Decision Making with Incomplete Information

Recent Researches in Communications, Electrical & Computer Engineering

Decision Making with Incomplete Information SYLVIA ENCHEVA Stord/Haugesund University College Faculty of Technology, Business and Maritime Sciences Bjørnsonsg. 45, 5528 Haugesund NORWAY [email protected] Abstract: The research described in this paper aims at facilitating the process of drawing conclusions while working with incomplete information. The approach is based on formal concept analysis and six valued logic. The Armstrong axioms are further applied to obtain knowledge in cases with incomplete information. Key–Words: Formal concept analysis, incomplete information, logic

1 Introduction

2 Related Work

Missing information is often treated as a negative response. Some models suggest ways to treat contradictory and incomplete information but the problem of extracting knowledge from incomplete information is not very well addressed. ”Therefore it is useful to have tools around to obtain as much of ’certain’ knowledge as possible in as many situations as possible. And in cases of incomplete knowledge it is desirable to be able to ’measure’ what is missing”, [2].

Intelligent systems usually reason quite successfully when they operate with well defined domains. If a domain is partially unknown a system has to extract and process new knowledge with adequate reasoning techniques. A methodology addressing the issues of automatically acquiring knowledge in complex domains called automatic case elicitation is presented in [13]. The method relies on real-time trial and error interaction. Case-based reasoning solves new problems exploring solutions of similar past problems. In order to begin case-based reasoning needs a set of training examples. It then creates some general cases and a solution is finally presented based on a comparison between one of the obtained general cases and the current one. Inspired by the Aristotle writing on propositions about the future - namely those about events that are not already predetermined, Lukasiewicz has devised a three-valued calculus whose third value, 21 , is attached to propositions referring to future contingencies [10]. The third truth value can be construed as ’intermediate’ or ’neutral’ or ’indeterminate’ [14]. A three-valued logic, known as Kleene’s logic is developed in [9] and has three truth values, truth, unknown and false, where unknown indicates a state of partial vagueness.These truth values represent the states of a world that does not change. A brief overview of a six-valued logic, which is a generalized Kleene’s logic, has been first presented in [11]. The six-valued logic was described in more detail in [8]. The six-valued logic distinguishes two types of unknown knowledge values - permanently or eternally

According to the principle of bounded rationality a decision-maker in a real-world situation will never have all information necessary for making an optimal decision. In [2] authors suggest treatment of incomplete knowledge based on the theory of formal concept analysis, [6]. Involvement of the three valued Kleene’s logic [5] is briefly mentioned. This threevalued logic has three truth values, truth, unknown and false, where unknown indicates a state of partial vagueness. These truth values represent the states of a world that does not change. The Kleene’s logic might be useful if one operates with one source of information. If however information is provided by two sources one needs six truth values. In this paper we present an approach to extract knowledge from incomplete data by applying the theory of formal concept analysis, six valued logic and Armstrong axioms. The rest of the paper is organized as follows. Related work and supporting theory may be found in Section 2. The model of the proposed system is presented in Section 3. The paper ends with a conclusion in Section 4. ISBN: 978-960-474-286-8

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true

knowledge

false

true

contradiction

unknown

unknown

contradictory

unknown t

f

t

unknown

unknown

f

unknown

truth

false

Figure 1: Partial ordering of truth values

Figure 2: Degree of truth

unknown value and a value representing current lack of knowledge about a state [7]. Two kinds of negation, weak and strong negation are discussed in [17]. Weak negation or negation-asfailure refers to cases when it cannot be proved that a sentence is true. Strong negation or constructable falsity is used when the falsity of a sentence is directly established. Let P be a non-empty ordered set. If sup{x, y} and inf {x, y} exist for all x, y ∈ P , then P is called a lattice [3]. In a lattice illustrating partial ordering of knowledge values, the logical conjunction is identified with the meet operation and the logical disjunction with the join operation. The meaning of the elements true, false, unknown, unknownt, unknownf , contradiction is

contradiction by degree of truth is presented in Fig. 2. The logical conjunction is identified with the meet operation and logical disjunction with the join operation. For describing six-valued logic we use notations as in [8]. Thus • t denotes true - it is possible to prove the truth of the formula (but not its falsity) • f denotes false - it is possible to prove the falsity of the formula (but not its truth) • ⊥ denotes unknown - it is not possible to prove the truth or the falsity of the formula (there is not enough information) • ⊥t denotes unknownt - intermediate level of truth between ⊥ and t

• true - possible to prove the truth of the formula only

• ⊥f denotes unknownf - intermediate level of truth between ⊥ and f

• false - possible to prove the falsity of the formula only • unknown - not possible to prove the truth or the falsity of the formula

• ⊤ denotes contradiction - it is possible to prove both the truth and the falsity of the formula

• unknownt - intermediate level of truth between unknown and true

The six-valued logic distinguishes two types of unknown knowledge values - permanently or eternally unknown value and a value representing current lack of knowledge about a state [7]. The epistemic value of formula when it is known that the formula may take on the truth value t is denoted by ⊥t and by ⊥f when it is known that the formula may take on the truth value f. Truth tables for the six-valued logic as shown in [7] and [8] are presented in Table 1, Table 2, and Table 3. The ’։’ implication shown in Table 4 is not an implication in the traditional sense since it does not satisfy modus ponens and the deduction theorem. These principals however hold for ’։’ whenever the premises are not contradictory.

• unknownf - intermediate level of truth between unknown and false • contradiction - possible to prove both the truth and the falsity of the formula A lattice showing a partial ordering of the elements true, false, unknown, unknownt, unknownf , contradiction by degree of knowledge is presented in Fig. 1. The logical conjunction is identified with the meet operation and logical disjunction with the join operation. A lattice showing a partial ordering of the elements true, false, unknown, unknownt, unknownf , ISBN: 978-960-474-286-8

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Armstrong axioms for arbitrary sets of attributes A, B, C and D to be applied: If α → γ, then α ∪ β → γ If α → β and β ∪ γ → δ, then α ∪ γ → δ The Armstrong axioms, [1] are sound in that they generate only functional dependencies in the closure of a set of functional dependencies (denoted as F + ) when applied to that set (denoted as F ). They are also complete in that repeated application of these rules will generate all functional dependencies in the closure F + , [1].

Table 1: Truth table for the connective ’¬’ in sixvalued logic ¬ t f f t ⊤ ⊤ ⊥t ⊥f ⊥f ⊥t ⊥ ⊥

3 The Decision Process Table 2: Truth table for the connective ’∧’ in sixvalued logic ∧ t f ⊤ ⊥t ⊥f ⊥

t t f ⊤ ⊥t ⊥f ⊥

f f f f f f f

⊤ ⊤ f ⊤ ⊤ ⊥f ⊥f

⊥t ⊥t f ⊤ ⊥t ⊥f ⊥

⊥f ⊥f f ⊥f ⊥f ⊥f ⊥f

⊥ ⊥ f ⊥f ⊥ ⊥f ⊥

Table 3: Truth table for the connective ’∨’ in sixvalued logic ∨ t f ⊤ ⊥t ⊥f ⊥

t t t t t t t

f t f ⊤ ⊥t ⊥f ⊥

⊤ t ⊤ ⊤ ⊥t ⊤ ⊥t

⊥t t ⊥t ⊥t ⊥t ⊥t ⊥t

⊥f t ⊥f ⊤ ⊥t ⊥f ⊥

⊥ t ⊥ ⊥t ⊥t ⊥ ⊥

Table 4: The ’։’ implication ։ f ⊥f ⊤ ⊥ ⊥t t

f ⊤ ⊤ ⊤ ⊥f ⊥f f

⊥f ⊤ ⊤ ⊤ ⊥f ⊥f ⊥f

ISBN: 978-960-474-286-8

⊥ ⊤ ⊤ ⊤ ⊥ ⊥ ⊥

⊤ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤

⊥t ⊤ ⊤ ⊤ ⊥t ⊥t ⊥t

t ⊤ ⊤ ⊤ ⊥t ⊥t t

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The decision making process is based on students’ responses to Web-based tests. A test consists of two questions requiring understanding of a single term (skill, rule, etc.) or several terms. A test considering a single term is denoted by a and b or c while a test considering two terms is denoted by ab and ac or bc respectively. Response options to a single question can be correct, incorrect or left unanswered. Thus the possible outcomes of a test are: tt - two correct responses f f - two incorrect responses uu - neither of the two questions is answered tf - one correct and one incorrect response tu - one correct response and the other question is left unanswered f u - one incorrect response and the other question is left unanswered Questions in a single test are not ordered since they are assumed to have similar level of significance. Once students’ responses are registered in a database we apply Armstrong axioms in order to extract maximal knowledge in the presence of incomplete information. Responses to tests concerning ab, ac, and bc before and after application of Armstrong axioms are presented in Table 6 and Table 8 in terms of six valued logic. The last columns in both tables illustrate a summary of these responses after ∧ operation. Fig. 3 and Fig. 4 are knowledge lattices for six valued logic visualizing the truth values for objects S1, ..., S12 with respect to the tests ab, ac, bc. The Armstrong axioms are applied to obtain new knowledge from incomplete information, Table 5. The results can be seen in Table 7. Note that Table 7 shows cases where entrances with incomplete information like f. ex uu or f u are filled with precise data. The initial data from Table 5 related to S5 and S12 does not allow drawing of conclusions according to the Armstrong axioms. The ini-


Table 5: Responses to optional tests with information a b c ab ac S1 tt uu tt tt tf S2 tu tt tf uu tt S3 tu tu tu tt uu S4 tf tu uu tu tt S5 tu tf tf tf tu S6 f u tu f u tf f u S7 f f f u tu f u f u S8 f f uu f f f f tu S9 f u f f tf uu f f S10 f u f u f u f f uu S11 f u f u uu f u f f S12 f u tu f u f u tf

Table 7: Results concerning responses to optional tests with decreased incomplete information a b c ab ac bc • S1 tt tt tt tt tf tt • S2 tu tt tf tt tt tt S3 tu tu tu tt tt • tt S4 tf tu tt • tu tt tt S5 tu tf tf tf tu uu S6 f f • tu uf tf fu ff S7 f f f u tu f f • f u f f S8 f f ff • f f f f tu f f S9 f u f f tf ff • f f f f S10 f u f u f u f f ff • f f S11 f u f u ff • f u f f f f S12 fu tu f u f u tf tu

incomplete bc tt tt tt tt uu ff ff ff ff ff ff tu

Table 6: Results for responses to optional tests in terms of six valued logic based on Table 5 ab ac bc ∧ S1 t ⊤ t ⊤ S2 ⊥ t t ⊥ S3 t ⊥ t ⊥ S4 ⊥t t t ⊥t S5 ⊤ ⊥t ⊥ ⊥f S6 ⊤ ⊥f f f S7 ⊤ ⊤ f f S8 f ⊥t f f S9 ⊥ f f f S10 f ⊥ f f S11 ⊥f f f f S12 ⊥f ⊤ ⊥t ⊥f

Figure 4: Positioning of objects S1, ..., S12 knowledge lattice based on Table 8

tial data from Table 5 related to S6 and S7 is used to fill in missing responses to single questions and in the rest of the cases missing responses to single tests have been deduced. Bullets in Table 7 indicate entrances with decreased incomplete information. The six valued logic is afterwords applied to interpret unprecise and contradictory information with respect to the values of ab, ac and bc, Table 8. A comparison of Fig. 3 and Fig. 4 shows that after applying the Armstrong axioms and six valued logic for objects S1, ..., S12 and attributes ab, ac, bc, objects S2 and S3 obtain new truth values.

4 Conclusion Figure 3: Positioning of objects S1, ..., S12 in a knowledge lattice based on Table 6

ISBN: 978-960-474-286-8

Incomplete data can be handled by the present approach. Optional tests are often partially answered by students. At the same time they provide information that can be used for suggesting appropriate help. 112


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Table 8: Results for responses to optional tests in terms of six valued logic based on Table 7 ab ac bc ∧ S1 t ⊤ t ⊤ S2 t t t t S3 t t t t S4 ⊥t t t ⊥t S5 ⊤ ⊥t ⊥ ⊥f S6 ⊤ ⊥f f f S7 ⊤ ⊥f f f S8 f ⊥t f f S9 f f f f S10 f f f f S11 ⊥f f f f S12 ⊥f ⊤ ⊥t ⊥f

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