[PPT] - Fuzzifying Modal Algebra Jules Desharnais, Bernhard M oller PowerPoint Presentation

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Fuzzy Modalities

Fuzzifying Modal Algebra

Jules Desharnais, Bernhard M¨

ller

Universit´ e Laval Qu´ ebec, University of Augsburg RAMiCS 2014

J. Desharnais/B. M¨
ller

RAMiCS 2014

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Why?

basic idea: bring together the concepts of fuzzy semirings (say, fuzzy relations or matrices) modal semirings (domain/codomain and box/diamond) domain and codomain “measure” enabledness in transition systems this motivated investigating modal fuzzy semirings not a new kind of algebraic “meta-system” for various fuzzy logics rather apply and re-use an existing well-established algebraic system for the particular case of fuzzy systems

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Basics of Fuzziness

fuzzy relations: mappings from pairs of elements into the interval [0, 1] values can be interpreted as transition probabilities or as capacities and in various other ways now take up the above idea of measuring enrich fuzzy semirings with domain/codomain operators apply the corresponding modal operators in the description and derivation of systems or algorithms in that realm seems to be a novel approach

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Basics of Fuzziness

adapt classical relational operators: (R ⊔ S)(x, y) = max (R(x, y), S(x, y)) (R ⊓ S)(x, y) = min (R(x, y), S(x, y)) (R ; S)(x, y) = sup

z

min (R(x, z), S(z, y)) with these operations, fuzzy relations form an idempotent semiring

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Basics of Fuzziness

weak notion of complementation R(x, y) =df 1 − R(x, y) main problem in transferring domain to fuzzy semirings: the original axiomatisation of domain used a Boolean subring of the overall semiring as the target set of the domain operator generally not present in the fuzzy case however, using weak negation and the concepts of t-norm and t-conorm (see below for the details), a substitute for Boolean algebra can be defined

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Overview

new axiomatisation of two variants of domain and codomain in the more general setting of idempotent left semirings avoiding complementation hence applicable to fuzzy relations for idempotent semirings such an axiomatisation has been given by Desharnais/Struth (2011) here: more general case of idempotent left semirings in which left distributivity of multiplication over addition and right annihilation

f zero are not required
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Overview

also, we weaken the domain axioms by requiring only isotony rather than distributivity over addition surprisingly, still a wealth of properties known from the semiring case persists in the more general setting however, it is no longer true that complemented subidentities are domain elements not really disturbing, though, because the fuzzy world has its own view of complementation anyway we show how the domain operators extend to matrices some applications of these are sketched

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Algebraic Basis

an idempotent left (or lazy) semiring, briefly an IL-semiring, is a quintuple (S, +, 0, ·, 1) with the following properties: (S, +, 0) is a commutative monoid and + is idempotent (S, ·, 1) is a monoid · is right-distributive over + and left-strict: (a + b) · c = a · c + b · c 0 · a = 0 · is right-isotone w.r.t. the subsumption order a ≤ b ⇔df a + b = b, which can be axiomatised as a · b ≤ a · (b + c)

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Algebraic Basis

an idempotent right semiring is defined symmetrically an I-semiring is a structure which is both an idempotent left and right semiring; hence its multiplication is both left and right distributive over its addition and its 0 is a left and right annihilator

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Predomain

general semiring elements abstractly model transition systems over states predicates on states can be modelled by sub-identities, i.e., by elements ≤ 1 multiplication p · a of a transition element a by a sub-identity p means restriction of a to the starting states characterised by p former approaches used tests, i.e., sub-identities with a complement relative to 1, and hence involved the Boolean

peration of negation

as mentioned in the introduction, we want to avoid that and hence give the following new axiomatisation of a (pre)domain

peration, whose range will replace the set of tests
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Predomain

a prepredomain IL-semiring is a structure (S, ), where S is an IL-semiring and the prepredomain operator : S → S satisfies

a ≤ 1

✭sub-id✮

0 ≤ 0

✭strict✮

a ≤ a · a

✭dom1✮

by isotony of · , Ax.

✭dom1✮ strengthens to the equality a =

a · a this means that restriction to all starting states is no actual restriction

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Predomain

is called a predomain operator if additionally

a ≤

(a + b)

✭isot✮

(

b · a) ≤ b

✭dom2✮

Ax.

✭isot✮ states that is isotone

Ax.

✭dom2✮ means that after restriction the remaining starting states

satisfy the restricting predicate

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Predomain

a predomain operator is called a domain operator if additionally it satisfies the locality axiom

(a ·

b) ≤ (a · b)

✭loc✮

Ax.

✭loc✮ again strengthens to an equality

it means that the domain of a · b is not determined by the inner structure or the final states of b; information about b in interaction with a suffices Mace4 shows that these axioms are independent by S we denote the image of S under , by p, q, . . . elements of S

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(Pre)Domain Calculus

as in the classical case, Axs.

✭dom1✮ and ✭dom2✮ combine into

a ≤ p ⇔ a ≤ p · a

✭llp✮

this characterises a as the least left-preserver of a in S and often allows a calculation of a using indirect equality unlike in the classical case, predomain is not characterised uniquely by the axioms however, if two domain operators have the same range, by the above remark they coincide

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(Pre)Domain Calculus

Theorem a selection of predomain laws in an IL-semiring:

p = p (stability)

predomain is fully strict, i.e., a = 0 ⇔ a = 0

S forms an upper semilattice with p ⊔ q =

(p + q) predomain preserves arbitrary existing suprema

(a + b) =

a ⊔ b we have the absorption laws p · (p ⊔ q) = p and p ⊔ (p · q) = p hence ( S, ·, ⊔) is a lattice

(a · b) ≤

(a · b)

(a · b) ≤

a predomain satisfies the partial import/export law (p · a) ≤ p · a p · q = (p · q); hence S is closed under ·

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(Pre)Domain Calculus

Lemma additional properties of a domain operator in an IL-semiring:

✭loc✮ strengthens to the equality

(a · b) = (a · b) domain satisfies the full import/export law (p · a) = p · a in an I-semiring, the lattice ( S, ·, ⊔) is distributive

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Fuzzy Domain Operators

we now apply this to the fuzzy setting first we generalise the notion of t-norms and pseudo-complementation to general IL-semirings, in particular to semirings that do not just consist of the interval [0, 1] (as, say, a subset of the real numbers) and where that interval is not necessarily linearly ordered

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Fuzzy Domain Operators

consider an IL-semiring S with the interval [0, 1] =df {x | x ≤ 1} a t-norm is a binary operator : [0, 1] × [0, 1] → [0, 1] that is isotone in both arguments, associative and commutative and has 1 as unit the definition implies p q ≤ p, q in a predomain IL-semiring the operator · restricted to [0, 1] is a t-norm a weak complement operator in an IL-semiring is a function ¬ : [0, 1] → [0, 1] that is an order-antiisomorphism, i.e., is bijective and satisfies p ≤ q ⇔ ¬q ≤ ¬p, such that additionally ¬¬p = p this implies ¬0 = 1 and ¬1 = 0

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Fuzzy Domain Operators

moreover, if the IL-semiring has a t-norm the associated t-conorm is defined as the analogue of the De Morgan dual of the t-norm: p q =df ¬(¬p ¬q)

Lemma assume a predomain IL-semiring with weak negation

p ≤ p q if p q is the infimum of p and q then p ⊔ q = p q

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Fuzzy Domain Operators

next, we deal with a special t-norm and its associated t-conorm

Lemma consider the sub-interval I =df [0, 1] of the real numbers

with x y =df min (x, y) and x y =df max (x, y) then (I, , 0, , 1) is an I-semiring and the identity function is a domain operator on I since this domain operator is quite boring, in a later section we will turn to matrices over I, where the behaviour becomes non-trivial

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Modal Operators

as in earlier approaches, in a predomain semiring we can define a forward diamond operator as | |a p =df (a · p) if the semiring has a weak complement the diamond can be dualised to a forward box operator by setting | |a] ] q =df ¬| |a ¬q the analogues of the classical properties of these operators still hold in the IL-semiring case

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Modal Operators

ne may wonder about the relation of these operators to those in other

systems of fuzzy modal logic these approaches usually deal only with algebras where the whole carrier set coincides with the interval [0, 1] this would, for instance, rule out the matrix semirings to be discussed in the next section

n the other hand, it would be interesting to see whether the use
f residuated lattices there could be carried over fruitfully to the

interval [0, 1] of general semirings this is left for future research

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Predomain and Domain in Matrix Algebras

we can use the elements of an IL-semiring as entries in matrices with pointwise addition and the usual matrix product the set of n × n matrices for some n ∈ I N becomes again an IL-semiring zero matrix acts as 0 diagonal unit matrix acts as 1

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Predomain and Domain in Matrix Algebras

assume a predomain operator for the underlying IL-semiring using the characteristic property (llp) we find the following formula for the predomain of a matrix (the 2 × 2 case generalises immediately):

a

b c d

=df
a ⊔

b

c ⊔

d

from this we obtain the following result

Lemma if an I-semiring S has a domain operator, then so does the

set of n × n matrices over S

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Application to Fuzzy Matrices

assume now that in [0, 1] we use the t-norm p q = p · q and that there is a weak complement operator ¬ then we obtain the following formula for the diamond from the domain formula:

a

b c d p q

=
|

|a p | |b q | |c p | |d q

a straightforward calculation shows
a

b c d p q

=
|

|a] ] p | |b] ] q | |c] ] p | |d] ] q

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Application to Fuzzy Matrices

ne potential application of this is the following

using the approach of Kawahara one can model a flow network as a matrix with the pipe capacities between the nodes as entries, scaled down to the interval [0,1] note that the entries may be arbitrary in [0, 1], not just 0 or 1 then the algebra with = min and = max is a domain semiring hence the set of fuzzy n × n matrices is, too for such a matrix C the expressions C and ¬ C′, where C′ is the componentwise negation of C, give for each node the maximum and minimum capacity emanating from that node

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Application to Fuzzy Matrices

to describe network shapes and restriction we can use crisp matrices, i.e., matrices with 0/1 entries only using crisp diagonal matrices P, we can express pre-/post-restriction by matrix multiplication on the appropriate side so if a matrix C gives the pipe capacities in a network, P · C and C · P give the capacities in the network in which all starting/ending points outside P are removed hence, if we take again = min and = max , the expression | |C P gives for each node the maximum outgoing capacity in the

utput restricted network C · P
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Application to Fuzzy Matrices

to explain the significance of | |C] ] P, we take a slightly different view of the fuzzy matrix model for flow analysis scaling down the capacities to [0, 1] could be done relative to a top capacity (not necessarily occurring in the network) then p ∈ [0, 1] would indicate how close the flow is to the top flow hence | |C] ] P would indicate the level of “non-leaking” outside of P if for instance | |C] ] P = 0, then the maximal flow outside of P is 1, i.e., leaking is maximal

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Application to Fuzzy Matrices

since on crisp matrices weak negation coincides with standard Boolean negation, we can, additionally, use these ideas to replay the algebraic derivation of the Floyd/ Warshall and Dijkstra algorithms by H¨

fner/M¨
ller

Elaborating on these examples will be the subject of further papers.

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What Else?

the paper proves the mentioned results using the novel notion of a restrictor, a common generalisation of tests and predomain elements we also parametrise the mentioned axioms for ((pre)pre)domain and thus investigate a whole family of related domain operators

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Conclusion

despite the weakness in assumptions, the generalised theory of predomain and domain has turned out to be surprisingly rich in results concerning applications, we certainly have just skimmed the surface and hope that others will join our further investigations

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ller