Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion Abstractness of Continuation Semantics for Asynchronous Concurrency Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Federated Logic Conference 2018 Workshop Domains Oxford, UK July 7-8, 2018 Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion Introduction 1 Continuations for Concurrency 2 Weak Abstractness 3 An Asynchronous Formalism 4 Conclusion 5 Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion Motivation and Aim A continuation is a semantic representation of the rest a computation [Stratchey and Wadsworth 1974] Traditional continuations can express: non-local exits, coroutines, even multitasking and ADA-like rendez-vous However, the traditional continuations do not work well enough in the presence of concurrency [Hieb, Dybvig and Anderson 1994] Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion Motivation and Aim In [Todoran 2000, Ciobanu and Todoran 2014] we introduced a continuation semantics for concurrency (CSC) CSC can express concurrent composition as well as various communication and synchronization mechanisms Intuitively, CSC is a denotational scheduler In the CSC approach continuations are application-specific structures of computations Rather than the functions to some answer used in the classic technique of continuations Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion Motivation and Aim In this talk we survey some applications of CSC and we investigate the abstractness of CSC We present an optimality criterion specific of continuation semantics that we name weak abstractness which Relaxes the completeness condition Preserves the correctness condition of the classic full abstractness criterion [Milner 1977] Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion Continuation Semantics for Traditional Concurrent Programming Concepts CSP-like synchronous communication Warren’s Andorra Model and asynchronous communication [Todoran and Papaspyrou 2000] [Todoran 2000] p ::= ( y = x ; ) ∗ x x ::= g | ≪ o ≫ | � l � | # � l � CSP Extended with Multiple | y | x � x Channels Communication g ::= a | fail l ::= ǫ | g ? x (+ g ? x ) ∗ [Ciobanu and Todoran 2015] o ::= ǫ | g : x (+ g : x ) ∗ j ::= c ? v | j & j a ::= v := e | c ! e | j s ::= a | y | s ; s | s + s | s � s ≪ o ≫� a � � a 1 ? x 1 + a 2 ? x 2 � � � l 1 � � · · · � � l n � c 1 ! e 1 � · · · � c n ! e n � ( c 1 ? v 1 & · · · c n ? v n ) ≡ ( v 1 , . . . , v n := e 1 , . . . , e n ) Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion Continuation Passing Semantics for Nature Inspired Formalisms Membrane Computing Spiking Neural P-Systems [Ciobanu and Todoran 2017] [Ciobanu and Todoran 2018] ρ ′ = ( D ′ , x ′ ) , ρ = ( D ; x ) , x = o 1 � o 4 x ′ = send ( � a 2 k − 1 � , { N 1 } ) � send ( a , { N 3 } ) D = membrane M 0 { [ o 1 , o 4 ] ⇒ o 2 � o 4 ; D ′ = [ o 2 ] ⇒ o 3 � new ( M 1 , l 1 , o 1 � o 5 ); neuron N 0 { r ǫ | { N 1 , N 2 , N 3 } } neuron N 1 { a + / [ a ] → a ; 2 | { N 2 } } [ o 2 ] ⇒ o 4 ; neuron N 2 { [ a k ] → a ; 1 | { N 3 } } [ o 3 ] ⇒ in ( l 1 , o 5 ) � o 5 ; [ o 4 , o 4 , o 5 ] ⇒ o 5 ; } ; neuron N 3 { [ a ] → a ; 0 | { N 0 } } } membrane M 1 { [ o 1 ] ⇒ o 4 � out ( o 4 ); [ o 5 , o 5 ] ⇒ δ } . } Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion On the Abstractness of Continuation Semantics The full abstractness condition is in general difficult to establish [Milner 1977] Even more difficult in continuation semantics We are not aware of any full abstractness result for a concurrent language designed with continuations Continuation-passing semantics for sequential languages are not fully abstract [Cartwright, Curien & Felleisen 1994] Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion On the Abstractness of Continuation Semantics Weak abstractness may be useful when full abstractness is difficult (or impossible) to achieve [ · ] We offer a denotational semantics [ ] for an asynchronous formalism; we use a domain of continuations D = Cont → P Cont = · · · D · · · The semantics is designed by using metric domains [De Bakker and Zucker 1982, America and Rutten 1989] Like the classic domains [Scott 1976, Scott 1982], metric spaces can also be used to express denotational semantics [ · ] ] is weakly abstract w.r.t. an O [ [ · ] We prove that [ ] Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion Classic Full Abstractness [Milner 1977] A denotational semantics D : L → D is said to be fully abstract with respect to a (corresponding) operational semantics O : L → O if D it is correct with respect to O ∀ x 1 , x 2 ∈ L [ D ( x 1 ) = D ( x 2 ) ⇒ ∀ S [ O ( S ( x 1 )) = O ( S ( x 2 ))]] D and complete with respect to O ∀ x 1 , x 2 ∈ L [ D ( x 1 ) � = D ( x 2 ) ⇒ ∃ S [ O ( S ( x 1 )) � = O ( S ( x 2 ))]] ( S is an L syntactic context) Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion Abstractness of Continuation Semantics In continuation semantics, D : L → D , D = Cont → F , the completeness condition (of full abstractness) is: ∀ x 1 , x 2 ∈ L [ ( ∃ γ ∈ Cont [ D ( x 1 ) γ � = D ( x 2 ) γ ]) ⇒ ( ∃ S [ O ( S ( x 1 )) � = O ( S ( x 2 ))]) ] In continuation semantics for concurrency (CSC) [Todoran 2000, Ciobanu and Todoran 2014] we cannot prove this condition for the whole domain of continuations A weaker condition can be established if we restrict the investigation to the class of denotable continuations Which is closed under arbitrary computations This condition we call weak completeness Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion The Class of Metric Domains for CSC A metric denotational domain of CSC is given as the unique solution of an equation [America and Rutten 1989] D ∼ 1 = Γ Γ → F Γ ∈ ) DCONT of domains for CSC is: The class (Γ Γ ::= 1 Γ 2 · D | M → Γ Γ | M × Γ Γ | M + Γ Γ | Γ Γ × Γ Γ | Γ Γ + Γ Γ ( M is an arbitrary set ( m ∈ ) M endowed with the discrete metric) Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
Introduction Continuations for Concurrency Weak Abstractness An Asynchronous Formalism Conclusion Resumptions and Denotable Continuations In the CSC approach a continuation is a structured configuration of (partially evaluated) denotations Following [De Bakker and De Vink 1996] we use the term resumption as an operational counterpart of the term continuation Let ( x ∈ ) X be a fixed set. We define a class ( R ∈ ) RES ( X ) of sets of resumptions for X by: R ::= X | M → R | M × R | M + R | R × R | R + R (here ( m ∈ ) M is an arbitrary set) Gabriel Ciobanu, Eneia Nicolae Todoran ”A.I. Cuza” University, Iasi, Technical University, Cluj-Napoca, Romania Abstractness of Continuation Semantics for Asynchronous Concurrency
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