory of levels, Medical
Bioinformatics and self-organizing, highly complex systems. Quantum Au-
tomata were defined in refs.[2] and [3] as generalized, probabilistic automata
with quantum state spaces [1]. Their next-state functions operate through
transitions between quantum states defined by the quantum equations of mo-
tions in the Schrödinger representation, with both initial and boundary condi-
tions in space-time. A new theorem is proven which states that the category of
quantum automata and automata–homomorphisms has both limits and colim-
its. Therefore, both categories of quantum automata and classical automata
(sequential machines) are bicomplete. A second new theorem establishes that
the standard automata category is a subcategory of the quantum automata
category. The quantum automata category has a faithful representation in the
category of Generalized (M,R)–Systems which are open, dynamic biosystem
networks [4] with defined biological relations that represent physiological func-
tions of primordial(s), single cells and the simpler organisms. A new category
of quantum computers is also defined in terms of reversible quantum automata
with quantum state spaces represented by topological groupoids that admit a
local characterization through unique, quantum Lie algebroids. On the other
hand, the category of n– ÃLukasiewicz algebras has a subcategory of centered n–
ÃLukasiewicz algebras (as proven in ref. [2]) which can be employed to design
and construct subcategories of quantum automata based on n–ÃLukasiewicz
diagrams of existing VLSI. Furthermore, as shown in ref.[2] the category of

1



BRAIN. Broad Research in Artificial Intelligence and Neuroscience,
ISSN 2067-3957, Volume 1, July 2010,
Special Issue on Complexity in Sciences and Artificial Intelligence,
Eds. B. Iantovics, D. Rǎdoiu, M. Mǎruşteri and M. Dehmer

centered n–ÃLukasiewicz algebras and the category of Boolean algebras are nat-
urally equivalent. A ‘no-go’ conjecture is also proposed which states that Gen-
eralized (M,R)–Systems complexity prevents their complete computability (as
shown in refs. [5]–[6]) by either standard, or quantum, automata.

Keywords: LM-logic algebra, algebraic category of LM-logic algebras,
fundamental theorems of LM-logic algebra, many-valued logics of highly com-
plex systems and Categorical Ontology, quantum automata categories, limits
and colimits, bicomplete categories, Quantum Relational Biology, generalized
metabolic-replication (M,R)–systems, complex bio-networks; quantum comput-
ers, computability of complex biological systems; centered n–ÃLukasiewicz alge-
bras categories of n–ÃLukasiewicz algebras, categories of Boolean algebras.

2000 Mathematics Subject Classification: OG320, OG330.

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Baianu et al. LM Many-Valued Logic Algebras

1. Algebraic Logic, Operational and Lukasiewicz Quantum Logic

As pointed out by Birkhoff and von Neumann as early as 1936, a logical
foundation of quantum mechanics consistent with quantum algebra is essen-
tial for both the completeness and mathematical validity of the theory. The
development of Quantum Mechanics from its very beginnings both inspired
and required the consideration of specialized logics compatible with a new
theory of measurements for microphysical systems. Such a specialized logic
was initially formulated by Birkhoff and von Neumann in 1936, and called
‘Quantum Logic’ (QL). However, in recent QL research several approaches
were developed involving several types of non-distributive lattice, and their
corresponding algebras, for n–valued quantum logics. Thus, modifications of
the ÃLukasiewicz logic algebras that were introduced in the context of alge-
braic categories [14] by Georgescu and Popescu [15]– also recently reviewed
and expanded by Georgescu [16]– can provide an appropriate framework for
representing quantum systems, or in their unmodified form, for describing the
activities of complex networks in categories of ÃLukasiewicz logic algebras [5].

There is nevertheless a serious problem remaining which is caused by the
logical inconsistency between any quantum algebra and the Heyting logic alge-
bra which has been suggested as a candidate for quantum logic. Furthermore,
quantum algebra and topological approaches that are ultimately based on set-
theoretical concepts and differentiable spaces (manifolds) also encounter seri-
ous problems of internal inconsistency. Since it has been shown that standard
set theory which is subject to the axiom of choice relies on Boolean logic there
appears to exist a basic logical inconsistency between the quantum logic–which
is not Boolean–and the Boolean logic underlying all differentiable manifold ap-
proaches that rely on continuous spaces of points, or certain specialized sets
of elements. A possible solution to such inconsistencies is the definition of a
generalized Topos concept, and more specifically, of an Extended Quantum
Topos (EQT) concept which is consistent with both QL and Quantum Alge-
braic, Logic, thus being potentially suitable for the developing a framework
that may unify quantum field theories with ultra-complex system modeling
and Complex Systems Biology (CSB).

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Baianu et al. LM Many-Valued Logic Algebras

2. Lattices and Von Neumann-Birkhoff (VNB) Quantum Logic:
Definition and Some Logical Properties

We commence here by giving the set-based Definition of a Lattice. An s–
lattice L, or a ‘set-based’ lattice, is defined as a partially ordered set that has
all binary products (defined by the s–lattice operation “

∧
”) and coproducts

(defined by the s–lattice operation “
∨

”), with the ”partial ordering” between
two elements X and Y belonging to the s–lattice being written as “X ¹ Y ”.
The partial order defined by ¹ holds in L as X ¹ Y if and only if X = X ∧ Y
(or equivalently, Y = X

∨
Y Eq.(3.1). A lattice can also be defined as a

category (see, for example, ref. [9]) subject to all ETAC axioms, (but not
subject, in general, to the Axiom of Choice usually encountered with sets
relying on (distributive) Boolean Logic), that has all binary products and all
binary coproducts, as well as the following ’partial ordering’ properties:

3. Quantum Logic (LQL), ÃLukasiewicz-Moisil (LM) and
Operator Algebras

With all truth ’nuances’ or assertions of the type << system A is excitable
to the i-th level and system B is excitable to the j-th level >> one can define a
special type of lattice that subject to the axioms introduced by Georgescu and
Vraciu [15] becomes a n-valued ÃLukasiewicz-Moisil, or LM -Algebra. Further
algebraic and logic details are provided in refs.[16] and [9].

In order to have the n-valued ÃLukasiewicz-Moisil logic (LML) algebra repre-
sent correctly the basic behavior of quantum systems [7], [17] –which is usually
observed through measurements that involve a quantum system interactions
with a macroscopic measuring instrument– several of these axioms have to be
significantly changed so that the resulting lattice becomes non-distributive and
also (possibly) non-associative. Several encouraging results in this direction
were recently obtained by Dala Chiara and coworkers. With an appropriately
defined quantum logic of events one can proceed to define Hilbert, or ‘nu-
clear’/Frechet, spaces in order to be able to utilize the ‘standard’ procedures
of quantum theories [17].

On the other hand, for classical systems, modeling with the unmodified
ÃLukasiewicz logic algebra can also include both stochastic and fuzzy behav-
iors. For examples of such models the reader is referred to a previous report

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Baianu et al. LM Many-Valued Logic Algebras

[5] where the activities of complex genetic networks are considered from a clas-
sical standpoint. One can also define as in [8] the ‘centers’ of certain types
of ÃLukasiewicz n-logic algebras; then one has the following important theo-
rem for such centered ÃLukasiewicz n-logic algebras which actually defines an
equivalence relation.

The Logic Adjointness Theorem (Georgescu and Vraciu [15], Georgescu
[16]):

There exists an Adjointness between the Category of Centered ÃLukasiewicz
n-Logic Algebras, CLuk–n, and the Category of Boolean Logic Algebras (Bl).

Remarks

(1) The logic adjointness relation between CLuk–n and (Bl is naturally
defined by the left- and -right adjoint functors between these two categories of
logic algebras.

(2) The natural equivalence logic classes defined by the adjointness re-
lationships in the above Adjointness Theorem define a fundamental, ‘logical
groupoid’ structure.

(3) In order to adapt the standard ÃLukasiewicz-Moisil logic algebra to the
appropriate Quantum ÃLukasiewicz-Moisil logic algebra, LQL, a few axioms of
LM-algebra need modifications, such as : N (N (X)) = Y 6= X (instead of the
restrictive identity N (N (X)) = X, whenever the context, ‘reference frame for
the measurements’, or ‘measurement preparation’ interaction conditions for
quantum systems are incompatible with the standard ‘negation’ operation N
of the ÃLukasiewicz-Moisil logic algebra; the latter remains however valid for
classical or semi–classical systems, such as various complex networks with n-
states (cf. [5]). Further algebraic and conceptual details were provided in a
rigorous review by George Georgescu [16], and also in the recently published
reports, [5]–[6].

Two new theorems are also noted in this context (albeit without proof).
Theorem 1- Completeness. The category of quantum automata and

automata–homomorphisms has both limits and colimits.

Theorem 2. The category of cllassical, finite automata is a subcategory
of the category of quantum automata.

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Baianu et al. LM Many-Valued Logic Algebras

4. Quantum Automata and Quantum Computation

Quantum computation and quantum ‘machines’ (or nanobots) were much pub-
licized in the early 1980’s by Richard Feynman (Nobel Laureate in Physics:
QED),and, subsequently, a very large number of papers- too many to cite all
of them here- were published on this topic by a rapidly growing number of
quantum theoreticians and some applied mathematicians.

Two such specific definitions are briefly considered next. Quantum au-
tomata were defined in refs. [2] and [3] as generalized, probabilistic automata
with quantum state spaces. Their next-state functions operate through tran-
sitions between quantum states defined by the quantum equations of motions
in the Schrödinger representation, with both initial and boundary conditions
in space-time.

A new theorem was proven which states that the category of quantum
automata and automata homomorphisms has both limits and colimits.

Therefore, both categories of quantum automata and classical automata
(sequential machines) are bicomplete. A second new theorem established that
the standard automata category is a subcategory of the quantum automata
category.

Definition 1. One obtains a simple definition of quantum automaton by
considering instead of the transition function of a classical sequential machine,
the (quantum) transitions in a finite quantum system with definite probabilities
determined by quantum dynamics. The quantum state space of a quantum
automaton is thus defined as a quantum groupoid over a bundle of Hilbert
spaces, or over rigged Hilbert spaces.

Formally, whereas a sequential machine, or state machine with state space
S, input set I and output set O, is defined as a quintuple: (S, I, O, δ : S ×S →
S, λ : S × I → O), a quantum automaton is defined by a triple (H, ∆ : H →
H, µ), where H is either a Hilbert space or a rigged Hilbert space of quantum
states and operators acting on H, and µ is a measure related to the quantum
logic, LM, and (quantum) transition probabilities of this quantum system.

Remark.
Quantum computation becomes possible only when macroscopic blocks of

quantum states can be controlled via quantum preparation and subsequent,
classical observation. Obstructions to actually building, or constructing quan-
tum computers are known to exist in dimensions greater than 2 as a result
of the standard K-S theorem. Subsequent definitions of quantum comput-
ers reflect attempts to either avoid or surmount such difficulties often without

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Baianu et al. LM Many-Valued Logic Algebras

seeking solutions through quantum operator algebras and their representations
related to extended quantum symmetries which define fundamental invariants
that are key to actual constructions of this type of quantum computers.

Definition 2. Alternatively, aquantum automaton is defined as a quantum
algebraic topology object– the triplet (Gd, H − RGd , Aut(G)), where Gd is a
locally compact quantum groupoid, H −RGd are the unitary representations of
Gd on rigged Hilbert spaces RGd of quantum states and quantum operators on
H, and Aut(Gd) is the transformation, or automorphism, groupoid of quantum
transitions.

Remark. Other definitions of quantum automata and quantum computa-
tions have also been reported that are closely related to recent experimental
attempts at constructing quantum computing devices.

Related Results: Quantum Automata Applications to Modeling
Complex Systems. The quantum automata category has a faithful rep-
resentation in the category of Generalized (M, R) -systems which are open,
dynamic bio-networks [6] with defined biological relations that represent phys-
iological functions of primordial(s), single cells and the simpler organisms. A
new category of quantum computers is also defined in terms of reversible quan-
tum automata with quantum state spaces represented by topological groupoids
that admit a local characterization through unique ‘quantum’ Lie algebroids.
On the other hand, the category of n-ÃLukasiewicz algebras has a subcategory
of centered n- ÃLukasiewicz algebras [15] (which can be employed to design
and construct subcategories of quantum automata based on n-ÃLukasiewicz di-
agrams of existing VLSI. Furthermore, as shown in ref. [15] the category of
centered n-ÃLukasiewicz algebras and the category of Boolean algebras are nat-
urally equivalent.

Variable machines with a varying transition function were previously dis-
cussed informally by Norbert Wiener as a possible model for complex biological
systems although how this might be achieved in Biocybernetics has not been
specifcally, or mathematically presented by Wiener.

A ‘no-go’ conjecture was also proposed which states that Generalized (M,R)–
Systems complexity prevents their complete computability by either standard
or quantum automata. The concepts of quantum automata and quantum com-
putation were initially studied and are also currently further investigated in
the contexts of quantum genetics, genetic networks with nonlinear dynamics,
and bioinformatics. In a previous publication [2]– after introducing the formal
concept of quantum automaton–the possible implications of this concept for

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Baianu et al. LM Many-Valued Logic Algebras

correctly modeling genetic and metabolic activities in living cells and organisms
were also considered. This was followed by a formal report on quantum and
abstract, symbolic computation based on the theory of categories, functors and
natural transformations [3]. The notions of topological semigroup, quantum
automaton,or quantum computer, were then suggested with a view to their
potential applications to the analogous simulation of biological systems, and
especially genetic activities and nonlinear dynamics in genetic networks. Fur-
ther, detailed studies of nonlinear dynamics in genetic networks were carried
out in categories of n-valued, ÃLukasiewicz Logic Algebras that showed signif-
icant dissimilarities [6] from the widespread Boolean models of human neural
networks that may have begun with the early publication of [18]. Molecu-
lar models in terms of categories, functors and natural transformations were
then formulated for uni-molecular chemical transformations, multi-molecular
chemical and biochemical transformations [7]. Previous applications of com-
puter modeling, classical automata theory, and relational biology to molecular
biology, oncogenesis and medicine were extensively reviewed and several impor-
tant conclusions were reached regarding both the potential and limitations of
the computation-assisted modeling of biological systems, and especially com-
plex organisms such as Homo sapiens sapiens [8], [9], [10]. Novel approaches
to solving the realization problems of Relational Biology models in Complex
System Biology are introduced in terms of natural transformations between
functors of such molecular categories. Several applications of such natural
transformations of functors were then presented to protein biosynthesis, em-
bryogenesis and nuclear transplant experiments. Other possible realizations in
Molecular Biology and Relational Biology of organisms were then suggested in
terms of quantum automata models of Quantum Genetics and Interactomics.
Future developments of this novel approach are likely to also include: fuzzy
relations in Biology and Epigenomics, Relational Biology modeling of Complex
Immunological and Hormonal regulatory systems, n-categories and generalized
LM –Topoi of ÃLukasiewicz Logic Algebras and intuitionistic logic (Heyting) al-
gebras for modeling nonlinear dynamics and cognitive processes in complex
neural networks that are present in the human brain, as well as stochastic
modeling of genetic networks in ÃLukasiewicz Logic Algebras (LLA).

Remark. Previous applications of computer modeling, classical automata
theory, and relational biology to molecular biology, neural networks, oncoge-
nesis and medicine were extensively reviewed in a previous monograph and
several important conclusions were reached regarding both the potential and

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Baianu et al. LM Many-Valued Logic Algebras

the severe limitations of the algorithm driven, recursive computation-assisted
modeling of complex biological systems [6].

5. Conclusions

Non-distributive varieties of many-valued, LM-logic algebras that are also non-
commutative open new possibilities for formal treatments of both complex
quantum systems and highly complex biological networks, such as genetic
nets, metabolic-replication systems (see for example refs. [19]–[21]), the in-
teractome and neural networks [6]. This novel approach that involves both
Algebraic Logic and Category Theory, provides an important framework for
understanding the complexity inherent in intelligent systems and their flexible,
adaptive behaviors. A consequence of the Logical Adjointness Theorem– which
defines categorically the natural equivalence between the category of centered
LM-logic algebras and that of Boolean logic algebras– is that one may be able
to define Artificial Intelligence analogs of neural networks based on centered
LM-logic algebras. In this process, higher dimensional algebra (HDA; [12]-[13])
and categorical models of human brain dynamics (refs. [8]–[11]) were predicted
to play a central role. These new approaches are also relevant for resolving the
tug-of-war between nature-vs.-nurture theories of human development and the
‘natural’ emergence through co-evolution of intelligence in the first H. sapiens
sapiens societies.

References

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[3] I.C. Baianu. 1971b. “Categories, Functors and Quantum Algebraic
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[4] I.C. Baianu. 1973. Some Algebraic Properties of (M,R)-Systems in
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[5] I.C. Baianu. 1977. A Logical Model of Genetic Activities in ÃLukasiewicz
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[16] G. Georgescu. 2006. N-valued Logics and ÃLukasiewicz–Moisil Alge-
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[17] N. P. Landsman : Mathematical topics between classical and quantum

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mechanics. Springer Verlag, New York, 1998.
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Automata., Bull. Math. Biol., 44: 661-668.

I. C. Baianu, PhD, Professor,
ACES College and NPRE Department,
University of Illinois at Urbana-Champaign,
Urbana IL 61801, USA
e-mail:ibaianu@illinois.edu

James. F. Glazebrook, PhD, Professor,
Department of Mathematics and Computer Science,
Eastern Illinois University,
Charleston IL 61920, USA
e-mail: jfglazebrook@eiu.edu

George Georgescu, PhD, Professor
Department of Mathematics,
Bucharest University, Romania
e-mail: ggeorgescu@ubuc.edu

Ronald Brown, PhD., Professor,
School of Informatics,
University of Wales,
Dean St. Bangor, Gwynedd LL57 1UT, UK
e-mail:r.brown@bangor.ac.uk

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