The Exceptional Type Theory: Overview As hinted before, we need to be call-by-name to feature full conversion. 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) raise B e The exceptional type theory extends vanilla CIC with p 10 / 44 raise E : □ : Π A : □ . E → A raise (Π x : A . B ) e ≡ λ x : A . raise B e match ( raise I e ) ret P with ⃗ ≡ raise ( P ( raise I e )) e where P : I → □ . Remark that in call-by-name, if M : A → B , in general M ( raise A e ) ̸≡ for otherwise we would not have ( λ x : A . M ) N ≡ M { x := N } .
P p t p f p e true P p t p f p e false P p t p f p e raise p e e Catch Me If You Can P b where catch p t catch p f P false catch e P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 b P true Remember that on functions: P false It means catching exceptions is limited to positive datatypes! For inductive types, this is a generalized induction principle . catch P P P true e E P raise e b P b rect 11 / 44 raise (Π x : A . B ) e ≡ λ x : A . raise B e
Catch Me If You Can Remember that on functions: 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) p f p t where 11 / 44 For inductive types, this is a generalized induction principle . It means catching exceptions is limited to positive datatypes! raise (Π x : A . B ) e ≡ λ x : A . raise B e catch B : Π P : B → □ . B rect : Π P : B → □ . P true → P true → P false → P false → (Π e : E . P ( raise B e )) → Π b : B . P b Π b : B . P b ≡ catch B P p t p f p e true ≡ catch B P p t p f p e false catch B P p t p f p e ( raise B e ) ≡ p e e
Mot d’Ordre : A Model It’s not just randomly coming up with syntax though. We want a justifjcation for what we are doing What about normalization? Subject reduction? Other nice properties? ... that’s called a model. We want a model of the exceptional type theory! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 12 / 44
Mot d’Ordre : A Model It’s not just randomly coming up with syntax though. We want a justifjcation for what we are doing What about normalization? Subject reduction? Other nice properties? ... that’s called a model. We want a model of the exceptional type theory! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 12 / 44
Mot d’Ordre : A Model It’s not just randomly coming up with syntax though. We want a justifjcation for what we are doing What about normalization? Subject reduction? Other nice properties? ... that’s called a model. We want a model of the exceptional type theory! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 12 / 44
Kardashian Functors, Anyone? Semantics of CIC has a fame of being horribly complex. I won’t lie: it is . But part of this fame is nonetheless due to its models. Set-theoretical models: because Sets are a (crappy) type theory. Pro: Sets! Con: Sets! Realizability models: construct programs that respect properties. Pro: Computational, computer-science friendly. Con: Not foundational (requires an alien meta-theory), not decidable. Categorical models: abstract description of type theory. Pro: Abstract, subsumes the two former ones. Con: Realizability + very low level, gazillion variants, intrisically typed, static. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 13 / 44
Kardashian Functors, Anyone? Semantics of CIC has a fame of being horribly complex. I won’t lie: it is . But part of this fame is nonetheless due to its models. Set-theoretical models: because Sets are a (crappy) type theory. Pro: Sets! Con: Sets! Realizability models: construct programs that respect properties. Pro: Computational, computer-science friendly. Con: Not foundational (requires an alien meta-theory), not decidable. Categorical models: abstract description of type theory. Pro: Abstract, subsumes the two former ones. Con: Realizability + very low level, gazillion variants, intrisically typed, static. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 13 / 44
Kardashian Functors, Anyone? Semantics of CIC has a fame of being horribly complex. I won’t lie: it is . But part of this fame is nonetheless due to its models. Set-theoretical models: because Sets are a (crappy) type theory. Pro: Sets! Con: Sets! Realizability models: construct programs that respect properties. Pro: Computational, computer-science friendly. Con: Not foundational (requires an alien meta-theory), not decidable. Categorical models: abstract description of type theory. Pro: Abstract, subsumes the two former ones. Con: Realizability + very low level, gazillion variants, intrisically typed, static. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 13 / 44
Kardashian Functors, Anyone? Semantics of CIC has a fame of being horribly complex. I won’t lie: it is . But part of this fame is nonetheless due to its models. Set-theoretical models: because Sets are a (crappy) type theory. Pro: Sets! Con: Sets! Realizability models: construct programs that respect properties. Pro: Computational, computer-science friendly. Con: Not foundational (requires an alien meta-theory), not decidable. Categorical models: abstract description of type theory. Pro: Abstract, subsumes the two former ones. Con: Realizability + very low level, gazillion variants, intrisically typed, static. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 13 / 44
Kardashian Functors, Anyone? Semantics of CIC has a fame of being horribly complex. I won’t lie: it is . But part of this fame is nonetheless due to its models. Set-theoretical models: because Sets are a (crappy) type theory. Pro: Sets! Con: Sets! Realizability models: construct programs that respect properties. Pro: Computational, computer-science friendly. Con: Not foundational (requires an alien meta-theory), not decidable. Categorical models: abstract description of type theory. Pro: Abstract, subsumes the two former ones. Con: Realizability + very low level, gazillion variants, intrisically typed, static. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 13 / 44
Curry-Howard Orthodoxy Friedman’s trick 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) CH (global monotonous cell) Forcing Markov’s rule (exceptions) classical logic ( callcc ) Instead, let’s look at what Curry-Howard provides in simpler settings. Double-negation On the logic side, extend expressivity through proof translation. Interpret mechanically efgectful programs (e.g. in Haskell) A type transformer T , two combinators, a few equations On the programming side, implement efgects using e.g. the monadic style. 14 / 44 Logical Interpretations ⇔ Program Translations
Curry-Howard Orthodoxy Friedman’s trick 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) CH (global monotonous cell) Forcing Markov’s rule (exceptions) classical logic ( callcc ) Instead, let’s look at what Curry-Howard provides in simpler settings. Double-negation On the logic side, extend expressivity through proof translation. Interpret mechanically efgectful programs (e.g. in Haskell) A type transformer T , two combinators, a few equations On the programming side, implement efgects using e.g. the monadic style. 14 / 44 Logical Interpretations ⇔ Program Translations
Curry-Howard Orthodoxy Instead, let’s look at what Curry-Howard provides in simpler settings. On the programming side, implement efgects using e.g. the monadic style. A type transformer T , two combinators, a few equations Interpret mechanically efgectful programs (e.g. in Haskell) On the logic side, extend expressivity through proof translation. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 14 / 44 Logical Interpretations ⇔ Program Translations Double-negation ⇒ classical logic ( callcc ) Friedman’s trick ⇒ Markov’s rule (exceptions) Forcing ⇒ ¬ CH (global monotonous cell)
CIC M CIC M Syntactic Models M 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) translation. implementing new terms given by the by going down to the CIC assembly language, Step 3: Expand A A Step 2: Flip views and actually pose Let us do the same thing with CIC: build syntactic models . A implies A M from it s.t. and derive on the syntax of Step 1: Defjne CIC . Step 0: Fix a theory 15 / 44
CIC M CIC M Syntactic Models A 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) translation. implementing new terms given by the by going down to the CIC assembly language, Step 3: Expand A Step 2: Flip views and actually pose M Let us do the same thing with CIC: build syntactic models . A implies A M from it s.t. and derive on the syntax of Step 1: Defjne 15 / 44 Step 0: Fix a theory T := CIC .
CIC M Syntactic Models A 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) translation. implementing new terms given by the by going down to the CIC assembly language, Step 3: Expand A M Let us do the same thing with CIC: build syntactic models . Step 2: Flip views and actually pose implies 15 / 44 Step 0: Fix a theory T := CIC . Step 1: Defjne [ · ] on the syntax of T and derive [ [ · ] ] from it s.t. ⊢ T M : A ⊢ CIC [ M ] : [ [ A ] ]
Syntactic Models Step 2: Flip views and actually pose 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) translation. implementing new terms given by the by going down to the CIC assembly language, Step 3: Expand Let us do the same thing with CIC: build syntactic models . 15 / 44 implies Step 0: Fix a theory T := CIC . Step 1: Defjne [ · ] on the syntax of T and derive [ [ · ] ] from it s.t. ⊢ T M : A ⊢ CIC [ M ] : [ [ A ] ] ∆ ⊢ T M : A = ⊢ CIC [ M ] : [ [ A ] ]
Syntactic Models Step 2: Flip views and actually pose 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) Let us do the same thing with CIC: build syntactic models . 15 / 44 implies Step 0: Fix a theory T := CIC . Step 1: Defjne [ · ] on the syntax of T and derive [ [ · ] ] from it s.t. ⊢ T M : A ⊢ CIC [ M ] : [ [ A ] ] ∆ ⊢ T M : A = ⊢ CIC [ M ] : [ [ A ] ] Step 3: Expand T by going down to the CIC assembly language, implementing new terms given by the [ · ] translation.
« CIC, the LLVM of Type Theory » P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 16 / 44
The Exceptional Implementation Let’s implement the exceptional type theory into CIC! Source is a CBN theory, so usual monadic encoding won’t work. We use a variant of our previous weaning translation. All typing and computations rules mentioned before hold for free. Let’s call the exceptional type theory to disambiguate it from CIC . Only parameter of the translation: a fjxed type of exceptions in the target. CIC P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 17 / 44
The Exceptional Implementation Let’s implement the exceptional type theory into CIC! Source is a CBN theory, so usual monadic encoding won’t work. We use a variant of our previous weaning translation. All typing and computations rules mentioned before hold for free. Let’s call the exceptional type theory to disambiguate it from CIC . Only parameter of the translation: a fjxed type of exceptions in the target. CIC P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 17 / 44
The Exceptional Implementation Let’s implement the exceptional type theory into CIC! Source is a CBN theory, so usual monadic encoding won’t work. We use a variant of our previous weaning translation. All typing and computations rules mentioned before hold for free. Only parameter of the translation: a fjxed type of exceptions in the target. CIC P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 17 / 44 Let’s call the exceptional type theory T E to disambiguate it from CIC .
The Exceptional Implementation Let’s implement the exceptional type theory into CIC! Source is a CBN theory, so usual monadic encoding won’t work. We use a variant of our previous weaning translation. All typing and computations rules mentioned before hold for free. Only parameter of the translation: a fjxed type of exceptions in the target. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 17 / 44 Let’s call the exceptional type theory T E to disambiguate it from CIC . ⊢ CIC E : □
CIC M CIC M The Exceptional Implementation, Negative case x x x M N M N x A M M A B If A then A . P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 e A Intuition: x Every exceptional type comes with its own implementation of failure! A A and A A A x A B x A B x A B e 18 / 44 ⊢ T E A : □ ⊢ CIC [ A ] : Σ A : □ . E → A . ⇝
CIC M CIC M The Exceptional Implementation, Negative case Intuition: 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) A . A then If x 18 / 44 Every exceptional type comes with its own implementation of failure! and ⊢ T E A : □ ⊢ CIC [ A ] : Σ A : □ . E → A . ⇝ [ [ A ] ] : □ := π 1 [ A ] [ A ] ∅ : E → [ [ A ] ] := π 2 [ A ] [ [Π x : A . B ] ] ≡ Π x : [ [ A ] ] . [ [ B ] ] [Π x : A . B ] ∅ e ≡ λ x : [ [ A ] ] . [ B ] ∅ e [ x ] ≡ [ M N ] ≡ [ M ] [ N ] [ λ x : A . M ] ≡ λ x : [ [ A ] ] . [ M ]
The Exceptional Implementation, Negative case and 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) x Intuition: 18 / 44 Every exceptional type comes with its own implementation of failure! ⊢ T E A : □ ⊢ CIC [ A ] : Σ A : □ . E → A . ⇝ [ [ A ] ] : □ := π 1 [ A ] [ A ] ∅ : E → [ [ A ] ] := π 2 [ A ] [ [Π x : A . B ] ] ≡ Π x : [ [ A ] ] . [ [ B ] ] [Π x : A . B ] ∅ e ≡ λ x : [ [ A ] ] . [ B ] ∅ e [ x ] ≡ [ M N ] ≡ [ M ] [ N ] [ λ x : A . M ] ≡ λ x : [ [ A ] ] . [ M ] If Γ ⊢ CIC M : A then [ [Γ] ] ⊢ CIC [ M ] : [ [ A ] ] .
The Exceptional Implementation, Failure A x A raise B e x A B e x B e x e x A B e P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 A B e raise It is straightforward to implement the failure operation. e E raise E A A Computational rules trivially hold! E e raise A A A A raise 19 / 44 : □ : Π A : □ . E → A
The Exceptional Implementation, Failure A x A raise B e x A B e x B e x e x A B e P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 A B e raise It is straightforward to implement the failure operation. Computational rules trivially hold! E raise 19 / 44 : □ : Π A : □ . E → A [ E ] : Σ A : □ . E → A [ E ] := ( E , λ e : E . e ) [ raise ] : Π A 0 : (Σ A : □ . E → A ) . E → π 1 A 0 [ raise ] := π 2
The Exceptional Implementation, Failure Computational rules trivially hold! 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) It is straightforward to implement the failure operation. 19 / 44 E raise : □ : Π A : □ . E → A [ E ] : Σ A : □ . E → A [ E ] := ( E , λ e : E . e ) [ raise ] : Π A 0 : (Σ A : □ . E → A ) . E → π 1 A 0 [ raise ] := π 2 [ raise (Π x : A . B ) e ] [ λ x : A . raise B e ] ≡ ≡ ≡ π 2 ((Π x : [ [ A ] ] . [ [ B ] ]) , ( λ ( e : E ) ( x : [ [ A ] ]) . π 2 [ [ B ] ] e )) [ e ] λ x : [ [ A ] ] . π 2 [ B ] [ e ] ≡
The Exceptional Implementation, Positive case The really interesting case is the inductive part of CIC. Could pose and take an arbitrary boolean for ... ... but that would not play well with computation, e.g. catch . Worse, what about ? P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 20 / 44 How to implement [ B ] ∅ : E → [ [ B ] ] ?
The Exceptional Implementation, Positive case The really interesting case is the inductive part of CIC. ... but that would not play well with computation, e.g. catch . Worse, what about ? P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 20 / 44 How to implement [ B ] ∅ : E → [ [ B ] ] ? Could pose [ [ B ] ] := B and take an arbitrary boolean for [ B ] ∅ ...
The Exceptional Implementation, Positive case The really interesting case is the inductive part of CIC. ... but that would not play well with computation, e.g. catch . P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 20 / 44 How to implement [ B ] ∅ : E → [ [ B ] ] ? Could pose [ [ B ] ] := B and take an arbitrary boolean for [ B ] ∅ ... Worse, what about [ ⊥ ] ∅ : E → [ [ ⊥ ] ] ?
P true P false The Exceptional Implementation, Positive case P 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) e e , reraise e using P b If b is an error If b is false , use second hypothesis If b is true , use fjrst hypothesis P b b P false P true b P b Very elegant solution: add a default case to every inductive type! P Pattern-matching is translated pointwise, except for the new case. 21 / 44 Inductive [ [ B ] ] := [ true ] : [ [ B ] ] | [ false ] : [ [ B ] ] | B ∅ : E → [ [ B ] ]
The Exceptional Implementation, Positive case Pattern-matching is translated pointwise, except for the new case. 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) Very elegant solution: add a default case to every inductive type! 21 / 44 Inductive [ [ B ] ] := [ true ] : [ [ B ] ] | [ false ] : [ [ B ] ] | B ∅ : E → [ [ B ] ] [ [Π P : B → □ . P true → P false → Π b : B . P b ] ] ≡ Π P : [ [ B ] ] → [ [ □ ] ] . P [ true ] → P [ false ] → Π b : [ [ B ] ] . P b If b is [ true ] , use fjrst hypothesis If b is [ false ] , use second hypothesis If b is an error B ∅ e , reraise e using [ P b ] ∅ e
☺ A type theory with efgects! ☺ Compiled away to CIC ! ☺ Features full conversion ☺ Features full dependent elimination 😗 Ah, yeah, and also, the theory is inconsistent. Shadok Logic Strikes Back Theorem The exceptional translation interprets all of CIC . It suffjces to raise an exception to inhabit any type. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 22 / 44
😗 Ah, yeah, and also, the theory is inconsistent. Shadok Logic Strikes Back Theorem The exceptional translation interprets all of CIC . It suffjces to raise an exception to inhabit any type. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 22 / 44 ☺ A type theory with efgects! ☺ Compiled away to CIC ! ☺ Features full conversion ☺ Features full dependent elimination
😗 Ah, yeah, and also, the theory is inconsistent. Shadok Logic Strikes Back Theorem The exceptional translation interprets all of CIC . It suffjces to raise an exception to inhabit any type. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 22 / 44 ☺ A type theory with efgects! ☺ Compiled away to CIC ! ☺ Features full conversion ☺ Features full dependent elimination
Shadok Logic Strikes Back Theorem The exceptional translation interprets all of CIC . It suffjces to raise an exception to inhabit any type. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 22 / 44 ☺ A type theory with efgects! ☺ Compiled away to CIC ! ☺ Features full conversion ☺ Features full dependent elimination 😗 Ah, yeah, and also, the theory is inconsistent.
Consistency: A Social Construct You can still use the CIC target to prove properties about 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) And now for a little ad before the second part of the show! You can prove that a program does not raise uncaught exceptions. Clifghanger programs! A Safe Target Framework An Impure Dependently-typed Programming Language E . e for some e raise , then M M If Theorem ( Exceptional Canonicity a.k.a. Progress a.k.a. Meaningless explanations) Do you whine about the fact that OCaml is logically inconsistent? 23 / 44
Consistency: A Social Construct An Impure Dependently-typed Programming Language Do you whine about the fact that OCaml is logically inconsistent? Theorem ( Exceptional Canonicity a.k.a. Progress a.k.a. Meaningless explanations) A Safe Target Framework You can still use the CIC target to prove properties about programs! Clifghanger You can prove that a program does not raise uncaught exceptions. And now for a little ad before the second part of the show! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 23 / 44 If ⊢ T E M : ⊥ , then M ≡ raise ⊥ e for some e : E .
Consistency: A Social Construct An Impure Dependently-typed Programming Language Do you whine about the fact that OCaml is logically inconsistent? Theorem ( Exceptional Canonicity a.k.a. Progress a.k.a. Meaningless explanations) A Safe Target Framework Clifghanger You can prove that a program does not raise uncaught exceptions. And now for a little ad before the second part of the show! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 23 / 44 If ⊢ T E M : ⊥ , then M ≡ raise ⊥ e for some e : E . You can still use the CIC target to prove properties about T E programs!
Consistency: A Social Construct An Impure Dependently-typed Programming Language Do you whine about the fact that OCaml is logically inconsistent? Theorem ( Exceptional Canonicity a.k.a. Progress a.k.a. Meaningless explanations) A Safe Target Framework Clifghanger You can prove that a program does not raise uncaught exceptions. And now for a little ad before the second part of the show! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 23 / 44 If ⊢ T E M : ⊥ , then M ≡ raise ⊥ e for some e : E . You can still use the CIC target to prove properties about T E programs!
Consistency: A Social Construct An Impure Dependently-typed Programming Language Do you whine about the fact that OCaml is logically inconsistent? Theorem ( Exceptional Canonicity a.k.a. Progress a.k.a. Meaningless explanations) A Safe Target Framework Clifghanger You can prove that a program does not raise uncaught exceptions. And now for a little ad before the second part of the show! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 23 / 44 If ⊢ T E M : ⊥ , then M ≡ raise ⊥ e for some e : E . You can still use the CIC target to prove properties about T E programs!
CIC P Informercial — Did You Know? types, 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) P Q . p CIC Q implies P p CIC If P and Q are The exceptional translation is just a principled Friedman’s A -translation! Friedman’s Trick in CIC . P type, then If P is a First-order purifjcation A A Informative double-negation As such, it can be used for classical proof extraction. 24 / 44
CIC P Informercial — Did You Know? Friedman’s Trick in CIC 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) P Q . p CIC Q implies P p CIC types, If P and Q are . The exceptional translation is just a principled Friedman’s A -translation! P type, then If P is a First-order purifjcation Informative double-negation As such, it can be used for classical proof extraction. 24 / 44 ] ∼ [ [ ¬¬ A ] = ([ [ A ] ] → E ) → E
Informercial — Did You Know? Friedman’s Trick in CIC 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) P Q . p CIC Q implies P p CIC types, If P and Q are 24 / 44 The exceptional translation is just a principled Friedman’s A -translation! First-order purifjcation Informative double-negation As such, it can be used for classical proof extraction. ] ∼ [ [ ¬¬ A ] = ([ [ A ] ] → E ) → E If P is a Σ 0 1 type, then ⊢ CIC [ [ P ] ] ↔ P + E .
Informercial — Did You Know? First-order purifjcation 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) Friedman’s Trick in CIC The exceptional translation is just a principled Friedman’s A -translation! 24 / 44 As such, it can be used for classical proof extraction. Informative double-negation ] ∼ [ [ ¬¬ A ] = ([ [ A ] ] → E ) → E If P is a Σ 0 1 type, then ⊢ CIC [ [ P ] ] ↔ P + E . If P and Q are Σ 0 1 types, ⊢ CIC Π p : P . ¬¬ Q implies ⊢ CIC Π p : P . Q .
Part II P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 25 / 44
If You Joined the Talk Recently The exceptional type theory is logically inconsistent! Clifghanger (cont.) You can prove that a program does not raise uncaught exceptions. Let’s call valid a program in that “does not raise exceptions”. For instance, there is no valid proof of the only valid booleans are true and false a function is valid if it produces a valid result out of a valid argument Validity is a type-directed notion! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 26 / 44
If You Joined the Talk Recently The exceptional type theory is logically inconsistent! Clifghanger (cont.) You can prove that a program does not raise uncaught exceptions. For instance, the only valid booleans are true and false a function is valid if it produces a valid result out of a valid argument Validity is a type-directed notion! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 26 / 44 Let’s call valid a program in T E that “does not raise exceptions”. there is no valid proof of ⊥
If You Joined the Talk Recently The exceptional type theory is logically inconsistent! Clifghanger (cont.) You can prove that a program does not raise uncaught exceptions. For instance, the only valid booleans are true and false a function is valid if it produces a valid result out of a valid argument Validity is a type-directed notion! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 26 / 44 Let’s call valid a program in T E that “does not raise exceptions”. there is no valid proof of ⊥
The Curry-Howard-Shadok Correspondence B 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) Zo ! Fools ! That’s parametricity . Come on. That’s intuitionistic realizability . What? That’s just logical relations . f x A x A x B A f 27 / 44 Let’s locally write M ⊩ A if M is valid at A .
What? That’s just logical relations . The Curry-Howard-Shadok Correspondence Come on. That’s intuitionistic realizability . Fools ! That’s parametricity . Zo ! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 27 / 44 Let’s locally write M ⊩ A if M is valid at A . f ⊩ A → B ≡ ∀ x : [ [ A ] ] . x ⊩ A → f x ⊩ B
What? That’s just logical relations . The Curry-Howard-Shadok Correspondence Come on. That’s intuitionistic realizability . Fools ! That’s parametricity . Zo ! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 27 / 44 Let’s locally write M ⊩ A if M is valid at A . f ⊩ A → B ≡ ∀ x : [ [ A ] ] . x ⊩ A → f x ⊩ B
What? That’s just logical relations . The Curry-Howard-Shadok Correspondence Come on. That’s intuitionistic realizability . Fools ! That’s parametricity . Zo ! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 27 / 44 Let’s locally write M ⊩ A if M is valid at A . f ⊩ A → B ≡ ∀ x : [ [ A ] ] . x ⊩ A → f x ⊩ B
What? That’s just logical relations . The Curry-Howard-Shadok Correspondence Come on. That’s intuitionistic realizability . Fools ! That’s parametricity . Zo ! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 27 / 44 Let’s locally write M ⊩ A if M is valid at A . f ⊩ A → B ≡ ∀ x : [ [ A ] ] . x ⊩ A → f x ⊩ B
What? That’s just logical relations . The Curry-Howard-Shadok Correspondence Come on. That’s intuitionistic realizability . Fools ! That’s parametricity . Zo ! P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 27 / 44 Let’s locally write M ⊩ A if M is valid at A . f ⊩ A → B ≡ ∀ x : [ [ A ] ] . x ⊩ A → f x ⊩ B
CIC M CIC M Making Everybody Agree 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) is the validity predicate. A where A M A A It’s actually folklore that these techniques are essentially the same. produce two sequents A M From Idea: We just have to adapt it to our exceptional translation. And there is already a parametricity translation for CIC! (Bernardy-Lasson) 28 / 44
CIC M CIC M Making Everybody Agree 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) is the validity predicate. A where A M A A It’s actually folklore that these techniques are essentially the same. produce two sequents A M From Idea: We just have to adapt it to our exceptional translation. And there is already a parametricity translation for CIC! (Bernardy-Lasson) 28 / 44
Making Everybody Agree produce two sequents 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) It’s actually folklore that these techniques are essentially the same. 28 / 44 And there is already a parametricity translation for CIC! (Bernardy-Lasson) We just have to adapt it to our exceptional translation. From Idea: ⊢ CIC [ M ] : [ [ A ] ] ⊢ M : A + ⊢ CIC [ M ] ε : [ [ A ] ] ε [ M ] where [ [ A ] ] ε : [ [ A ] ] → □ is the validity predicate.
CIC M CIC M Parametric Exceptional Translation (Sketch) Most notably, 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) M . A A then If Every pure term is now automatically parametric. 29 / 44 [ [Π x : A . B ] ] ε f ≡ Π( x : [ [ A ] ]) ( x ε : [ [ A ] ] ε x ) . [ [ B ] ] ε ( f x ) ∼ [ [ B ] ] ε b b = [ true ] + b = [ false ] = ∼ [ [ ⊥ ] ] ε s ⊥ =
Parametric Exceptional Translation (Sketch) Most notably, 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) Every pure term is now automatically parametric. 29 / 44 [ [Π x : A . B ] ] ε f ≡ Π( x : [ [ A ] ]) ( x ε : [ [ A ] ] ε x ) . [ [ B ] ] ε ( f x ) ∼ [ [ B ] ] ε b b = [ true ] + b = [ false ] = ∼ [ [ ⊥ ] ] ε s ⊥ = If Γ ⊢ CIC M : A then [ [Γ] ] ε ⊢ CIC [ M ] ε : [ [ A ] ] ε [ M ] .
A Few Nice Results Theorem (Syntax) 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) 30 / 44 Theorem (Consistency) Theorem (Canonicity) Let’s call T p E the resulting theory. It inherits a lot from CIC ! T p E is consistent. E M : N then M ⇝ ∗ ¯ n ∈ ¯ T p E enjoys canonicity, i.e if ⊢ T p N . T p E has decidable type-checking, strong normalization and whatnot.
What If There Were No Cake? Bernardy-Lasson parametricity is a conservative extension of CIC ... P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 31 / 44
What If There Were No Cake? Bernardy-Lasson parametricity is a conservative extension of CIC ... P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 31 / 44
What If There Were No Cake? Bernardy-Lasson parametricity is a conservative extension of CIC ... P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 31 / 44
p a semantical layer atop of it. p is the embodiement of Kreisel modifjed realizability in CIC . Less Is More is the unsafe Coq fragment, and 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) Actually ... as long as you prove they don’t escape! Spoiler ... but you can still raise them locally p raising uncaught exceptions is forbidden in Intuitively, 32 / 44 T p E is not a conservative extension of CIC .
p a semantical layer atop of it. p is the embodiement of Kreisel modifjed realizability in CIC . Less Is More is the unsafe Coq fragment, and 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) Actually ... as long as you prove they don’t escape! Spoiler ... but you can still raise them locally Intuitively, 32 / 44 T p E is not a conservative extension of CIC . raising uncaught exceptions is forbidden in T p E
p a semantical layer atop of it. p is the embodiement of Kreisel modifjed realizability in CIC . Less Is More is the unsafe Coq fragment, and 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) Actually ... as long as you prove they don’t escape! Spoiler ... but you can still raise them locally Intuitively, 32 / 44 T p E is not a conservative extension of CIC . raising uncaught exceptions is forbidden in T p E
p is the embodiement of Kreisel modifjed realizability in CIC . Less Is More Spoiler 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) Actually ... as long as you prove they don’t escape! ... but you can still raise them locally Intuitively, 32 / 44 T p E is not a conservative extension of CIC . raising uncaught exceptions is forbidden in T p E T E is the unsafe Coq fragment, and T p E a semantical layer atop of it.
Less Is More ... but you can still raise them locally 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) Spoiler ... as long as you prove they don’t escape! Intuitively, 32 / 44 T p E is not a conservative extension of CIC . raising uncaught exceptions is forbidden in T p E T E is the unsafe Coq fragment, and T p E a semantical layer atop of it. Actually T p E is the embodiement of Kreisel modifjed realizability in CIC .
Explaining the Analogy P m n 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) P n A n P n n A IP P n f n n f m Kreisel realizability n AC Kreisel realizability extends arithmetic with essentially two principles. CIC Logical meta-theory System T Programming language CIC Source theory 33 / 44 T p E HA or HA ω T E (“unsafe Coq”) HA ω
Explaining the Analogy Kreisel realizability 22/02/2018 Failure is Not an Option P.-M. Pédrot (MPI-SWS) Kreisel realizability extends arithmetic with essentially two principles. CIC Logical meta-theory System T Programming language CIC Source theory 33 / 44 T p E HA or HA ω T E (“unsafe Coq”) HA ω AC N : ( ∀ n : N . ∃ m : N . P ( m , n )) → ∃ f : N → N . ∀ n : N . P ( n , f n ) IP : ( ¬ A → ∃ n : N . P n ) → ∃ n : N . ¬ A → P n
Choice Not much to say here. In p , AC is a consequence of dependent elimination. The latter is in turn meta-theoretically justifjed by canonicity. In both cases, choice is built-in and a consequence of canonicity. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 34 / 44 AC N : ( ∀ n : N . ∃ m : N . P ( m , n )) → ∃ f : N → N . ∀ n : N . P ( n , f n ) In Kreisel realizability, AC N is a consequence of canonicity of System T.
Choice Not much to say here. The latter is in turn meta-theoretically justifjed by canonicity. In both cases, choice is built-in and a consequence of canonicity. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 34 / 44 AC N : ( ∀ n : N . ∃ m : N . P ( m , n )) → ∃ f : N → N . ∀ n : N . P ( n , f n ) In Kreisel realizability, AC N is a consequence of canonicity of System T. In T p E , AC N is a consequence of dependent elimination.
Choice Not much to say here. The latter is in turn meta-theoretically justifjed by canonicity. In both cases, choice is built-in and a consequence of canonicity. P.-M. Pédrot (MPI-SWS) Failure is Not an Option 22/02/2018 34 / 44 AC N : ( ∀ n : N . ∃ m : N . P ( m , n )) → ∃ f : N → N . ∀ n : N . P ( n , f n ) In Kreisel realizability, AC N is a consequence of canonicity of System T. In T p E , AC N is a consequence of dependent elimination.
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