Rank 3 Inhabitation of Intersection Types Revisited Andrej Dudenhefner Jan Bessai Boris D¨ udder Jakob Rehof Technical University of Dortmund, Germany May 20, 2016 1 / 26
Contents Intersection Type System 1 Intersection Type Inhabitation 2 HTM ≤ IHP 3 2 / 26
Intersection Type System (BCD) Characterizes normalization/strong normalization in λ -calculus [Pot80] Characterizes finite function tables [Sal+12] Framework for the study of semantic domains for the λ -calculus Undecidable type checking (does the given term have the given type) Undecidable typability (w/o rule ( ω )) (does the given term have any type) Undecidable inhabitation [Urz99] (is there any term having the given type) 3 / 26
Intersection Type System (BCD) Definition (Intersection Types T ) T ∋ σ, τ, ρ ::= a | ω | σ → τ | σ ∩ τ where a ∈ A Definition (Subtyping ≤ ) Least preorder (reflexive and transitive relation) over T such that σ ≤ ω, ω ≤ ω → ω, σ ∩ τ ≤ σ, σ ∩ τ ≤ τ, ( σ → τ 1 ) ∩ ( σ → τ 2 ) ≤ σ → τ 1 ∩ τ 2 , if σ ≤ τ 1 and σ ≤ τ 2 then σ ≤ τ 1 ∩ τ 2 , if σ 2 ≤ σ 1 and τ 1 ≤ τ 2 then σ 1 → τ 1 ≤ σ 2 → τ 2 4 / 26
Intersection Type System (BCD) Definition (Type Assignment) x : τ ∈ Γ ( Ax ) ( ω ) Γ ⊢ e : ω Γ ⊢ x : τ Γ , x : σ ⊢ e : τ Γ ⊢ e : σ Γ ⊢ e : τ ( ∩ I ) ( → I ) Γ ⊢ e : σ ∩ τ Γ ⊢ λ x . e : σ → τ Γ ⊢ e ′ : σ ( → E ) Γ ⊢ e : σ σ ≤ τ ( ≤ ) Γ ⊢ e : σ → τ Γ ⊢ ( e e ′ ) : τ Γ ⊢ e : τ 5 / 26
Intersection Type Inhabitation Definition ( ⊢ ? : τ ) Given a type τ is there a λ -term e such that ⊢ e : τ ? Definition (Rank [Lei83]) 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 )) ⊢ ? : τ with rank ( τ ) ≤ 2 is E XP S PACE -complete [Urz09] ⊢ ? : τ with rank ( τ ) ≥ 3 is undecidable [Urz09] 6 / 26
Which features of BCD contribute to undecidability of inhabitation? Can BCD proof search simulate a Turing machine directly ? Are there particular inhabitation instances which are hard to decide? 7 / 26
Approach in [BDS13] (Lambda Calculus with Types) EQA ≤ ETW ≤ WTG ≤ IHP (15 pages w/o EQA theory) EQA Emptiness problem for queue automata ETW Emptiness problem for typewriter automata WTG Problem of winning a “tree game” IHP Intersection type inhabitation problem [BDS13] Barendregt, Dekkers and Statman. “Lambda calculus with types”. Cambridge University Press, 2013. 8 / 26
Approach in [Sal+12; Loa01] WSTS ≤ LDF ≤ IHP (7+3 pages) WSTS Word problem in semi-Thue systems LDF λ -definability problem IHP Intersection type inhabitation problem [Loa01] Loader. “The undecidability of λ -definability”. Logic, Meaning and Computation. Springer Netherlands, 2001. [Sal+12] Salvati et al. “Loader and Urzyczyn are logically related”. Automata, Languages, and Programming. Springer Berlin Heidelberg, 2012. 9 / 26
Approach in [Urz09] ELBA ≤ SSTS1 ≤ HETM ≤ IHP (6 pages) ELBA Emptiness problem for linear bounded automata SSTS1 Problem of deciding whether there is a word that can be rewritten to 1s in a simple semi-Thue system HETM Halting problem for expanding tape machines IHP Intersection type inhabitation problem [Urz09] Urzyczyn. “Inhabitation of low-rank intersection types”. Typed Lambda Calculi and Applications. Springer Berlin Heidelberg, 2009. 10 / 26
Problematic aspects Introduced machinery is highly specialized Multiple degrees of non-determinism, alternation, parallelism Instructions create new instructions (higher order memory) λ -definability requires model theory Difficult to pinpoint necessary aspects 11 / 26
Goal HTM ≤ IHP 12 / 26
Intuition SSTS01 ≤ IHP Definition (Simple semi-Thue System, SSTS) A semi-Thue system over an alphabet Σ is simple , if each rule has the form ab ⇒ cd for some a , b , c , d ∈ Σ . Lemma (SSTS01) Given a simple semi-Thue system over Σ , it is undecidable whether ∃ n ∈ N . 0 n ։ 1 n 13 / 26
Simultaneous Set of Judgments Proof search algorithm [Bun08] uses Simultaneous Set of Judgments 1 Γ 1 ⊢ ? : τ 1 , . . . , Γ n ⊢ ? : τ n where dom (Γ 1 ) = . . . = dom (Γ n ) with transformations such as Γ 1 ⊢ ? : τ 1 , Γ 2 ⊢ ? : σ ∩ τ � Γ 1 ⊢ ? : τ 1 , Γ 2 ⊢ ? : σ, Γ 2 ⊢ ? : τ Γ 1 ⊢ ? : σ 1 → τ 1 , Γ 2 ⊢ ? : σ 2 → τ 2 � Γ 1 ∪ { x : σ 1 } ⊢ ? : τ 1 , Γ 2 ∪ { x : σ 2 } ⊢ ? : τ 2 where x is fresh Γ 1 ⊢ ? : τ 1 , Γ 2 ⊢ ? : τ 2 where x : σ 1 1 → σ 2 1 → τ 1 ∈ Γ 1 and x : σ 1 2 → σ 2 2 → τ 2 ∈ Γ 2 � Γ 1 ⊢ ? : σ 1 Γ 2 ⊢ ? : σ 1 2 and Γ 1 ⊢ ? : σ 2 Γ 2 ⊢ ? : σ 2 1 , 1 , 2 1 logically same as Intersection Synchronous Logic [PRR12] 14 / 26
Simple semi-Thue System Simulation Fix SSTS S over Σ with l , r , • �∈ Σ . Let Γ = { z : 1 } ∪ { x ab ⇒ cd : σ ab ⇒ cd | ab ⇒ cd ∈ S } where σ ab ⇒ cd = ( l → c → a ) ∩ ( r → d → b ) ∩ � ( • → e → e ) e ∈ Σ Let Γ 1 = Γ , Γ 2 = Γ , Γ 3 = Γ , Γ n − 2 = Γ , Γ n − 1 = Γ , Γ n = Γ , . . . y 1 : l y 1 : r y 1 : • . . . y 1 : • y 1 : • y 1 : • y 2 : • y 2 : l y 2 : r y 2 : • y 2 : • y 2 : • . . . . . . . . . . . . . . . . . . . . . . . . y n − 2 : • y n − 2 : • y n − 2 : • y n − 2 : • . . . y n − 2 : l y n − 2 : r y n − 1 : • y n − 1 : • y n − 1 : • . . . y n − 1 : • y n − 1 : l y n − 1 : r Intuitively: y : l , y : r in neighboring environments; y : • otherwise. 15 / 26
Simple semi-Thue System Simulation ab ⇒ cd tabu ⇒ tcdu for n = 4 is simulated by Γ 1 ⊢ ? : t , Γ 2 ⊢ ? : a , Γ 3 ⊢ ? : b , Γ 4 ⊢ ? : u using x ab ⇒ cd : ( l → c → a ) ∩ ( r → d → b ) ∩ � ( • → e → e ) e ∈ Σ Γ 1 ⊢ ? : t , Γ 2 ⊢ ? : c , Γ 3 ⊢ ? : d , Γ 4 ⊢ ? : u and Γ 1 ⊢ ? : • , Γ 2 ⊢ ? : l , Γ 3 ⊢ ? : r , Γ 4 ⊢ ? : • The second condition is satisfied iff l , r are inhabited in exactly the neighboring contexts. Intuitively: type environments encode rewrite rule and order information; inhabited atoms encode current string. 16 / 26
Simple semi-Thue System Simulation Γ 1 ⊢ ? : 1 , . . . , Γ n ⊢ ? : 1 is satisfied since z : 1 ∈ Γ i for 1 ≤ i ≤ n Lemma We have 0 n ։ 1 n iff Γ 1 ⊢ ? : 0 , . . . , Γ n ⊢ ? : 0 is satisfied. Next : construct Γ 1 ⊢ ? : 0 , . . . , Γ n ⊢ ? : 0 for arbitrary/unknown n 17 / 26
σ ∗ = (( • → ∗ ) → ∗ ) ∩ (( l → ∗ ) → #) ∩ (( r → #) ∩ ( • → $) → $) σ 0 = (( • → 0 ) → ∗ ) ∩ (( l → 0 ) → #) ∩ (( r → 0 ) → $) τ = σ ∗ → σ 0 → 1 → σ t 1 → . . . → σ t k → ( l → ∗ ) ∩ ( r → #) ∩ ( • → $) τ Γ ( l → ∗ ) ∩ ( r → #) ∩ ( • → $) Relative Tags l left y 1 : r y 1 : • y 1 : l r right ∗ # $ • other Absolute Tags • → ∗ l → ∗ ( r → #) ∩ ( • → $) $ last y 2 : • y 2 : r y 2 : • y 2 : l # next to last ∗ ∗ # $ ∗ other y 3 : • y 3 : • y 3 : r y 3 : l 0 0 0 0 18 / 26
Back to the Goal HTM ≤ IHP 19 / 26
HTM ≤ IHP Fix a TM M = (Σ , Q , q 0 , q f , δ ) where Σ : finite set of tape symbols with ∈ Σ Q : finite set of states with q 0 , q f ∈ Q q 0 : initial state q f : final state δ : Q × Σ → Q × Σ × { + 1 , − 1 } : transition function Let A = Σ ˙ ∪{ l , r , •} ˙ ∪{� q , a � | q ∈ Q , a ∈ Σ } ˙ ∪{◦ , ∗ , # , $ } The configuration ( q , 3 , abcd ) is represented as ab � q , c � d 20 / 26
TM simulation TM simulation using most n tape cells by Γ 1 ⊢ ? : � q 0 , � , Γ 2 ⊢ ? : , Γ n ⊢ ? : . . . , where � � σ f = a ∩ � q f , a � a ∈ Σ a ∈ Σ for t = (( q , c ) �→ ( q ′ , c ′ , + 1 )) ∈ δ ( • → a → a ) ∩ ( l → c ′ → � q , c � ) ∩ � � ( r → � q ′ , a � → a ) σ t = a ∈ Σ a ∈ Σ for t = (( q , c ) �→ ( q ′ , c ′ , − 1 )) ∈ δ ( • → a → a ) ∩ ( r → c ′ → � q , c � ) ∩ � � ( l → � q ′ , a � → a ) σ t = a ∈ Σ a ∈ Σ 21 / 26
Γ 1 , . . . , Γ n Initialization σ ∗ = (( • → ◦ ) → ◦ ) ∩ (( • → ∗ ) → ∗ ) ∩ (( l → ∗ ) → #) ∩ (( r → #) ∩ ( • → $) → $) σ 0 = (( • → � q 0 , � ) → ◦ ) ∩ (( • → ) → ∗ ) ∩ (( l → ) → #) ∩ (( r → ) → $) τ ⋆ = σ 0 → σ ∗ → σ f → σ t 1 → . . . → σ t k → ( l → ◦ ) ∩ ( r → #) ∩ ( • → $) where δ = { t 1 , . . . , t k } ◦ marks the first symbol to be initialized to � q 0 , � Lemma M halts starting with the empty tape iff there exists a λ -term e such that ∅ ⊢ e : τ ⋆ 22 / 26
Insights “Neighboring” judgments recognized using y : l and y : r TM simulation with fixed number of cells in rank 2 and order 2 Inhabitant directly encodes computation Initialization requires only one a ∩ b → c type in the environment to increase the number of simultaneous judgments τ ⋆ is of rank 3 and order 3 SSTS01 is convenient 23 / 26
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