Five Basic Concepts of. Axiomatic Rewriting Theory

Transcription

1 Five Basic Concepts of Axiomatic Rewriting Theory Paul-André Melliès Institut de Recherche en Informatique Fondamentale (IRIF) CNRS & Université Paris Denis Diderot 5th International Workshop on Confluence Obergurgl September 2016

2 Rewriting paths modulo homotopy An algebraic and topological notion of confluence 2

3 The λ-calculus with explicit substitutions Terms M ::= 1 MN λm M[s] Substitutions s ::= id M s s t Key idea: replace the β-rule of the λ-calculus (λx.m) N M [x := N] by the Beta-rule of the λσ-calculus (λ M) N M [N id] where the substitution is explicit and thus similar to a closure. 3

4 The eleven rewriting rules of the λσ-calculus Beta (λm)n M[N id] App (MN)[s] M[s]N[s] Abs (λm)[s] λ(m[1 (s )]) Clos M[s][t] M[s t] VarCons 1[M s] M VarId 1[id] 1 Map (M s) t M[t] (s t) IdL id s s Ass (s 1 s 2 ) s 3 s 1 (s 2 s 3 ) Shi f tcons (M s) s Shi f tid id 4

5 The eleven critical pairs of the λσ-calculus App + Beta (λm)[s](n[s]) App ((λm)n)[s] Beta M[N id][s] Clos + App (MN)[s t] Clos + Abs (λm)[s t] Clos + VarId 1[id s] Clos + VarCons 1[(M s) t] Clos + Clos M[s][t t ] Clos Clos Clos Clos (MN)[s][t] (λm)[s][t] 1[id][s] 1[M s][t] Clos M[s][t][t ] App Abs VarId VarCons (M[s](N[s]))[t] (λ(m[1 s ]))[t] 1[s] M[t] Clos M[s t][t ] Ass + Map (M s) (t t ) Ass + IdL id (s t) Ass + Shi f tid (id s) Ass + Shi f tcons ((M s) t) Ass + Ass (s s ) (t t ) Ass ((M s) t) t Map (M[t] s t) t Ass Ass Ass (id s) t ( id) s IdL Shi f tid Shi f tcons s t s ( (M s)) t s t Ass ((s s ) t) t Ass (s (s t)) t 5

6 A very dangerous critical pair ((λp)q)[s] App (λp)[s]q[s] Lam (λ(p[1 s ]))Q[s] Beta Beta P[Q id][s] P[1 s ][Q[s] id] Clos+Map Clos+Map P[Q[s] id s] P[Q[s] (s ) (Q[s] id)] IdL Ass+Shi f t P[Q[s] s] when s=m 1 M 2 M n id and the M is are in σ normal f orm P[Q[s] (s id)] This critical pair leads to a counter-example to strong normalization in the simply-typed λσ-calculus (TLCA 1995). 6

7 A fundamental problem Hence, one main challenge of Rewriting Theory: Classify the rewriting paths from a term P to its normal form Q f P Q g Very complicated in the case of the λσ-calculus... 7

8 I. Permutation tiles A geometric account of redex permutations 8

9 A key observation Theorem [Lévy 1978] In the λ-calculus, every two paths to the normal form f P Q are equal modulo a series of β-redex permutations: g f P Q g 9

10 A geometric intuition It is nice and clarifying to think of the redex permutation equivalence between f and g in a geometric way as a homotopy relation between rewriting paths This intuition can be made rigorous mathematically using Albert Burroni s notion of polygraph. 10

11 Permutation tiles [1] MQ MN v u u v P Q the redex u : M P and the redex v : N Q are independent P N 11

12 Permutation tiles [2] (λx.y)p (λx.y)m v u y the outer redex u : (λx.y) M y erases the inner redex v : M P u 12

13 Permutation tiles [3] (λx.xx)p (λx.xx)m u v MM v 1 u v 2 MP P P the outer redex u : (λx.xx) M MM duplicates the inner redex v : M P 13

14 Illustration of the theorem There is a 2-dimensional hole in the λ-calculus v (λx.x)(λy.y)z (λx.x)z (λy.y)z because the outer redex u u : (λx.x) (λy.y) z (λy.y) z is not equivalent modulo homotopy to the inner redex v : (λx.x) (λy.y) z (λx.x) z 14

15 Illustration of the theorem When one extends the two redexes u and v with w v (λx.x)(λy.y)z (λx.x)z w z the resulting rewriting paths u w and v w are normalizing and thus equivalent modulo homotopy! u 15

16 In the λσ-calculus... Critical pairs like ((λp)q)[s] App (λp)[s]q[s] Lam (λ(p[1 s ]))Q[s] Beta Beta P[Q id][s] P[1 s ][Q[s] id] Clos+Map Clos+Map P[Q[s] id s] P[Q[s] (s ) (Q[s] id)] IdL Ass+Shi f t P[Q[s] s] when s=m 1 M 2 M n id and the M is are in σ normal f orm P[Q[s] (s id)] generate 2-dimensional holes in the rewriting geometry and thus obstructions to homotopy equivalence... 16

17 A bridge between λσ and λ Key theorem [Abadi-Cardelli-Curien-Lévy 1990] Every rewriting path between λσ-terms f : P Q induces a rewriting path (modulo homotopy) σ( f ) : σ(p) σ(q) between the underlying λ-terms. Moreover, the translation preserves homotopy equivalence: f λσ g σ( f ) λ σ(g) 17

18 A remarkable consequence Fact. Two rewriting paths in the λσ-calculus f P Q are transported to the same homotopy class of rewriting paths g σ( f ) σ(p) λ σ(q) σ(g) when the λσ-term Q is in normal form. 18

19 What about head-normal forms? Can we classify the head-rewriting paths of the λσ-calculus? This requires to resolve two very serious difficulties: define a general notion of head-rewriting path for a term rewriting system admitting critical pairs establish that every head-rewriting path in the λσ-calculus f : P V is transported to a head-rewriting path σ( f ) : σ(p) σ(v) of the λ-calculus. 19

20 Axiomatic Rewriting Theory Main claim. This problem is arguably too difficult to resolve by working directly on the syntax of the λσ-calculus. One should move to a purely diagrammatic approach based on the 2-dimensional notion of permutation tile. The purpose of Axiomatic Rewriting Theory is to establish a number of important structural properties: standardisation theorem factorisation theorem stability theorem from the generic properties of permutation tiles in rewriting. 20

21 Axiomatic rewriting system Definition. A graph G = (V, E, 0, 1 ) defined by its source and target functions 0, 1 : E V together with a set of 2-dimensional tiles of the form Q v u M N u f P where the rewriting path f is of arbitrary length. 21

22 Reversible permutations Definition: a permutation tile Q v u M N u f P is called reversible when it has an inverse. Note that f is of length 1 in that specific case. 22

23 Reversible permutation tiles MQ v u MN P Q u v P N 23

24 Irreversible permutation tiles (λx.y)p v u (λx.y)m y u 24

25 Irreversible permutation tiles (λx.xx)p v u (λx.xx)m P P u MM v 1 v 2 MP 25

26 II. Standardisation cells Rewriting surfaces between rewriting paths 26

27 Key idea: let us track ancestors! MQ v u MN PQ u v PN 27

28 Key idea: let us track ancestors! λy.x P v u λy.x M y u 28

29 Key idea: let us track ancestors! v λx.x x P u λx.x x M u MM v 1 v MP P P 2 29

30 Standardisation cells (λx.(λy.x))mq b (λy.m)q c a M (λx.(λy.x))mn u (λy.m)n v 30

31 Illustration λx. λy. x MQ b λy. M Q c a M λx. λy. x MN u λy.m N v 31

32 Standardisation cells Definition. A standardisation cell θ : f g : M N is a triple ( f, g, ϕ) consisting of two coinitial and cofinal paths u 1 u 2 u p v 1 v 2 v q f = M N g = M N and of a function ϕ : {1,..., q} {1,..., p} called the ancestor function of the standardisation cell. 32

33 A 2-category of rewriting and standardisation Theorem. Every axiomatic rewriting system G induces a 2-category its objects are the terms, its morphisms are the rewriting paths, its cells are the standardisation cells. 33

34 III. Standard rewriting paths The normal forms of the 2-dimensional rewriting 34

35 Standard rewriting paths Definition. A rewriting path f : M N is called standard when every standardisation cell f g is reversible. 35

36 A diagrammatic standardisation theorem Theorem. For every rewriting path f, there exists a 2-dimensional cell f g transforming f into a standard path g. Moreover, the standard path g associated to the path f is unique, modulo reversible permutations. The 2-dimensional cell f g itself is unique, up to canonical 3-dimensional deformations. 36

37 Standardisation (λx.(λy.x))mq b (λy.m)q c a M (λx.(λy.x))mn u (λy.m)n v A two-dimensional process revealing the causal dependencies 37

38 IV. External rewriting paths The external-internal factorisation theorem 38

39 External rewriting paths Definition. A rewriting path M e P is called external when it satisfies the following property: P f Q e f standard = M P Q standard. 39

40 External rewriting paths The β-redex (λx.(λy.x))mn a (λx.(λy.x))mp is standard but not external in the diagram below: (λx.(λy.x))mq b (λy.m)q c a M (λx.(λy.x))mn u (λy.m)n v 40

41 Internal rewriting paths Definition. A rewriting path M m N is called internal when every factorization up to homotopy M e m P f N satisfies the following property: e is external = e is equal to the identity. 41

42 Factorization theorem [Existence] Suppose given an axiomatic rewriting system. Theorem. Every rewriting path M f N factors as e m M P N where the rewriting path e is external the rewriting path m is internal 42

43 Factorization theorem [uniqueness] For every commutative diagram M 1 P 1 N 1 u e 1 m 1 v e 1 and e 2 external m 1 and m 2 internal M 2 P 2 N 2 e 2 m 2 there exists a unique path h : P 1 P 2 such that e 1 m 1 M 1 P 1 N 1 u h v e 2 m 2 M 2 P 2 N 2 commutes. 43

44 Factorization theorem [uniqueness] For every commutative diagram (up to homotopy) M 1 P 1 N 1 u e 1 m 1 v e 1 and e 2 external m 1 and m 2 internal M 2 P 2 N 2 e 2 m 2 there exists a unique path h : P 1 P 2 (up to homotopy) such that e 1 m 1 M 1 P 1 N 1 u h v commutes (up to homotopy.) e 2 m 2 M 2 P 2 N 2 44

45 V. Head rewriting paths A universal cone of head-rewriting paths 45

46 Axiomatic set of values Definition. An axiomatic set H of values is a set of terms satisfying three properties: [1] the set H is closed under reduction: V H and V W = W H 46

47 Axiomatic set of values [2] In every reversible tile V 2 v u M W u v V 1 V 1 H and V 2 H = M H 47

48 Axiomatic set of values [3] In every irreversible tile V v u M W u f N V H = M H 48

49 Stability theorem Suppose given an axiomatic set H of values. Theorem. For every term M, there exists a cone of paths ( M e i ) V i i I satisfying the following property: for every rewriting path f : M W where W H there exists a unique index i I and a unique path such that h : V i W up to homotopy M f W e i h M V i W 49

50 A cone of head-rewriting paths This means that there is a unique head-rewriting path in the cone such that f factors as e i : M V i M f e i up to homotopy. V i h W 50

51 Application to the λσ-calculus Suppose that M is a λ-term seen as a λσ-term. Theorem. Every head rewriting path e i : M V i V i H λσ to the set H λσ of λσ-head-normal forms is transported to e : M σ(v i ) the unique head-rewriting path from M to the set H λ of head-normal forms in the λ-calculus. 51

52 Short bibliography On the λσ-calculus: Typed lambda-calculi with explicit substitutions may not terminate. Proceedings of TLCA 1995, LNCS 902, pp , The lambda-sigma calculus enjoys finite normalisation cones. Journal of Logic and Computation, vol 10 No. 3, pp , On axiomatic rewriting theory: A diagrammatic standardisation theorem. Processes, Terms and Cycles: Steps on the Road to Infinity. Jan Willem Klop Festschrift, LNCS 3838, A factorisation theorem in Rewriting Theory. Proceedings of CTCS 1997, LNCS 1290, A stability theorem in Rewriting Theory. Proceedings of LICS 1998, pp ,