Knowledge for tomorrow

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1 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück ÖÓÐ Ò º ÐÙ ÐÖº Deutsches Zentrum für Luft- und Raumfahrt Groningen, 28nd October 2018 Knowledge for tomorrow

2 Slide 1 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Applications of Idempotent Semirings Graph Theory - Berghammer, Stucke and Winter (RAMiCS 2015) - Brunet, Pous and Stucke (ITP 2016) - Glück (RAMiCS 2017) - Guttmann (RAMiCS 2017) Energy Optimization (Ésik, Fahrenberg, Leagy and Quaas at ATVA 2013) Language Problems (Backhouse, JLAP 2006) Fuzzy Logic (Kawahara, Furusawa and Winter at Various Occasions) Program Verification and Analysis - Armstrong, Struth and Weber (ITP 2013) - Glück and Krebs (RAMiCS 2015) - Michels, Joosten, Joosten, van der Woude (RAMiCS 2011) - Oliveira (JLAP 2014) Database Theory (Litak, Mikulás and Hiddders (RAMiCS 2014)) Tropical Optimization (Krivulin, Various Occasions)

3 Slide 2 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Alas, used data my be defective by - errors in measurement - estimated data - human shortcomings Classical mathematics/physics have error propagation numerical/statistical methods However, no such tools are known for idempotent semirings!

4 Slide 3 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Definition Idempotent Semiring Definition An idempotent semiring is a structure S s = (M s + s 0 s s 1 s) with internal operations + s and s and constants 0 s and 1 s in M s such that + s is commutative, associative and idempotent with neutral element 0 s, s is associative with neutral element 1 s and annihilator 0 s, and s distributes from both sides over + s. Ú s with x Ú s y df x + s y = y is the natural order of ss. Examples: tropical semiring Ê min + 0 = def (Ê 0 ½ min ½ + 0) max-min semiring Ê max min 0 = def (Ê 0 ½ max 0 min ½) semiring of finite languages, LAN fin Σ = def (Σ fin )

5 Slide 4 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Definition Measure System Definition A measure sysem is a structure S m = (M m + m 0 m m Ú m) with internal operations + m and m and a constant 0 m such that + m is commutative and associative with neutral element 0 m, m is associative and distributes from both sides over + m, Ú m is an order on M m with least element 0 m, and + m and Ú m are isotone in both arguments wrt. Ú m. No annihilation by 0 m, no neutral element of m! Examples: mê max + 0 = def (Ê 0 max 0 + ) mê max max 0 = def (Ê 0 ½ max 0 max ) mæ + 0 = def (Æ ) mlan fin Σ = def (Σ fin )

6 Slide 5 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Definition Distance Definition Given an idempotent semiring S s = (M s + s 0 s s 1 s) and a measure system S m = (M m + m 0 m m Ú m) we call a mapping d : M s M s M m an S m-distance on S s. It is called additive, if d(x 1 + s x 2 y 1 + s y 2 ) Ú m d(x 1 y 1 )+ m d(x 2 y 2 ), multiplicative, if d(x 1 s x 2 y 1 s y 2 ) Ú m d(x 1 y 1 ) m d(x 2 y 2 ), order preserving, if x Ú s y y Ú s z µ d(x y) Ú m d(x z), and strict, if d(x y) = 0 m x = y holds. A complete distance is an additive, multiplicative, order preserving and strict distance. Examples: d s(x y) = def x y is a complete mê max + 0 -distance on Ê min + 0. d m(x y) = def x y is a complete mê max max 0 -distance on Ê max min 0. d w(l 1 L 2 ) = def L 1 L 2 is an additive, order preserving and strict (but no multiplicative) mlanσ fin -distance on LANfin Σ.

7 Slide 6 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Distance Properties Lemma Let d be a complete S m-distance on S s. Then the following properties hold for all x y z ¾ M m, l n ¾ Æ 0 and mappings f g h : M s M s: n n n d( Σ x i i=1 Σ y i ) Ú m i=1 Σ d(x i y i ) i=1 d(x y + s z) Ú m d(x y)+ m d(x z) d(x y) Ú m d(x z) µ d(x n y n ) Ú m d(x z) n n n n d(x y) Ú m d(x z) µ d( Σ x i i=l Σ y i ) Ú m i=l Σ d(x z) i i=l If f and g are isotone with respect to Ú s and f(x) Ú s g(x) Ú s h(x) holds for all x ¾ M s then d(f n (x) g n (x)) Ú m d(f n (x) h n (x)) holds.

8 Slide 7 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 On the Way to Matrices applications of matrices over idempotent semirings: - shortest path: matrices over (Ê 0 ½ min ½ + 0) - maximum capacity path: matrices over (Ê 0 ½ max 0 min ½) - relations: matrices over ( ÇÇÄ ÄË ÌÊÍ ) - automata theory: matrices over (Σ fin ) therefore extension of distances to matrices writing S l m for l m-matrices over S matrix operations defined routinely (as in traditional linear algebra) order defined entrywise (provided S is ordered) folklore: if S is an idempotent semiring then so is S n n holds analogously for measure spaces

9 Slide 8 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Definition Matrix Distance Definition Given an idempotent semiring S s = (M s + s 0 s s 1 s), a measure system S m = (M m + m 0 m m Ú m), an S m-distance d on S s of any kind and two matrices A B ¾ Ms l n we define the distance d l n (A B) entrywise as a matrix of the type Mm l n, i.e., (d l n (A B)) ij = def d(a ij B ij ). How do properties of d carry over to d l n?

10 Slide 9 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Matrix Distance Properties Theorem Let d be an S m-distance on S s and consider the idempotent semiring Ss n n of n n-matrices over S s and let Sm n n be the set of n n-matrices over S m. Then the following properties hold: If d is additive then d n n is an additive S n n m -distance on S n n s. If d is order preserving then d n n is an order preserving S n n m -distance on S n n s. If d is strict then d n n is a strict Sm n n -distance on Ss n n. If d is both additive and multiplicative then d n n is a multiplicative Sm n n -distance on Ss n n.

11 Slide 10 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Application of Matrix Distance Consider an S m-distance d on S s. Then d n n is a complete Sm n n -distance on Ss n n. Assume three matrices A, Â and D from Sn n s with ds n n (A Â) D. n 1 n 1 n 1 Then we have ds n n ( Σ A k Σ Â k ) Σ D k. k=0 k=0 k=0 Applies to maximum capacity paths, shortest paths, maximum reliability paths,...

12 Slide 11 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Original Network 3 A 7 B E C 1 D

13 Slide 12 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Perturbated Network [2 4] [6 8] A B E [1 3] [3 5] [1 3] [6 8] [8 10] [1 3] D = C [0 2] D n 1 Σ D k depends on S m k=0

14 Slide 13 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Definition Norm Definition Given an idempotent semiring S s = (M s + s 0 s s 1 s) and a measure system S m = (M m + m 0 m m Ú m) we call a mapping : M s M m an S m-norm on S s. It is called additive if x + s y Ú m x + m y, multiplicative if x s y Ú m x m y, order preserving if x Ú s y µ x Ú m y, and strict if x = 0 m x = 0 s holds. A complete norm is an additive, multiplicative, order preserving and strict norm. Examples: For every idempotent semiring S s = (M s + s 0 s s 1 s) the identity is a complete S m-norm with S m = (M s + s 0 s s Ú s). L = def L is a complete mæ + 0 -norm on LANfin Σ.

15 Slide 14 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Induced Norm Theorem Let S s = (M s + s 0 s s 1 s) be an idempotent semiring and let d by an (additive, multiplicative, order preserving, strict) S m-distance on S s for some measure system S m = (M m + m 0 m m Ú m). Then the mapping d : M s M m, defined by x d = def d(0 s x), is an (additive, multiplicative, order preserving, strict) S m-norm on S s. It is called the norm induced by d. Example: on Ê n is the norm induced by d(x y) = df x y. The identity norm on LAN fin Σ is the norm induced by the distance dw = df L 1 L 2.

16 Slide 15 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Norm-Distributivity Definition An S m-distance d on S s of any kind is called norm-distributive if its induced norm fulfills the property d(x s y x s z) Ú m x d m d(y z) for all x, y and z. compare cx cy = c d(x y) or f(x) f(x + h) f ¼ (x) h d w is a norm-distributive distance on LAN fin Σ.

17 Slide 16 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Composing Norms and Distances Theorem Consider the distance d = def Æ d. Then the following holds: If d is additive and is additive and order preserving then d is additive. If d is multiplicative and is multiplicative and order preserving then d is multiplicative. If both d and are order preserving then d is order preserving. If d and are strict then d is strict. If d is norm-distributive and is multiplicative and order preserving then d is norm-distributive, too. Also well-behaved properties with respect to matrices.

18 Slide 17 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Final Example Consider again finite languages. Automata theory considers matrices of languages. Consider three such n n-matrices L 1, L 2 and L 3. Then we have d w(l 1 L 2 L 1 L 3 ) L 1 d w(l 2 L 3 ) and d N (L 1 L 2 L 1 L 3 ) L 1 d N (L 2 L 3 ).

19 Slide 18 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Further Work Kleene star eigenvalues/bideterminants randomized/stochastic errors topological considerations...

20 Slide 19 of 19 Distances, Norms and Error Propagation in Idempotent Semirings Roland Glück Groningen, 28nd October 2018 Review... some of his results may be "straightforward", some estimations "rough" and some proofs not particularly deep and rather tedious. Doing science is not quite an endless sequence of fireworks, ecstasy and champaign; by nature, there must be more moments of toil and frustration than insight and epiphany. We are so often pushed to present a distorted image or our work and focus on sell talk...