testing log log program favonia 1 Testing map map : . .( ) - PowerPoint PPT Presentation

testing log log program favonia 1

Testing map map : ∀α . ∀β .( α→β ) → list( α ) → list( β ) 2

Testing map map : ∀α . ∀β .( α→β ) → list( α ) → list( β ) α * = β * = ℕ identity function and [0, 1, …, n-1] 2

Testing exists exists : ∀α .( α→ 2) → list( α ) → 2 α * = 2 identity function and all lists 3

Testing pick pick : ∀α . α × α→α α * = 2 (true, false) 4

Theorem [Bernardy et al. ] p : ∀α . (F( α ) →α )×(G( α ) → K) → H( α ) pos pos pos α * = μ F if it exists ρ * : F( α *) →α * catamorphism “roll” 5

Theorem [Bernardy et al. ] p : ∀α . (F( α ) →α )×(G( α ) → K) → H( α ) pos pos pos project embed p : ∀α . P( α ) → H( α ) pos 6

My Problems with Embedding/Projection Type μ F buried under layers of existence theorems. Error-prone: e.g. , wrongly assuming list( α ) ⊆ ℕ ×( ℕ →α ). 7

p : ∀α . P( α ) → H( α ) 8

Logarithmic Conjecture p : ∀α . P( α ) → H( α ) α * = μα .log α P( α ) if it exists 8

Logarithmic Conjecture p : ∀α . P( α ) → H( α ) α * = μα .log α P( α ) if it exists Goal: simple and direct calculation 8

Testing pick pick : ∀α . α × α→α α * = log α ( α × α ) = log α ( α 2 ) = 2 “all locations of α ” “all ways to generate an α -element” 9

log α α = 1 log α K = 0 10

log α α = 1 log α K = 0 log α (A×B) = log α A + log α B log α (A+B) = log α A ∪ log α B A ∪ B ≈ “max(A,B)” can be A+B, ideally more optimized 10

log α α = 1 log α K = 0 log α (A×B) = log α A + log α B log α (A+B) = log α A ∪ log α B A ∪ B ≈ “max(A,B)” can be A+B, ideally more optimized log α (A ∪ B) = log α A ∪ log α B optional 10

log α α = 1 log α K = 0 log α (A×B) = log α A + log α B log α (A+B) = log α A ∪ log α B A ∪ B ≈ “max(A,B)” can be A+B, ideally more optimized log α (A ∪ B) = log α A ∪ log α B optional log α (A → B) = A × log α B “B A ” 10

Testing map map : ∀α . ∀β .( α→β ) → list( α ) → list( β ) 11

Testing map map : ∀α . ∀β .( α→β ) → list( α ) → list( β ) P( α ) = ( α→β ) × list( α ) 11

Testing map map : ∀α . ∀β .( α→β ) → list( α ) → list( β ) P( α ) = ( α→β ) × list( α ) log α P( α ) = log α ( α→β ) + log α list( α ) 11

Testing map map : ∀α . ∀β .( α→β ) → list( α ) → list( β ) P( α ) = ( α→β ) × list( α ) log α P( α ) = log α ( α→β ) + log α list( α ) = α × log α β = α × 0 ≅ 0 11

α Rules of Recursive Logarithm X ≅ P(X) X X log α X ≅ log α P(X) X X 12

α Rules of Recursive Logarithm X ≅ P(X) X X log α X ≅ log α P(X) X X log α ( μ X.P) = μ X ′ .(log α P)[ μ X.P/X] log α X = X ′ 12

Rules of Recursive Logarithm list( α ) = μ X. 1 + α × X log α (list( α )) = μ X ′ . 0 ∪ (1 + X ′ ) ≈ μ X ′ . 1 + X ′ = ℕ log α ( μ X.P) = μ X ′ .(log α P)[ μ X.P/X] log α X = X ′ I will assume 0 ∪ A = A from now on 13

Testing map map : ∀α . ∀β .( α→β ) → list( α ) → list( β ) P( α ) = ( α→β ) × list( α ) log α P( α ) = log α ( α→β ) + log α list( α ) ≈ ℕ ≅ 0 14

Logarithmic Conjecture p : ∀α . P( α ) → H( α ) α * = μα .log α P( α ) if it exists Goal: simple and direct calculation 15

Non-regular Recursion and Change of Bases / Chain Rules N( α ) ≅ 1 + α × N( α × α ) α α 2 α 4 α 8 1 example stolen from Categories of Containers [Abbott et al. ] 16

Non-regular Recursion and Change of Bases / Chain Rules N( α ) ≅ 1 + α × N( α × α ) α α 2 α 4 α 8 1 log α N( α ) ≅ 1 + “log α × α N( α × α )” × log α ( α × α ) N ′ ( α ) ≅ 1 + N ′ ( α × α ) × 2 example stolen from Categories of Containers [Abbott et al. ] 16

α Non-regular Recursion and Change of Bases / Chain Rules P P log α P(Q( α )) = P ′ (Q( α )) × Q ′ ( α ) Q Q 17

α Non-regular Recursion and Change of Bases / Chain Rules P P log α P(Q( α )) = P ′ (Q( α )) × Q ′ ( α ) Q Q Should work for arbitrary P with more equations log α (( α 2 ) 3 ) = 3 × 2 log α ( α 2 × α 2 × α 2 ) = 2 + 2 + 2 17

Non-regular Recursion and Change of Bases / Chain Rules F( α ) ≅ 1 + α + list( α ) × F( α 2 + α 3 ) × list( α ) ( fi nger trees) list( α ) may be optimized to α + α 2 + α 3 + α 4 18

Non-regular Recursion and Change of Bases / Chain Rules F( α ) ≅ 1 + α + list( α ) × F( α 2 + α 3 ) × list( α ) ( fi nger trees) F ′ ( α ) ≅ 1 ∪ ( ℕ + F ′ ( α 2 + α 3 ) × (2 ∪ 3) + ℕ ) list( α ) may be optimized to α + α 2 + α 3 + α 4 18

Corollary of Conjecture p : ∀α . (F( α ) →α )×(G( α ) → K) → H( α ) 19

Corollary of Conjecture p : ∀α . (F( α ) →α )×(G( α ) → K) → H( α ) log α ((F( α ) →α )×(G( α ) → K)) = log α (F( α ) →α ) + log α (G( α ) → K) ) ( 0 F α ≈ ≅ α * ≅ μα .(F( α ) + 0) ≅ μ F 19

Corollary of Conjecture p : ∀α . (F( α ) →α )×(G( α ) → K) → H( α ) log α ((F( α ) →α )×(G( α ) → K)) = log α (F( α ) →α ) + log α (G( α ) → K) ) ( 0 F α ≈ ≅ α * ≅ μα .(F( α ) + 0) ≅ μ F How to plug in ρ *, the catamorphism (“roll”)? 19

Upgraded Logarithmic Conjecture p : ∀α . P( α ) → H( α ) α * = μα .P ′ ( α ) if it exists 20

Upgraded Logarithmic Conjecture p : ∀α . P( α ) → H( α ) α * = μα .P ′ ( α ) if it exists ρ * : P ′ ( α *) → α * catamorphism “roll” 20

Upgraded Logarithmic Conjecture p : ∀α . P( α ) → H( α ) α * = μα .P ′ ( α ) if it exists ρ * : P ′ ( α *) → α * catamorphism “roll” wants: ρ + : something → P( α *) 20

has: ρ * : P ′ ( α *) → α * wants: ρ + : something → P( α *) 21

has: ρ * : P ′ ( α *) → α * wants: ρ + : something → P( α *) ρ * can fi ll in strictly positive positions 21

has: ρ * : P ′ ( α *) → α * wants: ρ + : something → P( α *) ρ * can fi ll in strictly positive positions exists : ∀α .( α→ 2) → list( α ) → 2 21

has: ρ * : P ′ ( α *) → α * wants: ρ + : something → P( α *) ρ * can fi ll in strictly positive positions exists : ∀α .( α→ 2) → list( α ) → 2 something: everything else 21

Killing the Strictly Positive α − = 1 K − = K (A → B) − = A → B − (A×B) − = A − × B − (A ∪ B) − = A − ∪ B − (A+B) − = A − + B − ( μβ .A) − = μβ .A − ρ + : P − ( α *) → P( α *) residual full 22

Corollary of Upgraded Conjecture p : ∀α . (F( α ) →α )×(G( α ) → K) → H( α ) = ((F( α ) →α ) × (G( α ) → K)) − = (F( α ) → 1) × (G( α ) → K) ≅ 1 ρ + (_, g) ≈ ( ρ *, g) with appropriate rules (omitted) ρ + puts ρ * at the right places 23

Problems: Suboptimal Types exists : ∀α .( α→ 2) → list( α ) → 2 α * = ℕ , but α = 2 with id seems better similarly “all”, “toString”, “ fi lter”, etc. 24

Problems: Suboptimal Types exists : ∀α .( α→ 2) → list( α ) → 2 α * = ℕ , but α = 2 with id seems better similarly “all”, “toString”, “ fi lter”, etc. length : ∀α .list( α ) → ℕ α * = ℕ , but α = 1 su ffi ces 24

Problems: Suboptimal Types exists : ∀α .( α→ 2) → list( α ) → 2 α * = ℕ , but α = 2 with id seems better similarly “all”, “toString”, “ fi lter”, etc. length : ∀α .list( α ) → ℕ α * = ℕ , but α = 1 su ffi ces Current: full distinctiveness Next: inspectability 24

Future Work 25

Future Work Conjectures → Theorems 25

Future Work Conjectures → Theorems Greatest fi xed points, etc. 25

Future Work Conjectures → Theorems Greatest fi xed points, etc. Unknown datatypes, not just functions Checker for parts of CMU 15-210 in 2015 General theory under development 25

Some Related Work Testing Polymorphic Properties Jean-Philippe Bernardy, Patrik Jansson, and Koen Claessen Categories of Containers Michael Abbott, Thorsten Altenkirch, and Neil Ghani A semantics for shape C. Barry Jay Species and Functors and Types, Oh My! Brent A. Yorgey 26

Unlike derivatives, logarithms di ff erentiate. credit to Danny Gratzer 27