We present a new framework for modelling users preferences in a fuzzy setting. Starting with a formal fuzzy context, the user enters so-called attribute dependency formulas based on his priorities. The method then yields the “interesting” formal concepts, that is, interesting from the point of view of the user. Our approach is designed for compounded attributes, i.e., attributes which include more than one trait. In this paper, after studying some properties of the formulas, we start investigating the computation of non-redundant bases for them. Such bases are wishful for a better overview of the preferences.
if A ∩ N 6= ∅, then B ∩ N 6= ∅.
A formal concept (C, D) ∈ B(G, M, I) satisfies A ⊑ B if D |= A ⊑ B.
These formulas were the starting point of our work. However, we develop a different kind of AD formulas, namely some that are appropriate for compounded attributes, i.e., attributes which include more than one trait. For instance the notion “wealth” is a compounded attribute consisting of “investment” and “fluency”. A person who is wealthy has to have high values on both investment and fluency. We develop such formulas for the fuzzy setting and automatically obtain the crisp case by choosing L = {0, 1} for the residuated lattice.
Some proofs are omitted due to lack of space but can be found in [
The paper is structured as follows: In Section 2 we give brief introductions to Fuzzy Sets, Fuzzy Logic and Formal Fuzzy Concept Analysis. Section 3 contains our new framework. Concluding remarks and further research topics are given in the last section. 2 2.1
In this subsection we present some basics about fuzzy sets and fuzzy logic. The
interested reader may find more details for instance in [
A complete residuated lattice with (truth-stressing) hedge is an
algebra L := (L, ∧, ∨, ⊗, →,∗ , 0, 1) such that: (L, ∧, ∨, 0, 1) is a complete lattice;
(L, ⊗, 1) is a commutative monoid; 0 is the least and 1 the greatest element; the
adjointness property, i.e., a ⊗ b ≤ c ⇔ a ≤ b → c, holds for all a, b, c ∈ L.
The hedge ∗ is a unary operation on L satisfying the following conditions:
i) a∗ ≤ a; ii) (a → b)∗ ≤ a∗ → b∗; iii) a∗∗ = a∗; iv) Vi∈I ai∗ = (Vi∈I ai)∗; for
every a, b, ai ∈ L (i ∈ I). Elements of L are called truth degrees, ⊗ and → are
(truth functions of) “fuzzy conjunction” and “fuzzy implication”. The hedge ∗
is a (truth function of) logical connective “very true”, see [
A common choice of L is a structure with L = [
G¨odel: a ⊗ b := min(a, b) with a → b := Product: a ⊗ b := ab with a → b := 1, a ≤ b b, a b , 1, a ≤ b b/a, a b .
Typical examples for the hedge are the identity, i.e., a∗ := a for all a ∈ L, and the globalisation, i.e., a∗ := 0 for all a ∈ L \ {1} and a∗ := 1 if and only if a = 1.
Let L be the structure of truth degrees. A fuzzy set (L-set) A in a universe U is a mapping A : U → L, A(u) being interpreted as “the degree to which u belongs to A”. We denote by u ∈ A the fact that A(u) = 1. If U = {u1, . . . , un}, then A can be denoted by A = {a1/u1, . . . , an/un} meaning that A(ui) equals ai for each i ∈ {1, . . . , n}. Let LU denote the collection of all L-sets in U . The operations with L-sets are defined component-wise. For instance, the intersection of L-sets A, B ∈ LU is an L-set A∩B in U s. t. (A∩B)(u) = A(u)∧B(u) for each u ∈ U , etc. Binary fuzzy relations (L-relations) between X and Y can be thought of as L-sets in the universe X × Y . For A, B ∈ LU , the subsethood degree, which generalises the classical subsethood relation ⊆, is defined as S(A, B) :=
subset of B. In particular, we write A ⊆ B iff S(A, B) = 1.
Fuzzy closure operators were introduced in [
Definition 1. Define on a set Y two mappings C, κ : LY → LY satisfying A ⊆ C(A), S(A1, A2)∗ ≤ S(C(A1), C(A2)), C(A) = C(C(A)), κ(A) ⊆ A, S(A1, A2)∗ ≤ S(κ(A1), κ(A2)), κ(A) = κ(κ(A)), for every A, A1, A2 ∈ LY . Then, C is called an L∗-closure operator and κ an L∗-kernel operator on Y . A system S := {Aj ∈ LY | j ∈ J } is a L∗-closure system if for each A ∈ LU it holds that [ (S(A, Aj)∗ ⊗ Aj) ∈ S. j∈J (1) (2) (3) (4) (5) The system S is called an L∗-kernel system if for each A ∈ LU it holds that For the globalisation, (2) becomes
A1 ⊆ A2 =⇒ C(A1) ⊆ C(A2),
A1 ⊆ A2 =⇒ κ(A1) ⊆ κ(A2),
and (4) and (5) become Tj∈J,A⊆Aj Aj and Sj∈J,A⊆Aj Aj, respectively.
Theorem 1. ([
Formal Fuzzy Concept Analysis
In the following we give brief introductions to Formal Fuzzy Concept Analysis
[
A triple (G, M, I) is called a formal fuzzy context if I : G × M → L is an L-relation between the sets G and M and L is the support set of some residuated lattice. Elements from G and M are called objects and attributes, respectively. The L-relation I assigns to each g ∈ G and each m ∈ M the truth degree I(g, m) ∈ L to which the object g has the attribute m. For L-sets A ∈ LG and B ∈ LM , the derivation operators are defined by
A↑(m) := ^ (A(g)∗ → I(g, m)), B↓(g) :=
^ (B(m)∗ → I(g, m))
(6)
g∈G
m∈M
for g ∈ G and m ∈ M . Then, A↑(m) is the truth degree of the statement “m
is shared by all objects from A”, and B↓(g) is the truth degree of “g has all
attributes from B”. The operators ↑,↓ form a so-called Galois connection with
hedges ([
Now we are ready to present our new framework. Given a fuzzy formal context, the user obtains the “interesting” concepts after entering a sort of order on the groups of attributes, and fix the truth values for their relevance. Recall, we designed this kind of AD formulas for compounded attributes. Thus, this notion is not the fuzzy equivalent one of the formulas presented in Section 1. For a straightforward fuzzification of those formulas see Remark 1.
Definition 2. A fuzzy attribute-dependency formula (fAD) over a set M of attributes is an expression A ⊑ B, where A, B ∈ LM are L-sets of attributes. A ⊑ B is true in an L-set N ∈ LM for α, β ∈ L \ {0} and α ≤ β, written N |=α,β A ⊑ B, if the following condition is satisfied: if S(A, N ) ≥ α, then S(B, N ) ≥ β. (7) For an fAD formula or a set T of fAD formulas, the values α and β are called the thresholds of A ⊑ B and T . An L-concept (C, D) ∈ B(G, M, I) satisfies A ⊑ B if D |=α,β A ⊑ B.
For notational simplicity we will sometimes omit α and β from |=α,β provided they are clear from the setting.
The set of all formal concepts from B(G, M, I) that satisfy a given set T of fAD formulas is denoted by BT (G, M, I). We call BT (G, M, I) together with the restricted concept order the fuzzy concept lattice of (G, M, I) constrained by T and denote it by BT (G, M, I).
The fAD formulas permit a two-sided modelling of the extracted L-concepts. On the one hand, α and β provide the thresholds to which an intent has to contain all elements of A and B. On the other hand, the truth degrees of the elements contained in A and B fix the thresholds to which we want the attributes to be contained in the intent of a concept satisfying the fAD formula. We will illustrate this fact in the forthcoming example.
In applications it is particularly useful to associate to the truth values of a residuated lattice L a Likert scale L. This allows the user to have a better understanding of the truth values. For instance let L = {0, 0.25, 0.5, 0.75, 1} be the support set of some residuated lattice with the associated Likert scale L = {not important, less important, important, very important, most important}, i.e, 0 =not important, 0.25 =less important, etc.
Example 1. Consider the fuzzy context given in Figure 1. It represents the evalgood team player
adaptive towards new confidential computer skills e v i t a r o b a ll o c : a 1 0 2 1 3 0.5 4 1 5 0 6 1 7 0 e v i t a n i m i r c s i d t o n : b 0.5 1 0.5 0.5 0.5 1 0.5 good organisational skills
i:tteeecaagnnmmm ll:rrseeoovbpdm lilii:ttceaaagykhnnn 0.5 1 1 1 1 1 0.5 1 1 1 1 1 0 0.5 0.5 0.5 1 1 0 0 0.5 t n e m n o r i v n e :f 0 1 0 0 0 1 0 t n e m n g i s s a : g 0 1 1 1 0 1 0 y t i r o i r p : h 0.5 1 1 1 0.5 1 0.5 uation of the employees of a small business regarding some qualities. Here each quality is compounded of two or more traits. For instance, an employee is a “good team player” if he/she is collaborative and not discriminative. The context has 44 L-concepts with the G¨odel logic which are far too many to be analysed by a busy manager. The manager however knows how good or bad the employees do their jobs and he is interested more in their collaboration than their organisational skills and more in their adaptivity than in their confidentiality. Therefore, he chooses the following two fAD formulas
{0.5/c, d, e} ⊑ {a, b} and {0.5/i, 0.5/j} ⊑ {f, g, h}, with α = 0.5 and β = 1. Then, he obtains 11 L-concepts which are given in Figure 2.
The manager realises that the company does neither send its employees to business trips nor to other companies and the employees should know their priorities. Therefore, he changes the second fAD formula into {0.5/i, 0.5/j} ⊑ {g, 0.5/h}. (8) (9) 1 2 3 4 5 6 7 8 9 10 11 1
2 Obviously the concepts from the first fAD formulas are a subset of the concepts from the second fAD formulas. The second couple of formulas yields 16 concepts, namely those from Figure 2 and 3.
1 2 3 Extent 4 5 6 7 a b c
Intent d e f g h i j k l
Proof. Clearly, BT (G, M, I) ⊆ B(G, M, I) and B(G, M, I) with the restricted concept order, is a partially ordered subset of B(G, M, I). Further note that BT (G, M, I) is bounded from below because the least L-concept of B(G, M, I) is (M ↓, M ), concept which is compatible with every fAD formula. Now, we have to show that BT (G, M, I) is closed under arbitrary suprema in B(G, M, I). To this end let (Aj , Bj ) ∈ BT (G, M, I) (j ∈ J ) be L-concepts. For any fAD formula A ⊑ B ∈ T , we have Bj |= A ⊑ B for every j ∈ J . Now, if there is j ∈ J such that Bj (a) < α for some a ∈ A, then ∩j∈J Bj (a) < α and we are done because then ∩j∈J Bj |= A ⊑ B. Contrary, if for all j ∈ J and all a ∈ A we have Bj (a) ≥ α, then ∩j∈J Bj (a) ≥ α for all a ∈ A. Since Bj |= A ⊑ B holds for all j ∈ J , we also have that Bj (b) ≥ β for all j ∈ J and b ∈ B. Due to the same argument as before, it holds ∩j∈J Bj (b) ≥ β for all b ∈ B and hence we have ∩j∈J Bj |= A ⊑ B, showing that BT (G, M, I) is closed under arbitrary suprema. ⊔⊓ In general BT (G, M, I) is not closed under arbitrary infima in B(G, M, I).
The fAD formulas are entered by the user. Thus, the chance that these formulas are redundant is very high. Wishful thinking suggests to have a set of non-redundant formulas because these are then easier to follow and to modify. Therefore, in the following we will develop methods for removing such redundancies.
Definition 3. An L-set N ∈ LM is a model of a set T of fAD formulas if we have N |= A ⊑ B for each A ⊑ B ∈ T . Let Mod(T ) denote the set of all models of T , i.e.,
Mod(T ) := {N ∈ LY | N |= A ⊑ B for each A ⊑ B ∈ T }.
An fAD formula A ⊑ B follows semantically from T , written T |= A ⊑ B, if for each N ∈ Mod(T ), we have N |= A ⊑ B.
Lemma 1. i) N |= A ⊑ {l1/y1, . . . , ln/yn} iff N |= A ⊑ {li/yi} holds for each i ∈ {1, . . . , n}. ii) For each set T of fAD formulas and each fAD formula ϕ, we have T |= ϕ iff ⌊T ⌋ |= ϕ, where ⌊T ⌋ := {A ⊑ {l/y} | A ⊑ B ∈ T and B(y) = l}.
Proof. i) If S(A, N ) < α, then we are done. Therefore, suppose S(A, N ) ≥ α.
have N ({li/yi}) ≥ β for all i ∈ {1, . . . , n} and thus N |= A ⊑ {li/yi} for all i ∈ {1, . . . , n}. ii) Follows by i), omitted due to lack of space. ⊔⊓
Thanks to Lemma 1 we may merge fAD formulas with the same left-hand side into a single fAD formula. The new formula is true in a model if and only if all its component fAD formulas are true in that model. This lemma allows us also to test semantic entailment in fAD formulas A ⊑ {l/y} rather than on the whole A ⊑ B.
In the following we will study the connection between the models of fAD formulas and L∗-closure systems. It will turn out that any L∗-closure system can be described by a set of fAD formulas.
Proposition 2. Let T be a set of fAD formulas. Then, Mod(T ) is an L∗-closure system with ∗ being the globalisation.
Proof. Let {Nj ∈ Mod(T ) | j ∈ J }. We will be showing that Mod(T ) is closed under arbitrary intersection, i.e., Tj∈J Nj is a model of T . For any fAD formula A ⊑ B ∈ T we have Nj |= A ⊑ B for every j ∈ J . Now, if there is j ∈ J such that Nj (a) < α for some a ∈ A, then ∩j∈J Nj (a) < α and we are done because then ∩j∈J Nj |= A ⊑ B. Contrary, if for all j ∈ J and all a ∈ A we have Nj (a) ≥ α, then ∩j∈J Nj (a) ≥ α for all a ∈ A. Since Nj |= A ⊑ B for all j ∈ J holds, we also have that Nj (b) ≥ β for all j ∈ J and b ∈ B. Due to the same argument as before, ∩j∈J Nj (b) ≥ β for all b ∈ B and hence we have ∩j∈J Nj |= A ⊑ B, showing that Mod(T ) is a closed under arbitrary intersection.
Mod(T ) is a closed under arbitrary intersection and due to Theorem 1 we just have to show that for any N ∈ Mod(T ) and any a ∈ L, a∗ → N is a model of T . However, this condition only holds if ∗ is the globalisation because, then we have
S(A, a∗ → N ) = a∗ → S(A, N ) =
1, a = 0,
S(A, N ), a = 1,
i.e., a∗ → N trivially satisfies any fAD formula if a = 0 or we do not gain
anything new to N in the case that a = 1. ⊔⊓
Remark 1. One may argue that due to Proposition 2 the fAD formulas are not
strong enough. However, we consider that this is not the case. In the crisp
setting, the AD formulas form a kernel system, and due to the connection between
AD formulas and attribute implications (for details see [
if A ∩ N α-true, then B ∩ N β-true, where an L-set X ∩ Y ∈ LM is α-true if there is at least one attribute m ∈ M such that (X ∩ Y )(m) ≥ α, where α ∈ L. We need the thresholds in order to ensure that the obtained concepts are indeed relevant. Now, the models of such formulas form a crisp like kernel system which can be shown in an analogous way to Proposition 2.
Lemma 2. For any L∗-closure system S in M there is a set T of fAD formulas over M such that S = Mod(T ).
Proof. Define a set T of fAD formulas by T := {A ⊑ CS (A) | A ∈ LM }, where
α = β = 1. Let N ∈ S, i.e., N = CS (N ). We have to show that N is a model of T , thus let N |= A ⊑ CS (A) for every A ⊑ CS (A) ∈ T . If S(A, N ) < 1, then N |= A ⊑ CS (A) and we are done. Now take S(A, N ) ≥ 1, i.e., A ⊆ N . Since CS is a closure operator we have CS (A) ⊆ CS (N ) = N , hence S(CS (A), N ) ≥ 1, i.e., N |= A ⊑ CS (A). Thus, N is a model of T and we have the first inclusion, namely S ⊆ Mod(T ). For the converse inclusion let N ∈ Mod(T ). We have if S(N, N ) ≥ 1, then S(CS (N ), N ) ≥ 1, where S(N, N ) ≥ 1 obviously holds and since N is a model of T we must also have S(CS (N ), N ) ≥ 1 yielding that N = CS (N ), i.e., N ∈ S and hence Mod(T ) ⊆ S. ⊔⊓
According to Proposition 2, Mod(T ) is an L∗-closure system, and therefore there must exist an L∗-closure operator CMod(T ) : LM → LM such that N = CMod(T )(N ) if and only if N ∈ Mod(T ). Hence, by definition CMod(T )(N ) is the least model in Mod(T ) which contains N . This definition of the L∗-closure operator does not provide a useful method to compute the closure of a given N . First, because one has to iterate over all models in Mod(T ) and second, such a iteration may be impossible if L is infinite because then Mod(T ) is infinite.
Similarly to (fuzzy) attribute implications we proceed as follows: For any set T of fAD formulas and for any L-set N ∈ LM of attributes, we define an L-set N T ∈ LM of attributes as follows:
N T := N ∪ [{β ⊗ B | A ⊑ B ∈ T, S(A, N ) ≥ α}.
Further, we define an L-set N Tn ∈ LY of attributes for each non-negative integer by (10) (11) (12) ⊔⊓
N Tn :=
N if n = 0, (N Tn−1 )T if n ≥ 1.
∞ clT(N ) := [ N Tn .
n=0 Proposition 3. For each N ∈ Mod(T ) we have clT(N ) = N .
Proof. Omitted due to lack of space.
The next lemma shows that the L∗-closure operator defined on the models of T coincides with the clT-operator defined in (12).
Lemma 3. Let T be a set of fAD formulas over M . Further let both M and L be finite. Then, clT is an L∗-closure operator such that for each N ∈ LM , CMod(T )(N ) = clT(N ).
the definition of clT we have N ⊆ clT(N ). We still have to show that clT(N ) belongs to Mod(T ) and that clT(N ) is the least model containing N . First of all note that the finiteness of L and M imply that LM is finite and that there exists a non-negative integer k such that clT(N ) = N Tk , where N Tk is given by (11).
We still have to show that clT(N ) ∈ Mod(T ), i.e., for any A ⊑ B ∈ T we have clT(N ) |= A ⊑ B. Suppose that S(A, clT(N )) ≥ α. Then, clT(N ) = N ∪ {β ⊗ B}. Obviously, S(B, N ∪{β⊗B}) ≥ β, proving that clT(N ) ∈ Mod(T ) which contains N . For any X ∈ Mod(T ) such that N ⊆ X we have to show that clT(N ) ⊆ X. This easily follows by the properties of closure operators and by Proposition 3.
In fact, we have that clT(N ) ⊆ clT(X) = X. ⊔⊓
Based on the previous result we present an algorithm for the computation of the closure CMod(T )(N ) of a fuzzy attribute set N ∈ LM provided L and M are finite.
1 repeat 2 take A ⊑ B ∈ T such that S(A, N ) ≥ α and S(B, N ) < β; 3 set N to N ∪ {β ⊗ B}; 4 until forall A ⊑ B ∈ T , (S(A, N ) < α) or (S(A, N ) ≥ α and S(B, N ) ≥ β); 5 return N
If we choose more restrictive values for α and β, we have the following connection between fuzzy implications and fAD formulas: Lemma 4. Let Imp(T ) := {A ⇒ B | for all A ⊑ B ∈ T } and T be a set of fAD formulas. If we choose α = β = 1, then the following holds
Mod(Imp(T )) ⊆ Mod(T ).
Proof. Omitted due to lack of space.
Definition 4. Two sets T1 and T2 of fAD formulas are called equivalent, written T1 ≡ T2, if for each ϕ1 ∈ T1 and ϕ2 ∈ T2 we have T1 |= ϕ2 and T2 |= ϕ1. Lemma 5. Let T1 and T2 be sets of fAD formulas. Then, the following are equivalent: i) Mod(T1) = Mod(T2), ii) For any fAD formula ϕ we have T1 |= ϕ ⇐⇒ T2 |= ϕ, iii) T1 ≡ T2.
Proof. Omitted due to lack of space.
Now we are prepared to introduce non-redundant bases.
Definition 5. A set T1 of fAD formulas is called a non-redundant base of T if T ≡ T1 and there is no T2 ⊂ T1 with T2 ≡ T . A set T1 of fAD formulas is called a minimal base of T if T ≡ T1 and for each T2 such that T ≡ T2, we have |T1| ≤ |T2|.
Obviously, if T1 is a minimal base of T , then T1 is a non-redundant base of T . The converse implication is not true in general.
For a given set T of fAD formulas we may compute a non-redundant base as follows: First note that if T1 := T \ {A ⊑ B} and T1 |= A ⊑ B, then T ≡ T1. We may then remove fAD formulas A ⊑ B from T step-by-step until there is no T1 ⊂ T such that T1 ≡ T . The computation of a non-redundant base with this method is quite laborious. In what follows, we present another connection between fuzzy attribute implications and fAD formulas which will considerably simplify this task. ⊔⊓ ⊔⊓ Lemma 6. Let T be a set of fAD formulas. We have Mod(T ) = Mod(Imp(T ∗)), where
Imp(T ∗) := {α ⊗ A ⇒ β ⊗ B | ∀A ⊑ B ∈ T, α, β thresholds of T }. (13) Proof. First note that an L-set N ∈ LM is a model of an attribute implication A ⇒ B in a fuzzy setting if ||A ⇒ B||N := S(A, N )∗ → S(B, N ) = 1.
Let N ∈ Mod(T ) and A ⊑ B ∈ T . We have two cases: i) S(A, N ) ≥ α and S(B, N ) ≥ β both hold. Consider its first part. Then, for every attribute m ∈ M , we have A(m) → N (m) ≥ α which by the adjointness property gives us α ⊗ A(m) ≤ N (m) and therefore S(α ⊗ A, N ) = 1 and thus S(β ⊗ B, N ) = 1. Hence, α ⊗ A ⇒ β ⊗ B||N = S(α ⊗ A, N )∗ → S(β ⊗ B, N ) = 1∗ → 1 = 1. ii) We have S(A, N ) < α which is equivalent to S(α ⊗ A, N ) < 1. Therefore, ||α ⊗ B ⇒ β ⊗ B||N = S(α ⊗ A, N )∗ → S(β ⊗ B, N ) = 0 → S(β ⊗ B, N ) = 1. Cases i) and ii) show that N is a model of Imp(T ∗).
For the converse inclusion let N ∈ Mod(Imp(T ∗)). Then, we have ||α ⊗ A ⇒ β ⊗ B||N = S(α ⊗ A, N )∗ → S(β ⊗ B, N ) = 1 (14) for any fuzzy implication A ⇒ B ∈ Imp(T ∗). Equation 14 holds if and only if one of the following situations appears: i) (S(α ⊗ A, N )∗ = 1 and S(β ⊗ B, N ) = 1) ⇐⇒ (S(A, N ) ≥ 1 and S(B, N ) ≥ 1). ii) S(α ⊗ A, N )∗ = 0 is equivalent to S(α ⊗ A, N ) < 1 which further is equivalent to S(A, N ) < α. The two cases prove N |=α,β A ⊑ B. ⊔⊓ Thus, a fAD formula A ⊑ B with thresholds α, β is satisfied if and only if the corresponding implication from Imp(T ∗), where ∗ is the globalisation, holds with truth value 1. With this link between fAD formulas and fuzzy attribute implications we may easily compute a minimal base for any set T of fAD formulas. First, we build the set Imp(T ∗) associated to T as given in (13). For this set we compute a minimal base of attribute implications BT ∗ . Finally, from BT ∗ we obtain a minimal base of fAD formulas for T by
BT := {A
⊑ B \ A | A ⇒ B ∈ BT ∗ }, where
A := _{C ∈ LM | α ⊗ C = α ⊗ A}, B := _{D ∈ LM | α ⊗ D = α ⊗ B}.
Note that, unlike the crisp case, in the fuzzy setting a formal context does
not have to have a unique stem base (see [
With the possibility of computing a non-redundant base, the user may review his choices and alter them conveniently. 4
We have presented two generalisations of crisp attribute dependency formulas into the fuzzy setting. Both variants allow the user to define a sort of order on the attributes. According to the entered constraints the user sees just a part of the concept lattice, namely the one containing the relevant concepts for him/her. Due to lack of space, the straightforward generalisation was just briefly presented in Remark 1.
The second approach was designed for compound attributes, such which incorporate more than one quality or specification. This time we required from the “interesting” concepts that if they contain all less important attributes with a threshold α, then they should also contain all more important attributes with a threshold β, were the thresholds are truth degrees such that α ≤ β. For such formulas, besides showing some of their properties, we focused mainly on the computation of their non-redundant bases.
Future work will focus on applying the method on real-world data and evaluating the outcomes by experts. Another research topic is the exploration of fAD formulas, where the user may alter the choices made without starting from scratch each time.