Graphical Linear Algebra PhD Open, University of Warsaw Pawel - - PowerPoint PPT Presentation
Graphical Linear Algebra PhD Open, University of Warsaw Pawel Sobocinski University of Southampton joint work with Filippo Bonchi, and Fabio Zanasi important contributions from Dusko Pavlovic, Fabio Gadducci, Aleks Kissinger, Brendan Fong and
Pawel Sobocinski University of Southampton
graphicallinearalgebra.net
joint work with Filippo Bonchi, and Fabio Zanasi
PhD Open, University of Warsaw
important contributions from Dusko Pavlovic, Fabio Gadducci, Aleks Kissinger, Brendan Fong and Paolo Rapisarda
than the classical concepts of matrices, bases, etc.
substitution) is good in situations where all operations have a coarity 1
n → X
in X are copyable and discardable
more careful with resources — here tree-like syntax can be less natural
algebra for this syntax
needed
categories called strict symmetric monoidal categories
⊕ -
stacking one on top of the other
= ⊕ = ⊕
associative but not commutative
;
the second
; = ; =
associative but not commutative
“stacking things and plugging them together”
⊕
⊕
⊕ ⊕ = ;
: (2, 2) : (4, 4) : (2, 0)
X : (k, l) Y : (m, n) X⊕Y : (k+m, l+n)
X : (k, l) Y : (l, m) X;Y : (k, m)
number of holes number of studs
X : (k, l) Y : (m, n) X⊕Y : (k+m, l+n)
⊕ = X : (k, l) Y : (l, m) X;Y : (k, m)
; = ?
; =
Pieces stretch and shrink as needed — the only important thing is the type (number of holes, number of studs)
“plugging things together” are very common
consists of two sets: objects and arrows
called domain and codomain, written C : k → m
a trivial “do-nothing” operation) Ik : k → k
composition
C : k → l D: l → m C ; D : k → m
— like the plugging in magic Lego
C : k → l D : l → m E : m → n (C ; D) ; E = C ; (D ; E) : k → n
not change the arrow, i.e. C : m → n Im ; C = C = C ; In
x→y iff x≤y.
is a preorder.
structure, consists of two sets: objects and arrows
called domain and codomain, written C : k → m
monoidal product
(k ⊕ m) ⊕ n = k ⊕ (m ⊕ n)
a trivial “do-nothing” operation)
(think of this as a unit type). This makes the set of
Ik : k → k k ⊕ I = I ⊕ k = k
composition
also called monoidal product.
C : k → l D: l → m C ; D : k → m
— like the plugging in magic Lego
C : k → l D: m → n C ⊕ D : k ⊕ m → l ⊕ n
— like the stacking in magic Lego
C : k → l D : l → m E : m → n (C ; D) ; E = C ; (D ; E) : k → n C : m → n D : m’ → n’ E : m’’ → n’’ (C ⊕ D) ⊕ E = C ⊕ (D ⊕ E) : m⊕m’⊕m’’ → n⊕n’⊕n’’
and preserve domains and codomains (c.f. morphism of digraphs)
F: X → Y C : m → n F(C) : F(m) → F(n) F(C ; D) = F(C) ; F(D) F(Ik) = IF(k)
equipped with a functor
⋅ : M × M → M ⊕ : C × C → C
(M×M is the cartesian product of sets) (C×C is the cartesian product of categories)
define of the cartesian product C×D of C and D?
can be composed with D. Then since ⊕, being a functor, preserves composition, we have: ⊕(A;B, C;D) = ⊕( (A,C) ; (B,D) ) = ⊕(A,C) ; ⊕(B,D) i.e. (A ; B) ⊕ (C ; D) = (A ⊕ C) ; (B ⊕ D)
This equation is sometimes called “middle-four interchange”
( A ⊕ Ir ) ; (Iq ⊕ B) = A ⊕ B = (Ip ⊕ B) ; (Is ⊕ A)
⊕(I(m,n)) = ⊕(Im, In) = Im⊕n i.e. Im ⊕ In = Im⊕n C: m → n Im ; C = C = C ; In
not change the arrow, i.e.
numbers of holes and studs)
categories
C m n
C : k → l D: l → m C ; D : k → m
C k l D m
C : k → l D: m → n C ⊕ D : k ⊕ m → l ⊕ n
C k l m D n
C : k → l D : l → m E : m → n (C ; D) ; E = C ; (D ; E) : k → n
k l m C D E n
C : m → n D : m’ → n’ E : m’’ → n’’ (C ⊕ D) ⊕ E = C ⊕ (D ⊕ E) : m ⊕ m’⊕ m’’ → n ⊕ n’⊕ n’’
C D E m m' m'' n n' n''
(A ; B) ⊕ (C ; D) = (A ⊕ C) ; (B ⊕ D)
A B C D
Stretching
m
C : m → n Im ; C = C = C ; In
= C = C C m n m n m n
( A ⊕ Ir ) ; (Iq ⊕ B) = A ⊕ B = (Ip ⊕ B) ; (Is ⊕ A) Sliding
A B = = A B A B p q r s p q r s p q r s
monoidal cat) stay as one dimensional things — wires
monoidal cat) become zero dimensional things (points, or boxes as we have been drawing)
connecting and stacking, we often want to permute the wires
categories to symmetric monoidal categories
m, n, there is a symmetry, or twist
twm,n: m⊕n → n⊕m
Crack Egg egg yolk white Whisk white white whisked whites yolk yolk sugar Beat yolky paste yolky paste mascarpone Stir thick paste whisked whites thick paste Fold crema di mascarpone Crack Egg egg white yolk Crack Egg egg white yolk Beat Whisk whisked whites thick paste mascarpone Stir crema di mascarpone sugar Fold yolky paste
Crack Egg egg yolk white Whisk white white whisked whites yolk yolk sugar Beat yolky paste yolky paste mascarpone Stir thick paste whisked whites thick paste Fold crema di mascarpone Crack Egg egg white yolk Crack Egg egg white yolk Beat Whisk whisked whites thick paste mascarpone Stir crema di mascarpone sugar Fold yolky paste
(C ⊕ C ⊕ id2) ; (id ⊕ tw ⊕ id3) ; (W ⊕ B ⊕ id) ; (id ⊕ S) ; F
F ⇒ G: X → Y αm : Fm → Gm Fm Gm Fn Gn
FC
GC
αm αn
commutes.
These are sometimes called naturality squares
follows, using string diagrams:
η ε F G F = F η ε G F G = G
F FGF
Fη
F
εF
G GFG
ηG Gε
G
functors C×C → C given by ⊕
second, with components twm,n : m⊕n→ n⊕m ⊕ = –1⊕–2: C×C → C ⊕’ = –2⊕–1: C×C → C
– stack the first on the second – stack the second on the first
twm,n: m⊕n → n⊕m
m m n n
p⊕n n⊕p q⊕n n⊕q
C⊕n
n⊕C
twp,n twq,n
C n n p q C n n p q =
naturality = sliding across twists m⊕p p⊕m m⊕q q⊕m
m⊕C
C⊕m
twm,p twm,q
C p q m m C p q m m =
invertible, and has itself as inverse, in the following sense
tightening (without wires tangling) twm,n ; twn,m= Im⊕n : m⊕n → m⊕n
=
m m m n n n m n In a braided monoidal category, the twist is invertible, but it is not, in general, it’s own inverse
monoidal category with a natural family of arrows twm,n : m⊕n → n⊕m, indexed by pairs of objects of C, such that twm,n ; twn,m= Im⊕n : m⊕n → m⊕n twm⊕n, p = (Im ⊕ twn,p) ; (twp,m ⊕ In): m⊕n⊕p→ p⊕m⊕n twp, m⊕n = (twp,m ⊕ In); (Im ⊕ twp,n): p⊕m⊕n→ m⊕n⊕p twI,m = Im = twm,I
instance of naturality of the twist
=
=
we get that “pure” wiring diagrams t⊕m → t⊕m are in 1-1 correspondence with permutations of the m-element set
permutations
natural number
from generators and variables
Example 1 - monoids Example 2 - abelian groups
ΣM = { ⋅:2, e:0 }
EM = { ⋅( ⋅(x, y), z ) = ⋅( x, ⋅(y, z) ), ⋅(x, e) = x, ⋅(e, x) = x }
ΣG = ΣM ∪ { i:1 }
EG = EM ∪ { ⋅(x, y) = ⋅(y, x), ⋅(x, i(x)) = e }
Universal Algebra
x ∈ Var x t1 t2 … tm σ ∈ Σ ar(σ) = m σ(t1, t2, …, tm) i.e. terms a trees with internal nodes labelled by the generators and the leaves labelled by variables
k → X
recursively compute the element of [[t]]α ∈ X which is the “meaning” of t
equation t1 = t2 in E, we have [[t1]]α = [[t2]]α as elements of X
categorically, due to Lawvere in the 1960s
“variable assignments” etc.
product if ∃ object X×Y and arrows π1: X×Y → X, π2: X×Y → Y so that the following universal property holds
product satisfies the universal property
the categorical product as monoidal product
X×Y Z X Y
π1 π2
f g h
for any object Z and arrows f: Z → X, g: Z → Y, ∃ unique h: Z → X×Y s.t. h ; π1 = f and h ; π2 = g
does it mean if X has (binary) categorical products?
product
and functors?
and homomorphisms?
x2, …, xm, modulo the equations of E
Examples in the theory of monoids 2 1
It is also possible (and elegant) to view L(Σ,E) as the free category with products on the data specified in (Σ,E)
(x1⋅x2)
2 1
(x2⋅x1)
1 1
(x1⋅e)
= 1 1
(x1)
m×n := m+n.
canonical arrow Δ: X→X×X called the diagonal.
describing it?
F(X×Y) = F(X) × F(Y)
preserving functor F: L(Σ,E) → Set Proof idea: since m = 1+1+…+1 (m times), to give a product preserving functor F from L(Σ,E) it is enough to say what F(1) is.
generalisation of classical universal algebra, with examples such as topological groups, etc.
need to be more careful with resources
1 2
(x1, x1)
2 1 (x1) 2 1 (x2) 1 2
c
= 1 2
(c1,c2)
theories (SMTs) which give rise to special kinds of symmetric monoidal categories called props
theories, a classical concept of universal algebra, but
than 1
both natural numbers
resources, we can’t use the standard notion of term (tree).
we will construct a la magic Lego
: (2, 1) Running example: the SMT of commutative monoids : (0, 1) we always have the following “basic tiles” around : (1, 1) : (2, 2)
and basic tiles, with the two operations of magic Lego
⊕ =
⊕ =
; =
A k l C m n A k l C m n = = A k l C m n
functoriality
A k l m m l = A k l m m k
naturality
i.e. pure wiring obeys the same equations as permutations
=
=
category SΣ on Σ
from generators, identity and twist, modulo diagrammatic reasoning
x y z x + y (x + y) + z
x y z y + z x + (y + z)
=
(Assoc)
x y x+y
x y y+x
=
(Comm)
x 0 + x
=
(Unit)
Note that all equations are of the form t1 = t2 : (m, n), that is, t1 and t2 must agree on domain and codomain
=
=
=
Equations Generators
Let’s call this SMT M, for monoid
= = = =
=
=
Equations Generators
=
=
=
symmetric monoidal category S(Σ,E)
permutation categories)
set {0, …, m-1} to the n element set {0, … , n-1} 3.The free strict symmetric monoidal category on one
a prop
symmetric monoidal functor
nose — easy peasy
generalisation of the notion of finite product preserving
and a sound and fully complete proof system to reason about things in F
thus its arrows are constructed inductively
a prop obtained from an SMT
7 ! 7 ! {} → {1} {1,2} → {1}
=
(Assoc)
=
(Comm)
=
(Unit)
Simple exercise: check the following hold in F
homomorphism [[–]] : M → F means that diagrammatic reasoning in M is sound for F
completeness?
enough for [[–]] to be an isomorphism? (i.e. invertible)
isomorphism it suffices to show that it is full and faithful
X such that F(f) = g
So full and faithful functor from a (free PROP on an) SMT = sound and fully complete equational charaterisation
constructed from the two basic building blocks together with permutations
multiplications followed by units, which corresponds to a factorisation of a function as an surjection followed by an injection. This factorisation is unique “up-to-permutation”.
property — it’s the “smallest” thing that contains X which satisfies the properties of “something”
X F G
strict symmetric monoidal functor X to Y such that the diagram below commutes
1 X Y functor functor
strict symmetric monoidal functor
(Hopf monoids)
monoids and comonoids
The monoid structure acts as addition/zero The comonoid structure acts as copying/discarding
x y x+y
x x x
x
responsible for all linear algebra
bimonoids, the equations of which show some of the ways that the interactions happen
= =
=
=
=
= =
=
=
=
numbers
5 : 2 → 1
✓ 3 15 ◆ : 1 → 2
✓ 1 2 3 4 ◆ : 2 → 2
A1 ⊕ A2 = ✓ A1 A2 ◆
diagrams, with the recursive definition below
:=
k+1
:=
k
+1 is “add one path”
m n m+n
=
m n nm
=
m m m
=
m m m
=
:=
k+1
:=
k
Given , prove 1. 2. 3. 4.
1 1 : 2 → 1
() : 0 → 1 ✓ 1 1 ◆ : 1 → 2 () : 1 → 0
Full - easy! Recursively define a syntactic sugar for matrices
Faithful - harder Use the fact that equations are a presentation of a distributive law, obtain factorisation of diagrams as comonoid structure followed by monoid structure - normal form
Recall: Since B is an SMT, suffices to say where generators go (and check that equations hold in the codomain)
from the jth port on the left to the ith port on the right
2 3 4
show that the monoidal product in B≅Mat is the categorical coproduct. When a monoidal product satisfies both the universal properties of products and coproducts, we say that it is a biproduct. In fact B≅Mat is the free category with biproducts on one object. Q3 (challenging). Given a category C, describe the free category with biproducts on C.
X+Y Z X Y
i1 i2
f g h
for any object Z and arrows f: X → Z, g: Y → Z, ∃ unique h: X+Y → Z s.t. i1 ; h = f and i2 ; h = g
(i.e. how ordinary syntax looks, with string diagrams)
σ . . .
(σ ∈ Σ)
=
=
=
and what else?
In particular, notice that B is isomorphic (as a symmetric monoidal category) to the Lawvere category of commutative monoids!
σ . . . = . . . σ σ . . . . . .
σ . . . = . . .
Exercise: show that the monoidal product now becomes a categorical product
=
=
=
=
=
= = = = = = = =
correspondence with natural numbers
correspondence with the integers
n n
=
:=
k+1
:=
k
:=
n
m n m+n
=
m n nm
=
free Z-modules
antipode appears an even number of times)
number of times)
✓ −1 1 ◆
1
() : 0 → 1 ✓ 1 1 ◆ : 1 → 2 () : 1 → 0
7! 7! 7! 7!
copying ; antipode ; adding
(−1) : 1 → 1
monoidal category, to the Lawvere category of commutative monoids.
components
contract that allows certain numbers to appear on the left and on the right
components
contract that allows certain numbers to appear on the left and on the right
The input/output framework is totally inappropriate for dealing with all but the most special system interconnections. [The input/output representation] often needlessly complicates matters, mathematically and conceptually. A good theory of systems takes the behavior as the basic notion. J.C. Willems, Linear systems in discrete time, 2009
x y , x+y () , 0 x , x x x , ()
x y , x+y () , 0 x , x x x , () x y x+y , 0 , () x x , x () , x
p q r p+q r p q+r
x y z x+y z x y+z
x = p+q z = q+r p=x+y r=y+z
Provided addition yields abelian group (i.e. there are additive inverses), the two are the same relation
y=-q
x+y x y x+y
since x and y are free, this is the identity relation
x
empty relation
x x x x x x x x x x
clearly both give the same relation
x x x x
identity relation
x
empty relation
=
= = =
+ special / strongly separable equations + “bone” equations
= =
=
= = = = = =
Frobenius monoid
= =
Snake lemma
=
(Frob)
=
(Unit)
=
(Counit)
Proof: “cup” “cap”
diagram can be factorised into comonoid structure ; monoid structure, this gave us centipedes
can be factored into monoid structure ; comonoid structure, these are often referred to as spiders
= = = = = = =
Special Frobenius monoid =
diagram is equal to one of the form
1 2 m 1 2 n
1 2 m 1 2 n
=
{( ✓ x y ◆ , ()) | x + y = 0 } {( ✓ x x ◆ , ())}
=
if multiplication on the left by p is injective (e.g. if p ≠ 0 in a field)
p
x px
p
px =
if multiplication on the left by p is surjective (e.g. if p ≠ 0 in a field)
p p
x px=py y =
(Bonchi, S., Zanasi, ’13, ’14)
= = =
Copying
= = =
Copyingop
= = =
Adding
= = = =
Adding meets Copying
= = =
Addingop
= = = =
Addingop meets Copyingop Antipode = = = = = Antipodeop = = = = =
H Hop
White monoid (adding) Black comonoid (copying) White comonoid (adding-op) Black monoid (copying-op)
Hopf Hopf Frobenius
Frobenius
= = = = = = = = = = = =
= =
The antipodes in H and Hop are formally different but we were slightly naughty with notation.
Adding meets Addingop
=
= = Copying meets Copyingop
=
= = = = Cups and Caps
p p p p (p ≠ 0)
= = Scalars
= = =
Copying
= = =
Copyingop
= = =
Adding
= = = =
Adding meets Copying
= = =
Addingop
= = = =
Addingop meets Copyingop Antipode = = = = = Antipodeop = = = = = Adding meets Addingop
=
= = Copying meets Copyingop
=
= = = = Cups and Caps
p p p p (p ≠ 0)
= = Scalars
Symmetry 1 - colour inversion Symmetry 2 - mirror image
IH
generators, e.g. Lemma
= = = =
so
= =
then p⋅u ∈ U
subspaces?
subspace
p q
if q ≠ 0:
p q p q
=
q q
=
q q q p
=
q p
q q
=
q q q q
=
q q q q q
=
q
suppose q,s ≠ 0:
⇒
p q r s
=
⇔
sp = qr
p s
=
p q q s r s q s r q s s
= =
r q
=
⟸
p q
=
p q s s
=
r q q s
=
r q s q r s
=
(q,s ≠ 0)
p q r s
=
p q r s s s q q sp sq qr qs sp qr sq
= = =
sp+qr sq
p q r s
=
p r s q
=
rp sq
as with proper rationals.
= =
= =
dimensional spaces (lines) of Q
2
Q
2, in particular:
(x, 1/2 x) (x, 2x)
Edalat and Potts suggested that two extra ‘numbers’, ∞ = 1/0 and ⊥ = 0/0, be adjoined to the set of real numbers (thus obtaining what in domain theory is called the ‘lifting’ of the real projective line) in order to make division always possible. In a seminar, Martin-Löf proposed that one should try to include these ‘numbers’ already in the construction of the rationals from the integers, by allowing not
with a field, but with a field with two extra elements.
Here we have three extra “numbers”!
Jesper Carlström, Wheels, On Division by Zero, 2001
:=
:=
∞
:=
integer matrices (Hopf monoids)
(Lack, Composing PROPs, 2004)
Theory of Monads, 1972)
O
δ0
← − A
δ1
− → O A ×O A ✏ / A
δ1
✏ A
δ0
/ O A ×O A
µ
− → A O
η
− → A
let’s call it “composition” let’s call it “identity”
P
Green PROP P
Q
Purple PROP Q
When can we understand P;Q as a PROP?
Q P P Q
λ
P Q P Q
PλQ
P P Q Q
distributive law is a 2-cell
multiplication and units in T and U in the obvious way (see diags)
λ : UT ⇒ TU
UUT
µUT
/
Uλ
✏ UT
λ
✏ UTU
λU / TUU T µU
/ TU UTT
UµT
/
λT
✏ UT
λ
✏ TUT
T λ / TTU µT U / TU
T
ηUT
}| | | | | | | |
T ηU
! B B B B B B B B UT
λ
/ TU
U
UηT
}{ { { { { { { {
ηT U
! C C C C C C C C UT
λ
/ TU
functions” (commutative comonoids)
Pullback!
2 × 2
π1
}zzzzzzzz ! D D D D D D D D
π2
! D D D D D D D D 2 " D D D D D D D D D 2 |zzzzzzzzz 1
2 × 2
π1
}zzzzzzzz ! D D D D D D D D
π2
! D D D D D D D D 2 " D D D D D D D D D 2 |zzzzzzzzz 1
(0,0) (0,1) (1,0) (1,1) 1 1
2 × 2
π1
}zzzzzzzz ! D D D D D D D D
π2
! D D D D D D D D 2 " D D D D D D D D D 2 |zzzzzzzzz 1
(0,0) (0,1) (1,0) (1,1) 1 1
2 × 2
π1
}zzzzzzzz ! D D D D D D D D
π2
! D D D D D D D D 2 " D D D D D D D D D 2 |zzzzzzzzz 1
(0,0) (0,1) (1,0) (1,1) 1 1
=
and cospan morphisms as 2-cells
= =
p p
=
IRSp ≅ Span(MatZ)
= =
=
= = =
(p ≠ 0)
IRSp
r r
=
r r r
=
r
IRCsp ≅ Cospan(MatZ) IRCsp
(p ≠ 0) p p
=
= = = = = =
r r
=
r r r
=
r
MatZ + MatZ
H + Hop Span(MatZ) IHSpan IHCospan Cospan(MatZ)
MatZ + MatZ
H + Hop Span(MatZ) IHSpan IHCospan Cospan(MatZ) LinRel IH
A B
m n k
C D
m n l
A A
m m n = m
A A
n n m = n
integer matrices (Hopf monoids)
solutions of a list of homogeneous equations
linear combinations
x+y=0 x y z 2y-z=0
2
x y z
x+y=0 2y-z=0
2
x y z
a[1, -1, 0] a
b[0, 1, 2]
2
b
a[1, -1, 0]+b[0,1,2]
2
a b
solutions of a list of homogeneous equations
linear combinations
x+y=0 x y z 2y-z=0
2
x y z
x+y=0 2y-z=0
2
x y z
Cospans
a[1, -1, 0] a
b[0, 1, 2]
2
b
a[1, -1, 0]+b[0,1,2]
2
a b
solutions of a list of homogeneous equations
linear combinations
x+y=0 x y z 2y-z=0
2
x y z
x+y=0 2y-z=0
2
x y z
Cospans
a[1, -1, 0] a
b[0, 1, 2]
2
b
a[1, -1, 0]+b[0,1,2]
2
a b
Spans
A A
AT AT
Injective matrices are the monos in MatZ
A A
=
⇒ ⇐
A F A G
= ⇒
A F A G
=
A A
⇒
F
=
G
? ? ? ?
A
? ? ? ?
A
is pullback in MatZ
A F A G
= ⇒
F
=
G
A A
= A A F G
= ⇒
F
=
G
Proof: Bizarro of last slide
⇒ ⇐
A A
=
A A
=
A A
=
A
= =
A
=
A A
=
A A
=
⇔
A
=
Proof: bizarro of last slide
A B
=
⇒ ⇐
so A is injective
A B A A
= =
bizarro argument yields other half
A A B B
= =
A A B
= =
decompositions, eigenvalues/eigenspaces, determinants
integer matrices (Hopf monoids)
PID R
to the graphical syntax together with equations
r1 r2 r1+r2
=
r1 r2
=
r2r1
1
= =
(polynomials with one variable, a PID) is especially interesting
(generatingfunctionology!)
(syntax) to Laurent series (semantics)
foundational structure in control and signal processing
MatQ[x] + MatQ[x]
HQ[x] + HQ[x]
Span(MatQ[x]) IHQ[x] Span IHQ[x] Cospan Cospan(MatQ[x]) LinRelQ(x) IHQ(x) MatQ[[x]] + MatQ[[x]]
Cospan(MatQ[[x]]) Span(MatQ[[x]]) LinRelQ((x))
isomorphisms faithful homomorphisms
In particular, IHQ[x] is sound and complete as a theory for LinRelQ((x))
1-x-x2 x
As linear relation over Q(x) is the space generated by
As linear relation over Q((x)) is the space generated by
(1 , x/(1-x-x2)) (1,0,0,… , 0,1,1,2,3,5,8,…)
k
− − →
k k k
− →
k
l
− − →
kl
k
x
l
k
− →
l
x
k
k l
− − →
k+l
− →
k k
− − →
k
− →
k
k
kl
− − →
l
k
x
l
l
− →
k
x
k
k+l
− − − →
k l
− →
k
− →
k k l
− − →
l k
s
u
− →
v
s0 t
v
− →
w t0
s ; t
u
− →
w s0 ; t0
s
u1
− − →
v1
s0 t
u2
− − →
v2
t0 s ⊕ t
u1 u2
− − − − →
v1 v2
s0 ⊕ t0
Bonchi, S., Zanasi, Full abstraction for signal flow graphs, POPL ‘15
Important point: all executions start with all registers initialised with 0!
:=
:=
x x
1 1 1 1
x x
1 2 1 1 2
x x
2 3 1 1 3
…
Operational semantics closely related to denotational semantics [linear relations over Q((x))] with some “implementation issues” in diagrams where signal flow is inconsistent e.g.
x x x x
k k
x x
1 2
x x
1 2
where the direction of signal flow is consistent
behaviour, given by operational semantics) coincides with denotational equality (the denoted linear relation)
1,2,3,4,…
Generating function Diagram Signal flow graph
1 (1 − x)2
x x (1-x)2
0,-4,0,-8,..
−4x (1 − x2)2
1,-2,3,-4,…
1 (1 + x)2
x x x x
x (1-x2)2
(1+x)2 x x
(C 1170 - C 1250)
1 2 3 5 8 13 21 34 55 89 144 233 377 A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth, and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the above written sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.
(extract from Liber Abaci, chapter 12, translated from Latin by Lawrence Sigler)
and recurrence relations 1, 2, 3, 5, 8, 13, … in modern presentations often given as
1, 1, 2, 3, 5, 8, …
0, 1, 1, 2, 3, 5, …
is an example of a recurrence relation here Fn+2 = Fn+1 + Fn
more generally, the nth term can be obtain as an expression featuring previous terms
val ffib : int list -> int list = <fun> # ffib [1;0;0;0;0;0;0;0];;
# ffib [1;1;1;1;1;1;1;1];;
# ffib [1;1;-3;1;-2;-4;1];;
Generalise the Fibonacci rabbit breeding rule from a sequence to a function of type
pairs in the rabbit pen at all times.
months? If the Fibonacci rule gives us a function of type to solve the sustainable rabbit breeding problem we could try to write the inverse
ffib: int list -> int list bfib: int list -> int list
graphicallinearalgebra.net