Combinatory Logic and Program Synthesis

Jakob Rehof

Technical University of Dortmund

Joint work w. B. D¨ udder, M. Martens (Dortmund) and P. Urzyczyn (Warsaw) and special thanks to Roger Hindley and the Torino λ-calculus group

September 2014

Combinatory Logic Synthesis (CLS)

A type-theoretic approach to component-oriented synthesis

Bottom-up specification Hoare logic

Classification Taxonomy

…

Types Component-oriented Synthesis

Synthesis relative to library (repository) of components Combinatory Logic Synthesis (CLS)

Libraries need classification systems to enable retrieval and composition

CLS

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References

I (CL)S-Framework: Combinatory Logic Synthesizer. With J. Bessai, A.

Dudenhefner, B. D¨ udder and M. Martens. To appear ISOLA 2014.

I Staged Composition Synthesis. With B. D¨ udder and M. Martens. ESOP 2014.

I Intersection Type Matching with Subtyping. With B. D¨ udder and M. Martens.

TLCA 2013.

I Towards Combinatory Logic Synthesis. BEAT 2013.

I Bounded Combinatory Logic. With B. D¨ udder, M. Martens and P. Urzyczyn.

CSL 2012.

I Using Inhabitation in Bounded Combinatory Logic with Intersection Types for Composition Synthesis. With B. D¨ udder, O.Garbe, M. Martens and P.

Urzyczyn. EPTCS 2012.

I The Complexity of Inhabitation with Explicit Intersection. With P. Urzyczyn.

R.L. Constable and A. Silva (Eds.): Logic and Program Semantics. Essays Dedicated to Dexter Kozen 2012.

I Finite Combinatory Logic with Intersection Types. With P. Urzyczyn. TLCA 2011.

I See also: Dagstuhl Seminar “Design and Synthesis from Components”, Schloss Dagstuhl, June 1–6 2014.

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Roadmap: Three Levels of Computation

L1 L2

CL Combinatory Logic

Compositional Metalanguage

Native Language

Composition-time

Inhabitation Composition-time

Reduction

Run-time

Execution Languages Notions

of computation

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Outline

Background on CLS

Intersection Types for Semantic Specification Bounded Combinatory Logic

Staged Composition Synthesis Conclusion and Future Work

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Outline

Background on CLS

Intersection Types for Semantic Specification Bounded Combinatory Logic

Staged Composition Synthesis Conclusion and Future Work

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Composition Synthesis

I Typed function composition (modus ponens) Γ ` F : τ <sup>0</sup> → τ Γ ` G : τ ⁰

Γ ` (F G ) : τ (→E) as logical model of applicative composition of named component interfaces (X : ρ) ∈ Γ from a repository Γ, satisfying goal τ

I Inhabitation problem as foundation for automatic synthesis:

∃e . Γ ` e : τ ? Notation Γ ` ? : τ

I

Does there exist a program composition e from repository Γ with Γ ` e : τ ? Inhabitation algorithm is used to construct (synthesize) e from Γ and τ

I CLS is inherently component-oriented

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Composition Synthesis

I Typed function composition (modus ponens) Γ ` F : τ <sup>0</sup> → τ Γ ` G : τ ⁰

Γ ` (F G ) : τ (→E) as logical model of applicative composition of named component interfaces (X : ρ) ∈ Γ from a repository Γ, satisfying goal τ

I Inhabitation problem as foundation for automatic synthesis:

∃e . Γ ` e : τ ? Notation Γ ` ? : τ

I

Does there exist a program composition e from repository Γ with Γ ` e : τ ? Inhabitation algorithm is used to construct (synthesize) e from Γ and τ

I CLS is inherently component-oriented

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Composition Synthesis

I Typed function composition (modus ponens) Γ ` F : τ <sup>0</sup> → τ Γ ` G : τ ⁰

Γ ` (F G ) : τ (→E) as logical model of applicative composition of named component interfaces (X : ρ) ∈ Γ from a repository Γ, satisfying goal τ

I Inhabitation problem as foundation for automatic synthesis:

∃e . Γ ` e : τ ? Notation Γ ` ? : τ

I

Does there exist a program composition e from repository Γ with Γ ` e : τ ? Inhabitation algorithm is used to construct (synthesize) e from Γ and τ

I CLS is inherently component-oriented

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Composition Synthesis

I Typed function composition (modus ponens) Γ ` F : τ <sup>0</sup> → τ Γ ` G : τ ⁰

Γ ` (F G ) : τ (→E) as logical model of applicative composition of named component interfaces (X : ρ) ∈ Γ from a repository Γ, satisfying goal τ

I Inhabitation problem as foundation for automatic synthesis:

∃e . Γ ` e : τ ? Notation Γ ` ? : τ

I

Does there exist a program composition e from repository Γ with Γ ` e : τ ? Inhabitation algorithm is used to construct (synthesize) e from Γ and τ

I CLS is inherently component-oriented

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Foundations in Combinatory Logic

Types τ ::= α | τ → τ ⁰ Terms e, e ⁰ ::= X | (e e ⁰ ) Rules

Γ, (X : τ ) ` X : S(τ ) (var)

Γ ` e : τ <sup>0</sup> → τ Γ ` e ⁰ : τ ⁰ Γ ` (e e ⁰ ) : τ (→ E)

Under Curry-Howard isomorphism, Hilbert-style presentation of minimal propositional logic (schematism + modus ponens)

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CL vs λ-calculus

I Fixed base Γ equivalent to λ-calculus:

{S : (α → β → γ) → (α → β) → α → γ, K : α → β → α}

I Inhabitation in SK-calculus is Pspace -complete (Statman)

I Proof/term enumeration (Ben-Yelles, Hindley)

I But fixed base is not the right model for composition synthesis, since repository (Γ) varies

I And λ-calculus as model is not component-oriented as is CL

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CL vs λ-calculus

I Fixed base Γ equivalent to λ-calculus:

{S : (α → β → γ) → (α → β) → α → γ, K : α → β → α}

I Inhabitation in SK-calculus is Pspace -complete (Statman)

I Proof/term enumeration (Ben-Yelles, Hindley)

I But fixed base is not the right model for composition synthesis, since repository (Γ) varies

I And λ-calculus as model is not component-oriented as is CL

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Relativized Inhabitation

I We consider the relativized inhabitation problem:

I

Given Γ and τ , does there exist e such that Γ ` e : τ?

I Relativized inhabitation is much harder

I

Undecidable: Linial-Post theorems, 1948 ff.

I The CLS view: Already in simple types, relativized inhabitation defines a Turing-complete logic programming language for component composition

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Relativized Inhabitation

I We consider the relativized inhabitation problem:

I

Given Γ and τ , does there exist e such that Γ ` e : τ?

I Relativized inhabitation is much harder

I

Undecidable: Linial-Post theorems, 1948 ff.

I The CLS view: Already in simple types, relativized inhabitation defines a Turing-complete logic programming language for component composition

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Relativized Inhabitation

I We consider the relativized inhabitation problem:

I

Given Γ and τ , does there exist e such that Γ ` e : τ?

I Relativized inhabitation is much harder

I

Undecidable: Linial-Post theorems, 1948 ff.

I The CLS view: Already in simple types, relativized inhabitation defines a Turing-complete logic programming language for component composition

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Turing-Completeness of Simple Types!

Two-counter automaton acceptance is undecidable. Two counter automaton A = hQ, q 0 , q _F , δi, control states Q, inital state q 0 , final state q _F , counters c ₁ , c ₂ ∈ N , transition relation δ given by (i = 1, 2):

I q : c _i := c _i + 1; goto p

I q : c _i := c _i − 1; goto p

I q : if (c _i = 0) then goto p else goto r

Configurations C = (q, n, m), q ∈ Q , n and m contents of counters c ₁ resp. c ₂ .

Types of the form [C ] = q → s ⁿ (0) → s ^m (0) will represent configurations C = (q, n, m)

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Encoding of A into Γ _A

I Fin : q _F → α → β

I q : c 1 := c 1 + 1; goto p:

Add 1 [q, p] : p → s(α) → β

→ q → α → β .

I q : c ₁ := c ₁ − 1; goto p:

Sub ₁ [q, p] : p → α → β

→ q → s (α) → β .

I q : if (c 1 = 0) then goto p else goto r :

I

Tst

[q, p] : p → 0 → β

→ q → 0 → β and

I

Tst

[q, r] : r → s(α) → β

→ q → s(α) → β .

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Reduction

Theorem

Let A be a two-counter automaton with initial configuration (q 0 , n 0 , m 0 ). A accepts if and only if there exists a term e with Γ A ` e : q ₀ → s ⁿ

(0) → s ^m

(0).

Lemma

Let C and C ⁰ be configurations in A. We have C → C ⁰ if and only if there is a term e with Γ A ` e : [C ⁰ ] → [C ].

Lemma

Let C be a configuration of A. C leads to acceptance in A if and only if there is a term e with Γ A ` e : [C ].

JR: BEAT 2013

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Reduction

Consider the two-counter automaton A = q 0 : c 1 := c 1 − 1; goto q 1

q 1 : if (c 1 = 0) then goto q _F else goto q 0

from initial state (q ₀ , 1, 0). Since Fin : q _F → 0 → 0

Tst ^Z ₁ [q 1 , q F ] : (q _F → 0 → 0) → (q 1 → 0 → 0) Sub ₁ [q ₀ , q ₁ ] : (q ₁ → 0 → 0) → (q ₀ → s(0) → 0) we get

Γ A ` Sub 1 [q ₀ , q 1 ] (Tst ^Z ₁ [q ₁ , q F ] Fin) : q ₀ → s(0) → 0

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Types as Logic Programs for Composition

I The input repository Γ is a logic program at the level of types

I Each combinator type is a rule in the program

I The inhabitation goal is the input goal to the program

I Search for inhabitants is the execution of the program

I Inhabitants are programs synthesized as solution space to the program

Broadly related (proof search as semantics of generalized logic programming):

D. Miller, G. Nadathur, F. Pfenning, A. Scedrov:Uniform Proofs as a Foundation for Logic Programming, Ann.

Pure App. Logic, 1991

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“Linial-Post Spectrum”

∞

Ptime co-NP Pspace Exptime 2Exptime

. . .

CPL IPL (S4)

T

R

?

Intermediate logics Subintuitionistic logics

R

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Outline

Background on CLS

Intersection Types for Semantic Specification Bounded Combinatory Logic

Staged Composition Synthesis Conclusion and Future Work

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Situation

We are given a repository of component names X _i with associated implementations C _i in a native implementation language L1

X 1 , C 1 : τ 1 , . . . , X n , C n : τ n

and an associated combinatory type environment C = { X 1 : φ ₁ , . . . , X n : φ _n } where φ i are enriched interface types with (φ i ) ^◦ ≡ τ i . We ask for combinatory compositions e such that

C ` e : φ and `

e [C _i /X _i ] : (φ) ^◦ The condition `

e [C i /X i ] : (φ) ^◦ is implementation type correctness.

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Situation

We are given a repository of component names X _i with associated implementations C _i in a native implementation language L1

X 1 , C 1 : τ 1 , . . . , X n , C n : τ n

and an associated combinatory type environment C = { X 1 : φ ₁ , . . . , X n : φ _n } where φ i are enriched interface types with (φ i ) ^◦ ≡ τ i . We ask for combinatory compositions e such that

C ` e : φ and `

e [C _i /X _i ] : (φ) ^◦ The condition `

e [C i /X i ] : (φ) ^◦ is implementation type correctness.

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Intersection Types for Semantic Specification

Semantic types t ::= a | t → t

| t ∩ t

Interface types φ ::= τ | φ∩t | φ → φ

| φ∩φ

Combinatory Logic with intersection types:

C , X : φ `

X : S(φ) (var) C `

e : φ → φ

C `

e

: φ C `

(e e

) : φ

(→E) C `

e : φ C `

e : φ

C `

e : φ∩φ

(∩I) C `

e : φ φ ≤ φ

C `

e : φ

(≤) (elimination rule contained in φ

∩ φ

≤ φ

).

Barendregt-Coppo-Dezani 1983, Dezani-Hindley 1992 Wells-Yakobowski 2005, Haack-Howard-Stoughton-Wells 2002

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Intersection Types for Semantic Specification

Semantic types t ::= a | t → t

| t ∩ t

Interface types φ ::= τ | φ∩t | φ → φ

| φ∩φ

Combinatory Logic with intersection types:

C , X : φ `

X : S(φ) (var) C `

e : φ → φ

C `

e

: φ C `

(e e

) : φ

(→E) C `

e : φ C `

e : φ

C `

e : φ∩φ

(∩I) C `

e : φ φ ≤ φ

C `

e : φ

(≤) (elimination rule contained in φ

∩ φ

≤ φ

).

Barendregt-Coppo-Dezani 1983, Dezani-Hindley 1992 Wells-Yakobowski 2005, Haack-Howard-Stoughton-Wells 2002

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Example Repository

Γ = {

O : TrObj

Tr : TrObj → D((R, R), R, R) pos : D((R, R), R, R) → ((R, R), R) cdn : ((R, R), R) → (R, R)

fst : (R, R) → R snd : (R, R) → R

tmp : D((R, R), R, R) → R cc2pl : ((R, R), R) → ((R, R), R) cl2fh : R → R

}

JR:Towards Combinatory Logic Synthesis, BEAT’13

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Semantic Type Structure

Trackdata

Pos Temp

Coord Time Cel Fh

Cart Polar Gpst Utc

Cx Cy Radius Angle

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Semantic Repository

C = {

O : TrObj

Tr : TrObj → D((R, R)∩Cart, R∩Gpst, R∩Cel) pos : D((R, R)∩a, R ∩a ⁰ , R) → ((R, R)∩a, R ∩a ⁰ )∩Pos cdn : ((R, R)∩a, R)∩Pos → (R, R)∩a

fst : ((R, R)∩Coord → R)∩

(Cart → Cx)∩(Polar → Radius) snd : ((R, R)∩Coord → R)∩

(Cart → Cy )∩(Polar → Angle) tmp : D((R, R), R, R ∩a) → R ∩a

cc2pl : (R, R)∩Cart → (R, R)∩Polar cl2fh : R∩Cel → R∩Fh

}

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Composition Synthesis via Inhabitation

C = {

O : TrObj

Tr : TrObj → D((R,R)∩Cart,R∩Gpst, R∩Cel) pos : D((R, R)∩a, R∩a

, R) → ((R,R)∩a, R∩a

)∩Pos cdn : ((R,R)∩a, R)∩Pos → (R, R)∩a

fst : ((R,R)∩Coord → R)∩

(Cart → Cx)∩(Polar → Radius ) snd : ((R,R)∩Coord → R)∩

(Cart → Cy)∩(Polar → Angle) tmp : D((R, R), R, R∩a) → R∩a cc2pl : (R,R)∩Cart → (R,R)∩Polar cl2fh : R∩Cel → R∩Fh

}

C `

? : R∩Fh C `

cl2fh (tmp (Tr O)) : R∩Fh

C `

? : R∩Radius C `

fst (cc2pl (cdn (pos (Tr O)))) : R∩Radius

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Composition Synthesis via Inhabitation

C = {

O : TrObj

Tr : TrObj → D((R,R)∩Cart,R∩Gpst, R∩Cel) pos : D((R, R)∩a, R∩a

, R) → ((R,R)∩a, R∩a

)∩Pos cdn : ((R,R)∩a, R)∩Pos → (R, R)∩a

fst : ((R,R)∩Coord → R)∩

(Cart → Cx)∩(Polar → Radius ) snd : ((R,R)∩Coord → R)∩

(Cart → Cy)∩(Polar → Angle) tmp : D((R, R), R, R∩a) → R∩a cc2pl : (R,R)∩Cart → (R,R)∩Polar cl2fh : R∩Cel → R∩Fh

}

C `

? : R∩Fh C `

cl2fh (tmp (Tr O)) : R∩Fh

C `

? : R∩Radius C `

fst (cc2pl (cdn (pos (Tr O)))) : R∩Radius

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Composition Synthesis via Inhabitation

C = {

O : TrObj

Tr : TrObj → D((R,R)∩Cart,R∩Gpst, R∩Cel) pos : D((R, R)∩a, R∩a

, R) → ((R,R)∩a, R∩a

)∩Pos cdn : ((R,R)∩a, R)∩Pos → (R, R)∩a

fst : ((R,R)∩Coord → R)∩

(Cart → Cx)∩(Polar → Radius ) snd : ((R,R)∩Coord → R)∩

(Cart → Cy)∩(Polar → Angle) tmp : D((R, R), R, R∩a) → R∩a cc2pl : (R,R)∩Cart → (R,R)∩Polar cl2fh : R∩Cel → R∩Fh

}

C `

? : R∩Fh C `

cl2fh (tmp (Tr O)) : R∩Fh

C `

? : R∩Radius C `

fst (cc2pl (cdn (pos (Tr O)))) : R∩Radius

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Composition Synthesis via Inhabitation

C = {

O : TrObj

Tr : TrObj → D((R,R)∩Cart,R∩Gpst, R∩Cel) pos : D((R, R)∩a, R∩a

, R) → ((R,R)∩a, R∩a

)∩Pos cdn : ((R,R)∩a, R)∩Pos → (R, R)∩a

fst : ((R,R)∩Coord → R)∩

(Cart → Cx)∩(Polar → Radius ) snd : ((R,R)∩Coord → R)∩

(Cart → Cy)∩(Polar → Angle) tmp : D((R, R), R, R∩a) → R∩a cc2pl : (R,R)∩Cart → (R,R)∩Polar cl2fh : R∩Cel → R∩Fh

}

C `

? : R∩Fh C `

cl2fh (tmp (Tr O)) : R∩Fh

C `

? : R∩Radius C `

fst (cc2pl (cdn (pos (Tr O)))) : R∩Radius

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Composition Synthesis via Inhabitation

C = {

O : TrObj

Tr : TrObj → D((R,R)∩Cart,R∩Gpst, R∩Cel) pos : D((R, R)∩a, R∩a

, R) → ((R,R)∩a, R∩a

)∩Pos cdn : ((R,R)∩a, R)∩Pos → (R, R)∩a

fst : ((R,R)∩Coord → R)∩

(Cart → Cx)∩(Polar → Radius ) snd : ((R,R)∩Coord → R)∩

(Cart → Cy)∩(Polar → Angle) tmp : D((R, R), R, R∩a) → R∩a cc2pl : (R,R)∩Cart → (R,R)∩Polar cl2fh : R∩Cel → R∩Fh

}

C `

? : R∩Fh C `

cl2fh (tmp (Tr O)) : R∩Fh

C `

? : R∩Radius C `

fst (cc2pl (cdn (pos (Tr O)))) : R∩Radius

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Outline

Background on CLS

Intersection Types for Semantic Specification Bounded Combinatory Logic

Staged Composition Synthesis Conclusion and Future Work

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Bounded Combinatory Logic bcl ^k (→, ∩)

• Levels

`(a) = 0, for a ∈ A ;

`(τ → σ) = 1 + max{`(τ ), `(σ)};

`( T

i=1

τ

) = max{`(τ

) | i = 1, . . . , n}.

`(S) = max{`(S(α)) | S(α) 6= α}

• bcl

(→, ∩), k ≥ 0 (and finite CL, fcl , with S = id ).

[`(S) ≤ k]

Γ, X : τ `

X : S(τ ) (var) Γ `

e : τ → σ Γ `

e

: τ Γ `

(e e

) : σ (→E)

Γ `

e : τ Γ `

e : σ

Γ `

e : τ ∩ σ (∩I) Γ `

e : τ τ ≤ σ Γ `

e : σ (≤)

D¨udder, Martens, Rehof, Urzyczyn CSL 2012

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Bounded Combinatory Logic bcl ^k (→, ∩)

• Levels

`(a) = 0, for a ∈ A ;

`(τ → σ) = 1 + max{`(τ ), `(σ)};

`( T

i=1

τ

) = max{`(τ

) | i = 1, . . . , n}.

`(S) = max{`(S(α)) | S(α) 6= α}

• bcl

(→, ∩), k ≥ 0 (and finite CL, fcl , with S = id ).

[`(S) ≤ k]

Γ, X : τ `

X : S(τ ) (var) Γ `

e : τ → σ Γ `

e

: τ Γ `

(e e

) : σ (→E)

Γ `

e : τ Γ `

e : σ

Γ `

e : τ ∩ σ (∩I) Γ `

e : τ τ ≤ σ Γ `

e : σ (≤)

D¨udder, Martens, Rehof, Urzyczyn CSL 2012

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Complexity for Finite and Bounded CL

Theorem (RU TLCA 2011) For finite combinatory logic fcl :

1. Relativized inhabitation in fcl (→) is in Ptime

2. Relativized inhabitation in fcl(→, ∩) is Exptime-complete Theorem (DMRU CSL 2012)

For bounded combinatory logic bcl k :

1. Relativized inhabitation in bcl k (→) is Exptime -complete for all k

2. Relativized inhabitation in bcl k (→, ∩) is (k + 2)- Exptime -complete

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Upper Bound ATM for bcl ^k (→, ∩): Aspace (exp _{k +1} (n))

Input : Γ, τ, k Γ = {f : (0 → 1) ∩ (1 → 0),

x : (α → β) → (β → γ) → (α → γ)}

τ = (0 → 0) ∩ (1 → 1) loop :

1 choose (x : σ) ∈ Γ; σ

= (0 → 0) → (0 → 0) → (0 → 0) ∩ · · · ∩ 2 σ

:= T {S(σ) | S ∈ S

}; (1 → 1) → (1 → 1) → (1 → 1) 3 choose m ∈ {0, . . . , kσ

k}; (0 → 1)→(1 → 0)→(0 → 0)∩

4 choose P ⊆ P

(σ

); (1 → 0)→(0 → 1)→(1 → 1) 5 if ( T

π∈P

tgt

(π) ≤ τ) then (0 → 0)∩(1 → 1)≤ τ 6 if (m = 0) then accept ;

7 else

8 forall(i = 1 . . . m)

9 τ := T

π∈P

arg

(π); τ :=(0 → 1)∩(1 → 0) τ :=(1 → 0)∩(0 → 1)

10 goto loop;

11 else reject;

(x f ) f : (0 → 0) ∩ (1 → 1)

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Upper Bound ATM for bcl ^k (→, ∩): Aspace (exp _{k +1} (n))

Input : Γ, τ, k Γ = {f : (0 → 1) ∩ (1 → 0),

x : (α → β) → (β → γ) → (α → γ)}

τ = (0 → 0) ∩ (1 → 1) loop :

1 choose (x : σ) ∈ Γ; σ

= (0 → 0) → (0 → 0) → (0 → 0) ∩ · · · ∩ 2 σ

:= T {S(σ) | S ∈ S

}; (1 → 1) → (1 → 1) → (1 → 1) 3 choose m ∈ {0, . . . , kσ

k}; (0 → 1)→(1 → 0)→(0 → 0)∩

4 choose P ⊆ P

(σ

); (1 → 0)→(0 → 1)→(1 → 1) 5 if ( T

π∈P

tgt

(π) ≤ τ) then (0 → 0)∩(1 → 1)≤ τ 6 if (m = 0) then accept ;

7 else

8 forall(i = 1 . . . m)

9 τ := T

π∈P

arg

(π); τ :=(0 → 1)∩(1 → 0) τ :=(1 → 0)∩(0 → 1)

10 goto loop;

11 else reject;

(x f ) f : (0 → 0) ∩ (1 → 1)

24 / 46

Upper Bound ATM for bcl ^k (→, ∩): Aspace (exp _{k +1} (n))

Input : Γ, τ, k Γ = {f : (0 → 1) ∩ (1 → 0),

x : (α → β) → (β → γ) → (α → γ)}

τ = (0 → 0) ∩ (1 → 1) loop :

1 choose (x : σ) ∈ Γ; σ

= (0 → 0) → (0 → 0) → (0 → 0) ∩ · · · ∩ 2 σ

:= T {S(σ) | S ∈ S

}; (1 → 1) → (1 → 1) → (1 → 1) 3 choose m ∈ {0, . . . , kσ

k}; (0 → 1)→(1 → 0)→(0 → 0)∩

4 choose P ⊆ P

(σ

); (1 → 0)→(0 → 1)→(1 → 1) 5 if ( T

π∈P

tgt

(π) ≤ τ) then (0 → 0)∩(1 → 1)≤ τ 6 if (m = 0) then accept ;

7 else

8 forall(i = 1 . . . m)

9 τ := T

π∈P

arg

(π); τ :=(0 → 1)∩(1 → 0) τ :=(1 → 0)∩(0 → 1)

10 goto loop;

11 else reject;

(x f ) f : (0 → 0) ∩ (1 → 1)

24 / 46

Upper Bound ATM for bcl ^k (→, ∩): Aspace (exp _{k +1} (n))

Input : Γ, τ, k Γ = {f : (0 → 1) ∩ (1 → 0),

x : (α → β) → (β → γ) → (α → γ)}

τ = (0 → 0) ∩ (1 → 1) loop :

1 choose (x : σ) ∈ Γ; σ

= (0 → 0) → (0 → 0) → (0 → 0) ∩ · · · ∩ 2 σ

:= T {S(σ) | S ∈ S

}; (1 → 1) → (1 → 1) → (1 → 1) 3 choose m ∈ {0, . . . , kσ

k}; (0 → 1)→(1 → 0)→(0 → 0)∩

4 choose P ⊆ P

(σ

); (1 → 0)→(0 → 1)→(1 → 1) 5 if ( T

π∈P

tgt

(π) ≤ τ) then (0 → 0)∩(1 → 1)≤ τ 6 if (m = 0) then accept ;

7 else

8 forall(i = 1 . . . m)

9 τ := T

π∈P

arg

(π); τ :=(0 → 1)∩(1 → 0) τ :=(1 → 0)∩(0 → 1)

10 goto loop;

11 else reject;

(x f ) f : (0 → 0) ∩ (1 → 1)

24 / 46

Upper Bound ATM for bcl ^k (→, ∩): Aspace (exp _{k +1} (n))

Input : Γ, τ, k Γ = {f : (0 → 1) ∩ (1 → 0),

x : (α → β) → (β → γ) → (α → γ)}

τ = (0 → 0) ∩ (1 → 1) loop :

1 choose (x : σ) ∈ Γ; σ

= (0 → 0) → (0 → 0) → (0 → 0) ∩ · · · ∩ 2 σ

:= T {S(σ) | S ∈ S

}; (1 → 1) → (1 → 1) → (1 → 1) 3 choose m ∈ {0, . . . , kσ

k}; (0 → 1)→(1 → 0)→(0 → 0)∩

4 choose P ⊆ P

(σ

); (1 → 0)→(0 → 1)→(1 → 1) 5 if ( T

π∈P

tgt

(π) ≤ τ) then (0 → 0)∩(1 → 1)≤ τ 6 if (m = 0) then accept ;

7 else

8 forall(i = 1 . . . m)

9 τ := T

π∈P

arg

(π); τ :=(0 → 1)∩(1 → 0) τ :=(1 → 0)∩(0 → 1)

10 goto loop;

11 else reject;

(x f ) f : (0 → 0) ∩ (1 → 1)

24 / 46

Upper Bound ATM for bcl ^k (→, ∩): Aspace (exp _{k +1} (n))

Input : Γ, τ, k Γ = {f : (0 → 1) ∩ (1 → 0),

x : (α → β) → (β → γ) → (α → γ)}

τ = (0 → 0) ∩ (1 → 1) loop :

1 choose (x : σ) ∈ Γ; σ

= (0 → 0) → (0 → 0) → (0 → 0) ∩ · · · ∩ 2 σ

:= T {S(σ) | S ∈ S

}; (1 → 1) → (1 → 1) → (1 → 1) 3 choose m ∈ {0, . . . , kσ

k}; (0 → 1)→(1 → 0)→(0 → 0)∩

4 choose P ⊆ P

(σ

); (1 → 0)→(0 → 1)→(1 → 1) 5 if ( T

π∈P

tgt

(π) ≤ τ) then (0 → 0)∩(1 → 1)≤ τ 6 if (m = 0) then accept ;

7 else

8 forall(i = 1 . . . m)

9 τ := T

π∈P

arg

(π); τ :=(0 → 1)∩(1 → 0) τ :=(1 → 0)∩(0 → 1)

10 goto loop;

11 else reject;

(x f ) f : (0 → 0) ∩ (1 → 1)

24 / 46

Lower bound

Generic reduction by simulation of exp _k+1 (n)-space bounded alternating Turing machines.

Given ATM M, we construct, in polynomial time, an environment Γ such that M is accepting if and only if Γ ` _k ? : Tape is solvable.

25 / 46

Lower bound - main ideas

For any fixed level-parameter K :

I Intersection type numerals hi i _K : we can represent numbers 0 ≤ i ≤ exp _K+1 (n) − 1 as intersection types.

I We can represent ATM configurations C of size exp _K+1 (n) as intersection types [ C ]

I Exploiting K -bounded polymorphism, we can represent these types implicitly in polynomial sized types

I ATM sequences C 1 C 2 · · · C m coded by reverse implications [C i+1 ] → [C i ] in Γ

26 / 46

Lower bound - main ideas

For any fixed level-parameter K :

I Intersection type numerals hi i _K : we can represent numbers 0 ≤ i ≤ exp _K+1 (n) − 1 as intersection types.

I We can represent ATM configurations C of size exp _K+1 (n) as intersection types [ C ]

I Exploiting K -bounded polymorphism, we can represent these types implicitly in polynomial sized types

I ATM sequences C 1 C 2 · · · C m coded by reverse implications [C i+1 ] → [C i ] in Γ

26 / 46

Lower bound - main ideas

For any fixed level-parameter K :

I Intersection type numerals hi i _K : we can represent numbers 0 ≤ i ≤ exp _K+1 (n) − 1 as intersection types.

I We can represent ATM configurations C of size exp _K+1 (n) as intersection types [ C ]

I Exploiting K -bounded polymorphism, we can represent these types implicitly in polynomial sized types

I ATM sequences C 1 C 2 · · · C m coded by reverse implications [C i+1 ] → [C i ] in Γ

26 / 46

Lower bound - main ideas

For any fixed level-parameter K :

I Intersection type numerals hi i _K : we can represent numbers 0 ≤ i ≤ exp _K+1 (n) − 1 as intersection types.

I We can represent ATM configurations C of size exp _K+1 (n) as intersection types [ C ]

I Exploiting K -bounded polymorphism, we can represent these types implicitly in polynomial sized types

I ATM sequences C 1 C 2 · · · C m coded by reverse implications [C i+1 ] → [C i ] in Γ

26 / 46

Lower bound: intersection type numerals

B = {0, 1}, B [n] = {0

, . . . , 0

} ∪ {1

, . . . , 1

} I N

= { T

i=1

b

| b

∈ {0

, 1

} for i = 1 . . . n}

I N

= { T

τ∈N_k

(τ → b

) | b

∈ {0, 1}, for τ ∈ N

} I J

T

i=1

b

K = P

i=1

J b

K × 2

, with J 0

K = 0 and J 1

K = 1 I J

T

σ∈N_k

(τ → b

) K = P

τ∈N_k

b

× 2

I J 0

∩ 1

∩ 0

∩ 1

K = 2 + 8 = 10

I J ((0

∩ 0

) → 0) ∩ ((0

∩ 1

) → 1) ∩ ((1

∩ 0

) → 0) ∩ ((1

∩ 1

) → 1) K = 10 I σ ∈ N

: 0 ≤ J σ K ≤ exp

(n) − 1, uniquely

27 / 46

Lower bound: intersection type numerals

B = {0, 1}, B [n] = {0

, . . . , 0

} ∪ {1

, . . . , 1

} I N

= { T

i=1

b

| b

∈ {0

, 1

} for i = 1 . . . n}

I N

= { T

τ∈N_k

(τ → b

) | b

∈ {0, 1}, for τ ∈ N

} I J

T

i=1

b

K = P

i=1

J b

K × 2

, with J 0

K = 0 and J 1

K = 1 I J

T

σ∈N_k

(τ → b

) K = P

τ∈N_k

b

× 2

I J 0

∩ 1

∩ 0

∩ 1

K = 2 + 8 = 10

I J ((0

∩ 0

) → 0) ∩ ((0

∩ 1

) → 1) ∩ ((1

∩ 0

) → 0) ∩ ((1

∩ 1

) → 1) K = 10

I σ ∈ N

: 0 ≤ J σ K ≤ exp

(n) − 1, uniquely

27 / 46

Lower bound: intersection type numerals

B = {0, 1}, B [n] = {0

, . . . , 0

} ∪ {1

, . . . , 1

} I N

= { T

i=1

b

| b

∈ {0

, 1

} for i = 1 . . . n}

I N

= { T

τ∈N_k

(τ → b

) | b

∈ {0, 1}, for τ ∈ N

} I J

T

i=1

b

K = P

i=1

J b

K × 2

, with J 0

K = 0 and J 1

K = 1 I J

T

σ∈N_k

(τ → b

) K = P

τ∈N_k

b

× 2

I J 0

∩ 1

∩ 0

∩ 1

K = 2 + 8 = 10

I J ((0

∩ 0

) → 0) ∩ ((0

∩ 1

) → 1) ∩ ((1

∩ 0

) → 0) ∩ ((1

∩ 1

) → 1) K = 10

I σ ∈ N

: 0 ≤ J σ K ≤ exp

(n) − 1, uniquely

27 / 46

Configurations

The explicit representation of an ATM configuration C is

[ C ] =

exp

(n)−1

\

i=0

Cell(a _i , q, hmi _K , hii _K )

where

I hii _K is the address of the cell

I hmi _K is the address of the cell under the r/w-head

I q is the control state

I a _i is the symbol in the cell

We will represent configurations implicitly as Cell(α, β, γ, δ)

28 / 46

Configurations

The explicit representation of an ATM configuration C is

[ C ] =

exp

(n)−1

\

i=0

Cell(a _i , q, hmi _K , hii _K )

where

I hii _K is the address of the cell

I hmi _K is the address of the cell under the r/w-head

I q is the control state

I a _i is the symbol in the cell

We will represent configurations implicitly as Cell(α, β, γ, δ)

28 / 46

Corollary: n.e. with uniform (refinement) types

Fix X ⊆ A and define uniform types Ξ

(τ) for τ ∈ T

:

Ξ

(0) = X

Ξ

(τ → σ) = (Ξ

(τ ) ⇒ Ξ

(σ))

Note: With such types we can represent any finite function f : A → B by T

x∈A

x → f (x ).

Corollary

Relativized inhabitation with uniform types refining monomorphic native types is nonelementary recursive.

Proof.

Upper bound: every problem Γ `? : φ

is decidable within bcl

(→, ∩) with k = max{`(φ

) | φ ∈ rn(Γ)}.

Lower bound: notice that all constructions in l.b. for bcl

(→, ∩) can be carried out with uniform types.

29 / 46

Corollary: Henkin’s theory Ω in bcl ^k (→, ∩)

Satisfiability of formulae

Φ ::= 0 ∈ x

| 1 ∈ x

| x

∈ y

| ¬Φ | ∀x

.Φ | Φ ∧ Φ

where x

ranges over D

with D

= {0, 1}, D

= P(D

).

L. Henkin: A theory of propositional types, Fundamenta Mathematicae 52 (1963) 323–344.

Representation in bcl

(→, ∩) (for sufficiently large k):

I A set variable x

is represented by a type variable x

. I Membership predicate Mem

Num

(x

) → Num

(y

) → In

(x

, y

) → Mem

(x

, y

) where In

(x

, x

→ 1) and NotIn(x

, x

→ 0) are axioms.

I Use alternation to code quantifiers as usual (Urzyczyn 1997).

30 / 46

Ongoing: optimization & algorithm engineering

B. D¨ udder: Automatic Synthesis of Component & Connector-Software Architectures with Bounded Combinatory Logic, Diss. Dortmund, Aug. 2014.

31 / 46

Intersection Type Subtyping

• Intersection type subtyping [Barendregt, Coppo, Dezani 1983] is the least preorder generated by the axioms A. Types are taken modulo associativity, commutativity and idempotence of ∩.

σ ≤

ω, ω ≤

ω → ω, σ ∩ τ ≤

σ, σ ∩ τ ≤

τ, σ ≤

σ ∩ σ;

(σ → τ ) ∩ (σ → ρ) ≤

σ → τ ∩ ρ;

If σ ≤

σ

and τ ≤

τ

then σ ∩ τ ≤

σ

∩ τ

and σ

→ τ ≤

σ → τ

.

• Matching: Given τ, σ, does there exist S with S(τ) ≤

σ?

Theorem (DMR TLCA 2013)

Intersection type matching is NP -complete.

32 / 46

Open Problems

I Is intersection type unification induced by ≤ _A decidable?

I

Close (but no cigar!):

Anantharaman, Narendran, Rubinowitch. Unification Modulo ACUI Plus Distributivity Axioms. Journal of Automated Reasoning (33) 2004.

I Complexity of inhabitation in bcl k (→, ∩) w/o (∩I )

I

Compare RU: The Complexity of Inhabitation with Explicit Intersection. Essays dedicated to Dexter Kozen, 2012.

33 / 46

Outline

Background on CLS

Intersection Types for Semantic Specification Bounded Combinatory Logic

Staged Composition Synthesis Conclusion and Future Work

33 / 46

SCS (DMR ESOP 2014)

Goal: Introduce metalanguage L2 into composition synthesis

I L1 may be limited, e.g.:

I

L1 might be very low-level

I

L1 might not have a notion of function application

I L2 should contain λ-calculus

I

Special-purpose composition operators

I

Meta-level computation over L1-code

I

Higher-order abstraction

I Challenges

I

How to expose language distinction in types?

I

How to ensure implementation type correctness?

I

How to ensure staged composition?

34 / 46

SCS (DMR ESOP 2014)

Goal: Introduce metalanguage L2 into composition synthesis

I L1 may be limited, e.g.:

I

L1 might be very low-level

I

L1 might not have a notion of function application

I L2 should contain λ-calculus

I

Special-purpose composition operators

I

Meta-level computation over L1-code

I

Higher-order abstraction

I Challenges

I

How to expose language distinction in types?

I

How to ensure implementation type correctness?

I

How to ensure staged composition?

34 / 46

SCS (DMR ESOP 2014)

Goal: Introduce metalanguage L2 into composition synthesis

I L1 may be limited, e.g.:

I

L1 might be very low-level

I

L1 might not have a notion of function application

I L2 should contain λ-calculus

I

Special-purpose composition operators

I

Meta-level computation over L1-code

I

Higher-order abstraction

I Challenges

I

How to expose language distinction in types?

I

How to ensure implementation type correctness?

I

How to ensure staged composition?

34 / 46

SCS (DMR ESOP 2014)

Goal: Introduce metalanguage L2 into composition synthesis

I L1 may be limited, e.g.:

I

L1 might be very low-level

I

L1 might not have a notion of function application

I L2 should contain λ-calculus

I

Special-purpose composition operators

I

Meta-level computation over L1-code

I

Higher-order abstraction

I Challenges

I

How to expose language distinction in types?

I

How to ensure implementation type correctness?

I

How to ensure staged composition?

34 / 46

Main Ideas in SCS

Use modal types φ (“code of type φ”) to expose language distinction to compostion synthesis.

I Introduction of modal intersection types

I Davies and Pfenning’s calculus λ ^,→ _e as L2 implementation language

I Challenges of staging solved by theory of λ ^,→ _e

I Challenge of implementation type correctness solved by conservative extension theorem

I Extension of inhabitation (semi-)algorithm for modal intersection types

I Extension of (CL)S-framework and experiments

35 / 46

Main Ideas in SCS

Use modal types φ (“code of type φ”) to expose language distinction to compostion synthesis.

I Introduction of modal intersection types

I Davies and Pfenning’s calculus λ ^,→ _e as L2 implementation language

I Challenges of staging solved by theory of λ ^,→ _e

I Challenge of implementation type correctness solved by conservative extension theorem

I Extension of inhabitation (semi-)algorithm for modal intersection types

I Extension of (CL)S-framework and experiments

35 / 46

Metalanguage L2

Modal types σ ::= τ | σ → σ

, where τ ranges over L1-types Metalanguage L2 (adapted from λ

) over L1:

∆; (Γ, x : σ) `

x : σ (var)

∆; (Γ, x : σ) `

M : σ

∆; Γ `

λx : σ.M : σ → σ

(→I) ∆; Γ `

M

: σ → σ

∆; Γ `

M

: σ

∆; Γ `

(M

M

) : σ

(→E)

∆; ∅ `

T : τ

∆; Γ `

box T : τ (I) ∆; Γ `

M

: τ (∆, u : τ); Γ `

M

: σ

∆; Γ `

letbox u : τ = M

in M

: σ (E)

Reduction

letbox u = box T in M 7−→ M[u := T]

Davies and Pfenning:A Modal Analysis of Staged Computation, JACM, 2001

36 / 46

Combinatory Logic C2

Semantic types s ::= b | s → s

| s ∩ s

| t Interface types ψ ::= σ | ψ∩s | φ | ψ → ψ

| ψ∩ψ

Combinatory Logic C2 over C1

C = {box X i : φ} ∪ {Y _j : ψ} (φ by rule of necessitation)

Logically, contains modal logic S4

@ , λF : (α → β).λX : α.

letbox f : α → β = F in letbox x : α = X in

box f (x)

@ : (α → β) → α → β

eval , λX : α. letbox x : α = X in x eval : α → α

37 / 46

Combinatory Logic C2

Semantic types s ::= b | s → s

| s ∩ s

| t Interface types ψ ::= σ | ψ∩s | φ | ψ → ψ

| ψ∩ψ

Combinatory Logic C2 over C1

C = {box X i : φ} ∪ {Y _j : ψ} (φ by rule of necessitation)

Logically, contains modal logic S4

@ , λF : (α → β).λX : α.

letbox f : α → β = F in letbox x : α = X in

box f (x)

@ : (α → β) → α → β

eval , λX : α. letbox x : α = X in x eval : α → α

37 / 46

Combinatory Logic C2

Semantic types s ::= b | s → s

| s ∩ s

| t Interface types ψ ::= σ | ψ∩s | φ | ψ → ψ

| ψ∩ψ

Combinatory Logic C2 over C1

C = {box X i : φ} ∪ {Y _j : ψ} (φ by rule of necessitation)

Logically, contains modal logic S4

@ , λF : (α → β).λX : α.

letbox f : α → β = F in letbox x : α = X in

box f (x)

@ : (α → β) → α → β

eval , λX : α. letbox x : α = X in x eval : α → α

37 / 46

Stratified logics and languages

Mmap , λF : (α → β).λA : Array(α).

letbox f : α → β = F in letbox a : Array(α) = A in

box(for (i := 0) to sizeof(a) − 1{a[i] := f(a[i]); }) Mmap : (α → β) → Array(α) → Array(β)

38 / 46

Stratified logics and languages

Ppar , λP

: Proc.λP

: Proc.

letbox p

: Proc = P

in letbox p

: Proc = P

in

box(p

k p

) Ppar : Proc → Proc → Proc

Ppar : (Proc ∩ A

) → (Proc ∩ A

) → (Proc ∩ A

× A

)

Ppar : ((Proc ∩ A

) → (Proc ∩ A

) → (Proc ∩ A

× A

)) ∩ ParComp

39 / 46

Example

C = {

O , box O : TrObj

Tr , box Tr : (TrObj → D((R, R) ∩ Cart,R ∩ Gpst, R ∩ Cel)) tmp , box tmp : (D((R,R),R,R ∩ a) → R ∩ a ∩ ms)

} D = {

@ : (α → β) → α → β

• : (β → γ) → (α → β) → (α → γ) cl2fh : ((R ∩ Cel) → (R ∩ Fh))∩Conv

♦ : (α ∩ ms) → (α → β)∩Conv → (β ∩ ms) }

40 / 46

Example

Combinator bindings for D :

• , λG : (β → γ).λF : (α → β).

letbox f : α → β = F in

letbox g : β → γ = G in box(fn y : α => (g (f y)))

cl2fh , λz : R.

letbox u : R = z in

box(let x : R = u in x ∗ (9 div 5) + 32)

♦ , λz : α.λF : α → β.(F z)

41 / 46

Example

I Query: C ∪ D ` _C2 ? : (TrObj → (R ∩ Cel ∩ ms))

I Solution: tmp • Tr : (TrObj → (R ∩ Cel ∩ ms ))

I L2-reduction:

tmp • Tr 7−→ ^∗

box(fn y : TrObj => (tmp (Tr y)))

42 / 46

Example

I Query: C ∪ D ` _C2 ? : (TrObj → (R ∩ Cel ∩ ms))

I Solution: tmp • Tr : (TrObj → (R ∩ Cel ∩ ms ))

I L2-reduction:

tmp • Tr 7−→ ^∗

box(fn y : TrObj => (tmp (Tr y)))

42 / 46

Example

I Query: C ∪ D ` _C2 ? : (TrObj → (R ∩ Cel ∩ ms))

I Solution: tmp • Tr : (TrObj → (R ∩ Cel ∩ ms ))

I L2-reduction:

tmp • Tr 7−→ ^∗

box(fn y : TrObj => (tmp (Tr y)))

42 / 46

Example

I Query: C ∪ D ` _C2 ? : (R ∩ Fh ∩ ms)

I Solution: (tmp @ (Tr @ O)) ♦ cl2fh

I L2-reduction:

(tmp @ (Tr @ O)) ♦ cl2fh 7−→ ^∗

box(let x : R = tmp (Tr O) in x ∗(9 div 5) + 32)

43 / 46

Example

I Query: C ∪ D ` _C2 ? : (R ∩ Fh ∩ ms)

I Solution: (tmp @ (Tr @ O)) ♦ cl2fh

I L2-reduction:

(tmp @ (Tr @ O)) ♦ cl2fh 7−→ ^∗

box(let x : R = tmp (Tr O) in x ∗(9 div 5) + 32)

43 / 46

Example

I Query: C ∪ D ` _C2 ? : (R ∩ Fh ∩ ms)

I Solution: (tmp @ (Tr @ O)) ♦ cl2fh

I L2-reduction:

(tmp @ (Tr @ O)) ♦ cl2fh 7−→ ^∗

box(let x : R = tmp (Tr O) in x ∗(9 div 5) + 32)

43 / 46

Example

Let us add to C

c2f : ((R ∩ Cel ∩ ms ) → (R ∩ Fh ∩ ms))

I Query: C ; D ` _C2 ? : (R ∩ Fh ∩ ms)

I Solution: c2f @ ((tmp • Tr) @ O)

I After L2-reduction:

box((c2f ((fn y : TrObj => (tmp (Tr y))) O))

44 / 46

Example

Let us add to C

c2f : ((R ∩ Cel ∩ ms ) → (R ∩ Fh ∩ ms))

I Query: C ; D ` _C2 ? : (R ∩ Fh ∩ ms)

I Solution: c2f @ ((tmp • Tr) @ O)

I After L2-reduction:

box((c2f ((fn y : TrObj => (tmp (Tr y))) O))

44 / 46

Example

Let us add to C

c2f : ((R ∩ Cel ∩ ms ) → (R ∩ Fh ∩ ms))

I Query: C ; D ` _C2 ? : (R ∩ Fh ∩ ms)

I Solution: c2f @ ((tmp • Tr) @ O)

I After L2-reduction:

box((c2f ((fn y : TrObj => (tmp (Tr y))) O))

44 / 46

Outline

Background on CLS

Intersection Types for Semantic Specification Bounded Combinatory Logic

Staged Composition Synthesis Conclusion and Future Work

44 / 46

Some Main Points

I A new approach to component-oriented synthesis

I An “Aristotelian” rather than “Fregean” approach

I Types as logic programs that compute compostions

I Easily applicable to native API’s

I Very large set of application scenarios

I Robust foundations in proof theory

I Implemented in (CL)S-framework (B. D¨ udder, M. Martens)

I Applied in several experiments

45 / 46

Ongoing & Future Work

I Application to connector synthesis (B. D¨ udder: ArchiType)

I Application to OO-synthesis (traits, dependency injection)

I Combinatory process synthesis

I Algorithm engineering

I Constrained types Q (~ α) ⇒ φ

I Automatic software configuration

I Stratified logics

I Component-oriented synthesis in theorem-proving?

I . . .

46 / 46