Bounding Principles for Decision Problems with Intersection Types

Transcription

1 Bounding Principles for Decision Problems with Intersection Types Jakob Rehof Technical University of Dortmund, Germany TYPES 2017, May 30th, Budapest, Hungary

2 Acknowledgements My students in Dortmund (former and present) including Jan Bessai, Boris Düdder (now Copenhagen), Andrej Dudenhefner, Moritz Martens (now in industry) Collaborators, including Paweł Urzyczyn (Warsaw), George Heineman (WPI Boston), Ugo de Liguoro (Torino) Colleagues, including Mariangiola Dezani, Simona Ronchi Della Rocca, Mario Coppo and the Torino λ-calculus group, Roger Hindley (Swansey), Aleksy Schubert and Warsaw group TYPES (Warsaw 2010, Bergen 2011, Novi Sad 2016)

3 Intersection Types and Decision Problems Intersection types [CDC80; CDCV81; BCDC83] allow terms to be assigned multiple types Γ M : τ 1... τ n Due to their enormous expressive power, the classical decision problems, typability and inhabitation, are undecidable Compare with simple types: Typability is in linear time (unification) Inhabitation (provability in minimal intuitionistic logic) is Pspace-complete [Sta79]

4 Intersection Types Definition ([CDCV80],[BCDC83],..., [BDS13]) Types T : σ, τ ::= a σ τ σ τ, Atoms a A Γ, x : τ x : τ (Var) Γ, x : σ M : A ( I) Γ λx.m : σ A Γ M : σ τ Γ N : σ ( E) Γ M N : τ Γ M : τ 1 Γ M : τ 2 ( I) Γ M : τ 1 τ 2 Γ M : τ 1 τ 2 ( E) Γ M : τ i

5 Bounding principles We are concerned with understanding the borderline between decidability and undecidability for these decision problems Motivations can be practical, purely theoretical, or (preferably) both

6 Bounding principles We are concerned with understanding the borderline between decidability and undecidability for these decision problems Motivations can be practical, purely theoretical, or (preferably) both We are especially interested in generic bounding principles such that: The rules (logic) of the system remain(s) intact Interesting levels of expressiveness are retained (depending on bound), up to complete recovery, e.g., S = k S k Decidability, complexity, algorithms pivot around the bound New (fine) structure of the decision problems is exposed

7 Some examples... a very non-exhaustive list... time(k): In many cases not very revealing space(k) regular languages time( f(n)), space( f(n)) complexity theory Quantifier alternation Logical hierarchies Bounded depth Frege systems Proof complexity Transfinite induction up to α Gentzen ɛ 0, ordinal analysis......

8 In λ-calculus Type Systems and This Talk Pervasive principles Functional order and rank of a logical operator (after Leivant [Lei83]) For intersection types: Refinement After Freeman and Pfenning, PLDI 1991 [FP91] Semantic types Bounded combinatory logic, TLCA 2011, CSL 2012 ff. [RU11; Düd+12] Non-idempotence Bucciareli, Kesner, Ronchi Della Rocca, TCS 2014 [BKRDR14] Norm and dimension D&R, POPL 2017, LICS 2017 [DR17a; DR17b]

9 Rank Rank-bound wrt. a logical operator (after Leivant [Lei83]): Rank = maximal order (nesting depth to the left of ) at which the operator can appear. Example: system F (operator ) Example: intersection types (operator ) Definition (Intersection type rank) rank(τ) = 0 if τ is a simple type rank(σ τ) = max(rank(σ) + 1, rank(τ)) rank(σ τ) = max(1, rank(σ), rank(τ))

10 Typability and Inhabitation Given a (standard) type system Γ M : τ we consider decision problems Typability. Given M, does there exist Γ and τ such that Γ M : τ? Inhabitation. Given Γ and τ, does there exist M such that Γ M : τ?

11 Typability and Inhabitation Given a (standard) type system Γ M : τ we consider decision problems Typability. Given M, does there exist Γ and τ such that Γ M : τ? Inhabitation. Given Γ and τ, does there exist M such that Γ M : τ? Assuming the standard property (for Γ = {x i : τ i } (i=1...n) ) Γ M : τ λx 1... λx n.m : τ 1 τ n τ we can consider rank k-bounded versions of the problems by asking for M : σ, where σ is rank k.

12 Typability and Inhabitation (in rank k)

13 System F Inhabitation is undecidable M. H. Löb, Embedding First Order Predicate Logic in Fragments of Intuitionistic Logic, JSL 1976 [L 76] See also P. Urzyczyn, Inhabitation in typed lambda-calculi (A syntactic approach), TLCA 1997 [Urz97] Typability is undecidable J. Wells, Typability and Type-Checking in the Second-Order lambda-calculus are Equivalent and Undecidable, LICS 1994 [Wel94] Typability is decidable for rank 2 und undecidable for all ranks k > 2 A. Kfoury and J. Wells, A Direct Algorithm for Type Inference in the Rank-2 Fragment of the Second-Order lambda-calculus, LFP 1994 [KW99]

14 Intersection Types: Typability Typability is undecidable Follows from characterization of normalization and solvability properties Coppo, Dezani, Venneri [CDCV81], Pottinger [Pot80], see also Ghilezan [Ghi96] and Barendregt, Dekkers, Statman [BDS13] Rank-bounded fragments Decidable in all bounded-rank fragements: A. Kfoury and J. Wells, Principality and Decidable Type Inference for Finite-Rank Intersection Types, POPL 1999 [KW99] Complexity exponential in rank: Kfoury, Mairson, Turbak and Wells, Relating Typability and Expressiveness in Finite-Rank Intersection Type Systems, ICFP 1999 [Kfo+99]

15 Intersection Types: Inhabitation Inhabitation is undecidable P. Urzyczyn, The Emptiness Problem for Intersection Types, JSL 1999 [Urz99] via reduction from halting problems for queue automata using rank 4 intersections Related to the λ-definability problem S. Salvati, Recognizability in the Simply Typed Lambda-Calculus, WoLLIC 2009 [Sal09] Salvati, Manzonetto, Gehrke and Barendregt, Urzyczyn and Loader are logically related, ICALP 2012 [Sal+12] Undecidability of λ-definability: Loader 1993 [Loa01]

16 Intersection Types: Rank-Bounded Inhabitation Rank 2-inhabitation is decidable Equivalent to rank 2-polymorphism: H. Yokouchi, Embedding a Second Order Type System into an Intersection Type System, I&C 1995 [Yok95] Exptime-hard: D. Kuśmierek, The Inhabitation Problem for Rank Two Intersection Types, TLCA 2007 [Kus07] Rank 2-inhabitation is Expspace-complete and rank k-inabitation is undecidable for all ranks k > 2 P. Urzyczyn, Inhabitation of Low-Rank Intersection Types, TLCA 2009 [Urz09] Proof techniques via bus machines, an alternating, expanding instruction device, also used to show Expxpace-completeness of inhabitation with explicit intersection [RU12]

17 Combinatory Logic Synthesis (CLS) A type-theoretic approach to component-oriented synthesis Component-oriented Synthesis Synthesis relative to library (repository) of components Combinatory Logic Synthesis (CLS) Libraries need classification systems to enable retrieval and composition Classification Taxonomy Types CLS Bottom-up specification Hoare logic

18 Combinatory Logic Synthesis (CLS) A type-theoretic approach to component-oriented synthesis

19 Combinatory Logic Synthesis (CLS) A type-theoretic approach to component-oriented synthesis Can we use inhabitation in combinatory logic with intersection types as a foundation for component-oriented, type-based synthesis? Typed combinators X : τ as named interfaces Automated composition synthesis via inhabitation Intersection types as semantic types (cf. also Haack,Wells,Yakobowski et al. [Haa+02; WY05]) for specification Beyond purely functional composition via meta-programming compose a meta-program which, when executed, computes (say) a Java program

20 Combinatory Logic Synthesis (CLS) This idea was developed in a series of papers, including: Finite Combinatory Logic with Intersection Types, TLCA 2011 [RU11] First mention of CL with intersection types as semantic specifications for synthesis via inhabitation Bounded Combinatory Logic, CSL 2012 [Düd+12] Complexity hierarchy for relativized inhabitation with CL Using Inhabitation in Bounded Combinatory Logic with Intersection Types for Composition Synthesis, ITRS 2012 [Düd+12] First short experiments Towards Combinatory Logic Synthesis, BEAT 2013 [Reh13] Outline of research program Design and Synthesis from Components (Dagstuhl Seminar 14232), 2014 (organized by Rehof and Vardi) [RV14] The idea of synthesis from components Staged Composition Synthesis, ESOP 2014 [DMR14] Moving to the meta-level via modal types Mixin Composition Synthesis Based on Intersection Types, TLCA 2015 [Bes+15] First study of object-oriented features The idea of using intersection types as foundation for type-based synthesis also taken up for λ-calculus inhabitation, see Frankle, Osera, Walker, Zdancewic, POPL 2016 [Fra+16]

21 Combinatory Logic Synthesis (CLS) It is currently being developed into a meta-programming framework based on a combinatory extension to Scala led by Jan Bessai (TU Dortmund), Boris Düdder (U Copenhagen), George Heineman (WPI Boston) Combinatory Logic Synthesizer, ISOLA 2014 [Bes+14] Synthesizing Type-safe Compositions in Feature Oriented Software Designs using Staged Composition, ModSyn-PL 2015 [DRH15] Towards Migrating Object-Oriented Frameworks to Enable Synthesis of Product Line Members, SPLC 2015 [Hei+15] Combinatory Synthesis of Classes using Feature Grammars, FACS 2015 [Bes+16b] A Long and Winding Road Towards Modular Synthesis, ISOLA 2016 [Hei+16] Combinatory Process Synthesis, ISOLA 2016 [Bes+16a] Component-Oriented Synthesis via Algebras and Combinatory Logic, in preparation

22 Example Repository Γ = { customerform : (String java.net.url OptionSelection Form) dropdownselector : (java.net.url OptionSelection) radiobut tonselector : (java.net.url OptionSelection) companytitle : String databaselocation : java.net.url logolocation : java.net.url alternatelogolocation : java.net.url } From: Scala integrated framework for CLS by Bessai, Düdder, Dudenhefner, Heinemann

23 Example Repository Γ = { customerform : (String java.net.url OptionSelection Form) (Title Location(Logo) ChoiceDialog(α) OrderMenu(α)) dropdownselector : (java.net.url OptionSelection) (Location( Database) ChoiceDialog( DropDown)) radiobut tonselector : (java.net.url OptionSelection) (Location( Database) ChoiceDialog(RadioBut tons)) companytitle : String Title databaselocation : java.net.url Location( Database) logolocation : java.net.url Location(Logo) alternatelogolocation : java.net.url Location(Logo) }

24 Combinator Interface in Scala

25 Combinator Implementation in Scala

26 Variations

27 Web Interface to Inhabitation Service

28 Combinatory Logic With Intersection Types cl(, ) Γ, X : τ X : S(τ) (var) Γ e : τ σ Γ e : τ ( E) Γ (e e ) : σ Γ e : τ Γ e : σ ( I) Γ e : τ σ Γ e : τ τ σ ( ) Γ e : σ Types are taken modulo associativity, commutativity and idempotence of, and is least preorder containing (cf. [BCDC83]) 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. Note The SKI-calculus has been studied with intersection types (Dezani and Hindley [DCH92]) But, in CLS, the combinatory theory Γ represents an arbitrary repository (basis not fixed)

29 CLS World View CL over arbitrary bases is a general theory of component collections (repositories) Each repository is a combinatory theory The special implementation theory {S, K, I} is one, very special case Cellarius Harmonia Macrocosmica Planisphaerium Copernicanum

30 Relativized Inhabitation We consider the relativized inhabitation problem: Given a set of typed combinators Γ and τ, does there exist combinatory expression e such that Γ e : τ? 1 See also talk on Wednesday by A. Dudenhefner, Lower End of the Linial-Post Spectrum

31 Relativized Inhabitation We consider the relativized inhabitation problem: Given a set of typed combinators Γ and τ, does there exist combinatory expression e such that Γ e : τ? Inhabitation for fixed base {S, K, I} is Pspace-complete in simple types (Statman s theorem) Relativized inhabitation is much harder Undecidable in simple types: Linial-Post theorems, 1948ff. [LP49] 1 1 See also talk on Wednesday by A. Dudenhefner, Lower End of the Linial-Post Spectrum

32 Relativized Inhabitation We consider the relativized inhabitation problem: Given a set of typed combinators Γ and τ, does there exist combinatory expression e such that Γ e : τ? Inhabitation for fixed base {S, K, I} is Pspace-complete in simple types (Statman s theorem) Relativized inhabitation is much harder Undecidable in simple types: Linial-Post theorems, 1948ff. [LP49] 1 The CLS view: Already in simple types, relativized inhabitation defines a Turing-complete logic programming language for component composition Reduction from 2-counter automata [Reh13] Similar idea used to prove undecidability for synthesis in ML relative to library of functions [BSWC16] 1 See also talk on Wednesday by A. Dudenhefner, Lower End of the Linial-Post Spectrum

33 Linial-Post Spectrum... 2Exptime? Exptime Pspace co-np Ptime R T R Subintuitionistic logics IPL (S4) Intermediate logics CPL

34 Bounded Combinatory Logic bcl k (, ) Levels l(a) = 0, for a A; l(τ σ) = 1 + max{l(τ), l(σ)}; l( n i=1 τ i) = max{l(τ i ) i = 1,..., n}. l(s) = max{l(s(α)) S(α) α}

35 Bounded Combinatory Logic bcl k (, ) Levels l(a) = 0, for a A; l(τ σ) = 1 + max{l(τ), l(σ)}; l( n i=1 τ i) = max{l(τ i ) i = 1,..., n}. l(s) = max{l(s(α)) S(α) α} bcl k (, ), k 0 (and finite CL, fcl, with S = id). [l(s) k] Γ, X : τ k X : S(τ) (var) Γ k e : τ σ Γ k e : τ ( E) Γ k (e e ) : σ Γ k e : τ Γ k e : σ ( I) Γ k e : τ σ Γ k e : τ τ σ ( ) Γ k e : σ Bounded Combinatory Logic, CSL 2012 [Düd+12]

36 Complexity for Finite and Bounded CL Theorem (TLCA 2011 [RU11]) For finite combinatory logic fcl: 1 Relativized inhabitation in fcl( ) is in Ptime 2 Relativized inhabitation in fcl(, ) is Exptime-complete Theorem (CSL 2012 [Düd+12]) 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

37 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)

38 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 := {S(σ) S S (Γ,τ,k) x }; (1 1) (1 1) (1 1)

39 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 := {S(σ) S S (Γ,τ,k) x }; (1 1) (1 1) (1 1) 3 choose m {0,..., σ }; (0 1) (1 0) (0 0) 4 choose P P m (σ ); (1 0) (0 1) (1 1)

40 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 := {S(σ) S S (Γ,τ,k) x }; (1 1) (1 1) (1 1) 3 choose m {0,..., σ }; (0 1) (1 0) (0 0) 4 choose P P m (σ ); (1 0) (0 1) (1 1) 5 if ( π P tgt m (π) τ) then (0 0) (1 1) τ 6 if (m = 0) then accept;

41 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 := {S(σ) S S (Γ,τ,k) x }; (1 1) (1 1) (1 1) 3 choose m {0,..., σ }; (0 1) (1 0) (0 0) 4 choose P P m (σ ); (1 0) (0 1) (1 1) 5 if ( π P tgt m (π) τ) then (0 0) (1 1) τ 6 if (m = 0) then accept; 7 else 8 9 forall(i = 1... m) τ := π P arg i (π); τ :=(0 1) (1 0) τ :=(1 0) (0 1) 10 goto loop; 11 else reject;

42 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 := {S(σ) S S (Γ,τ,k) x }; (1 1) (1 1) (1 1) 3 choose m {0,..., σ }; (0 1) (1 0) (0 0) 4 choose P P m (σ ); (1 0) (0 1) (1 1) 5 if ( π P tgt m (π) τ) then (0 0) (1 1) τ 6 if (m = 0) then accept; 7 else 8 9 forall(i = 1... m) τ := π P arg i (π); τ :=(0 1) (1 0) τ :=(1 0) (0 1) 10 goto loop; 11 else reject; (x f) f : (0 0) (1 1)

43 Lower bound - main ideas 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

44 Lower bound - main ideas 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 For any fixed level-parameter K: Intersection type numerals i K : we can represent numbers 0 i exp K+1 (n) 1 as intersection types. We can represent ATM configurations C of size exp K+1 (n) as intersection types [C ] Exploiting K-bounded polymorphism, we can represent these types implicitly in polynomial sized types ATM sequences C 1 C 2 C m coded by reverse implications [C i+1 ] [C i ] in Γ

45 Refinement (after [FP91]) Definition Let T o be simple types over an atom o. Fix X A and define uniform types U X (τ) for τ T o : U X (o) = X U X (τ σ) = (U X (τ) U X (σ))

46 Refinement (after [FP91]) Definition Let T o be simple types over an atom o. Fix X A and define uniform types U X (τ) for τ T o : U X (o) = X U X (τ σ) = (U X (τ) U X (σ)) With such types we can represent any finite function f : A B at the type level by a A(a f(a))

47 Refinement (after [FP91]) Definition Let T o be simple types over an atom o. Fix X A and define uniform types U X (τ) for τ T o : U X (o) = X U X (τ σ) = (U X (τ) U X (σ)) With such types we can represent any finite function f : A B at the type level by a A(a f(a)) We can express finite abstract interpretations, e.g., succ : (Nat Nat) (zero pos) (pos pos) (even odd) (odd even)

48 Refinement (after [FP91]) Definition Let T o be simple types over an atom o. Fix X A and define uniform types U X (τ) for τ T o : U X (o) = X U X (τ σ) = (U X (τ) U X (σ)) With such types we can represent any finite function f : A B at the type level by a A(a f(a)) We can express finite abstract interpretations, e.g., succ : (Nat Nat) (zero pos) (pos pos) (even odd) (odd even) Typability (λ-calculus) is in linear time over simple types, and type inference is computable [FP91]

49 Refinement (after [FP91]) Definition Let T o be simple types over an atom o. Fix X A and define uniform types U X (τ) for τ T o : U X (o) = X U X (τ σ) = (U X (τ) U X (σ)) With such types we can represent any finite function f : A B at the type level by a A(a f(a)) We can express finite abstract interpretations, e.g., succ : (Nat Nat) (zero pos) (pos pos) (even odd) (odd even) Typability (λ-calculus) is in linear time over simple types, and type inference is computable [FP91] Inhabitation (λ-calculus) is undecidable.

50 Refinement (after [FP91]) Definition Let T o be simple types over an atom o. Fix X A and define uniform types U X (τ) for τ T o : U X (o) = X U X (τ σ) = (U X (τ) U X (σ)) With such types we can represent any finite function f : A B at the type level by a A(a f(a)) We can express finite abstract interpretations, e.g., succ : (Nat Nat) (zero pos) (pos pos) (even odd) (odd even) Typability (λ-calculus) is in linear time over simple types, and type inference is computable [FP91] Inhabitation (λ-calculus) is undecidable. Proof: Note that [Sal+12] uses only uniform types for λ-definability.

51 CL(, ) over Uniform (Refinement) Types Definition Let T o be simple types over an atom o. Fix X A and define uniform types U X (τ) for τ T o : U X (o) = X U X (τ σ) = (U X (τ) U X (σ)) Corollary Relativized inhabitation with uniform types is nonelementary recursive. Proof. Upper bound: every problem Γ? : σ is decidable within bcl k (, ) with k = max{l(τ) σ rn(γ) U X (τ)}. Lower bound: notice that all constructions in l.b. for bcl k (, ) can be carried out with uniform types.

52 Corollary: Henkin s theory Ω in bcl k (, ) Satisfiability of formulae Φ ::= 0 x 1 1 x 1 x k y k+1 Φ x k.φ Φ Φ where x k ranges over D k with D 0 = {0, 1}, D k+1 = P(D k ). L. Henkin: A theory of propositional types, Fundamenta Informaticae 52 (1963) Representation in bcl k (, ) (for sufficiently large k): A set variable x k is represented by a type variable x k. Membership predicate Mem k Num k (x k ) Num k+1 (y k+1 ) In k (x k, y k+1 ) Mem k (x k, y k+1 ) where In k (x k, x k 1) and NotIn(x k, x k 0) are axioms. Use alternation to code quantifiers as usual (Urzyczyn 1997).

53 Ongoing: optimization & algorithm engineering From B. Düdder: Automatic Synthesis of Component & Connector-Software Architectures with Bounded Combinatory Logic, Diss. Dortmund, Aug

54 Inhabitation in Bounded Dimension Pieter Brueghel the Elder - The Dutch Proverbs - Google Art Project.jpg 1559 Intersection Type Calculi of Bounded Dimension, POPL 2017 [DR17a]

55 Strict Intersection Types Definition (Strict Intersection Types) T s A, B ::= a σ A T σ, τ ::= A 1 A n n 1 Definition (Strict Type Assignment [Bak11](Def. 5.1)) 1 i n (Var) Γ, x : Γ s M : A i for i = 1... n n ( I) A i=1 i s x : A i Γ s M : n i=1 A i Γ s M : σ A Γ s N : σ ( E) Γ s M N : A Γ, x : σ s M : A ( I) Γ s λx.m : σ A

56 Notational Variant Definition (Strict Intersection Types) A, B ::= a σ A σ, τ ::= [A 1,..., A n ] n 1 Definition (Strict Type Assignment [Bak11](Def. 5.1)) 1 i n Γ, x : [A 1,..., A n ] s x : [A i ] (Var) Γ s M : [A i ] for i = 1... n ( I) Γ s M : [A i,..., A n ] Γ s M : [σ A] Γ s N : σ ( E) Γ s M N : [A] Γ, x : σ s M : [A] Γ s λx.m : [σ A] ( I)

57 Set Theoretic Elaborations Definition (Elaborations) P, Q, R ::= x S (λx.p) S (P Q) S, S = [A 1,..., A n ], n 1 Definition (P Q, defined for P Q ) x S x S x S S (λx.p) S (λx.q) S (λx.p Q) S S (PQ) S (P Q ) S ((P P )(Q Q )) S S Definition (Norm ) x S = S (λx.p) S = max{ P, S } (PQ) S = max{ P, Q, S }

58 Set Theoretic Elaborations Definition (Elaborations) P, Q, R ::= x S (λx.p) S (P Q) S, S = [A 1,..., A n ], n 1 Definition (P Q, defined for P Q ) Definition (Norm ) x S x S x S S (λx.p) S (λx.q) S (λx.p Q) S S (PQ) S (P Q ) S ((P P )(Q Q )) S S x S = S (λx.p) S = max{ P, S } (PQ) S = max{ P, Q, S } Non-negativity : P > 0 Subadditivity : P Q P + Q for P Q

59 Set Theoretic Elaboration System Definition (Γ P : σ) 1 i n Γ, x : [A 1,..., A n ] x [A i ] : [A i ] (Var) Γ, x : σ P : [A] Γ (λx.p) [σ A] : [σ A] ( I) Γ P : [σ A] Γ Q : σ ( E) Γ (P Q) [A] : [A] Γ P i : [A i ] for i = 1... n ( I) Γ n i=1 P i : [A 1,..., A n ]

60 Set Theoretic Elaboration System Definition (Γ P : σ) 1 i n Γ, x : [A 1,..., A n ] x [A i ] : [A i ] (Var) Γ, x : σ P : [A] Γ (λx.p) [σ A] : [σ A] ( I) Γ P : [σ A] Γ Q : σ ( E) Γ (P Q) [A] : [A] Γ P i : [A i ] for i = 1... n ( I) Γ n i=1 P i : [A 1,..., A n ] Intuition The operation n i=1 P i allows us to measure usage of ( I) as a logical resource under norm

61 Set Theoretic Elaboration System Definition (Γ P : σ) 1 i n Γ, x : [A 1,..., A n ] x [A i ] : [A i ] (Var) Γ, x : σ P : [A] Γ (λx.p) [σ A] : [σ A] ( I) Γ P : [σ A] Γ Q : σ ( E) Γ (P Q) [A] : [A] Γ P i : [A i ] for i = 1... n ( I) Γ n i=1 P i : [A 1,..., A n ] Definition Write Γ M P : σ iff Γ P : σ with M P. Clearly, Γ s M : σ iff P. Γ M P : σ

62 Intersection Type Calculus of Bounded Dimension Definition (λ [ ] n ) Γ n M : σ iff P. Γ M P : σ with P n Lemma Γ s M : σ iff Γ n M : σ for some n > 0 Definition (Dimension) The set theoretic dimension of a term M at Γ and σ is dim σ Γ = min{n Γ n M : σ}

63 Subject Reduction in Bounded Dimension Terms can be elaborated in non-increasing norm under β-reduction: Theorem (Subject Reduction for λ [ ] n ) If Γ M P : τ and M β M, then there exists R with R P such that Γ M R : τ Consequences: Each dimensional fragment λ [ ] n is a meaningful type system. Inhabitation in bounded dimension for λ [ ] n can be limited to search for normal forms.

64 Inhabitation in Bounded Set Theoretic Dimension Problem (Inhabitation for λ [ ] ) Given environment Γ, type σ and number n > 0: is there a term M such that Γ n M : σ?

65 Inhabitation in Bounded Set Theoretic Dimension Problem (Inhabitation for λ [ ] ) Given environment Γ, type σ and number n > 0: is there a term M such that Γ n M : σ? Theorem The inhabitation problem for λ [ ] is undecidable. Proof. By subject reduction and normalization it suffices to search for normal forms in norm n. Let N be the size of input. By the subformula property [BCDC83] (Lemma 4.5), inhabitation in bounded norm N is equivalent to inhabitation. Proposition For n = 1 set-theoretic inhabitation is Pspace-complete ([RU12] Cor. 22).

66 Multiset Elaboration System Definition Treat types σ = [A 1,..., A n ] and sets S as multisets s and let denote multiset union. Definition ( P : s) 1 i n, x : [A 1,..., A n ] x [A i ] : [A i ] (Var), x : s P : [A] (λx.p) [σ A] : [s A] ( I) P : [s A] Q : s ( E) (P Q) [A] : [A] P i : [A i ] for i = 1... n ( ) ( I) n P i=1 i : [A 1,..., A n ] where ( ) is for each x s in n i=1 P i if x free in M, then s (x).

67 Examples Example (I λx.x) x [a] x [b] ( I) ( I) (λx.x [a] ) [a a] (λx.x [b] ) [b b] ( I) P := (λx.x [a, b] ) [a a, b b] P = 2 Example (ω λx.xx) x [a a] x [a] ( E) (x [a a] x [a] ) [a] P := (λx.(x [a a] x [a] ) [a] ) [[a, a a] a] P = 1 ( I) Note that the system is non-idempotent and non-linear.

68 Examples Example (x(y(λz.z)) Let A [a a, b b] e, B [b b, c c] f, = {x : [e, f] g, y : [A, B]}. We have P : [e], where P is x [[e, f] g] (y [A, B] (λz.z a b b c ) a a b b b b c c )

69 Beyond Rank 2 Inhabitation The rank hierarchy jumps from Expspace in rank 2 to infinity at ranks k 3, as was shown by urzyczyn [Urz09] But we can show [DR17a] that dimensional bound provides a rank-independent stratification, decidable in each dimensional bound, and subsuming the rank 2 fragment in Expspace Camille Flammarions 1833 (excerpt)

70 Inhabitation in Bounded Multiset Dimension Definition Γ n M : σ iff, P, s. ( M = P : s with Γ = and σ = s and P n) where ( ) collapses multisets to underlying sets.

71 Inhabitation in Bounded Multiset Dimension Definition Γ n M : σ iff, P, s. ( M = P : s with Γ = and σ = s and P n) where ( ) collapses multisets to underlying sets. Problem (Γ n? : σ) Given Γ, σ and n > 0: is there a term M such that Γ n M : σ?

72 Inhabitation in Bounded Multiset Dimension Definition Γ n M : σ iff, P, s. ( M = P : s with Γ = and σ = s and P n) where ( ) collapses multisets to underlying sets. Problem (Γ n? : σ) Given Γ, σ and n > 0: is there a term M such that Γ n M : σ? Theorem Inhabitation in bounded multiset dimension is Expspace-complete. For each dimensional bound d > 0, inhabitation is in ATIME(N 2d ) where N denotes the size of the input Γ and σ.

73 Inhabitation in Bounded Multiset Dimension Definition Γ n M : σ iff, P, s. ( M = P : s with Γ = and σ = s and P n) where ( ) collapses multisets to underlying sets. Problem (Γ n? : σ) Given Γ, σ and n > 0: is there a term M such that Γ n M : σ? Theorem Inhabitation in bounded multiset dimension is Expspace-complete. For each dimensional bound d > 0, inhabitation is in ATIME(N 2d ) where N denotes the size of the input Γ and σ. Corollary For each fixed n inhabitation in multiset dimension n is Pspace-complete.

74 Dimensional Analysis of Rank 2 Typings Proposition Suppose we can derive N = P : [A 1,..., A n ] in rank 2, where N is a normal form. Then P = n.

75 Dimensional Analysis of Rank 2 Typings Proposition Suppose we can derive N = P : [A 1,..., A n ] in rank 2, where N is a normal form. Then P = n. Consequence Substantial generalization of inhabitation in rank 2 fragment, generalizing across all ranks within Expspace.

76 Dimensional Analysis of Rank 2 Typings Proposition Suppose we can derive N = P : [A 1,..., A n ] in rank 2, where N is a normal form. Then P = n. Consequence Substantial generalization of inhabitation in rank 2 fragment, generalizing across all ranks within Expspace. Remark Compare to linear, non-idempotent system of Bucciareli, Kesner, Ronchi Della Rocca [BKRDR14]: Inhabitation is decidable [BKRDR14] and NP-complete [DR17a] Typability is undecidable.

77 The World Cellarius Harmonia Macrocosmica Planisphaerium Copernicanum

78 The World 1 N

79 Multiset Dimension and Parallel Constraints Luca Pacioli, attr. Jacopo de Barbari 1495 (excerpt)

80 Inhabitation (after Urzyczyn [Urz97])

81 Inhabitation

82 Inhabitation

83 Inhabitation

84 Inhabitation

85 Inhabitation

86 Inhabitation

87 Inhabitation

88 Intersection Type Inhabitation (after Urzyczyn [Urz09])

89 Intersection Type Inhabitation

90 Intersection Type Inhabitation

91 Intersection Type Inhabitation

92 Intersection Type Inhabitation

93 Intersection Type Inhabitation

94 Multiset Norm and Dimension

95 Multiset Norm and Dimension See also our talk at TYPES 2016, Rank 3 Inhabitation of Intersection Types Revisited [Bes+16c] and extended version at arxiv.

96 Typability in Bounded Dimension Five-set Venn diagram using congruent ellipses in a 5-fold rotationally symmetrical arrangement devised by Branko Grünbaum. Typability in Bounded Dimension, LICS 2017 [DR17b]

97 Elaboration Types

98 Elaboration Types

99 Elaboration Types

100 Elaboration Types

101 Subformula Filtration The classical subformula property ([BCDC83], Lemma 4.5) says that, on normal forms, every decoration type is a subformula of some observable type We can show that, on all typable terms, one can obtain the subformula fitration property: Every constituent subformula is also a decoration type For X T s, we induce a filtration function F X : T T which removes subformula components of intersection types not appearing in X

102 Subformula Filtration The classical subformula property ([BCDC83], Lemma 4.5) says that, on normal forms, every decoration type is a subformula of some observable type We can show that, on all typable terms, one can obtain the subformula fitration property: Every constituent subformula is also a decoration type For X T s, we induce a filtration function F X : T T which removes subformula components of intersection types not appearing in X Proposition Let T P denote the decoration types in P and suppose T P X. Then In particluar, taking T P = X we have Γ P : σ F X (Γ) F X (P) : F X (σ) Γ P : σ F TP (Γ) F TP (P) : F TP (σ)

103 Subformula Filtration Example For P (λx.x a ) [a, b] a we have T P = {a, [a, b] a} and since b T P we get F TP (P) = (λx.x a ) [a] a

104 Bounded Width Theorem Definition (Width) a = 1 σ A = max{ σ, A } A 1 A m = max{m, A 1,..., A m } Lift to environments, elaborations, and derivations by taking maximal width over all types appearing. Theorem (Bounded Width Property) Let a derivation D Γ M P : σ be given with P d. Then there exists a derivation D Γ M P : σ such that D d M and P = P. Proof. By filtration with F TP and using F TP (D T P together with T P P M d M

105 Typability in Bounded Dimension Problem (Typability in bounded set-theoretic dimension) Given a λ-term M and a dimension d, does there exist Γ and σ such that Γ d M : σ? (Recall: Γ d M : σ iff P. Γ P : σ, P M, P d)

106 Typability in Bounded Dimension Problem (Typability in bounded set-theoretic dimension) Given a λ-term M and a dimension d, does there exist Γ and σ such that Γ d M : σ? (Recall: Γ d M : σ iff P. Γ P : σ, P M, P d) Theorem The typability problem in bounded set-theoretic dimension is Pspace-complete.

107 Typability in Bounded Dimension Problem (Typability in bounded set-theoretic dimension) Given a λ-term M and a dimension d, does there exist Γ and σ such that Γ d M : σ? (Recall: Γ d M : σ iff P. Γ P : σ, P M, P d) Theorem The typability problem in bounded set-theoretic dimension is Pspace-complete. Remark The upper bound is constructed by nondeterministic reduction to standard unification.

108 Conclusion We can probably do much more (algorithmically) with intersection types than (I, at least) thought possible! Principles of bounding complementary to that of rank exist and are both useful and logically interesting

109 Ongoing & Future Work Further work on Scala-integrated, combinatory meta-programming framework and applications to software composition synthesis Type inference and theory of principal elaborations in bounded dimension Theory of abstract vector space structure of elaborations (semimodule over semiring of type substitution actions) More speculative: Integration of CLS with generate-and-test loops Applications in probabilistic program induction and learning Applications to theorem proving: automatic composition of proof tactics?

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114 Bibliography V Mario Coppo, Mariangiola Dezani-Ciancaglini, and Betti Venneri. Principal Type Schemes and Lambda-Calculus Semantics. In: To H. B. Curry. Essays on Combinatory Logic, Lambda-calculus and Formalism. Ed.: R. Hindley and J. Seldin. Accademic Press, London, 1980, pp M. Coppo, M. Dezani-Ciancaglini, and B. Venneri. Functional Characters of Solvable Terms. In: Zeitschrift für mathematische Logik und Grundlagen der Mathematik (1981), pp M. Dezani-Ciancaglini and R. Hindley. Intersection Types for Combinatory Logic. In: Theoretical Computer Science (1992), pp

115 Bibliography VI Boris Düdder, Moritz Martens, and Jakob Rehof. Staged Composition Synthesis. In: ESOP 2014, Proceedings of European Symposium on Programming, Grenoble, France Vol LNCS. Springer, 2014, pp Andrej Dudenhefner and Jakob Rehof. Intersection Type Calculi of Bounded Dimension. In: POPL 2017, Proceedings of the 44th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Paris, France, January Andrej Dudenhefner and Jakob Rehof. Typability in Bounded Dimension. In: LICS 2017, Proceedings of the 32nd ACM/IEEE Symposium on Logic in Computer Science, Reykjavik, Iceland, June

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117 Bibliography VIII Boris Düdder et al. Using Inhabitation in Bounded Combinatory Logic with Intersection Types for Composition Synthesis. In: Proceedings Sixth Workshop on Intersection Types and Related Systems, ITRS 2012, Dubrovnik, Croatia, 29th June Ed. by Stéphane Graham-Lengrand and Luca Paolini. Vol EPTCS. 2012, pp doi: /EPTCS url: T. Freeman and F. Pfenning. Refinement Types for ML. In: Proceedings of PLDI 91. ACM, 1991, pp Jonathan Frankle et al. Example-Directed Synthesis: A Type-Theoretic Interpretation. In: POPL 16. ACM. 2016, pp S. Ghilezan. Strong Normalization and Typability with Intersection Types. In: Notre Dame Journal of Formal Logic 37.1 (1996), pp doi: /ndjfl/

118 Bibliography IX Christian Haack et al. Fully Automatic Adaptation of Software Components Based on Semantic Specifications. In: AMAST. Vol LNCS. Springer, 2002, pp George T. Heineman et al. Towards Migrating Object-Oriented Frameworks to Enable Synthesis of Product Line Members. In: Proceedings of the 19th International Conference on Software Product Line, SPLC 2015, Nashville, TN, USA, July 20-24, , pp doi: / url:

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120 Bibliography XI Dariusz Kusmierek. The Inhabitation Problem for Rank Two Intersection Types. In: TLCA 2007, Proceedings of Typed Lambda Calculus and Applications. Vol LNCS. Springer, 2007, pp A. J. Kfoury and J. B. Wells. Principality and Decidable Type Inference for Finite-Rank Intersection Types. In: POPL 1999, Proceedings of the 26th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, San Antonio, TX, USA, January 20-22, , pp doi: / Martin H. Löb. Embedding First Order Predicate Logic in Fragments of Intuitionistic Logic. In: J. Symbolic Logic 41.4 (1976), pp

121 Bibliography XII D. Leivant. Polymorphic Type Inference. In: POPL 1983, Proceedings of the 10th ACM Symposium on Principles of Programming Languages. 1983, pp doi: / Ralph Loader. The undecidability of lambda definability. In: Logic, Meaning and Computation: Essays in Memory of Alonzo Church (2001), pp Samuel Linial and Emil L. Post. Recursive Unsolvability of the Deducibility, Tarski s Completeness and Independence of Axioms Problems of Propositional Calculus. In: Bulletin of the American Mathematical Society 55 (1949), p. 50. G. Pottinger. A Type Assignment for the Strongly Normalizable Lambda-Terms. In: To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Ed. by J. Hindley and J. Seldin. Academic Press, 1980, pp

122 Bibliography XIII Jakob Rehof. Towards Combinatory Logic Synthesis. In: BEAT 2013, 1st International Workshop on Behavioural Types. ACM, Jakob Rehof and Paweł Urzyczyn. Finite Combinatory Logic with Intersection Types. In: TLCA 2011, Proceedings of Typed Lambda Calculus and Applications. Vol LNCS. Springer, 2011, pp Jakob Rehof and Paweł Urzyczyn. The Complexity of Inhabitation with Explicit Intersection. In: Logic and Program Semantics - Essays Dedicated to Dexter Kozen on the Occasion of His 60th Birthday. Ed. by Robert L. Constable and Alexandra Silva. Vol LNCS. Springer, 2012, pp

123 Bibliography XIV Jakob Rehof and Moshe Y. Vardi. Design and Synthesis from Components. Dagstuhl Seminar In: vol Dagstuhl Reports S. Salvati et al. Urzyczyn and Loader are logically related. In: ICALP 2012, Proceedings of Automata, Languages, and Programming - 39th International Colloquium. Vol LNCS. Springer, 2012, pp doi: / _34. S. Salvati. Recognizability in the Simply Typed Lambda-Calculus. In: WoLLIC 2009, Proceedings of Workshop on Logic, Language, Information and Computation. Ed. by H. Ono, M. Kanazawa, and R. J. G. B. de Queiroz. Vol LNCS. Springer, 2009, pp

124 Bibliography XV Richard Statman. Intuitionistic Propositional Logic is Polynomial-space Complete. In: Theoretical Computer Science 9 (1979), pp doi: / (79) P. Urzyczyn. Inhabitation of Low-Rank Intersection Types. In: TLCA 2009, Proceedings of Typed Lambda Calculus and Applications. Vol LNCS. Springer, 2009, pp doi: / _26. P. Urzyczyn. Inhabitation in Typed Lambda-Calculi (A Syntactic Approach). In: TLCA 97, Typed Lambda Calculi and Applications, Proceedings. Vol LNCS. Springer, 1997, pp doi: / P. Urzyczyn. The Emptiness Problem for Intersection Types. In: Journal of Symbolic Logic 64.3 (1999), pp doi: /

125 Bibliography XVI J. B. Wells. Typability and Type-Checking in the Second-Order lambda-calculus are Equivalent and Undecidable. In: Proceedings of the Ninth Annual Symposium on Logic in Computer Science (LICS 94), Paris, France, July 4-7, , pp doi: /LICS Joe B. Wells and Boris Yakobowski. Graph-Based Proof Counting and Enumeration with Applications for Program Fragment Synthesis. In: LOPSTR Vol LNCS. Springer, 2005, pp Hirofumi Yokouchi. Embedding a Second Order Type System into an Intersection Type System. In: Inf. Comput (1995), pp doi: /inco