ALGEBRAIC NONDETERMINISMAND
TRANSITION SYSTEMS
by
PRITI SINHA
under guidance of
Dr Wagish Shukla and Dr Sanjiva Prasad
Department of Mathematics Department of CSE
Submitted
in fulfillment of the requirements of the degree of Doctor of Philosophy
to the
INDIAN INSTITUTE OF TECHNOLOGY, DELHI
February 2005
Chapter 1
Introduction
Nondeterminism of course has a long history but we take the Zadeh fuzzy theory
as a starting point. The well known definition Xp−→ [0, 1] of a fuzzy set thinks
of elements of an ordered set, namely the interval [0, 1], as a set of truth-values.
In the seventies this idea was shown by Manes to be a semantics of the pivotal
construction of a pair of adjoint functors which arise practically everywhere in
mathematics. This freed nondeterminism from the requirement of the ordered
set of truth-values so that the boolean logic of the probabilistic set theory is not a
partially ordered set [[Man82b], section 6]. On the other hand, the idea of a set
of truth-values as a poset, indeed as a complete Heyting algebra, has vigorously
persisted leading to various developments in fuzzy set theory.
Side by side, there is the question of nondeterminism in machines, embodied in
the notion of a transition system. The questions that we examine are centered
on various incarnations of these two concepts, namely fuzzy sets and transition
systems, in mathematical literature and their interrelations.
1
2 CHAPTER 1. INTRODUCTION
1.1 This thesis: the general outline
The thesis is divided into three parts.
Part 1, titledAlgebraic nondeterminism for sets, consists of four chapters, namely:
1. What is a fuzzy set?, 2. Algebraic theories, 3. Fuzzy sets as Ω-sets: “Totally
fuzzy sets” , 4. The Chu option: a category Qua.
In the first chapter, we talk about various types of fuzzy sets in use and place them
in more general contexts. In the second chapter, the definition of fuzzy theory
is stated. These are algebraic theories of Lawvere which have been called ‘fuzzy
theories’ byManes when the category considered isSet. Essentially a fuzzy theory
is a Kleisli category in Set. We integrate the various types of fuzzy sets mentioned
in chapter 1 into instances of fuzzy theories.
In the third chapter, we discuss another interpretation of fuzzy sets, namely, Ω-
sets ( where Ω is a complete Heyting algebra ). These are also known as “totally
fuzzy sets”. We also work out many computational details ofΩ-sets which are not
readily available in addition to discussing the Zadeh fuzzy sets along these lines.
In the last chapter of this part, we construct a category Quawhich subsumes most
of the constructions in the preceding chapters besides generalizing constructions
in many other branches, in particular certain models of linear logic. It may be
noted that a connection between fuzzy logic and linear logic was established by
Kreinovich, Nguyen and Wojciechowsk [KNW] and Barr [Bar96].
Part 2, titled Algebraic nondeterminism for machines, consists of three chapters,
namely: 1. What is a transition system?, 2. Lawvere metric spaces and transi-
tion systems, 3. Non-commutative fuzzy logic. In the first chapter we examine,
rewrite and generalize transition systems using several techniques. In the second
chapter we establish that transitions systems support a “generalized logic” using
1.2. THIS THESIS: CHAPTER DETAILS AND CONTRIBUTIONS 3
the “metric space approach” introduced by Lawvere [Law73]. In the last chapter
of this part we examine the logical frame which was employed in the preceeding
chapter and extend it to the case of non-commutative conjunctions; here “frame”
can also be understood in the technical sense of “complete Heyting algebra” for
which it is sometimes used as a synonym.
Part 3, titled Fuzzy setsdTransition systems = ? is an attempt to extract the
commonalities between fuzzy sets and fuzzy machines as discussed in part 1 and
part 2 respectively. It consists of only one chapter, namely, ‘Qua as an answer’, in
which we suggest that the category Qua may support both fuzzy sets and fuzzy
machines and therefore can be taken to be a partial answer to the question raised
in the title of this part.
The thesis closes with conclusion and some ruminations on future work which
could be taken up.
1.2 This thesis: chapter details and contributions
We now provide chapter details and contributions of each chapters separately.
Chapter 2: What is a fuzzy set?
The first section, “fuzzy sets classically”, examines the concept of (Type-1) fuzzy
sets, Type-2 fuzzy sets and Level-2 fuzzy sets as they appear in the standard fuzzy
set literature. Essentially, in this section we have recorded our disagreement with
this kind of classification by noting that Type-2 fuzzy sets are simply fuzzy binary
relations ( as Type-n fuzzy sets are n-ary fuzzy relations) on a set and Level-2 fuzzy
4 CHAPTER 1. INTRODUCTION
sets are simply (Type-1) fuzzy sets with [0, 1][0,1] in place of [0, 1] as the algebra of
truth-values.
We also suggest that the algebra of truth-values should be taken to be a complete
Heyting algebra rather than the closed interval [0, 1]. Wehavemade our suggestion
with the motivation that truth-values should model possible states of knowledge.
However, there are many reasons why one should work with a complete Heyting
algebra.
When Zadeh [Zad65] introduced fuzzy sets, he regarded fuzzy sets essentially as
“crisp” sets with a [0, 1]-valued membership-degree function. When we choose
a lattice H of truth values for membership-degrees, meets and joins in H define
standard conjunction and disjunction. IfH is complete, infimum and supremum
in H define universal and existential quantification which generalize standard
conjunction and disjunction. From basic principles, implication should be right
adjoint to conjunction, and this right adjoint exists if H is a complete Heyting
algebra. Other connectives like t-norms can be defined in a complete Heyting
algebra which permit a “fuzzy logic” such as the Lukasiewicz logic to be brought
in. This is in addition to the standard (intuitionistic) Heyting logic.
There is a natural tie-up between a fuzzy set X −→ 0, 1 and the topos Set0,1 via
“α-cuts”. This is explored in the section “A different perspective: Fuzzy sets as
functors” where we take a look at Type-1 fuzzy sets from a different angle. We
observe that it is a functor from [0, 1] to Setin, the category of sets and injections.
In the next section ‘Categories of fuzzy sets’, we define a category of fuzzy setswith
objects as Type-1 fuzzy sets seen as functors as outlined above, and morphisms
as natural transformations. This motivates our definition of generalized Type-1
fuzzy set. We note that most of the available categories of fuzzy sets do not form
toposes. By contrast, viewing fuzzy sets as functors as proposed in the preceding
1.2. THIS THESIS: CHAPTER DETAILS AND CONTRIBUTIONS 5
section, provides a topos environment.
Thus in this chapter we have reviewed, examined and modified the definition of
classical fuzzy sets.
Chapter 3: Algebraic theories
From an entirely different perspective, Manes proposed his “distributional set
theory” [Man82a] as the proper universe for studying “fuzzy sets”. Explicitly,
algebraic theories (of Lawvere) are called fuzzy theories byManes [Man82b] in the
context of the category Set.
There are three forms of algebraic theories, all statable in terms of each other,
namely, “clone form”, “monoid form” and “extension form”. In this chapter we
show that various types of fuzzy set theories (discussed in chapter 1) are nothing
but instances of algebraic theories. This re-purposes (classical) fuzzy sets. For
some generic examples we have supplied computations for all the three forms
since in order to understand a particular theory, it is generally always necessary to
explictly see it in all the three forms.
We also add to the list of examples given by Manes and many others, enabling us
to look at some other relevant constructions as instances of fuzzification.
In particular, we construct two examples we believe to be new: possibilistic poset
theory and algebraic theory of monoid-labelled K-relations. The latter is in fact a
mother example subsuming many known important example-classes.
Besides these examples, we observe in section 3.2.1 that if we limit ourselves to
matrix theories, fuzzy set theory is nothing but linear algebra over a semiring.
We end this chapter by including a section on composition of algebraic theories
which essentially reports the recent work in [EGOAV00]. The result was given
6 CHAPTER 1. INTRODUCTION
there without an explicit proof for the general case.
Chapter 4: Fuzzy sets as Ω-sets: “Totally fuzzy sets”
Another interpretation of fuzzy sets is Ω-sets, where Ω is a complete Heyting
algebra. We examine it in this chapter.
Ω-sets and Ω-functions constitute the canonical example of a topos. Toposes are
relevant to our study in many ways.
A transition system [ see chapter 6 ] has a natural definition in a topos; indeed the
fact that the transition systemQ×Σ×Q −→ 0, 1 can be considered as a coalgebra
Q −→ P(Σ × Q) = D(Q) [ see section 6.2.2 ], where D is the composition of the
power set functor with (Σ × −), is transferable to any category which has finite
products and power set objects, in particular to a topos.
A topos is the same as a tabular allegory [ see appendix ] so that in a topos we have
sets, functions and relations; a topos is the framework for sets and functions, and an
allegory is the framework for sets and relations. Thus the contemporary switchover
from a transition system as a subset of Q × Σ × Q to a coalgebra Q −→ D(Q) is
directly available in a topos.
Therefore our main interest in Ω-sets centers on to Ω-relations rather than Ω-
functions. For this reason we have to extensively work out many computational
details of Ω-sets which are not readily available. Thus this chapter is in the
character of a tutorial on Ω-sets.
Nevertheless we believe that the computational workouts on Ω-relations and in
particular the observation thatΩ-sets andΩ-relations actually constitute a unitary
allegory will be welcome by readers, as will the motivations from the comparison
with the concept of a metric space.
1.2. THIS THESIS: CHAPTER DETAILS AND CONTRIBUTIONS 7
Ω-sets have been called “totally fuzzy sets” because in them not only the member-
ship but the equality is also fuzzified.
So now there are at least two toposes for fuzzy sets; generalized Type-1 fuzzy sets
and Ω-sets. This opens up the possibility of extending the concept of transition
systems as relations in these toposes rather than just relations between sets. We do
not attempt it in this thesis.
Chapter 5: The Chu option: a category Qua
In this chapter, which concludes Part I, we construct a category Qua, which par-
ticularizes to almost all existing categories of fuzzy sets in various ways, besides
generalizing constructions in many other areas.
We show that the category of L-fuzzy sets defined by Goguen [Gog67], category
RelL defined by Barr [Bar96], the category ML(C) (for the case C = Set) defined
by de Paiva [dP89], the category PV defined by Blass [Bla95] and the category
of games, GameK defined by Lafont [LS91] are special cases of the category Qua.
The difference in our approach is that all the categories discussed above except
the one by Barr, consider ‘function-induced’ morphisms only. In our formulation,
morphisms are relation-pairs rather than function-pairs.
In our context, working with Chu spaces / Games is fruitful because when we
view a “fuzzy relation” Xα−→ T(Y), where T is a matrix theory functor over a
semiringK as a function X×Yα−→ K, it turns out that a “fuzzy relation” is nothing
but a “game over K” or a “K-Chu space”. Since GameK is proposed by Lafont as
“generalized linear algebra”, we obtain a confirmation of our view of “fuzzy sets
as linear algebra” proposed in section 3.2.1.
We further construct a generalization of the category Qua. This particularizes to a
category of “poset-valued sets” which is a generic technique for building models
8 CHAPTER 1. INTRODUCTION
of linear logic [SdP04].
Chapter 6: What is a transition system?
In this chapter, which starts Part 2, we begin by discussing different versions
of transition systems already present in literature, proposing different ways of
generalizing them and writing them in terms of various fuzzy theories.
The basic observations include: a transition system is the same as (i) a dialgebra
F(Q) −→ T(Q) inK , (ii) an algebra F(Q) Q in the Kleisli category KT, and (iii) a
coalgebra Q F(Q) in the Kleisli category KT. With F = (Σ × −) and T = 0, 1(−),
K = Set, this is the classical case of labelled transition systems; of course we are
occasionally working with “relational (algebras and) coalgebras” here, which are
currently being favoured over the functional coalgebraic approach to transition
systems. Recalling that Σ × Q = F(Q) −→ Q is an automaton, we have a slogan
“algebra F(Q) Q is an automaton and coalgebra Q F(Q) is a transition
system”, which is no surprise when we observe that what is known as a labelled
transition system was called Σ-automaton by Eilenberg [Eil74].
Apart from these, we have introduced the concept of a transition system as a
certainmonoidmorphism. This particularizes to at least three known instances, (1)
DeterministicΣ-automata of Eilenberg, (2) Fuzzyfinite statemachines ofMordeson
and Malik and (3) Fuzzy automata of Mockor.
We next come to bisimulations.
We start with reviewing the coalgebraic definition of bisimulation ( and later
include the dialgebraic version). Then we propose fuzzy bisimulations in the
context of a semiring as the set of truth-values anddemonstrate that this is the same
as the coalgebraic definition of bisimulation when the semiring is multiplicatively
idempotent.
1.2. THIS THESIS: CHAPTER DETAILS AND CONTRIBUTIONS 9
Our point of view is however, that it is much more robust to think of transition
systems as F-models where F is an endofunctor on Set. The F-models were
introduced byM. Barr [Bar70], who called them “relational pre-algebras”. Further,
bisimulations should be thought of as morphisms of F-models.
We have therefore formulated transition systems and bisimulations in these terms
and demonstrated that “functional bisimulations”, “relational bisimulations” and
their coalgebraic versions (including the generalization by Rutten [Rut00] ) are all
captured by our F-model approach.
Chapter 7: Lawvere metric spaces and transition systems
We start this chapter by noting that a transition system Q ×Σ ×Q −→ 0, 1 can be
written asQ×Qµ
−→ 0, 1Σ. So a transition systemmeasures “the distance between
two states q1, q2, as an element µ(q1, q2) ∈ 0, 1Σ , with σ ∈ µ(q1, q2) iff q1
σ−→ q2”.
This reading encourages us to read a transition system as “a generalized metric
space” and replace 0, 1Σ by L. Of course the Frechet axioms are not expected
to hold: For example µ(q1, q2) cannot be expected to be same as µ(q2, q1). It turns
out that “transition systems = generalized metric spaces” means “simulations =
non-expansive functions”.
In order to motivate and contextualize this point of view, we supply a self-
contained account of generalized metric spaces as introduced by Lawvere in
[Law73] but with Q [ see section 7.11 ] in place of his [0,∞]. This is necessary
in view of our interpretation that µ(q1, q2) is the possible state of knowledge “
there exists a transition from q1 to q2”. It forces us to develop the theory from the
beginning because only the logical properties of the closed interval [0,∞] ( or [0, 1]
) are distilled into Q.
We also introduce an algebraic theory in Lawvere metric spaces whose Kleisli cat-
10 CHAPTER 1. INTRODUCTION
egory is the category of “metric relations” or “profunctors /modules” (as Lawvere
calls them [Law73, Law86] ).
We further note that the theory of metric transition systems and metric bisimula-
tions developed by Rutten [Rut00] in the context of [0, 1]-valued metric spaces is a
particular case within this approach.
Chapter 8: Non-commutative fuzzy logic
In the previous chapter we have taken Q, the set of truth-values in which the
distances were evaluated with a commutative conjunction. There are reasons why
many people working in fuzzy logic have found it desirable to deal with non-
commutative conjunctions. In particular non-commutative t-norms and systems
like weak pseudo-Wajsberg algebras have been investigated.
In this chapter we have tried to systematically investigate the formal laws that are
going to come up in an algebra of truth-values aspiring tomodel non-commutative
conjunctions. We have also paid attention to some degenerate useful situations
like integrality.
Chapter 9: Qua as an answer
In this chapter we seek to respond to the question asked in the title of Part III,
namely “Fuzzy setsdTransition systems = ?”.
We say thatQua is an answer since categories of fuzzy sets (andmanymore) are in-
stances ofQua (demonstrated in Part I) and categories of transition systems are also
instances of Qua. More precisely, when we particularize Qua (by taking X∗ = X∗),
we obtain two “categories of transition systems” as subcategories of Qua; (i) with
morphisms as simulations and, (ii) with morphisms as strong bisimulations.
1.2. THIS THESIS: CHAPTER DETAILS AND CONTRIBUTIONS 11
Chapter 10: Conclusion and future work
This chapter closes the thesis. We have commented very briefly on our work
but our emphasis in this closing chapter is on a few problems that stem from
this dissertation in particular or the philosophy used in general. Specifically, we
formulate problems on looking at Qua as a 2-category, the question “what is a
member of a fuzzy set”, and some other directions including the possibility of
“noncommutative Lawvere metric spaces”, that is, Lawvere metric spaces whose
distance function takes values in a noncommutative fuzzy logic.
The style of exposition in this dissertation will perhaps be considered by some
readers as excessively detailed. In our experience, we have found it inconvenient
to read documents which refer to various other documents for the necessary defi-
nitions and basic results, or skip too many proofs under the remark “it is obvious”.
We have chosen therefore, to provide practically every definition used and have
supplied detailed proofs for all the results established. Some of the definitions,
which may be regarded as a part of standard vocabulary, but nevertheless may be
unfamiliar to some readers are supplied in the appendix.
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