Observational Semantics for a Concurrent Lambda Calculus with Reference Cells and Futures
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1 Observaional Semanics for a Concurren Lambda Calculus wih Reference Cells and Fuures David Sabel J. W. Goehe-Universiy Frankfur, Germany join work wih: Joachim Niehren (Lille, France), Manfred Schmid-Schauß (Frankfur, Germany), Jan Schwinghammer (Saarbrücken, Germany) MFPS XXIII, 2007
2 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Ouline 1 The Calculus λ(fu) 2 Observaional Semanics 3 Correcness Proofs 4 Resuls & Fuure work J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 2 Semanics for a Concurren λ-calculus wih Cells and Fuures
3 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Synax Small-Sep Reducion The Calculus λ(fu) Properies and Feaures of λ(fu) proposed by Niehren, Schwinghammer, Smolka 2006, TCS core-language of Alice ML call-by-value λ-calculus concurrency (a collecion of hreads / processes) synchronisaion via handles and fuures (eager as well as lazy) reference cells (value exchange beween hreads) J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 3 Semanics for a Concurren λ-calculus wih Cells and Fuures
4 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Synax Small-Sep Reducion Our Conribuion Observaional Semanics based on may- and mus-convergence sensible noion for equivalence of processes equivalence of expressions implies equivalen behavior, e.g. disinguishes erroneous and error-free programs Program Transformaions invesigae correcness of program ransformaions proof echniques for reasoning abou correcness of ransformaions J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 4 Semanics for a Concurren λ-calculus wih Cells and Fuures
5 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Synax Small-Sep Reducion Relaed Work Ong 1993, LICS: Non-deerminism in a funcional seing Kuzner, Schmid-Schauß 1998, ICFP: A Non-Deerminisic Call-by-Need Lambda Calculus (may & mus-convergence) (diagrams) Moran, Sands, Carlsson 1999, COORDINATION: (conex lemma) Erraic Fudges: A semanic heory for an embedded coordinaion language Pis 2002, Applied Semanics: (conex. equiv. for ML wih local sae) Operaional Semanics and Program Equivalence Carayol, Hirschkoff, and Sangiorgi 2005, TCS: (oher mus-convergence) On he represenaion of McCarhy s amb in he π-calculus J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 5 Semanics for a Concurren λ-calculus wih Cells and Fuures
6 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Synax Small-Sep Reducion Two-Lel-Synax of λ(fu) Layer of Processes p Process ::= p 1 p 2 (νx)p x e x susp = e x c v y h x y h Layer of λ-expressions e Exp ::= x c λx.e e 1 e 2 exch(e 1, e 2 ) c Cons ::= uni cell hread handle lazy v Val ::= x c λx.e x, y, z Var J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 6 Semanics for a Concurren λ-calculus wih Cells and Fuures
7 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Synax Small-Sep Reducion Two-Lel-Synax of λ(fu) Layer of Processes p Process ::= p 1 p 2 (νx)p x e x susp = e x c v y h x y h Layer of λ-expressions e Exp ::= x c λx.e e 1 e 2 exch(e 1, e 2 ) c Cons ::= uni cell hread handle lazy v Val ::= x c λx.e x, y, z Var Srucural Congruence p 1 p 2 p 2 p 1 (p 1 p 2 ) p 3 p 1 (p 2 p 3 ) (νx)(νy)p (νy)(νx)p (νx)(p 1 ) p 2 (νx)(p 1 p 2 ) if x fv(p 2 ) J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 6 Semanics for a Concurren λ-calculus wih Cells and Fuures
8 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Synax Small-Sep Reducion Evaluaion Relaion Small-Sep Reducion (local) β-cbv L () E[(λy.e) v] (νy)(e[e] y v) hread.new() E[hread v] (νz)(e[z] z v z) lazy.new() E[lazy v] (νz)(e[z] z = susp v z) lazy.rigger() F [x] x = susp e F [x] x e fu.deref() F [x] x v F [v] x v cell.new() E[cell v] (νz)(e[z] z c v) cell.exch() E[exch(z, v 1 )] z c v 2 E[v 2 ] z c v 1 handle.new() E[handle v] (νy)(νx)(e[v y x] x h y) handle.bind() E[x v] x h y E[uni] y v x h E::=x e E e E::=[ ] e E e v e E exch( e E, e) exch(v, e E) F ::=x e E[[ ] v] x e E[exch([ ], v)] J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 7 Semanics for a Concurren λ-calculus wih Cells and Fuures
9 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Synax Small-Sep Reducion Evaluaion Relaion Small-Sep Reducion (D-closed) β-cbv L () hread.new() lazy.new() lazy.rigger() fu.deref() cell.new() D[E[(λy.e) v]] D[(νy)(E[e] y v)] D[E[hread v]] D[(νz)(E[z] z v z)] D[E[lazy v]] D[(νz)(E[z] z = susp v z)] D[F [x] x = susp e] D[F [x] x e] D[F [x] x v] D[F [v] x v] D[E[cell v]] D[(νz)(E[z] z c v)] cell.exch() D[E[exch(z, v 1 )] z c v 2 ] D[E[v 2 ] z c v 1 ] handle.new() handle.bind() D[E[handle v]] D[(νy)(νx)(E[v y x] x h y)] D[E[x v] x h y] D[E[uni] y v x h ] process conexs D::=[ ] p D D p (νx)d J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 7 Semanics for a Concurren λ-calculus wih Cells and Fuures
10 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Successful Processes May- and Mus-Convergence Conexual Equivalence Discussion Successful Processes Succesful Processes A process p is successful iff p well-formed and x e: x is bound o a consan, absracion, cell, lazy fuure, handle or handled fuure. bound includes chains of indirecions x x 1 x 1 x 2... x n 1 x n Examples: successful x λy.y x y y z z c uni Examples: no successful x x x yx y xy J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 8 Semanics for a Concurren λ-calculus wih Cells and Fuures
11 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Successful Processes May- and Mus-Convergence Conexual Equivalence Discussion Successful Processes Succesful Processes A process p is successful iff p well-formed and x e: x is bound o a consan, absracion, cell, lazy fuure, handle or handled fuure. bound includes chains of indirecions x x 1 x 1 x 2... x n 1 x n Examples: successful x λy.y x y y z z c uni Examples: no successful x x x yx y xy J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 8 Semanics for a Concurren λ-calculus wih Cells and Fuures
12 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Successful Processes May- and Mus-Convergence Conexual Equivalence Discussion May- and Mus-Convergence May-Convergence p iff p : p p p successful Mus-Divergence p iff p Mus-Convergence p iff p : p p = p... includes weak divergences, i.e. processes ha have an infinie aluaion, bu all successors w.r.. are may-convergen p May-Divergence ω p iff p J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 9 Semanics for a Concurren λ-calculus wih Cells and Fuures
13 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Successful Processes May- and Mus-Convergence Conexual Equivalence Discussion Conexual Equivalence (of Processes) Two Conexual Preorders based on may-convergence p 1 p 2 iff D : D[p 1 ] = D[p 2 ] based on mus-convergence p 1 p 2 iff D : D[p 1 ] = D[p 2 ] ess may- / mus- convergence in all conexs Conexual Preorder / Conexual Equivalence = = J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 10 Semanics for a Concurren λ-calculus wih Cells and Fuures
14 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Successful Processes May- and Mus-Convergence Conexual Equivalence Discussion Why Weak Divergences are Included in Mus-Convergence? Former Approach oal mus-convergence: p oal iff all aluaions of p erminae successfully. Example of Carayol, Hirschkoff, Sangiorgi, 2005 Y λf.(choice I f) I I I I J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 11 Semanics for a Concurren λ-calculus wih Cells and Fuures
15 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Successful Processes May- and Mus-Convergence Conexual Equivalence Discussion Why Weak Divergences are Included in Mus-Convergence? Former Approach oal mus-convergence: p oal iff all aluaions of p erminae successfully. Example of Carayol, Hirschkoff, Sangiorgi, 2005 Y λf.(choice I f) I I I I wih oal mus-convergence: I oal Y λf.(choice I f) oal choice I wih our mus-convergence: I Y λf.(choice I f) choice I J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 11 Semanics for a Concurren λ-calculus wih Cells and Fuures
16 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Successful Processes May- and Mus-Convergence Conexual Equivalence Discussion Why Weak Divergences are Included in Mus-Convergence? Fairness fair aluaion: ery possible redex will be reduced enually Wih our mus-convergence convergence predicaes unchanged if aluaions are resriced o fairness fair = and fair = Hence: fair = J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 12 Semanics for a Concurren λ-calculus wih Cells and Fuures
17 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Proving Correcness of a Transformaion Le be a (D-closed) ransformaion on processes Proof Plan (show ) Correcness is correc iff Show for all p 1, p 2 wih p 1 p 2 : preserves may-convergence: p 1 = p 2 p 2 = p 1 preserves mus-convergence: p 1 = p 2 p 2 = p 1 J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 13 Semanics for a Concurren λ-calculus wih Cells and Fuures
18 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Forking and Commuing Diagrams for Transformaion Diagrams are mea-rewriing rules r, r f relaions on processes. r f r forking diagram r r f commuing diagram J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 14 Semanics for a Concurren λ-calculus wih Cells and Fuures
19 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Forking and Commuing Diagrams for Transformaion Diagrams are mea-rewriing rules r, r f relaions on processes. r f r se of forking diagrams is complee iff for ery p 1 p 3 p 2 here is an applicable diagram r r f se of commuing diagrams is complee iff for ery p 1 p 2 p 3 here is an applicable diagram J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 14 Semanics for a Concurren λ-calculus wih Cells and Fuures
20 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Preservaion of May-Convergence prove p 1 = p 2 p 1 p 2 p 1 successful J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 15 Semanics for a Concurren λ-calculus wih Cells and Fuures
21 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Preservaion of May-Convergence prove p 1 = p 2 p 1 p 1 successful p 2 J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 15 Semanics for a Concurren λ-calculus wih Cells and Fuures
22 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Preservaion of May-Convergence prove p 1 = p 2 p 1 p 2 p 1 p 2 successful successful J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 15 Semanics for a Concurren λ-calculus wih Cells and Fuures
23 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Preservaion of May-Convergence prove p 1 = p 2 p 1 p 2 p 1 p 2 successful successful prove p 2 = p 1 p 1 p 2 p 2 successful J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 15 Semanics for a Concurren λ-calculus wih Cells and Fuures
24 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Preservaion of May-Convergence prove p 1 = p 2 p 1 p 2 p 1 p 2 successful successful prove p 2 = p 1 p 1 p 2 p 1 p 2 successful successful J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 15 Semanics for a Concurren λ-calculus wih Cells and Fuures
25 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Proving Correcness of a Transformaion Proof Plan Show for all p 1, p 2 wih p 1 p 2 : preserves may-convergence: p 1 = p 2 p 2 = p 1 preserves mus-convergence: p 1 = p 2 p 2 = p 1? J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 16 Semanics for a Concurren λ-calculus wih Cells and Fuures
26 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Proving Correcness of a Transformaion Proof Plan Show for all p 1, p 2 wih p 1 p 2 : preserves may-convergence: p 1 = p 2 p 2 = p 1 preserves mus-convergence (= preserves may-divergence): p 1 = p 2 equivalen o p 2 = p 1 p 2 = p 1 equivalen o p 1 = p 2 p equivalen o p : p p p J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 16 Semanics for a Concurren λ-calculus wih Cells and Fuures
27 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Preservaion of Mus-Convergence prove p 1 = p 2 p 1 p 2 p 1 p 2 prove p 2 = p 1 p 1 p 2 p 1 p 2 musdivergen musdivergen musdivergen musdivergen J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 17 Semanics for a Concurren λ-calculus wih Cells and Fuures
28 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Diagram Mehod diagrams may be more complex... fu.deref() fu.deref fu.deref fu.deref fu.deref() fu.deref fu.deref() does no work: inducion on he lengh of he reducion sequence soluion: well-founded measure which is decreased by ery diagram J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 18 Semanics for a Concurren λ-calculus wih Cells and Fuures
29 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Correcness of Transformaions Proof Technique Transformaions on Expression Transformaions on Expressions Conexual Equivalence of Expressions e 1 e 2 iff C : C[e 1 ] = C[e 2 ] e 1 e 2 iff C : C[e 1 ] = C[e 2 ] = = Correcness ransformaion on expressions is correc if Conex Lemma D, E :(D[E[e 1 ]] = D[E[e 2 ]] D[E[e 1 ]] = D[E[e 2 ]] ) = e 1 e 2 resrics he number of conexs needed o be aken ino accoun! (λx.e) v cbv-β e[v/x]: prove correcness of D[E[(λx.e) v]] D[E[e[v/x]]] J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 19 Semanics for a Concurren λ-calculus wih Cells and Fuures
30 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Resuls Conclusion Fuure Work Resuls Correc Program Transformaions all reducions excep for cell.exch() deerminisic cell exchange (νx)(e[exch(x, v 1 )] x c v 2 ) (νx)(e[v 2 ] x c v 1 ) arbirary copying of values C[x] x v C[v] x v garbage collecion if x bv(c) p (νy 1 )... (νy n )p p if p successful & y 1,..., y n conain all process variables of p call-by-value β (wihou sharing)... (λx.e) v e[v/x] J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 20 Semanics for a Concurren λ-calculus wih Cells and Fuures
31 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Resuls Conclusion Fuure Work Resuls Incorrec Program Transformaions cell.exch() E[exch(z, v 1 )] z c v 2 E[v 2 ] z c v 1 call-by-name β (λx.e) e e[e /x] lazy.rigger( ) C[x] x susp = e C[x] x e cell.new( ) C[cell v] (νz)(c[z] z c v) hread.new( ) C[hread v] (νz)(c[z] z v z) lazy.new( ) C[lazy v] (νz)(c[z] z susp = v z) handle.new( ) C[handle v] (νy)(νx)(c[v y x] x h y) handle.bind( ) C[x v] x h y C[uni] y v x h means ha conex C is no an E conex J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 21 Semanics for a Concurren λ-calculus wih Cells and Fuures
32 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Resuls Conclusion Fuure Work Conclusion we have presened an observaional equivalence for λ(fu) based on may- as well as mus-convergence enables o reason abou he correcness of ransformaions of saeful and concurren compuaions he used proof mehods are successful in paricular we proved correcness of parial aluaion which is used in compilers J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 22 Semanics for a Concurren λ-calculus wih Cells and Fuures
33 The Calculus λ(fu) Observaional Semanics Correcness Proofs Resuls & Fuure work Resuls Conclusion Fuure Work Fuure Work exensions of he calculus: ypes, case-expressions and daa consrucors (e.g. liss) invesigae saic analyses (e.g. ouch-analysis) and prove correcness of he relaed opimisaions maybe our mehods are applicable o oher process calculi like he π-calculus J. Niehren, D. Sabel, M. Schmid-Schauß, J. Schwinghammer 23 Semanics for a Concurren λ-calculus wih Cells and Fuures
Observational Semantics for a Concurrent Lambda Calculus with Reference Cells and Futures
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