„LSP w zastosowaniu do reprezentacji wiedzy i wnioskowania w warunkach niepewności”

„LSP w zastosowaniu do reprezentacji

wiedzy i wnioskowania w warunkach

niepewności”

Zbigniew Suraj

Instytut Informatyki

Uniwersytet Rzeszowski

zsuraj@univ.rzeszow.pl

Seminarium Zakładu ISWD Instytutu Informatyki PP i Sekcji ISWD i OE Komitetu Informatyki PAN, Poznań, 15 stycznia 2013

1.

 

Informal introduction to

LP

-nets

2.

 

Formal definitions of

LP

-nets

3.

 

Net knowledge representation

4.

 

Algorithm

5.

 

Modelling with

LP

-nets

6.

 

Conclusion and further work

•

 

Petri nets are a good graphical and mathematical tool for

modelling of concurrent systems

•

 

These nets have an origin in Petri’s PhD thesis published

in 1962.

C.A. Petri

Carl Adam Petri (1926-2010) – German mathematician and computer

scientist, honorary professor at the University of Hamburg. A creator of the general net theory.

Kommunikation mit Automaten (Communication with automata), Bonn: Institut für Instrumentelle Mathematik, Schriften des IIM Nr. 2, 1962

Petri nets

(contd.)

(a) A marking before firing a transition t (b) A marking after firing a transition t t

•

•

t

•

p₁ p₃ p₂ p₁ p₂ p₃ (a) (b)

•  History of Petri nets

Beginning

C.A. Petri: Kommunikation mit Automaten, Bonn: Institut für Instrumentelle Mathematik, Schriften des IIM Nr. 2, 1962.

•  David, R., Alla, H., Petri Nets and Grafcet. Tools for modeling

discrete event systems, Prentice-Hall, London 1992.

•  Jensen, K., Coloured Petri Nets. Basic concepts, analysis methods

and practical use, vol. 1-3, Springer, Berlin 1992-1997.

•  Murata, T., Petri Nets: Properties, Analysis and Applications,

Proceedings of IEEE, Vol. 77, No. 4, April 1989.

•  Peterson, J.L., Petri Net Theory and the Modeling of Systems,

Prentice-Hall, Englewood Cliffs, NJ, 1981.

•  Reisig, W., Petri nets. An Introduction, Springer, Berlin 1985.

•  Starke, P.H., Petri-Netze. Grundlagen, Anwendungen, Theorie, VEB

Deutscher Verlag der Wissenschaften, Berlin 1980.

•  V. E. Kotov, Petri Nets [in Russian], Nauka, Moscow 1984.

•  Suraj, Z., Komarek, B., GRAF. A graphical system for construction

and analysis of Petri nets [in Polish], Akademicka Oficyna

Wydawnicza PLJ, Warsaw 1994.

•  Suraj, Z., Szpyrka, M., Petri nets and PN-tools. Tools for

construction and analysis of Petri nets [in Polish], Pedagogical

University Press, Rzeszów 1999.

•  Szpyrka, M., Petri nets in the modelling and analysis of concurrent

systems [in Polish], WNT, Warsaw 2008. •  The Petri nets world:

http://www.informatik.uni-hamburg.de/TGI/PetriNets/

•

 

There are a formal tool for knowledge representation

and modelling reasoning in KBS.

•

 

Logical Petri nets (

LP

-nets) comprise Petri nets and

fuzzy sets introduced by L.A. Zadeh in 1965.

C.G. Looney

Carl G. Looney (1934 - ) – American computer scientist and engineer, professor emeritus at the University of Nevada, Reno. A creator of fuzzy Petri net theory.

Fuzzy Petri Nets for Rule-Based Decision-making, IEEE Transaction on Systems, Man, and Cybernetics, Vol. 18, No. 1, Jun-Feb. 1988, 178-83.

L.C. Zadeh

Lotfi C. Zadeh (1921 - ) – American computer scientist, professor at the University of California, Berkeley, USA. A creator of the fuzzy set theory.

The first article:

LP-nets

(contd.)

•

 

They are good for intuitive and graphical knowledge representation

and for modelling reasoning (forward, backward) in rule knowledge

bases

•

 

They are suitable for quality evaluation and reasoning efficiency in

rule knowledge bases

•

 

They can be used for detecting anomalies in rule knowledge bases

•

 

They can be used for modelling hierarchical structures of rule

knowledge bases

•

 

They allow graphical ways for conclusion explanation

•

 

They are easy for computer implementation (also parallel and

LP-nets

(contd.)

•

 

History of LP-

nets

Beginning

C.G. Looney: Fuzzy Petri Nets for Rule-Based Decision-making, 1988.

•

 

Fuzzy Petri nets (

FP

-nets)

Propositions at places Transitions interpreted as implications Logical operators:

In = min | max,

Out₁ = *, Out₂ = max Truth degree of proposition: [0,1]

Truth degree of implication: [0,1]

t 0.90 „Humidity is low” „It is Monday”

•

1.0

•

0.9 „It is cold” t 0.81

•

„Humidity is low” 0.90 „It is Monday”

•

1.0

•

0.9 „It is cold”

LP-nets

(contd.)

•

 

Extended fuzzy Petri nets (

EFP-

nets)

Adding to the net model a threshold function with values in [0,1]

Chen, S.M., Ke, J.S., and Chang, J.F.: Knowledge representation using fuzzy Petri nets. IEEE Trans. on Knowledge and Data Engineering 2(3), Sept. 1990, 311-319.

t 0.90 „Humidity is low” „It is Monday”

•

1.0

•

0.9 „It is cold” t 0.81

•

„Humidity is low” 0.90 „It is Monday”

•

1.0

•

0.9 „It is cold” 0.3 0.3

S.-M. Chen

Shyi-Ming Chen (1960 - ) – computer scientist and engineer,

professor at the National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C.

Chen, S.M., Ke, J.S., and Chang, J.F.: Knowledge representation using fuzzy Petri nets. IEEE Trans. on Knowledge and Data Engineering 2(3), Sept. 1990, 311-319

LP-nets

(contd.)

•

 

Binary Petri nets (

BP-

nets)

Limited version of EFP-nets: •  truth degree of proposition: 0(false), 1(true)

•  truth degree of implication: 0(false) – omitted, 1(true)

•  treshold function – constant (=1)

•  logical operators: In = and | or Out₁ = and Out₂ = or t 1.0 „Humidity is low” „It is Monday”

•

1.0

•

1.0 „It is cold” t 1.0

•

„Humidity is low” 1.0 „It is Monday”

•

1.0

•

1.0 „It is cold” 1.0 1.0

Extensions of LP-nets

•

 

Generalised fuzzy Petri nets (

GFP-

nets)

Extending the set of operators:

In = any t-norm | s-norm In example: T₁(a,b) = a*b

Out₁ = any t-norm T₂(a,b) = min(a,b)

Out₂ = any s-norm S(a,b) = max(a,b)

Suraj, Z.: Generalised Fuzzy Petri Nets for Approximate Reasoning in Decision Support

Systems. In: Proc. of CS&P'2012, Sept. 28-30, 2012, Humboldt University, Berlin, 2012, pp.

370-381. t 0.90 „Humidity is low” „It is Monday”

•

1.0

•

0.9 „It is cold” t 0.9

•

„Humidity is low” 0.90 „It is Monday”

•

1.0

•

0.9 „It is cold” 0.3 0.3 (T₁(.), T₂(.), S(.)) (T₁(.), T₂(.), S(.))

Extensions of LP-nets

(contd.)

•

 

Parameterised fuzzy Petri nets (

PFP-

nets)

Further extending the set of operators: In example:

In = any param. family of sums | products T_H1_{(a,b) = a*b}

Out₁ = any param. family of products S_H1_{(a,b) = a+b-a*b}

Out₂ = any param. family of sums

Suraj, Z.: Parameterised Fuzzy Petri Nets for Approximate Reasoning in Decision Support

Systems. In: Proc. of AMLTA'2012, Dec. 8-10, 2012, Cairo, Egypt, Comm. in Comp. and

Infor. Sci. series, Vol. 322, Springer, 2012, pp. 33-42.

t 0.90 „Humidity is low” „It is Monday”

•

1.0

•

0.9 „It is cold” t 0.81

•

„Humidity is low” 0.90 „It is Monday”

•

1.0

•

0.9 „It is cold” 0.3 0.3 (T_H1_{(.), T} H1(.), SH1(.)) (TH1(.), TH1(.), SH1(.))

DEFINITION 1. A tuple

N

= (

P

,

T

,

I

,

O

,

M

₀

) is a (

classical

) Petri net

(

P

-net), if:

•

 P

= {

p

₁

,

p

₂

, …,

p

_n

}

- a finite set of

places

,

n

> 0

•

 T

= {

t

₁

,

t

₂

, …,

t

_m

}

- a finite set of

transitions

,

m

>0

•

 P

and

T

-

disjoint

•

 I

:

T

→ 2

P

- the

input function,

a mapping from

transitions to the family of all

place subsets

•

 O

:

T

→ 2

P

- the

output function,

a mapping from

transitions to the family of all place

subsets

•

 M

₀

:

P

→

IN -

an

initial marking

of

N,

where

IN =

{0,1, …}

Example 1.

Let

N

= (

P

,

T

,

I

,

O

,

M

₀

) be a

P

-net in Fig. 1

.

Fig. 1. A P-net N with the initial marking M₀ = (2,1,3,0,0)

t2 t1 p4 p1 p2 p3 p5 P = {p1, p2, …, p5} T = {t1, t2} I: I(t1) = {p1, p2}, I(t2) = {p2, p3} O: O(t1) = {p4}, O(t2) = {p5}

A binary Petri net

– definition

DEFINITION 2. A tuple

N

_B

= (

P

,

T

,

S

,

I

,

O

,

α

,

M

₀

) is a binary Petri

net (

BP

-net), if:

•

 P, T, I, O

-

the same meaning as in DEFINITION 1

•

 

S = {s

₁

, s

₂

, …, s

_n

}

- a finite set of statements

•

 P

,

T

,

S

- pairwise disjoint and

card(P) = card(S)

•

 

α: P → S

- the statement binding function, a bijection

from places to statements

•

 

M

₀

:

P

→ {0,1}

- the initial marking, a mapping from places

to the set {0,1}

Example 2.

Let

N

_B

= (

P

,

T

,

S

,

I

,

O

,

α

,

M

₀

) be a

BP

-net in Fig. 2.

Fig. 2. A BP-net with the initial marking M₀ = (1,1,1,0,0)

s4 s1 s3 s5 t2 t1 s2 p4 p1 p2 p3 p5 P, T, I, O - defined as in Example 1 S = {s1, s2, …, s5} α: α(p1) = s1, α(p2) = s2, α(p3) = s3, α(p4) = s4, α(p5) = s4,

A fuzzy Petri net

– definition

DEFINITION 3. A tuple

N

_F

= (

P

,

T

,

S

,

I

,

O

,

α

,

β

,

γ

,

M

₀

) is

a fuzzy Petri net

(

FP

-net), if:

•

 P, T, S

,

I, O, α

-

the same meaning as in DEFINITION 2

•

 

β

:

T

→ [0,1] - the

truth degree function,

a mapping

from transitions to [0,1]

•

 

γ

:

T

→ [0,1] - the

threshold function,

a mapping

from transitions to [0,1]

•

 

M

₀

:

P

→ [0,1]

- the initial marking, a mapping from

places to [0,1]

Example 3.

Let

N

_F

= (

P

,

T

,

S

,

I

,

O

,

α

,

β

,

γ

,

M

₀

) be a

FP

-net in Fig. 3

Fig. 3. A FP-net with the initial marking M₀ = (0.6,0.4,0.7,0,0)

s4 s1 s3 s5 t2 t1 s2 p4 p1 p2 p3 p5 0.6 0.4 0.7 0.3 0.4 0.8 0.7 P, T, S, I, O, α - defined as in Example 2 β: β(t1) = 0.7, β(t2) = 0.8 γ: γ(t1) = 0.4, γ(t2) = 0.3

Triangular norms

A

t

-norm is defined as

t

: [0,1]

2

_→

_[0,1]

_{such that for each}

a, b, c from [0,1]:

(1)

 

it has 1 as the unit element,

i.e., t(a,1) = a;

(2)

 

it is monotone,

i.e., if a ≤ b then t(a, c) ≤ t(b, c)

(3) it is commutative,

i.e., t(a, b) = t(b, a)

(4) it is associative,

i.e., t(t(a, b), c) = t(a, t(b, c))

Examples of t-norms:

T

_M

(a,

b) = min(a,

b)

,

T

_P

(a,

b) = a

*

b,

Triangular norms

(contd.)

A s-norm (or t-conorm) is defined as s: [0,1]

2

_→

_{[0,1] such}

that for each a, b, c from [0,1]:

(1)

 

it has 0 as the unit element,

i.e., s(a,0) = a;

(2)

 

it is monotone,

i.e., if a ≤ b then s(a, c) ≤ s(b, c)

(3) it is commutative,

i.e., s(a, b) = s(b, a)

(4) it is associative,

i.e., s(s(a, b), c) = s(a, s(b, c))

Examples of s-norms:

S

_M

(a,

b) = m

ax

(a,

b)

,

S

_P

(a,

b) =

a + b -

a

*

b,

Relationships between the

selected

t

- and

s

-norms

PROPOSITION 1. Let T

_D

, T

_M

, T

_L

, T

_P

be t-norms, and S

_D

,

S

_M

, S

_L

, S

_P

- s-norms as above. Then:

T

_D

≤

T

_L

≤

T

_P

≤

T

_M

≤

S

_M

≤

S

_P

≤

S

_L

≤

S

_D

where:

T

_D

(a,b) = a, if b = 1;

S

_D

(a,b) = a, if b = 0;

T

_D

(a,b) = b, if a = 1;

S

_D

(a,b) = b, if a = 0;

T

_D

(a,b) = 0, otherwise.

S

_D

(a,b) = 1, otherwise.

Generalised fuzzy Petri net

–

definition

DEFINITION 4. A tuple N

_G

= (P, T, S, I, O, α, β,

γ

,

Op

,

δ

, M

₀

) is

a generalised fuzzy Petri net (GFP-net), if:

•

 

P, T, S, I, O, α, β,

γ

, M

₀

- the same meaning as in

DEFINITION 3

•

 

Op

- the

set of operators

, a finite set of

t-

norms and

s

-norms

•

 

δ

:

T

→

Op

3

- the

operator binding function,

a mapping from transitions to

Op

3

_{, i.e., the set of all triples of}

Generalised fuzzy Petri net

(contd.)

•

 

In

-

the input operator

;

In belongs to

one of the classes

t-norms or

s-norms

•

 

Out

₁

and Out

₂

- the output operators; Out

₁

belongs to

the class of

t

-norms, and Out

₂

belongs

to the class of

s

-norms

Example 4.

Let

N

_G

= (

P

,

T

,

S

,

I

,

O

,

α

,

β

,

γ

,

Op

,

δ

,

M

₀

) be

GFP

-net in Fig.

1.

Fig. 4. GFP-net with the initial marking M₀ = (0.6,0.4,0.7,0,0)

s4 s1 s3 s5 t2 t1 s2 p4 p1 p2 p3 p5 0.6 0.4 0.7 0.3 0.4 0.8 0.7 (max, *, max) (min, *, max) P, T, S, I, O, α, β, γ - defined as in Example 3 Op = { min, max, * } δ(t1) = ( max, *, max } δ(t2) = ( max, *, max }

Parameterised families of sums

and products

TABLE 1. An exemplary list of parameterised families of sums and products v v v _b a_{) (1 ) 1)]}1/ 1 ( , 0 ( [max 1_{− − + − − max (0,a}v _bv _1)]1/v − +

Sum S

_i

(a,b,v)

Product

T

_i

(a,b,v)

Range

)

,

(

_−∞

_∞

ab v ab v b a ) 1 ( 1 ) 2 ( − − − − + ) )( 1 ( v a b ab v ab − + − + ( ∞0, ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − + − − − 1 ) 1 )( 1 ( 1 log 1 1 1 v v v a b v ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − + 1 ) 1 )( 1 ( 1 log v v va b v ] ) ( , 1 [ min _{a +}v _bv 1 v/ _{1 min[1, ((1 a)}v _{(1 b)}v₎1 v/ _]

( ∞

0

,

)

− + − −

)

,

0

( ∞

) , 1 , 1 max( ) 1 , , ( min v b a v b a ab b a − − − − − + ) , , ( max a b v ab ) 1 , 0 ( v v v b a / 1 ) 1 1 ( ) 1 1 ( 1 1 1 ⎥⎦ ⎤ ⎢⎣ ⎡ − + − + − v v v b a / 1 ) 1 1 ( ) 1 1 ( 1 1 ⎥⎦ ⎤ ⎢⎣ ⎡ − + − + ) , 0 ( ∞ SS

i

H DP F Y D

PROPERTIES:

S

_SS

(a,b,1) = S

_L

(a,b)

T

_SS

(a,b,1) = T

_L

(a,b)

S

_H

(a,b,1) = S

_P

(a,b)

S

_Y

(a,b,1) = S

_L

(a,b)

S

_D

(a,b,1) = S

_H

(a,b,0)

S

_DP

(a,b,?) = S

_?

(a,b)

S

_F

(a,b,0) = S

_M

(a,b)

T

_H

(a,b,1) = T

_P

(a,b)

T

_Y

(a,b,1) = T

_L

(a,b)

T

_D

(a,b,1) = T

_H

(a,b,0)

T

_DP

(a,b,?) = T

_?

(a,b)

T

_F

(a,b,0) = T

_M

(a,b)

H - Hamacher, SS – Schweizer-Sklar, DP – Dubois i Prade, F – Frank, Y – Yager, D – Dombi

PROPERTIES:

∞ −∞

=

≥

≥

=

≥

=

≥

=

_{M SS P SS L SS}v _{D SS} SS

T

T

T

T

T

T

T

T

T

0 1 ∞ −∞

=

≤

≤

=

≤

=

≤

=

D SS v SS L SS P SS M SS

S

S

S

S

S

S

S

S

S

0 1 ∞

=

≥

≥

=

_{D D}v _{M D} D

S

S

S

S

S

0 ∞

=

≥

≥

=

_{M F}v _{L F} F

T

T

T

T

T

0 ∞

=

≤

≤

=

_{D D}v _{M D} D

T

T

T

T

T

0 ∞

=

≤

≤

=

≤

=

_{D Y L Y}v _{M Y} Y

T

T

T

T

T

T

T

0 1 ∞

=

≥

≥

=

≥

=

_{D Y L Y}v _{M Y} Y

S

S

S

S

S

S

S

0 1 ∞

=

≤

≤

=

_{M F}v _{L F} F

S

S

S

S

S

0 ∞

=

≥

≥

=

_{P H}v _{D H} H

T

T

T

T

T

1 ∞

=

≤

≤

=

_{P H}v _{D H} H

S

S

S

S

S

1 1. 2. 3. 4. 5.

S_D T_D S_L S_P T_L T_P T_M S_M Sv SS Sv F Sv Y Tv F Tv Y Tv SS Sv H Sv D Tv D Tv H = Sw D = S-wSS= SwY= S0F Sw H = S0D= SwSS= S0Y = S1 SS = S1Y= SwF T1 SS = T1Y= TwF Tw H = T0D= TwSS= T0Y = Tw D = T-wSS= TwY= T0F= T1 H = T0SS= S1 H = S0SS=

Parameterised fuzzy Petri net –

definition

DEFINITION 5. A tuple N

_P

= (P, T, S, I, O, α, β,

γ

,

Op

,

δ

, M

₀

) is

a parameterised fuzzy Petri net (PFP-net), if:

•

 

P, T, S, I, O, α, β,

γ

,

δ

, M

₀

- the same meaning as in

DEFINITION 4

•

 

Op

- a finite set of

parameterised

families of sums and

Example 5.

Let

N

_p

= (

P

,

T

,

S

,

I

,

O

,

α

,

β

,

γ

,

Op

,

δ

,

M

₀

) be

PFP

-net in Fig.

7.

Fig. 5. PFP-net with the initial marking M₀ = (0.6,0.4,0.7,0,0)

s4 s1 s3 s5 t2 t1 s2 p4 p1 p2 p3 p5 0.6 0.4 0.7 0.3 0.4 0.8 0.7 ( S(.), T(.), S(.) ) ( T(.), T(.), S(.) ) P, T, S, I, O, α, β, γ - defined as in Example 3 Op = { S(.), T(.) } δ(t1) = ( S(.),T(.), S(.) ) δ(t2) = ( T(.),T(.), S(.) )

Markings and firing rule

Let N

_P

be a PFP-net.

A mapping M: P → [0,1] is a

marking

of N

_P

.

A transition t

is enabled by M and a parameter v, if

In(v)(M(p

_i1

), …, M(p

_ik

}) ≥

γ

(t) > 0 for p

_ij

in I(t), j=1,…,k

Firing rule

Let N_P = (P, T, S, I, O, α, β, γ, Op, δ, M₀) be a PFP-net, I(t) = {p_i1, p_i2,…, p_ik} – a set of input places of t, β(t) – a value of the truth degree function β for t

such that 0 < β(t) <= 1, In(v), Out₁(v), Out₂(v) – parameterised input/output operators for t.

Mode 1. If

M

with

v

is a marking of

N

_P

enabling

t

and

M

_v

’

is the

marking derived from

M

with

v

by firing

t

, then for each

p

in

P

:

⎪

⎩

⎪

⎨

⎧

∈

∈

=

otherwise

p

M

t

O

p

if

p

M

t

p

M

p

M

v

In

v

Out

v

Out

t

I

p

if

p

M

k i i v

)

(

)

(

)),

(

)),

(

)),

(

...,

),

(

(

)

(

)(

(

)(

(

)

(

,

0

)

(

'

1 1 2

β

Firing rule (contd.)

Mode 2. If

M

with

v

is a marking of

N

_P

enabling

t

and

M

_v

’

is the

marking derived from

M

with

v

by firing

t

, then for each

p

in

P

:

⎩

⎨

⎧

∈

=

otherwise

p

M

t

O

p

if

p

M

t

p

M

p

M

v

In

v

Out

v

Out

p

M

i ik v

)

(

)

(

)),

(

)),

(

)),

(

...,

),

(

)(

(

)(

(

)(

(

)

('

2 1 1

β

Example 6.

Fig 6. A marking of PFP-net after firing t₁ (Mode 1); the transition t₂

is not enabled by M₀. s4 s1 s3 s5 t2 t1 s2 p4 p1 p2 p3 p5 0.53 0.7 0.3 0.4 0.8 0.7 s4 s1 s3 s5 t2 t1 s2 p4 p1 p2 p3 p5 0.6 0.4 0.7 0.3 0.4 0.8 0.7 ( S(.), T(.), S(.) ) ( T(.), T(.), S(.) ) _{( T(.), T(.), S(.) )} ( S(.), T(.), S(.) ) ab v ab v b a v b a SH ) 1 ( 1 ) 2 ( ) , , ( − − − − + = ) )( 1 ( ) , , ( ab b a v v ab v b a TH − + − + =

Let v = 1. Then

SH(a,b,1) = a+b−ab TH(a,b,1) = ab

Example 7.

Fig 7. A marking of PFP-net after firing t₁ (Mode 2).

s4 s1 s3 s5 t2 t1 s2 p4 p1 p2 p3 p5 0.53 0.7 0.3 0.4 0.8 0.7 s4 s1 s3 s5 t2 t1 s2 p4 p1 p2 p3 p5 0.6 0.4 0.7 0.3 0.4 0.8 0.7 ( S(.), T(.), S(.) ) ( T(.), T(.), S(.) ) _{( T(.), T(.), S(.) )} ( S(.), T(.), S(.) )

Let v = 1. Then

0.6 0.4 ab v ab v b a v b a SH ) 1 ( 1 ) 2 ( ) , , ( − − − − + = TH(a,b,v) _{v (1 v)(}ab_{a b ab)} − + − + = ab b a b a SH( , ,1) = + − TH(a,b,1) =ab

Rule types and their net

representation

Type 0:

IF s

_j

THEN s

_k

(CF = q

_i

)

Fig. 8(a). A marking before firing t_i. Fig. 8(b). A marking after firing t_i.

Rule types and their net

representation

Type 1:

IF s

_j1

and/or s

_j2

and/or … and/or s

_jn

THEN s

_k

(CF = q

_i

)

Fig. 9(a). A marking before firing t_i. Fig. 9(b). A marking after firing t_i.

y_k = Out₁(Inv_(y

Rule types and their net

representation

Type 2:

IF s

_j

THEN s

_k1

and s

_k2

and … and s

_kn

(CF = q

_i

)

Fig. 10(a) A marking before firing t_i. Fig. 10(b) A marking after firing t_i.

REMARKS:

1. For non-zero markings of output places, we should take into account

the output operator Out₂. Thus, in each formula presented above a final token value y’k should be computed as follows:

y’_k = Out₂(y_k, M(p))

where y_k denotes the token values computed by formulas

presented above for suitable types of rules, and M(p) is a marking of output place p.

2. We can consider combinations of above types of rules. Each

statement from rules is represented by one place in a LP-net model. However, each rule is represented by one transition. Firing a given transition corresponds to firing a rule which is represented by this transition.

ALGORITHM: A construction of LP-net Input: A set R of rules

Output: A LP-net N

F:= 0; //the empty set

for r in R do

if r is a rule of Type 0 then

construct a subnet N_r as shown in Fig. 8(a); end

if r is a rule of Types 1 then

construct a subnet N_r as shown in Fig. 9(a); end

if r is a rule of Type 2 then

construct a subnet N_r as shown in Fig. 10(a);

end

F:= F u {N_r };

end

Integrate all subnets from a family F on joint places and create a result net N;

Illustrative Example

Let us

consider the following situation:

a

train B waits at a certain

station for a train

A

to arrive in order to allow some passengers to

change train

A

to train

B

. Now, a conflict arises when the train

A

is late. In this situation, the following alternatives can be taken

into consideration:

•

 

Train

B

waits for train

A

to arrive. In this case, train

B

will

depart with delay.

•

 

Train

B

departs in time. In this case, passengers disembarking

train

A

have to wait for a later train.

•

 

Train

B

departs in time, and an additional train is employed for

Illustrative Example

(contd.)

To make a decision, several inner conditions have to be taken

into account, for example:

•

 

the delay period,

•

 

the number of passengers changing trains, etc.

The discussion regarding an optimal solution to the problem

of divergent aims such as:

•

 

minimization of delays throughout the traffic network,

•

 

warranty of connections for the customer satisfaction,

•

 

efficient use of expensive resources, etc.

FACTS: s1, s2, s3, s4 RULE BASE:

r1: IF s2 THEN s6 r3: IF s1 AND s4 AND s6 THEN s7

r2: IF s3 THEN s6 r4: IF s4 AND s5 THEN s8

where:

s1: Train B was the last train in this direction on this day.

s2: The delay of train A is large.

s3: There is an urgent need for the track of train B.

s4: Many passengers would like to change for train B.

s5: The delay of train A is small.

s6: (Let) train B depart according to schedule.

s7: Employ an additional train C (in direction of train B).

s8: Let train B wait for train A.

Ex.: Specification of Rule Controller for

Net Model of Rule Controller for

Train-Traffic Administration

Version 1:

r1: IF s2 THEN s6 r2: IF s3 THEN s6

r3: IF s1 AND s4 AND s6 THEN s7 r4: IF s4 AND s5 THEN s8 s1 s5 s2 s4 s3 s6 _s7 t4 t3 t2 t1 s8

•

•

•

•

•

(Tv 1(.),Tv2(.),Sv(.)) (Tv 1(.),Tv1(.),Sv(.)) (--,Tv 1(.),Sv(.)) (--, Tv 1 (.),Sv (.)) 0.9 0.7 0.8 0.4 0.5 1.0 0.7 0.9 0.3 0.3 0.4 0.5 0.8

The symbol „– „ denotes lack of Input operator

Start places

Net Model of Rule Controller for

Train-Traffic Administration

r1: IF s2 THEN s6 r2: IF s3 THEN s6 r12: IF s2 OR s3 THEN s6 Version 2: r12: IF s2 OR s3 THEN s6

r3: IF s1 AND s4 AND s6 THEN s7 r4: IF s4 AND s5 THEN s8 s1 s5

•

s2 s4 s3 s6 s7 t4 t3 t12 s8

•

•

•

•

(Tv 1(.),Tv2(.),Sv(.)) (Tv 1(.),Tv1(.),Sv(.)) (Sv _(.),Tv 1(.),Sv(.)) 0.8 0.3 0.9 0.4 0.7 0.5 0.6 0.7 0.4 0.9 0.5

Net Model of Rule Controller for

Train-Traffic Administration

Substitution: S_M(a,b) = max(a,b) T_P(a,b) = min(a,b) Approximate reasoning 1: s1 s5

•

s2 s4 s3 s6 s7 t4 t3 t12 s8

•

•

•

•

(T_P(.),T_P(.),S_P(.)) (T_P(.),T_P(.),S_P(.)) (S_P(.),T_P(.),S_P(.)) 0.8 0.3 0.6 0.4 0.7 0.5 1.0 0.7 0.4 0.9 0.5

A final marking of goal places after firing t₄ (Mode 2).

0.5

•

Final decision:

s7: Employ an additional train C.

Net Model of Rule Controller for

Train-Traffic Administration

Substitution:

S_P(a,b) = a+b – a*b T_P(a,b) = a * b Approximate reasoning 2: s1 s5

•

s2 s4 s3 s6 s7 t4 t3 t12 s8

•

•

•

•

(T_P(.),T_P(.),S_P(.)) (T_P(.),T_P(.),S_P(.)) (S_P(.),T_P(.),S_P(.)) 0.8 0.3 0.6 0.4 0.7 0.5 1.0 0.7 0.4 0.9 0.5

A final marking of goal places after firing t₄ (Mode 2). A sequence t₁₂t₃ is not firable.

0.45

•

Final decision:

Net Model of Rule Controller for

Train-Traffic Administration

Substitution: S_L(a,b) = min(a+b,1) T_L(a,b) = max(0,a+b-1) Approximate reasoning 3: s1 s5

•

s2 s4 s3 s6 s7 t4 t3 t12 s8

•

•

•

•

(T_L(.),T_L(.),S_L(.)) (T_L(.),T_L(.),S_L(.)) (S_L(.),T_L(.),S_L(.)) 0.8 0.3 0.6 0.4 0.7 0.5 1.0 0.7 0.4 0.9 0.5

A final marking of goal places as in figure. A sequence t₁₂t₃ and t₄ are not firable.

Final decision:

Conclusion

•

 

W

e have

presented

a

brief survey of L

P

-nets

.

•

 

W

e have also provided

an approach to construction of

LP-nets.

•

 

It seems that L

P

-nets are

suitable for the design and

Further work

•

 

Interesting problem arises when we want to know how

to choose the relevant triangular norms for optimizing

reasoning process.

•

 

Verify the proposed approach on real-life data.

•  Chen, S.-M., Ke, J.-S., Chang, J.-F.: Knowledge Representation

Using Fuzzy Petri Nets, Transactions on Knowledge and Data

Engineering, Vol. 2, No.3, Sept. 1990, 311-319.

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Using Matrix Representation of Extended Fuzzy Petri Nets,

Fundamenta Informaticae 60(1-4), 2004, 143-157.

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IEEE Transaction on Systems, Man, and Cybernetics, vol.18, no.1,

Jun-Feb. 1988, pp.178-83.

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Concurrency, Specification, and Programming (CS&P'2012)", Sept. 28-30, 2012, Humboldt University, Berlin, 2012, pp. 370-381.

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Generalised Fuzzy Petri Nets. In: Proc. of Int. Conf. on Intelligent

Systems Design and Applications (ISDA'2012), Nov. 27-29, 2012, Kochi, India, IEEE Press, 2012, pp. 101-106.

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