A Rewrite System for Strongly Normalizable Terms

Transcription

1 A Rewrite System for Strongly Normalizable Terms IRIF Seminar Olivier Hermant & Ronan Saillard CRI, MINES ParisTech, PSL University June 28, 2018 O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

2 Intersection Types (x : F) Γ (Axiom) Γ x : F Γ X : F G Γ Y : F ( E ) Γ X Y : G Γ, x : F X : G ( I ) Γ λx.x : F G Γ X : F G ( E1 ) Γ X : F Γ X : F G ( E2 ) Γ X : G Γ X : F Γ X : G ( I Γ X : F G λx.(x x) : (A (A A)) A if X is typable, it is SN if X is SN, it is typable O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

3 Statman s System [2012] higher-order predicate/new connective D : ι o o o, a discriminator D v F G properties : D 0 F G is F and D 1 F G is G intuition D is an if...then...else... operator v DvFG encodes F G meaning given by rewrite rules D0FG F (0) D1FG G (1) Dt(F G)(H K) (DtFH) (DtGK) ( ) Dt( xf)g x(dtfg) ( 1 ) DtF( xg) x(dtfg) ( 2 ) xf F ($ ) X yf y xf ($$) ( ) no variable is captured (including those of t) ( ) x does not appear in F O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

4 Typing Rules (x : F) Γ (Axiom) Γ x : F Γ X : F F G (Conv) Γ X : G Γ X : F G Γ Y : F ( E ) Γ X Y : G Γ, x : F X : G ( I ) Γ λx.x : F G Γ X : F v : ι v fv(γ) Γ X : v.f ( I ) Γ X : v.f t : ι t free for v in F ( E ) Γ X : F[v/t] Figure Typing Rules of Minimal Natural Deduction with Conversion if X is typable, it is SN if X is SN, it is typable O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

5 Example Give a type to λx.(x x) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

6 Goals 1 down to first-order intuitionistic logic 2 prove with rewrite rules suitable framework : Deduction Modulo Theory if X is typable, it is SN if X is SN, it is typable 3 (further work) the superconsistency conjecture (G. Dowek) : a type system is superconsistent iff it is SN ( ) : proof by Dowek ( ) :? this system is a candidate to disprove the conjecture O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

7 Method a type system with conversion is totally acceptable conversion is now defined otherwise remind Statman s system D0FG F (0) D1FG G (1) Dt(F G)(H K) (DtFH) (DtGK) ( ) Dt( xf)g x(dtfg) ( ) DtF( xg) x(dtfg) ( ) xf F ($ ) X yf y xf ($$) we need to get rid of D at least at the propositional level what can we save from this system in Deduction Modulo Theory? O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

8 term rewrite rule : A A Deduction Modulo Theory Rewrite Rule A term (resp. proposition) rewrite rule is a pair of terms (resp. formulæ) l r, where F V(l) F V(r) and, in the propositiona case, l is atomic. Examples : proposition rewrite rule : A B x x A x B Conversion modulo a Rewrite System We consider the congruence generated by a set of proposition rewrite rules R and a set of term rewrite rules E (often implicit) Example : A A x x A x A O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

9 Typing Rules (x : F) Γ (Axiom) Γ x : F Γ X : F F G (Conv) Γ X : G Γ X : F G Γ Y : F ( E ) Γ X Y : G Γ, x : F X : G ( I ) Γ λx.x : F G Γ X : F v : ι v fv(γ) Γ X : v.f ( I ) Γ X : v.f t : ι t free for v in F ( E ) Γ X : F[v/t] Figure Typing Rules of Minimal Natural Deduction Modulo Theory O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

10 Proof of A A with and without DM without (Γ := z : A A x(x A x A)) : (ax) E2 Γ z : A A x(x A x A) Γ z 2 : x(x A x A) A A Γ (z 2 (λy.y)) : A A. Γ λy.y : x(x A x A) A A with (ax) y : x A y : x A λy.y : x A x A I λy.y : x(x A x A) (Conv) λy.y : A A as if we replaced z1 and z 2 with λα.α (see also Polarized Deduction Modulo Theory) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

11 Statman s System in First-Order Minimal DMT Analysis D0FG F (0) D1FG G (1) Dt(F G)(H K) (DtFH) (DtGK) ( ) Dt( xf)g x(dtfg) ( ) DtF( xg) x(dtfg) ( ) xf F ($ ) X yf y xf ($$) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

12 Statman s System in First-Order Minimal DMT Analysis D0FG F (0) D1FG G (1) Dt(F G)(H K) (DtFH) (DtGK) ( ) Dt( xf)g x(dtfg) ( ) DtF( xg) x(dtfg) ( ) xf F ($ ) X yf y xf ($$) get rid of D reflect it as a term (techniques to embed HOL) introduce the following terms :,, D, 0, 1 and a unique unary predicate ε O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

13 Statman s System in First-Order Minimal DMT Analysis D0FG F (0) D1FG G (1) Dt(F G)(H K) (DtFH) (DtGK) ( ) Dt( xf)g x(dtfg) ( ) DtF( xg) x(dtfg) ( ) xf F ($ ) X yf y xf ($$) get rid of D reflect it as a term (techniques to embed HOL) introduce the following terms :,, D, 0, 1 and a unique unary predicate ε combine terms properly (e.g. forbid ) simple types, with ι (0 and 1) and o (propositional terms) 0, 1 : ι D : ι o o o : (ι o) o : o o o add one propositional symbol ṗ : o propositional rewriting : ε(a B) ε(a) ε(b) ε( A) x.ε(a x) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

14 Statman s System in First-Order Minimalist Deduction Modulo Theory Statman s System : D0FG F (0) D1FG G (1) Dt(F G)(H K) (DtFH) (DtGK) ( ) Dt( xf)g x(dtfg) ( ) DtF( xg) x(dtfg) ( ) xf F ($ ) X yf y xf ($$) we can readily define three rewrite rules : D0FG F (0) D1FG G (1) Dt(F G)(H K) (DtFH) (DtGK) ( ) still to be defined Dt( F)G? ( 1 ) DtF( G)? ( 2 ) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

15 How to Abstract Need an equivalent of Dt( xf)g x(dtfg) at term level Dv( F)G no free variable x hence no freshness constraint (good) nevertheless need to define something like for some fresh x Dv( F)G (λx.(dv(fx)g)) two solutions exist in Deduction modulo theory 1 allow λ-abstraction in the simply-typed term language 2 replace this by a combinatorial calculus choice : Solution 1 cumbersome : explicit substitutions interfere with D Solution 2 cumbersome too could we have dropped explicit substitutions? O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

16 Combinatorial Calculus SKI we introduce and application symbols : I : ι ι K : τ ι τ S : (ι τ α) (ι τ) ι α denote in those slides as white space usual reduction rules I x x (I) K x y x (K) S x y z x z (y z) (S) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

17 Combinatorial Calculus SKI defining abstraction it is possible (see textbooks) we only need Dv( F)G (λx.(dv(fx)g)) with x fresh. so, define λx.(dv(fx)g) as S (K DvF) (S (KG) I) similarly for DvF( G) (λx.(dvf(gx))) note : no new (rewriting) redex is created some redex might be destroyed (take 0 or 1 for v) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

18 The Final Rewriting System encoding the logic : ε(a B) ε(a) ε(b) (ε ) I x x (I) ε( A) x.ε(a x) (ε ) K x y x (K) S x y z x z (yz) (S) encoding Statman s rules : D0FG F (0) D1FG G (1) Dt(F G)(H K) (DtFH) (DtGK) ( ) Dv( F)G (λx.(dv(fx)g)) ( 1 ) DvF( G) (λx.(dvf(gx))) ( 2 ) no way to encode ($) - pruning unnecessary quantifiers, ($$) - permuting quantifiers nonterminating rules but we need confluence! 7 critical pairs : Dv( F)( G) (needs ($$)) D0( F)G (reducing by 1 freezes the (0)) impossible for terms, weak confluence at the ε-level (sweat) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

19 Termination First encoding the logic : ε(a B) ε(a) ε(b) (ε ) I x x (I) ε( A) x.ε(a x) (ε ) K x y x (K) S x y z x z (yz) (S) encoding Statman s rules : D0FG F (0) D1FG G (1) Dt(F G)(H K) (DtFH) (DtGK) ( ) Dv( F)G (λx.(dv(fx)g)) ( 1 ) DvF( G) (λx.(dvf(gx))) ( 2 ) termination 1 ε reduces the number of, 2 typed-restricted S and K imply no duplication of and 3 goes up in terms 4 goes up and decreases 5 simply-typed SKI terminates automatically prove termination? O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

20 Confluence impossible at the term level D0( F)G (λx.(d0(f x)g) F fixed at the proposition level ε(d0( F)G) ε( (λx.(d0(f x)g)) ε( F) x.ε(f x) still problematic for D0F( G) : and Dv( F)( G) : yε(f) ε(f) x yε(dv(fx)(gy)) y xε(dv(fx)(gy)) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

21 Confluence up to equivalence but we can have confluence up to : variable renaming, pruning and inversion of quantifiers does not fit in a term equational theory E Definition A B : if A and B normal, and either A = x.ε(ta ), B = y.ε(t B ) and ε(t A ) ε(t B ) or A = x.(a1 A 2 ), B = y.(b 1 B 2 ) and A 1 B 1 and A 2 B 2. Lemma B A A0 B0 (in NF) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

22 Proof of the Lemma : Strategy B A (prove) B A (IH) C0 B 0 (NF) A 0 get rid of no rewrite step from B to A or to B0 prove the existence of C0 induction on : the height of the reduction tree of B, noted B easy if A B B does not involve a critical pair let us see the F D0( F)G λx.(d0(f x)g) case O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

23 we have B = K[D0( F)G] One Critical Pair A = K[ F] B = K[ λx.(d0(f x)g)] B 0 (NF) long time to wait before joining at the ε level introduce the inductive invariant t 1 t 2 if there exist a context K and two terms θ 1, θ 2 such that : A t 1 = K[ θ 1 ], B t 2 = K[ θ 2 ] B 0, P(θ 1, θ 2 ) where P(u 1, u 2 ) is : u 2 x u 1 x and if u 2 u 2 then for some u 1 u 1, P(u, 1 u ) 2 P( F, λx.(d0(f x)g)) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

24 Critical Pair : Interesting Subcases t 1 t 2 if there exist a context K and two terms θ 1, θ 2 such that : A t 1 = K[ θ 1 ], B t 2 = K[ θ 2 ] B 0, P(θ1, θ 2 ) where P(u 1, u 2 ) is : u 2 x u 1 x and if u 2 u 2 then for some u 1 u 1, P(u, 1 u ) 2 proof the existence of C0 by induction on t 2 B 0 t2 = L[ε( θ 2 )] L[ x.ε(θ 2 x)] Confluence case. Saved! (big IH as t 2 < B ) t2 = L[Dv( θ 2 )Z] L[ λ x (Dv(θ 2 x)z)] IH, since P(λx.(Dv(θ 1 x)z), λx.(dv(θ 2 x)z)) t2 = K[ θ 2 ] K [ θ 2 ] : fits t2 = K[ θ 2 ] K[ θ 2 ] : fits O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

25 Digging Terms confluence up to at the proposition level! with intersection types, merging derivations : Γ X : F Γ X : G ( I ) Γ X : F G derivation transformations in Statman s system Derivation Merge (Statman) If Γ 1 X : F and Γ 2 X : G then DvΓ 1 Γ 2 X : DvFG we must prove the result perform this on terms (D is a term) Proposition Reification γ(ε(t)) := t γ(f G) := γ(f) γ(g) γ( x.f) := (λx.(γ(f))) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

26 Reification Proposition Reification γ(f) noted Ḟ. γ(ε(t)) := t γ(f G) := γ(f) γ(g) γ( x.f) := (λx.(γ(f))) ε(γ(f)) F we need (for conversions) if F F, G G then DvḞĠ DvḞ Ġ problem : not preserved by γ! Counter-example : F = x.ε(d0ab) and F = x.ε(a) Ḟ is not convertible with F When F contains quantifiers : redexes frozen by λ O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

27 Reification Proposition Reification γ(f) noted Ḟ. We actually need Merges are Convertible γ(ε(t)) := t γ(f G) := γ(f) γ(g) γ( x.f) := (λx.(γ(f))) if F F, G G then ε(dvḟġ) ε(dvḟ Ġ ) Works separately for F1 G 1 and F 2 G 2, but not for deeper combinations : ε(dv(ḟ 1 Ḟ 2 )ṗ) not convertible with ε(dv(ġ1 Ġ2)ṗ) ṗ is not implicational : ε cannot expose the structure O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

28 Digging Proposition Reification γ(f) noted Ḟ. γ(ε(t)) := t γ(f G) := γ(f) γ(g) γ( x.f) := (λx.(γ(f))) ε(dv(ḟ1 Ḟ 2 )ṗ) not convertible with ε(dv(ġ1 Ġ2)ṗ) idea : dig out the implicational structure replace all ṗ with ṗ ṗ potentially, n times notation t{n} Merge Conversion If F 1 F 2 and G 1 G 2 then, for some n, ε(dvf 1 {n} G 1 {n}) ε(dvf 2 {n} G 2 {n}). O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

29 All SN terms are Typable finally able to follow Statman s line derivations are organized in segments Γ X : F ( I ),(Conv),( E ) Γ X : G that we can organize as Γ X : F ( E ) Γ X : G (Conv) Γ X : G ( I ) Γ X : H we can merge two segments : Γ 1 X : F Γ 2 X : G (seg) Γ 1 X : F Γ 2 X : G with some digging into Γ X : ε(dvḟ{n}ġ{n}) (seg) we can also merge two derivations of the same term X O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

30 All SN terms are Typable, and conversely SN typable : 1 follow more or less Statman s way (with more explanations) 2 All terms in NF are typable 3 if a reduct of a head β-redex is typable, so is the redex 4 if X is SN, it is typable typable SN find the worse reduction strategy, F (M) (Barendregt) prove that it terminates or use Reducibility Candidates? (cf. plain intersection types) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29

31 Conclusion we have intersection types in Deduction Modulo Theory no new connectives, etc rewrite rules instead interesting confluence property regain nice properties and some extensionality : lot of plumbing further work do we have a model with values on the reducibility candidates? one technical detail to fix (variable renaming) O. Hermant (MINES ParisTech) Intersection Types in DMT June 28, / 29