Distances, Norms and Error Propagation in Idempotent Semirings

Abstract

Error propagation and perturbation theory are well-investigated areas of mathematics dealing with the influence of errors and perturbations of input quantities on output quantities. However, these methods are restricted to quantities relying on real numbers under traditional addition and multiplication. We aim to present first steps of an analogous theory on idempotent semirings, so we define distances and norms on idempotent semirings and matrices over them. These concepts are used to derive inequalities characterizing the influence of changes in the input quantities on the output quantities of some often used semiring expressions.

A Deferred Proofs and Remarks

Example 3.3 and Continuation. First we show additive subdistributivity of ${{\varvec{d}}}_{s}$ which means $|\min \{x_1,x_2\}-\min \{y_1,y_2\}|\le \max \{|x_1-y_1|,|x_2-y_2|\}$. W.l.o.g. we assume $x_1\le x_2$ and hence $|\min \{x_1,x_2\}-\min \{y_1,y_2\}|=|x_1-\min \{y_1,y_2\}|$. Now we distinguish the following cases:

• 1. $y_1\le y_2$: Then we have $\min \{y_1,y_2\}=y_1$ and hence $|\min \{x_1,x_2\}-\min \{y_1,y_2\}|=|x_1-y_1|$. Now the claim is obvious due to $|x_1-y_1|\le \max \{|x_1-y_1|,|x_2-y_2|\}$.

• 2. $y_2<y_1$: Then we have $\min \{y_1,y_2\}=y_2$ and hence $|\min \{x_1,x_2\}-\min \{y_1,y_2\}|=|x_1-y_2|$. We distinguish the following cases:

  • (2a) $y_2\le x_1$: By assumption we have $x_1\le x_2$ and hence $|x_1-y_2|\le |x_2-y_2|$. Together with $|x_2-y_2|\le \max \{|x_1-y_1|,|x_2-y_2|\}$ we obtain the claim.

  • (2b) $y_2>x_1$: By assumption we have $y_1>y_2$ and hence $|x_1-y_2|<|x_1-y_1|$, and the claim follows analogously to above.

For multiplicative subdistributivity we have to show $|(x_1+x_2)-(y_1+y_2)|\le |x_1-y_1|+|x_2-y_2|$. But this is an easy consequence of elementary calculus and the triangle inequality $|x+y|\le |x|+|y|$.

To show order preservation we have to prove that $x\ge y$ and $y\ge z$ imply $|x-y|\le |x-z|$ (note that the natural order of coincides with $\ge$). The proof of this obvious property and strictness is left to the reader.$\square$

Example 3.4 and Continuation. Considering additive subdistributivity of ${{\varvec{d}}}_{m}$ we have to show $|\max \{x_1,x_2\}-\max \{y_1,y_2\}|\le \max \{|x_1-y_1|,|x_2-y_2|\}$. Here we assume w.l.o.g. $x_2\le x_1$ and hence $|\max \{x_1,x_2\}-\max \{y_1,y_2\}|=|x_1-\max \{y_1,y_2\}|$. Now we have the following cases:

• 1. $y_1\ge y_2$: Then we have $\max \{y_1,y_2\}=y_1$ and hence $|\max \{x_1,x_2\}-\max \{y_1,y_2\}|=|x_1-y_1|$. This implies the claim due to $|x_1-y_1|\le \max \{|x_1-y_1|,|x_2-y_2|\}$.

• 2. $y_1<y_2$: Then we have $\max \{y_1,y_2\}=y_2$ and hence $|\max \{x_1,x_2\}-\max \{y_1,y_2\}|=|x_1-y_2|$. Again, we distinguish two cases:

  • (2a) $y_2\le x_1$: Then we have $y_1<y_2\le x_1$ and hence $|x_1-y_2|<|x_1-y_1|$, and $|x_1-y_1|\le \max \{|x_1-y_1|,|x_2-y_2|\}$ implies the claim.

  • (2b) $y_2>x_1$: Recalling the assumption $x_2\le x_1$ we have $|x_1-y_2|\le |x_2-y_2|$; here $|x_2-y_2|\le \max \{|x_1-y_1|,|x_2-y_2|\}$ implies the claim.

This shows additive subdistributivity; multiplicative subdistributivity corresponds to additive subdistributivity from Example 3.3. Order preservation and strictness are also left to the reader.$\square$

Example 3.5 and continuations First we show additive subdistributivity of ${{\varvec{d}}}_{w}$ via the inclusion

$$\begin{aligned} (A\cup B)\triangle (C\cup D)\subseteq (A\triangle C)\cup (B\triangle D). \end{aligned}$$

(1)

To this purpose, we expand the left side of Inequation 1 and obtain

$$\begin{aligned} ((A\cup B)\cap \overline{(C\cup D)})\cup ((C\cup D)\cap \overline{(A\cup B)}). \end{aligned}$$

(2)

Analogously, its right side can be rewritten as

$$\begin{aligned} (A\cap \overline{C})\cup (C\cap \overline{A})\cup (B\cap \overline{D})\cup (D\cap \overline{B}). \end{aligned}$$

(3)

Due to symmetry of Expressions 2 and 3 in the involved variables, it suffices to show that $(A\cup B)\cap \overline{(C\cup D)}$ is contained in Expression 3. However, $(A\cup B)\cap \overline{(C\cup D)}$ equals $(A\cap \overline{(C\cup D)})\cup (B\cap \overline{(C\cup D)})$, and $A\cap \overline{(C\cup D)}$ is contained in $A\cap \overline{C}$ (and hence in 3) due to antitony of the complement. An analogous argument shows that $(B\cap \overline{(C\cup D)})$ is contained in 3.

Strictness and order preservation are here a simple consequences of set theory so we give an example that this distance is not multiplicative. To this purpose, we consider the three languages $L_1=_{def}\{a\}$, $L_2=_{def}\{b\}$ and $L_3=_{def}\{c\}$. Then we have $(L_1\cdot L_2)\triangle (L_1\cdot L_2)= \{ab,ac\}\nsubseteq \emptyset = (L_1\triangle L_1)\cdot (L_2\triangle L_3)$.

For norm-distributivity of ${{\varvec{d}}}_{w}$ we first note that its induced norm corresponds to the identity (due to $\emptyset \triangle L=L$) so we have to show $(L_1L_2)\triangle (L_1L_3)\subseteq L_1(L_2\triangle L_3)$ (for better readability, we omit the $\cdot$ for concatenation). By expanding the symmetric difference and distributivity of concatenation over union this leads to the inclusion $(L_1L_2\backslash L_1L_3)\cup (L_1L_3\backslash L_1L_2)\subseteq L_1(L_2\backslash L_3)\cup L_1(L_3\backslash L_2)$. Obviously, it suffices to show $L_1L_2\backslash L_1L_3\subseteq L_1(L_2\backslash L_3)$, so we reason as follows:

Additivity, strictness and order preservation of the -norm $|\cdot |$ on $\mathbf m LAN_{\varSigma }^{\mathbf{fin }}$ follow from basic facts about set cardinality. For multiplicative subdistributivity we consider two finite languages $L_1$ and $L_2$ and note that the mapping $\mathbf{concat }:L_1\times L_2\rightarrow L_1\cdot L_2$, defined by $\mathbf{concat }((w_1,w_2))=_{def}w_1\cdot w_2$ is surjective. Now we have $|L_1\cdot L_2|\le |L_1\times L_2|= |L_1|\cdot |L_2|$ by surjectivity of $\mathbf{concat }$ and elementary combinatorics. All properties of ${{\varvec{d}}}_{N}$ follow now from Theorem 4.6.$\square$

Example 4.3. Recall that a fuzzy relation according to [20] is a mapping $\alpha :X\times X\rightarrow [0,1]$ for some set X. Addition of two fuzzy relations $\alpha$ and $\beta$ is defined by $(\alpha \oplus \beta )(x_1,x_2)=_{def}\max \{\alpha (x_1,x_2),\beta (x_1,x_2)\}$ and multiplication by $(\alpha \odot \beta )(x_1,x_2)=_{def}\max _{x_3\in X}\min \{\alpha (x_1,x_3),\beta (x_3,x_2)\}$ whereas the cardinality $|\alpha |$ of a fuzzy relation $\alpha$ is given by $|\alpha |=_{def}\sum \limits _{x_1,x_2\in X}\alpha (x_1,x_2)$.

Consider now the set $X=\{x_1,x_2\}$ and the two fuzzy relations $\alpha$ and $\beta$ with $\alpha (x_1,x_2)= 1= \beta (x_2,x_1)$ and $\alpha (x_i,x_j)= 0= \beta (x_i,x_j)$ otherwise. Then $\alpha \oplus \beta$ is given by $(\alpha \oplus \beta )(x_1,x_2)= 1= (\alpha \oplus \beta )(x_2,x_1)$ and $(\alpha \oplus \beta )(x_1,x_1)= 0= (\alpha \oplus \beta )(x_2,x_2)$. Hence we have $|\alpha \oplus \beta |=2$ but on the other hand $\max \{|\alpha |,|\beta |\}= \max \{1,1\}= 1$, so the fuzzy cardinality is no additive -norm.

Writing finite fuzzy relations in a canonical way as matrices we consider now the the fuzzy relations $\alpha = \begin{pmatrix} 0 &{} 0 &{} 0 \\ 0.9 &{} 0 &{} 0\\ 0.9 &{} 0 &{} 0 \end{pmatrix}$ and $\beta = \begin{pmatrix} 0.9 &{} 0.9 &{} 0.9 \\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 \end{pmatrix}$. Then we have $|\alpha |+|\beta |=4.5$ but also $\alpha \odot \beta = \begin{pmatrix} 0 &{} 0 &{} 0 \\ 0.9 &{} 0.9 &{} 0.9\\ 0.9 &{} 0.9 &{} 0.9 \end{pmatrix}$ and hence $|\alpha \odot \beta |=5.4$ so the fuzzy cardinality is no multiplicative -norm either.

However, the fuzzy cardinality is a strict and order preserving norm which can be easily seen by isotony of addition and the equivalence $\sum \limits _{x\in X}x=0\Leftrightarrow \forall x\in X:x=0$, provided the fact $x\ge 0$ for all $x\in X$ (note that the range of a fuzzy relation is the interval [0, 1]).$\square$