DL π A Linear π -Calculus with Dependent Types in Agda Luca Ciccone Luca Padovani 5 June 2020 Luca Ciccone, Luca Padovani DL π 5 June 2020 1 / 33
Motivation and Goal Motivation and Goal 1 DL π - Language 2 DL π - Agda Formalization 3 Examples 4 Encoding 5 Conclusions 6 Luca Ciccone, Luca Padovani DL π 5 June 2020 2 / 33
Motivation and Goal Motivation Session types = linear channels + pairs + sums [Dardha et al., 2017] Session = chain of one-shot communications Messages are pairs : payload + continuation channel Luca Ciccone, Luca Padovani DL π 5 June 2020 3 / 33
Motivation and Goal Motivation Session types = linear channels + pairs + sums [Dardha et al., 2017] Session = chain of one-shot communications Messages are pairs : payload + continuation channel Dependent session types = linear channels + dependent pairs? Value-dependent session types [Toninho et al., 2011] Liquid Pi [Griffith and Gunter, 2013] Dependent protocols in Idris [Brady, 2017] Dependent session-typed processes [Toninho and Yoshida, 2018] Label-dependent session types [Thiemann and Vasconcelos, 2020] Luca Ciccone, Luca Padovani DL π 5 June 2020 3 / 33
Motivation and Goal Motivation Session types = linear channels + pairs + sums [Dardha et al., 2017] Session = chain of one-shot communications Messages are pairs : payload + continuation channel Dependent session types = linear channels + dependent pairs? Value-dependent session types [Toninho et al., 2011] Liquid Pi [Griffith and Gunter, 2013] Dependent protocols in Idris [Brady, 2017] Dependent session-typed processes [Toninho and Yoshida, 2018] Label-dependent session types [Thiemann and Vasconcelos, 2020] Yes! Luca Ciccone, Luca Padovani DL π 5 June 2020 3 / 33
Motivation and Goal Goal Develop a minimal, Agda-based, linear π -calculus with dependent pairs ( DL π ) in which dependent session types can be encoded Luca Ciccone, Luca Padovani DL π 5 June 2020 4 / 33
Motivation and Goal Goal Develop a minimal, Agda-based, linear π -calculus with dependent pairs ( DL π ) in which dependent session types can be encoded Analogous expressiveness of Brady [2017]’s DSL and Toninho and Yoshida [2018]’s calculus, but: different type structure linear channels + dependent pairs instead of session types Agda mechanisation of the metatheory we lift from Agda all the machinery related to dependent types computation of data-dependent types and processes Luca Ciccone, Luca Padovani DL π 5 June 2020 4 / 33
DL π - Language Motivation and Goal 1 DL π - Language 2 DL π - Agda Formalization 3 Examples 4 Encoding 5 Conclusions 6 Luca Ciccone, Luca Padovani DL π 5 June 2020 5 / 33
DL π - Language DL π - Processes M , N ::= p pure term Terms | u name | M , N pair Processes P , Q ::= idle inaction | u ( x ) . P input | u � M � output | let x , y = M in P pair splitting | P | Q parallel composition | ( a ) P restriction | ∗ P replication Luca Ciccone, Luca Padovani DL π 5 June 2020 6 / 33
DL π - Language DL π - Processes M , N ::= p pure term Terms | u name | M , N pair Processes P , Q ::= idle inaction | u ( x ) . P input | u � M � output | let x , y = M in P pair splitting | P | Q parallel composition | ( a ) P restriction | ∗ P replication Pure terms injected in DL π Luca Ciccone, Luca Padovani DL π 5 June 2020 6 / 33
DL π - Language DL π - Types Domains A , B ∈ A pure types σ , ρ ∈ { 0 , 1 , ω } multiplicities t , s ::= A pure type Types σ , ρ [ t ] | channel type | Σ( x : t ) s linear dependent pair Luca Ciccone, Luca Padovani DL π 5 June 2020 7 / 33
DL π - Language DL π - Types Domains A , B ∈ A pure types σ , ρ ∈ { 0 , 1 , ω } multiplicities t , s ::= A pure type Types σ , ρ [ t ] | channel type | Σ( x : t ) s linear dependent pair Multiplicities can be combined with a sum Luca Ciccone, Luca Padovani DL π 5 June 2020 7 / 33
DL π - Language DL π - Types Domains A , B ∈ A pure types σ , ρ ∈ { 0 , 1 , ω } multiplicities t , s ::= A pure type Types σ , ρ [ t ] | channel type | Σ( x : t ) s linear dependent pair Multiplicities can be combined with a sum 1 + 1 = ω , ω + ω = ω Luca Ciccone, Luca Padovani DL π 5 June 2020 7 / 33
DL π - Language DL π - Types Domains A , B ∈ A pure types σ , ρ ∈ { 0 , 1 , ω } multiplicities t , s ::= A pure type Types σ , ρ [ t ] | channel type | Σ( x : t ) s linear dependent pair Multiplicities can be combined with a sum 1 + 1 = ω , ω + ω = ω Types can be combined Luca Ciccone, Luca Padovani DL π 5 June 2020 7 / 33
DL π - Language DL π - Types Domains A , B ∈ A pure types σ , ρ ∈ { 0 , 1 , ω } multiplicities t , s ::= A pure type Types σ , ρ [ t ] | channel type | Σ( x : t ) s linear dependent pair Multiplicities can be combined with a sum 1 + 1 = ω , ω + ω = ω Types can be combined Partial operation (in Agda represented as a relation ) Luca Ciccone, Luca Padovani DL π 5 June 2020 7 / 33
DL π - Language DL π - Types Domains A , B ∈ A pure types σ , ρ ∈ { 0 , 1 , ω } multiplicities t , s ::= A pure type Types σ , ρ [ t ] | channel type | Σ( x : t ) s linear dependent pair Multiplicities can be combined with a sum 1 + 1 = ω , ω + ω = ω Types can be combined Partial operation (in Agda represented as a relation ) Pairs are linear Luca Ciccone, Luca Padovani DL π 5 June 2020 7 / 33
DL π - Language Parallel composition t-par Γ 1 ⊢ P Γ 2 ⊢ Q Γ 1 + Γ 2 ⊢ P | Q Luca Ciccone, Luca Padovani DL π 5 June 2020 8 / 33
DL π - Language Parallel composition t-par Γ 1 ⊢ P Γ 2 ⊢ Q Γ 1 + Γ 2 ⊢ P | Q Resources are combined Γ 1 + Γ 2 = Γ 1 , Γ 2 dom(Γ 1 ) ∩ dom(Γ 2 ) = ∅ ( u : t , Γ 1 ) + ( u : s , Γ 2 ) = ( u : t + s ) , (Γ 1 + Γ 2 ) Luca Ciccone, Luca Padovani DL π 5 June 2020 8 / 33
DL π - Language Parallel composition t-par Γ 1 ⊢ P Γ 2 ⊢ Q Γ 1 + Γ 2 ⊢ P | Q Resources are combined Γ 1 + Γ 2 = Γ 1 , Γ 2 dom(Γ 1 ) ∩ dom(Γ 2 ) = ∅ ( u : t , Γ 1 ) + ( u : s , Γ 2 ) = ( u : t + s ) , (Γ 1 + Γ 2 ) Sum of types 1 , 0 [ t ] + 0 , 1 [ t ] = 1 , 1 [ t ] Luca Ciccone, Luca Padovani DL π 5 June 2020 8 / 33
DL π - Language Receive - Send t-input t-output Γ 1 ⊢ u : 1 , 0 [ t ] Γ 1 ⊢ u : 0 , 1 [ t ] Γ 2 , x : t ⊢ P Γ 2 ⊢ M : t Γ 1 + Γ 2 ⊢ u ( x ) . P Γ 1 + Γ 2 ⊢ u � M � Resources must be combined Luca Ciccone, Luca Padovani DL π 5 June 2020 9 / 33
DL π - Language Receive - Send t-input t-output Γ 1 ⊢ u : 1 , 0 [ t ] Γ 1 ⊢ u : 0 , 1 [ t ] Γ 2 , x : t ⊢ P Γ 2 ⊢ M : t Γ 1 + Γ 2 ⊢ u ( x ) . P Γ 1 + Γ 2 ⊢ u � M � Resources must be combined Linear channels Input 1 , 0 Output 0 , 1 Luca Ciccone, Luca Padovani DL π 5 June 2020 9 / 33
DL π - Language Receive - Send t-input t-output Γ 1 ⊢ u : 1 , 0 [ t ] Γ 1 ⊢ u : 0 , 1 [ t ] Γ 2 , x : t ⊢ P Γ 2 ⊢ M : t Γ 1 + Γ 2 ⊢ u ( x ) . P Γ 1 + Γ 2 ⊢ u � M � Resources must be combined Linear channels Input 1 , 0 Output 0 , 1 Luca Ciccone, Luca Padovani DL π 5 June 2020 9 / 33
DL π - Language Dependent Pairs t-pair Γ 1 ⊢ M : t Γ 2 ⊢ N : s { � M � / x } Γ 1 + Γ 2 ⊢ M , N : Σ( x : t ) s Luca Ciccone, Luca Padovani DL π 5 June 2020 10 / 33
DL π - Language Dependent Pairs t-pair Γ 1 ⊢ M : t Γ 2 ⊢ N : s { � M � / x } Γ 1 + Γ 2 ⊢ M , N : Σ( x : t ) s Filter function introduced Luca Ciccone, Luca Padovani DL π 5 June 2020 10 / 33
DL π - Language Dependent Pairs t-pair Γ 1 ⊢ M : t Γ 2 ⊢ N : s { � M � / x } Γ 1 + Γ 2 ⊢ M , N : Σ( x : t ) s Filter function introduced Map from DL π terms to pure terms � p � = p � x � = x � a � = tt � M , N � = � M � , � N � Luca Ciccone, Luca Padovani DL π 5 June 2020 10 / 33
DL π - Language Dependent Pairs t-pair Γ 1 ⊢ M : t Γ 2 ⊢ N : s { � M � / x } Γ 1 + Γ 2 ⊢ M , N : Σ( x : t ) s Filter function introduced Map from DL π terms to pure terms � p � = p � x � = x � a � = tt � M , N � = � M � , � N � From the point of view of types � σ , ρ [ t ] � = ⊤ � A � = A � Σ( x : t ) s � = Σ( x : � t � ) � s � Luca Ciccone, Luca Padovani DL π 5 June 2020 10 / 33
DL π - Language Dependent Pairs t-pair Γ 1 ⊢ M : t Γ 2 ⊢ N : s { � M � / x } Γ 1 + Γ 2 ⊢ M , N : Σ( x : t ) s Filter function introduced Map from DL π terms to pure terms � p � = p � x � = x � a � = tt � M , N � = � M � , � N � From the point of view of types � σ , ρ [ t ] � = ⊤ � A � = A � Σ( x : t ) s � = Σ( x : � t � ) � s � Channels are erased Luca Ciccone, Luca Padovani DL π 5 June 2020 10 / 33
DL π - Language Dependent Pairs t-pair Γ 1 ⊢ M : t Γ 2 ⊢ N : s { � M � / x } Γ 1 + Γ 2 ⊢ M , N : Σ( x : t ) s Filter function introduced Map from DL π terms to pure terms � p � = p � x � = x � a � = tt � M , N � = � M � , � N � From the point of view of types � σ , ρ [ t ] � = ⊤ � A � = A � Σ( x : t ) s � = Σ( x : � t � ) � s � Channels are erased No dependency on channels Luca Ciccone, Luca Padovani DL π 5 June 2020 10 / 33
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