Signatures – Example

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Synthesis Jan Bessai

Algebra

Signatures Terms

Synthesis

Type System Sort Translations Inhabitation Target Translation

Demo (Scala) Conclusion References

Synthesis

Jan Bessai

2017-06-21

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Algebra

Signatures Terms

Synthesis

Last Time: Pandoras Box

Real Programming Languages

▶ mostly informal specifications

▶ complex rule systems (ML: approx. 200 rules)

▶ lots and lots of variation (c.f. Pure Type Systems)

▶ sometimes ”broken”

(c.f. Java Unsoundness [1])

F.S.Church - Opening up Pandoras Box, source: wikimedia.org

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Today: Equipment for Ghostbusters

source: wikimedia.org

▶ Language Independence with Algebra

▶ mathematical abstraction of interfaces

▶ implementation in any target language

▶ Synthesis

▶ pick an inhabitation friendly target language

▶ translate results

▶ Demo

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Algebra

Signatures Terms

Synthesis

Signatures – Formal Interfaces

Definition (Term Signatures)

Given a map 𝕊 ∶ Set → Setand a finite set of variables𝕍, the term signature Σ_𝕊,𝕍= (𝕆,arity,domain,range)is defined by:

▶ A set𝕆 of operation symbols

▶ A function arity∶ 𝕆 → ℕ⁰

▶ A family domain∶ (∏^{arity(𝑜𝑝)}_𝑖=1 𝕊(𝕍))_{𝑜𝑝∈𝕆}

▶ A function range∶ 𝕆 → 𝕊(𝕍)

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Signatures – Example

Informally: Σ_𝕊,𝕍 = {allUsers∶ List(String);

newUser∶ String;

allUserIds∶ List(int);

newUserId∶ int;

append∶ (𝛼, List(𝛼)) ↠ List(𝛼);

first∶ List(𝛼) ↠ 𝛼 } Or formally:

▶ 𝕍 ∶ Set 𝕍 = {𝛼}

▶ 𝕊 ∶ Set → Set

𝕊(𝑋) = 𝑋 ∪ {String, int} ∪ {List(𝑥) ∣ 𝑥 ∈ 𝕊(𝑋)}

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Signatures – Example

newUser∶ String;

newUserId∶ int;

▶ 𝕆 ∶ Set

𝕆 = {allUsers,newUser,allUserIds, newUserId,append,first}

▶ arity∶ 𝕆 → ℕ⁰

arity= {allUsers↦ 0,newUser↦ 0,allUserIds↦ 0, newUserId↦ 0,append↦ 2,first↦ 1}

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Signatures – Example

newUser∶ String;

newUserId∶ int;

▶ domain∶ (∏^{arity(𝑜𝑝)}

𝑖=1 𝕊(𝕍))_{𝑜𝑝∈𝕆}

domain= {allUsers↦ (),newUser↦ (), allUserIds↦ (),newUserId↦ (), append↦ (𝛼, List(𝛼)),first↦ List(𝛼)}

▶ range∶ 𝕆 → 𝕊(𝕍)

range= {allUsers↦ List(String),newUser↦ String, allUserIds↦ List(int),newUserId↦ int, append↦ List(𝛼),first↦ 𝛼}

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Signatures – Substitutions

Definition (Substitution)

Given a map 𝕊 ∶ Set → Setand a set of variables 𝕍, we lift substitutions 𝑆 ∶ 𝕍 → 𝕊(∅) to𝑆⁺∶ 𝕊(𝕍) → 𝕊(∅) by applying a substitution map _⁺∶ (𝕍 → 𝕊(∅)) → 𝕊(𝕍) → 𝕊(∅).

▶ We call𝑠 ∈ 𝕊(∅)a closed sort and require closed sorts to be countable.

▶ Open sorts𝑠 ∈ 𝕊(𝕍) are used to provide parametric polymorphism and closed sorts can be created from open sorts by applying substitutions.

▶ Example:

▶ 𝕍 ∶ Set 𝕍 = {𝛼}

▶ 𝕊 ∶ Set → Set

𝕊(𝑋) = 𝑋 ∪ {String, int} ∪ {List(𝑥) ∣ 𝑥 ∈ 𝕊(𝑋)}

▶ _⁺∶ (𝕍 → 𝕊(∅)) → 𝕊(𝕍) → 𝕊(∅)

𝑆⁺= { 𝛼 ↦ 𝑆(𝛼), List(𝑥) ↦ List(𝑆⁺(𝑥)), String ↦ String, int ↦ int}

▶ 𝑆 ∶ 𝕍 → 𝕊(∅) 𝑆(𝛼) = int

▶ 𝑆⁺(List(List(𝛼)) = List(List(𝑖𝑛𝑡))) _{6 / 27}

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Signatures – Extensions

▶ subsorting: define a preorder≤_𝕊⊆ 𝕊(∅) × 𝕊(∅) on closed sorts

▶ reflexivity: 𝑠₁≤_𝕊 𝑠₁

▶ transitivity: 𝑠₁≤_𝕊 𝑠₂and𝑠₂≤_𝕊𝑠₃ imply𝑠₁≤_𝕊 𝑠₃

▶ well-formed substitution spaces: restrict the space of valid substitions, s.t. 𝑆 ∈WF⊆ 𝕍 → 𝕊(∅)

Example:

𝕊(𝑋) = 𝑋 ∪ {Odd, Even, Nat} 𝕍 = {𝛼, 𝛽}

Σ_𝕊,𝕍 = {one∶ Odd;

succ∶ 𝛼 ↠ 𝛽 }

▶ Even ≤_𝕊Natand Odd ≤_𝕊Nat

▶ WF⊆ 𝕍 → 𝕊(∅) WF= { {𝛼 ↦ Odd, 𝛽 ↦ Even}, {𝛼 ↦ Even, 𝛽 ↦ Odd}

{𝛼 ↦ Nat, 𝛽 ↦ Nat}}

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Algebra

Signatures Terms

Synthesis

Connecting Interface and Implementation

Definition (Subsorted Σ_𝕊,𝕍-Algebra)

Given a term signature Σ_𝕊,𝕍= (𝕆,arity,domain,range), a sort preorder ≤_𝕊 and a function spaceWF⊆ 𝕍 → 𝕊(∅)of well-formed substitutions, a subsorted Σ_𝕊,𝕍-Algebra is

▶ A carrierℂ, which is a family of sets indexed by closed sorts𝑠 ∈ 𝕊(∅)and

▶ A𝕊(∅) indexed family of morphisms𝑚_𝑠∶ 𝔽_𝑠(ℂ) → ℂ_𝑠 where

𝔽_𝑠(ℂ) = ∑

𝑆∈WF

∑

𝑜∈ranged(𝑠,𝑆) arity(𝑜)

∏

𝑖=1

ℂ_𝑆+(𝜋_𝑖(domain_𝑜))

with ranged∶ 𝕊(∅) ×WF→ Setdefined by ranged(𝑠, 𝑆) = {𝑜 ∈ 𝕆 ∣ 𝑆⁺(range(𝑜)) ≤_𝕊𝑠}.

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Synthesis

Algebra – Example (Carrier)

Σ_𝕊,𝕍= {allUsers∶ List(String);

newUser∶ String;

newUserId∶ int;

first∶ List(𝛼) ↠ 𝛼 } ℂ_𝑠=

{exp∣ expis a Java expression s.t. program

import java.util.List;

public class C { private translate(𝑠) x = exp; }

compiles for

translate = { int ↦int, String ↦String, List(𝑥) ↦List<translate(𝑥)>}}

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Algebra

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Synthesis

Algebra – Example (morphisms)

newUser∶ String;

newUserId∶ int;

first∶ List(𝛼) ↠ 𝛼 } 𝑚_𝑠∶ 𝔽_𝑠(ℂ) → ℂ_𝑠 where

𝔽_𝑠(ℂ) = ∑_𝑆∈WF ∑𝑜∈ranged(𝑠,𝑆)∏^arity(𝑜)_𝑖=1 ℂ_𝑆+(𝜋_𝑖(domain_𝑜))

with ranged(𝑠, 𝑆) = {𝑜 ∈ 𝕆 ∣ 𝑆⁺(range(𝑜)) ≤_𝕊 𝑠}

▶ 𝑚List(String)(𝑆,allUsers, ()) =

new java.util.Arrays.asList("bob", "jack")

▶ 𝑚List(String)({𝛼 ↦ String},append, (𝑥, 𝑙)) = l.add(x)

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Algebra

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Synthesis

Algebra – Example (morphisms)

newUser∶ String;

newUserId∶ int;

𝔽_𝑠(ℂ) = ∑_𝑆∈WF ∑𝑜∈ranged(𝑠,𝑆)∏^arity(𝑜)_𝑖=1 ℂ_𝑆+(𝜋_𝑖(domain_𝑜))

▶ 𝑚_List(int)(𝑆,allUserIds, ()) =

new java.util.Arrays.asList(1, 2)

▶ 𝑚_List(int)({𝛼 ↦ int},append, (𝑥, 𝑙)) = l.add(x)

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Algebra – Example (morphisms)

newUser∶ String;

newUserId∶ int;

𝔽_𝑠(ℂ) = ∑

𝑆∈WF ∑

𝑜∈ranged(𝑠,𝑆)∏^arity(𝑜)

𝑖=1 ℂ_𝑆+(𝜋_𝑖(domain_𝑜))

▶ 𝑚_String(𝑆,newUser, ()) ="trudy"

▶ 𝑚_String({𝛼 ↦ String},first, 𝑙) =l.get(0)

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Algebra – Example (morphisms)

newUser∶ String;

newUserId∶ int;

𝔽_𝑠(ℂ) = ∑

𝑆∈WF ∑

𝑜∈ranged(𝑠,𝑆)∏^arity(𝑜)

𝑖=1 ℂ_𝑆+(𝜋_𝑖(domain_𝑜))

▶ 𝑚_int(𝑆,newUserId, ()) =3

▶ 𝑚_int({𝛼 ↦ int},first, 𝑙) =l.get(0)

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Synthesis

Algebra – Language Independence

By exchanging the carrierℂ we could have compiledΣ_𝕊,𝕍 to any other language. Even untyped languages are no

problem, e.g. unary natural numbers:

Σ_𝕊,𝕍 = {one∶ Odd;

succ∶ 𝛼 ↠ 𝛽 }

ℂ_Odd∋ 𝑜 ∶∶=I∣I𝑒; ℂ_Even∋ 𝑒 ∶∶=I𝑜; ℂ_Nat ∋ 𝑛 ∶∶= 𝑒 ∣ 𝑜 𝑚_Even({𝛼 ↦ Odd, 𝛽 ↦ Even},succ, 𝑜) =I𝑜

𝑚_Odd(𝑆,one, ()) =I

𝑚_Odd({𝛼 ↦ Even, 𝛽 ↦ Odd},succ, 𝑒) =I𝑒 𝑚_Nat = 𝑚_Even∪ 𝑚_Odd

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Synthesis – Goal

Given a signatureΣ_(𝕍,𝕊) (an interface), a well-formedness restriction WF, a subsort relation≤_𝕊, an algebra (𝔻, ℎ)(an implementation), and a sort 𝑠(a goal), find all elements of 𝔻_𝑠 (programs), which can be generated from the signature only using ℎ.

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Synthesis – Battle Plan

1. Pick a well-understood small type system

▶ Combinatory Logic with Intersection Types, Distributing Covariant Constructors and Finite Substitution Spaces 2. For the desired𝕊 find injective maps

J_K∶ (𝕊(𝑉 ) → 𝕋_𝑉)_{𝑉 ∈{𝕍,∅}}, from open and closed sorts to types𝕋_∅ and type schemas 𝕋_𝕍, such that

▶ the maps respect subsorting/subtyping

▶ commute with substitutions on sorts/type schemas

▶ generated type (schemas) use∩only in argument positioins of arrows

▶ top-level arrows of generated type (schemas) are protected by type-constructors

3. GenerateΓ with entries

op∶J𝑠₁K𝕍 →J𝑠₂K𝕍→ … →J𝑠K𝕍 for each op∶ (𝑠₁, 𝑠₂, … , 𝑠_𝑛) ↠ 𝑠 ofΣ_(𝕊,𝕍)

4. Use inhabitationΓ ⊢ ? ∶J𝑠K∅to produce terms of carrier ℂ_𝑠= {𝑀 ∣ Γ ⊢ 𝑀 ∶J𝑠K∅}

5. Translate results to desired target carrier 𝔻_𝑠

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Combinatory Logic with Intersection Types

Γ(𝑥) = 𝜏 𝑆 ∈WF

(Var) Γ ⊢ 𝑥 ∶ 𝑆^∗(𝜏 )

Γ ⊢ 𝑀 ∶a→b Γ ⊢ 𝑁 ∶a (→E) Γ ⊢ (𝑀 𝑁 ) ∶b

Γ ⊢ 𝑀 ∶a a≤b Γ ⊢ 𝑀 ∶b (≤)

Γ ⊢ 𝑀 ∶a Γ ⊢ 𝑀 ∶b (∩I) Γ ⊢ 𝑀 ∶a∩b

▶ _^∗∶ (𝕍 → 𝕋_∅) → (𝕋_𝕍→ 𝕋_∅) lifts substitutions to type schemas, just like _⁺∶ (𝕍 → 𝕊_∅) → (𝕊_𝕍→ 𝕊_∅) on sorts.

▶ WFis a translation of the well-formed space we specify with our signature

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Intersection Types and Distributing Covariant Constructors

𝐶₁≤_ℂ𝐶₂ |𝐶₁| = |𝐶₂| ∀𝑛 ∶a_𝑛≤b_𝑛

(CAx) 𝐶₁(a₁,a₂, … ,a_|𝐶₁_|) ≤ 𝐶₂(b₁,b₂, … ,b_|𝐶₂_|)

(≤ 𝜔)

a≤ 𝜔 (→ 𝜔)

𝜔 ≤ 𝜔 → 𝜔

(CDist) 𝐶₁(a₁,a₂, … ,a_|𝐶₁_|) ∩ 𝐶₁(b₁,b₂, … ,b_|𝐶₁_|) ≤

𝐶₁(a₁∩b₁,a₂∩b₂, … ,a_|𝐶₁_|∩b_|𝐶₁_|) a₂≤a₁ b₁≤b₂

(Sub) a₁→b₁≤a₂→b₂

(Dist) (a→b₁) ∩ (a→b₂) ≤a→b₁∩b₂

(Idem)

a≤a∩a a₁≤a₂ a₂≤a₃

(Trans) a₁≤a₃

a≤b₁ a≤b₂

(GLB) a≤b₁∩b₂

(LUB₁)

a∩b≤a (LUB₂)

a∩b≤b 18 / 27

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Synthesis

Constructors – Examples

(CDist) Pair(URL, Protocol) ∩ Pair(DatabaseLocation, Network) ≤

Pair(URL ∩ DatabaseLocation, Protocol ∩ Network)

List ≤_ℂList |List| = 1

Even ≤_ℂNat |Even| = 0 = |Nat|

(CAx) Even ≤ Nat

(CAx) List(Even) ≤ List(Nat)

Exercise:

Show 𝜔 ≤a→ 𝜔 and

(a₁→b₁) ∩ (a₂→b₂) ≤a₁∩a₂→b₁∩b2

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Sort Translations

We can find the sort translation J_K for all sorts isomorphic to regular trees:

𝔗_𝕍 ∋ 𝑡₁, 𝑡₂… , 𝑡_𝑛∶∶= 𝑙(𝑡₁, … , 𝑡_arity(𝑙)) ∣ 𝛼 respecting variance labeled subtyping:

𝑙₁≤_𝕃𝑙₂ arity(𝑙₁) = arity(𝑙₂)

∀𝑛 ∶ v_𝑙₁(𝑛) = v_𝑙₂(𝑛) 𝑡_𝑛≤_𝔗𝑡^′_𝑛 ifv_𝑙₁(𝑛) =co 𝑡^′_𝑛≤_𝔗𝑡_𝑛ifv_𝑙₁(𝑛) =contra 𝑡_𝑛≤_𝔗𝑡^′_𝑛∧ 𝑡^′_𝑛≤_𝔗𝑡_𝑛ifv_𝑙

1(𝑛) =in 𝑙₁(𝑡₁, … , 𝑡_𝑛) ≤_𝔗𝑙₂(𝑡^′₁, … , 𝑡^′_𝑛)

Subtyping example:

arity(Arrow) = 2, v_Arrow(0) =contra, v_Arrow(1) =co Arrow(Nat, Even) ≤_𝔗 Arrow(Even, Nat)

Translation idea:

▶ J𝑙(𝑡₁, 𝑡₂, … 𝑡_𝑛)K𝕍 =

■((𝑙(J𝑡₁K^v^𝑙,J𝑡₂K^v^𝑙, … ,J𝑡_𝑛K^v^𝑙) → •) → •)

▶ J𝑡K^co= (J𝑡K𝕍→ •) → •J𝑡K^contra= (• →J𝑡K𝕍) → • J𝑡Kⁱⁿ = (J𝑡K𝕍 →J𝑡K𝕍) → •

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Inhabitation – Combinatory Logic with Intersection Types

For finite well-formed substitution spaces we have

▶ Type checking (Γ ⊢ 𝑀 ∶a?) is decidable

▶ Type inhabitation (Γ ⊢ ? ∶a) is decidable

▶ the system is a slight extension of [3]

▶ we also get a grammar describing all inhabitants

▶ When we have specifications Γ₁,WF₁,a and Γ₂,WF₂,busing completely unrelated constructor symbols we can use

Γ(𝑥) = Γ₁(𝑥) ∩ Γ₂(𝑥),WF=WF₁⊎WF₂ and get Γ ⊢ 𝑀 ∶a∩b⇔ Γ₁⊢ 𝑀 ∶aand Γ₂⊢ 𝑀 ∶b

▶ we can refine the search for inhabitants using multiple specifications in addition to signatures

▶ warning: this only holds for unrelated constructor symbols!

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Target Translation

Fs(D) Cs Ds

hs

fs

▶ We want a function 𝑓_𝑠to translate inhabitation results ℂ_𝑠= {𝑀 ∣ Γ ⊢ 𝑀 ∶J𝑠K∅} to some target language carrier𝔻_𝑠, for which we have an algebraℎ_𝑠

▶ We can proof that there is a coalgebra 𝑚⁻¹_𝑠 ∶ ℂ_𝑠→ 𝔽_𝑠(ℂ):

▶ for a term𝑀 =op𝑀₁𝑀₂… 𝑀_{arity(𝑜𝑝)} with typeJ𝑠K∅

we get𝑚⁻¹_𝑠 (𝑀) = (𝑆,op, (𝑀₁, 𝑀₂, … , 𝑀_{arity(𝑜𝑝)}))

▶ We construct𝑓_𝑠by solving the fixpoint equation: 𝑓 = (ℎ_𝑠∘ 𝔽(𝑓)_𝑠∘ 𝑚⁻¹_𝑠 )

𝑠∈𝕊(∅)

▶ base case: the size of𝑀 is 1, hence𝑀 =op,

arity(op) = 0and𝑚⁻¹_𝑠 (𝑀) = (𝑆,op, ()): solve byℎ_𝑠.

▶ recursion: the size is𝑛 + 1we have

𝑚⁻¹_𝑠 (𝑀) = (𝑆,op, (𝑀₁, 𝑀₂, … , 𝑀_arity(op)))with smaller𝑀_𝑖: solve byℎ_𝑠 and𝑓_𝑠 on smaller parts.

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Synthesis

Target Translation

Fs(C) Fs(D) Cs Ds

hs

m⁻¹_s

fs

▶ We construct𝑓_𝑠by solving the fixpoint equation: 𝑓 = (ℎ_𝑠∘ 𝔽(𝑓)_𝑠∘ 𝑚⁻¹_𝑠 )

𝑠∈𝕊(∅)

𝑚⁻¹_𝑠 (𝑀) = (𝑆,op, (𝑀₁, 𝑀₂, … , 𝑀_arity(op)))with smaller𝑀_𝑖: solve byℎ_𝑠 and𝑓_𝑠 on smaller parts.

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Synthesis

Target Translation

Fs(C) Fs(D) Cs Ds

F(f)_s hs

m⁻¹_s

fs

▶ We construct𝑓_𝑠 by solving the fixpoint equation:

𝑓 = (ℎ_𝑠∘ 𝔽(𝑓)_𝑠∘ 𝑚⁻¹_𝑠 )

𝑠∈𝕊(∅)

𝑚⁻¹_𝑠 (𝑀) = (𝑆,op, (𝑀₁, 𝑀₂, … , 𝑀_arity(op)))with

smaller𝑀_𝑖: solve byℎ_𝑠 and𝑓_𝑠 on smaller parts. 22 / 27

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Target Translation – Properties

Fs(C) Fs(D) Cs Ds

F(f)s

h_s m⁻¹_s

f_s

The construction is (up to choice of well-formed sort substitutions):

▶ Sound: all generated𝔻_𝑠 terms areℎ_𝑠 interpretations of signature operations

▶ Complete: by listing allℂ_𝑠 we get all 𝔻_𝑠 which areℎ_𝑠 interpretations of signature operations

▶ Unique: all possible ways of constructing a translation 𝑓_𝑠 yield identical results to our construction

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Synthesis

Related Approaches

▶ Traditionally, multi-sorted signatures don’t use sub-typing and variables

▶ if we leave them out, the signature specifications are compatible e.g. with those in lectures by Prof.

Padawitz [4]

▶ the fixpoint constrution of𝑓_𝑠 is related to Lambek’s Lemma ([6], Lemma 1)

▶ Without additions one can also use SMT Solving [5]

▶ An ad-hoc solving approach was used in [2]

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Demo (Scala)

Conclusion References

Demo (Scala)

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Conclusion

▶ Signatures are generic interface descriptions and algebras implementations

▶ Algebraic bureaucracy is powerful enough to cut through the programming language chaos

▶ We can connect (almost arbitrary) signatures with combinatory logic

▶ Synthesis can be done outside the ”simple” world of𝜆^→

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References

[1] Nada Amin and Ross Tate.“Java and Scala’s Type Systems are Unsound: The Existential Crisis of Null Pointers”.In:Proceedings of the 2016 ACM SIGPLAN International Conference on Object-Oriented Programming, Systems, Languages, and Applications.

ACM. 2016, pp. 838–848.

[2] Luís Eduardo de Souza Amorim, Sebastian Erdweg, Guido Wachsmuth, and Eelco Visser.“Principled Syntactic Code Completion Using Placeholders”.In:

Proceedings of the 2016 ACM SIGPLAN International Conference on Software Language Engineering. SLE 2016. Amsterdam, Netherlands: ACM, 2016, pp. 163–175.isbn: 978-1-4503-4447-0.doi:

10.1145/2997364.2997374.url:

http://doi.acm.org/10.1145/2997364.2997374.

[3] Boris Düdder, Moritz Martens, Jakob Rehof, and Pawel Urzyczyn.“Bounded Combinatory Logic”.In:

Computer Science Logic (CSL’12) - 26th International Workshop/21st Annual Conference of the EACSL, CSL 2012, September 3-6, 2012, Fontainebleau, France.

2012, pp. 243–258.doi:

10.4230/LIPIcs.CSL.2012.243.url:http:

//dx.doi.org/10.4230/LIPIcs.CSL.2012.243.

[4] Peter Padawitz.“From Grammars and Automata to Algebras and Coalgebras”.In:Algebraic Informatics - 4th International Conference, CAI 2011, Linz, Austria, June 21-24, 2011. Proceedings. 2011, pp. 21–43.doi:

10.1007/978-3-642-21493-6_2.url:http:

//dx.doi.org/10.1007/978-3-642-21493-6_2.

[5] Andrew Reynolds, Viktor Kuncak, Cesare Tinelli, Clark Barrett, and Morgan Deters.“Refutation-based synthesis in SMT”.In:Formal Methods in System Design(Feb. 16, 2017), p. 1.issn: 1572-8102.doi:

10.1007/s10703-017-0270-2.url:

http://dx.doi.org/10.1007/s10703-017-0270-2.

[6] Michael B. Smyth and Gordon D. Plotkin.“The Category-Theoretic Solution of Recursive Domain Equations”.In:SIAM J. Comput.11.4 (1982), pp. 761–783.doi:10.1137/0211062.