Linguistic Summaries in Evaluating Elementary Conditions, Summarizing Data and Managing Nested Queries

3.1Evaluation of Optional Atomic Conditions

In database queries, the usual way of selecting the relevant entities is realized via the conditions expressed by conjunction (AND operator) and disjunction (OR operator). The former cannot cover the aggregation of mandatory and optional requirements, because all the requirements are mandatory. If only one atomic condition from a larger set is rejected, the overall matching degree is zero (zero is the absorbing element in conjunction). The latter is based on the substitutability principle, i.e. one satisfied atomic requirement is sufficient.

Let us recall the standard classification of aggregation functions (Dubois and Prade, 2004). Conjunctive aggregation functions are characterized by A(x)⩽min(x), disjunctive by A(x)⩾max(x), averaging by min(x)⩽A(x)⩽max(x), and remaining aggregation functions are called mixed, where x is a vector, x=(x1,…,xn).

Thus, the following problems in conjunctive aggregation might appear. First, the presence of mandatory and optional requirements, which was addressed by the asymmetric AND IF POSSIBLE conjunction (Dujmović, 1975; Bosc and Pivert, 2012) and axiomatized in Hudec and Mesiar (2020). Second, the aforementioned empty answer problem, i.e. cases when not a single record meets a larger set of atomic conditions. Third, all atomic predicates might be optional where the principle the more the better holds, i.e. an entity is preferred over another one if it satisfies more predicates. Thus, to exclude weakly performing entities, and on the other side to mitigate the empty answer problem, conjunction is relaxed by the quantifier most of, that is, most of atomic predicates should be satisfied.

The query relaxation by the fuzzy relative quantifier: the most of atomic conditions should be satisfied is initially suggested by Kacprzyk and Ziółkowski (1986). It is formalized by the quantified summaries (2), (3) as follows:

This approach deals with the presumption that all atomic conditions are optional, but the majority of them should be met. When all atomic conditions are met very weakly, then due to the quantified evaluation aggregated by the quantifier (3) the solution is zero. Hence, the aggregation for low values behaves like a nilpotent conjunction (e.g. Łukasiewicz t-norm Tl(x)=max(0,∑i=1nxi−(n−1)), where for instance Tl(0.2,0.1,0.2,0.3)=0) and for high values behaves like nilpotent disjunction (e.g. Łukasiewicz t-conorm Sl(x)=min(0,∑i=1nxi), where for instance Sl(0.9,0.8,0.9,0.8)=1). The disjunctive nilpotent observation holds only when the majority of atomic conditions is significantly met, e.g. y⩾n (see, Fig. 1 and (4)). Thus, value 1 is not automatically the absorbing element when several elementary conditions are met. The result for (0.2,0.8,0.4,1) is 0.29 ( m=0.5, n=0.85). Similarly, for (0,0.5,0.9,0.9) the solution is 0.21. In these cases, the behaviour is averaging. Clearly, this aggregation is a mixed one. These observations can be further axiomatized into the frame of the standard classification of aggregation functions, which is a future research topic.

This aggregation is illustrated in Table 1, where the entity e2 is rejected, even though it meets weakly all the atomic conditions (the property of conjunctive nilpotency). Entities e1 and e3 fail to meet one of the atomic requirements, but are evaluated as two acceptable ones because they meet significantly all the other requirements, with e3 being almost the ideal solution. These cases do not behave like disjunctive ones, because few fully satisfied predicates do not guarantee straightforwardly the solution equal to 1. Finally, entity e5 is evaluated as the ideal one due to very significant satisfaction of all requirements, even though not a single requirement is fully met (the property of disjunctive nilpotency, which appears only when most of the predicates are met significantly). For completeness, a solution by the arithmetic mean is shown in the last column. The arithmetic mean is not able to exclude entities e2 and e4. Clearly, by (4) all requirements are considered as optional and compensative when the proportion of (fully or partially) satisfied elementary requirements is high, usually greater than 0.5, for covering the natural meaning of terms majority or most of. Entities e6 and e7 show that the boundary conditions of aggregation functions are met.

3.2Evaluation of Mandatory and Optional Atomic Conditions

In many cases, several atomic requirements are mandatory, while the other ones are optional, and moreover, if a higher proportion of optional requirements is satisfied, then the entity is more suitable. In order to cover this aggregation requirement, we have modified the Eq. (4) in the following way

A suitable method for formalizing AND connective (conjunction) in fuzzy logic is by triangular norms (or in short t-norms), because of the desirable properties (monotonicity, associativity, symmetry and the presence of a neutral element) (Hájek, 1998). The four basic t-norms are (Klement et al., 2005): minimum t-norm, product t-norm, Łukasiewicz t-norm and drastic product. The least suitable is drastic product due to its very restrictive nature and non-continuity. The product t-norm and Łukasiewicz t-norm have a downward reinforcement property, whereas the minimum t-norm has the property of idempotency (Beliakov et al., 2007). For instance, when using the Łukasiewicz t-norm, the solution is greater than zero when both mandatory and quantified parts are significantly satisfied. We have two conjunctions in Eq. (5): among mandatory and between mandatory and quantified requirements. In addition, there exist conjunctive functions which do not meet all the axioms of t-norms, e.g. C(a,b)=av·bw (Beliakov et al., 2007; Hudec and Vuc˘etić, 2019), where v>1 and w>1 indicate the importance of predicates. Observe that for v=w=1 we get the product t-norm and for v<1, w<1 and v+w=1 we get the geometric mean. Although the other functions could be examined, this work is focused on the minimum t-norm.

An illustrative example is searching for suitable accommodation units. The atomic requirements can be low price, high safety rating, altitude above sea level around 1500 meters, low pollution, small population density, short distance to the nearest grocery store, high rating of the accommodation unit and the like. It is highly presumable that none of the evaluated units meets all predicates in a pure conjunctive way. However, the user might express that the low price and high safety ratings are mandatory requirements, whereas the others are optional.

3.3Evaluation of Nested Queries

This class of queries is convenient for the 1 : N relationships in a database such as district-municipality and customer-invoice. An example of a nested query condition might be as follows: select districts where the most of municipalities have high pollution and high number of respiratory diseases. The answer should be a list of districts which fully or partially meet the condition ranked downward from the best by the intensity of matching degrees.

The procedure for calculating validities is created straightforwardly as the extension of (2) in the following way (Hudec, 2016):

This method integrates the vertical aggregation explained in Section 2 and the horizontal one discussed in Section 3.1. Further, we may have a quantified condition on a higher hierarchical level and a quantified summary on a lower level, e.g. select districts where the most of {P1…Pn} are met and most of the respective municipalities have high value of Rg, where P1…Pn are atomic requirements posed on districts’attributes, whereas Rg is a requirement related to the municipalities’ attribute.

In the case of aggregating the conjunction of atomic predicates on the “1” side and the quantified condition on the “N” side, the formalization is as follows:

The right part of (7) aggregates on the set of entities U2 (1), while the left part aggregates on the set of attributes A1.

The next section is devoted to examples and to a discussion illustrating the suggested approach.

4.1Illustrative Examples

This part consists of the following three examples: (i) evaluation of the mandatory and optional atomic conditions, (ii) summarization of the particular attributes for all records and (iii) nested query for hierarchical data.

An example of aggregating the mandatory and optional atomic conditions

Hypothetical twelve accommodation units and matching degrees of ten atomic requirements relevant for a hypothetical guest are shown in Table 2. In the example, low price ( P1) and high safety rating ( P2) are mandatory requirements, whereas the majority of the remaining ones: altitude above the sea level around 1500 meters ( P3), low pollution ( P4), small population density ( P5), high rating of the accommodation unit ( P6), short distance to the nearest grocery store ( P7), high number of sunny days ( P8), short distance to the funicular ( P9) and high cleanness ( P10), should be met. The minimum t-norm was adopted for both conjunctions in (5), among the mandatory requirements and the aggregation with the quantified part.

When we consider all the requirements to be mandatory, then not a single unit is selected, since we can find a non-satisfied elementary predicate for all the units. The probability of the empty answer problem increases when the number of atomic requirements increase, as is the case in this example. Thus, the user either abandons the holiday (a less probable alternative), or relaxes the requirement. It can usually be realized by reducing the number of atomic conditions or relaxing some of the atomic predicates (e.g. for the predicate short distance the user will accept a bit longer distance). The option suggested in this work is the quantified relaxation. In the relaxed quantified condition, if the price or the safety is not satisfied, then the unit is rejected regardless of the degrees of satisfaction of the other requirements. A non-satisfied requirement from the set {P3,…,P10} is not a reason for rejection. In Table 2 we can see that the most suitable unit is u6, followed by u3. Although not a single unit fully satisfies all the expectations, two units are very close to meeting them, and therefore are the most preferable, although not the ideal alternatives.

An example of summarized information

Let us have the same data as in Table 2. A hypothetical manager in a tourism agency wishes to know whether the most of accommodation units have a high cleanness rate, and also whether the most of accommodation units have a short distance to the nearest grocery store. In this example, we have two summarization tasks related to one attribute each. Focus in the calculations is on the predicates high cleanness, P10 and short distance, P7, respectively, where the matching degrees are shown in respective columns.

By applying Eq. (2), where n=12 (number of entities) and parameters of the quantifier most of (Fig. 1) are 0.5 and 0.85 we obtain the validity of the sentence the most of accommodation units have a high cleanness rate equal to 0.94. The high truth value indicates that this sentence is accepted. However, the validity of the second sentence is 0 (the proportion 1n∑i=1nμP(xi) is equal to 0.46), and therefore this sentence is rejected.

Moreover, linguistic summaries are able to offer an alternative answer when the initial quantified sentence is of an insufficient validity (Hudec et al., 2018). The proportion of the afore evaluated second sentence (0.46) indicates that the most suitable quantifier is about half, and therefore the answer is not only that the validity of the initial sentence is zero, but we can provide an alternative summary: about half of accommodation units have short distance to the nearest grocery store. Similarly, the user can examine the remaining predicates of interest.

An example of a nested query

An analyst is interested in revealing highly attractive districts where the most of accommodation units have a low rating, i.e. districts, which might have a significant increase in the number of visits if the accommodation units improved their quality.

Let us have hypothetical accommodation units situated in four districts as is shown in Tables 3 and 4, where we have already calculated the matching degrees to atomic predicates. For instance, the attribute rating assumes values from the [0,100] interval. The predicate low rating is formalized by a L fuzzy set (see Fig. 2) with parameters m=10 and b=30. Accommodation unit u4 has an aggregated rating 13 and therefore belongs to the predicate low rating with the degree of 0.85. The matching degrees of the other units are calculated in the same way.

The analyst sees that the most problematic district is d1, followed by d3, whereas d2 meets this requirement very weakly and d4 is not a problematic district. So, the analyst has recognized the districts where the improvements in the quality of the accommodation units might bring a significant increase in the number of visits.

Similarly, the analyst can evaluate the other attributes of accommodation units such as distances to relevant points and prices, as well as the attributes of the districts, where the accommodation units are situated, such as pollution and safety.

4.2Discussion

The proposed approach has a high applicability potential. Quantification of optional requirements and their aggregation by the conjunction with the mandatory ones augments the database querying possibilities by providing further ways of expressing the users’ requirements in diverse tasks and mitigates the empty answer problem. This holds especially for the cases, when the users pose a higher number of atomic requirements, but are not sure which of them might be excluded in the case of an empty answer, or they are interested in seeing which entities satisfy at least the majority of optional requirements.

For these tasks, we adopted the equation for the calculation of the validities of quantified sentences (2) in such a way that instead of the proportion of entities that meet the summarizer, we have the proportion of atomic requirements met by the evaluated entities (4). Further, we aggregated this equation with the mandatory requirements by a conjunctive function (5). In this work, we adopted the minimum t-norm for conjunction. The future research work should be focused on examining the suitability of the other conjunctive functions between the mandatory and quantified part, and among the mandatory predicates.

The initial equation for linguistic summaries (2) remains suitable for expressing knowledge about the attribute in the entire data set. Less statistically literate users (e.g. domain experts on the business intelligence strategic dashboards, and general public on the official statistics data dissemination websites (Hudec et al., 2018; Schield, 2011) or the eInforming stage of smart cities (Terán et al., 2016)) might benefit especially from the summarization by short quantified sentences.

Business intelligence visualization considers strategic, operational and analytical dashboards (Vaisman and Zimányi, 2014). Strategic dashboard is a collection of multiple visual components (e.g. charts and key performance indicators) on a single view so the main messages can be monitored at a glance (Few, 2006). Hence, the main goal is to explain what is going on, not to explain the reasons for such a behaviour. Therefore, the linguistic summaries in their initial sense (2) are suitable for strategic dashboards. They are able to express the model of behaviour linguistically, and are especially suitable for the top-managers. Further, a pattern such as the most of visits from French speaking countries is in spring months cannot be easily interpreted graphically. As we see, the solutions of linguistically summarized sentences do not explain the reasons for such behaviour, but indicate what is going on and emphasize the most perspective or the most problematic cases.

Evaluation of entities by quantified aggregation (5) and (7) is suitable for analytical dashboards to support the tasks focusing on the recognition of the perspective or problematic entities by the conditions expressed via the natural language expressions in order to cover the users’ uncertainties regarding the conditions expressed by a larger number of requirements.

Users searching for the most suitable entities, like the accommodation units in the above-mentioned illustrative examples, and for the summarized information regarding the various aspects of our society (e.g. key statistical figures released by the national statistical institutes) might benefit from this approach. A promising application field is empowering the concept of smart cities by flexible summarizing of the developments in the city and by supporting the search for the most suitable entities such as dwelling units, the most acceptable locations and similar participants for the discussion groups. The next task for the future work will be the development of intuitive user-friendly interfaces.

We have used the sigma-counts method within the computing with words methodology for the construction of fuzzy predicates and relative quantifiers. Adjectives such as high and quantifiers such as most of are always expressed by increasing functions, regardless of their translation to other human languages. The same holds for the adjective low, which is always expressed by a decreasing function. We have also used methods for aggregating entities and attributes into compound user requirements.

The data are stored in databases as numbers that in many cases pretend a precision, since the real data are frequently not available as precise numbers, but they are more or less non-precise or fuzzy. The internal trustworthiness shows that the results are less sensitive to the factors such as a small imprecision in the data and the user’s linguistic requirements formalized as fuzzy sets. Fuzzy logic queries allow similar entities to be similarly evaluated, or to similarly contribute to the summaries. Next, the boundary conditions, the key requirements in the quantified aggregation, are satisfied (see Table 1, entities e6 and e7). The boundary condition is by definition satisfied for the linguistic summary (2) and the conjunction, and therefore it holds for (5) and (7). And the monotonicity (if the matching degree of one atomic predicate increases, whereas the other remains the same, the solution remains the same or increases) is a matter of direct verification.

External trustworthiness generalizes the other situations and data sets. If the user query contains a large number of elementary predicates for evaluation or a large number of entities for summarization, the limitation is the computational capacity. Mathematically, aggregation is not limited. When we consider, for instance, business intelligence dashboards, the set of attributes for evaluation is not large. Regarding the summarization on an increasing number of records, for example, in the fact tables in data warehouses, the proportion in (2), or the quantified part in (7) is a distributive operation (in fact, the addition is, and therefore, previous sum is used). So, we can build results on the previously calculated proportions.

Next, data incompleteness also occurs in databases (the value is simply unknown or inconvenient for the provider to offer). It influences the conjunction in the same way as in the classical SQL. In the quantified condition, it is not so problematic, because when we replace the missing value by the matching degree 0, we do not automatically exclude any records. Adjustment of the missingness-tolerant evaluation suggested for the logic scores of preferences (Dujmović, 2018) is a topic for the future research. Generally, we can assign values from 0 (full penalty for missingness) to 1 (full tolerance). This issue is more relevant to the conjunctive and quantified evaluation of attributes than to the summarization of a large number or records.