FLABloM: Functional linear algebra with block matrices Adam - PowerPoint PPT Presentation

FLABloM: Functional linear algebra with block matrices Adam Sandberg Eriksson Patrik Jansson Chalmers University of Technology, Sweden { saadam,patrikj } @chalmers.se May 26, 2016

Functional linear algebra with block matrices ◮ Inspired by work on parallel parsing by Bernardy & Jansson ◮ Matrices in Agda ◮ Reflexive, transitive closure of matrices � �  1 0 1   � 0 � 1    � � 1   0 0 1

Matrices Desirable: ◮ Easy to program with ◮ Easy to write proofs with

Matrices Desirable: ◮ Easy to program with ◮ Easy to write proofs with Possibilities: ◮ Vectors of vectors: Vec ( Vec a n ) m ◮ Functions from indices: Fin m → Fin n → a ◮ . . .

Matrices: shapes A type for shapes: data Shape : Set where L : Shape B : ( s 1 s 2 : Shape ) → Shape

Matrices: shapes A type for shapes: data Shape : Set where L : Shape B : ( s 1 s 2 : Shape ) → Shape Shapes for one dimension: (a vector/row matrix) B B L L L 1 3 16

Matrices: building blocks Matrices are indexed by two shapes: data M ( a : Set ) : ( rows cols : Shape ) → Set

Matrices: building blocks Matrices are indexed by two shapes: data M ( a : Set ) : ( rows cols : Shape ) → Set  � �   � � � �  . ... ... . .         � �     [ a ] , [ · · · ] [ · · · ]     , ,  � �   � � � �  . ... ...  .    .     M a L ( B c 1 c 2 ) M a ( B r 1 r 2 ) L M a ( B r 1 r 2 ) ( B c 1 c 2 ) M a L L

Matrices: a datatype data M ( a : Set ) : ( rows cols : Shape ) → Set where One : a → M a L L Col : { r 1 r 2 : Shape } → M a r 1 L → M a r 2 L → M a ( B r 1 r 2 ) L Row : { c 1 c 2 : Shape } → M a L c 1 → M a L c 2 → M a L ( B c 1 c 2 ) Q : { r 1 r 2 c 1 c 2 : Shape } → M a r 1 c 1 → M a r 1 c 2 → M a r 2 c 1 → M a r 2 c 2 → M a ( B r 1 r 2 ) ( B c 1 c 2 )

Rings A hierarchy of rings as Agda records: ◮ SemiNearRing ≃ , +, · , 0 (+ is associative and commutes, 0 identity of + and zero of · , · distributes over +) ◮ SemiRing 1 (1 identity of · , · is associative) ◮ ClosedSemiRing an operation ∗ with w ∗ ≃ 1 + w · w ∗ .

Lifting matrices We take a semi-(near)-ring and lift it to square matrices. A lifting function Square for each Shape and ring structure. : Shape → SemiNearRing → SemiNearRing Square Square ′ : Shape → SemiRing → SemiRing Square ′′ : Shape → ClosedSemiRing → ClosedSemiRing

Lifting matrices (Parts of) lifted equivalence: ≃ S : ∀ { r c } → M s r c → M s r c → Set ( One x ) ≃ S ( One x 1 ) = x ≃ s x 1 ( Row m m 1 ) ≃ S ( Row n n 1 ) = ( m ≃ S n ) × ( m 1 ≃ S n 1 ) (Parts of) lifted multiplication: · : ∀ { r m c } → M s r m → M s m c → M s r c S · = One ( x · s y ) One x S One y Row m 0 m 1 · S Col n 0 n 1 = m 0 · S n 0 + S m 1 · S n 1

Proofs: reflexivity reflS : ∀ { r c } → ( X : M s r c ) → X ≃ S X reflS ( One x ) = refl s { x } reflS ( Row X X 1 ) = reflS X , reflS X 1 reflS ( Col X X 1 ) = reflS X , reflS X 1 reflS ( Q X X 1 X 2 X 3 ) = reflS X , reflS X 1 , reflS X 2 , reflS X 3

Closure for matrices Computing the reflexive, transitive closure: [ a ] ∗ = [ a ∗ ] 11 · A 12 · ∆ ∗ · A 21 · A ∗   ∗   11 + A ∗ 11 · A 12 · ∆ ∗ A 11 A 12 A ∗ A ∗ 11 =     ∆ ∗ · A 21 · A ∗ ∆ ∗ A 21 A 22 11 (with ∆ = A 22 + A 21 · A ∗ 11 · A 12 ) with proof that it satisfies w ∗ ≃ 1 + w · w ∗

Reachability example 1 2 3 4 � 0 � 0 ∗ � �  0 0  0 0 0 1      � 0 � 0    � � 1 0   0 0 0 0

Reachability example 1 2 3 4 � 0 � 0 � 1 � 0 ∗ � � � �  0 0   0 0  0 0 0 1 0 1 0 1         =  � 0 � 0   � 0 � 1      � � � � 1 0 1 1     0 0 0 0 0 0 0 1 1 2 3 4

Wrapping up Conclusions, further work, et.c. ◮ This matrix definition is useable... ◮ A more flexible matrix definition: sparse? fewer constructors? ◮ Automation (of proofs)! ◮ Generalisation to closed semi-near-ring for parsing applications. ◮ Agda development available at https://github.com/DSLsofMath/FLABloM .