5. Conclusion

competitors (subjects s1, …, skÞ are not sufficient for the whole order completion. In this case, more rational may be the second approach when order is splitted between non-antagonistic (cooperating) competitors in such a way that the total amount of objects produced is distributed among subjects s1, …, sk. This techniques may be modeled by unitary multimetagrammar Sq, which scheme Rq includes unitary

<sup>1</sup> � <sup>a</sup><sup>1</sup>

…

<sup>1</sup> � <sup>a</sup><sup>k</sup>

as well as all URs, having place in the scheme, constructed higher by application

which is directly induced by order q ¼ f g n � a and boundary conditions, defining

n o

and thus values n1, …, nk are parts of order q ¼ f g n � a , distributed among subjects s1, …, sk, respectively: s<sup>1</sup> will produce n<sup>1</sup> objects a, s<sup>2</sup> � n<sup>2</sup> objects a, up to sk,

first ! <sup>1</sup> � car, <sup>2800</sup> � usd, <sup>2800</sup> � usd‐1, <sup>1</sup> � accessories‐set,

second ! <sup>1</sup> � car, <sup>2500</sup> � usd, <sup>2500</sup> � usd‐2, <sup>1</sup> � accessories‐set,

third ! <sup>1</sup> � car, <sup>2200</sup> � usd, <sup>2200</sup> � usd‐3, <sup>1</sup> � accessories‐set: If order q ¼ f g 10 � car , then Fq ¼ car ¼ 10; 0≤γ<sup>1</sup> ≤ 10; 0≤γ<sup>2</sup> ≤10; 0≤γ f g <sup>3</sup> ≤10

usd‐<sup>3</sup> <sup>≥</sup>6600g: According to such Fq, set of terminal multisets, generated by UMMG

� �, may contain element of the form <sup>f</sup><sup>10</sup> � car, <sup>25600</sup> � usd,

<sup>14000</sup> � usd‐1, <sup>5000</sup> � usd‐2, <sup>6600</sup> � usd‐3, <sup>5</sup> � <sup>γ</sup>1, <sup>2</sup> � <sup>γ</sup>2, <sup>3</sup> � <sup>γ</sup>3, …g,

<sup>q</sup> may contain boundary and optimizing conditions, selecting

<sup>q</sup> <sup>¼</sup> <sup>f</sup>usd <sup>¼</sup> min; usd‐1≥2800; usd‐2≥5000;

∑ k i¼1

Example 8. Let us transform set R from Example 7 to the following:

order ! γ<sup>1</sup> � first, γ<sup>2</sup> � second, γ<sup>3</sup> � third,

<sup>1</sup> � b i <sup>1</sup>; …; m<sup>i</sup>

<sup>1</sup>, …, n<sup>1</sup>

<sup>1</sup> , …, nk

<sup>l</sup><sup>1</sup> � <sup>a</sup><sup>1</sup> l1 ,

lk � ak lk ,

<sup>b</sup><sup>1</sup> ! <sup>1</sup> � a, n<sup>1</sup>

bk ! <sup>1</sup> � a, nk

of the first approach, excluding (34). Filter Fq contains boundary condition

As seen, terminal multisets, generated by UMMG S ¼ aq; Rq; Fq

<sup>n</sup> � <sup>a</sup>; <sup>n</sup><sup>1</sup> � <sup>γ</sup>1; …; nk � <sup>γ</sup>k; <sup>m</sup><sup>i</sup>

aq ! γ<sup>1</sup> � b1, …, γ<sup>k</sup> � bk, (38)

a ¼ n, (40)

0≤γ<sup>i</sup> ≤n: (41)

li � b i li

ni ¼ n, (43)

� �, have form

, (42)

(39)

metarule

and unitary rules

Enhanced Expert Systems

domains of variables γ1, …, γk:

where, according to (38)–(40),

which will produce nk objects a.

terminal multisets, for example, F<sup>0</sup>

∪ F<sup>0</sup>

102

q, where F<sup>0</sup>

Sq ¼ aq; Rq; Fq

Presented primary survey of multigrammatical knowledge representation along with brief consideration of its possible applications is, of course, only a background for future development, which most valuable directions may be:


Some of the listed directions are already developed by the author and his colleagues; some are waiting their time, being targeted to the creation of unified framework for the intellectual (knowledge-based) digital economy. This way is leading us to the Big Knowledge paradigm being generalization of the Big Data one, which is already everyday reality. The author will be glad, if this paper will be of any interest for some scholars working in the related areas.

#### Acknowledgements

The author is grateful to Prof. Fred Roberts for useful discussions and support, and to Prof. Jeffrey Ullman, whose useful remarks on the primary version of this

work contributed to its essential upgrade. A significant incentive for the development of the proposed approach was its positive assessment by Prof. Noam Chomsky, which early works on syntactic structures formed a conceptual background of the described mathematical toolkit.

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