Conditional Rewriting

Transcription

1 Conditional Rewriting Bernhard Gramlich ISR 2009, Brasilia, Brazil, June 22-26, 2009 Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

2 Outline Introduction Basics in Conditional Rewriting Termination Confluence Transforming CTRSs Into TRSs Systems with Extra Variables (3-Ctrss) Transforming CTRSs Into TRSs Further Topics (expressive power, modularity,... ) Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

3 Lecture 1 Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

4 Introduction Examples x z = true = x y = true, y z = true a = b = a c, a = c = a b (a = b a = c) Hence: minimal models non-unique. Consequence (here): no negation in conditions. f (x, y) g(x) = P(x, y) (built-in predicates, not here) a b = c = d a b = c d a b = c d a b = c d a b = a c x z true = x y true, y z true s(x) + y s(z) = x + y z s(x) + y s(z) = x + y z Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

5 Basics in Conditional Rewriting Format of CESs / CTRSs conditional (term) equational system (CES) R: axioms of shape l = r = s 1 = t 1,..., s n = t n ; leads to conditional equational logic (CEL): = R conditional (term) rewrite system (CTRS) R: rules of shape l r = s 1 t 1,..., s n t n ; interpretation of equality in conditions: = : semi-equational system = : join system = : oriented system =, all t 1 R u -irreducible ground terms: normal (join) system (where R u = {l r l r = c R}) Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

6 Basics in Conditional Rewriting Join CTRSs as non-left-linear normal (join) CTRSs replace every l r = s 1 t 1,..., s n t n by l r = eq(s 1, t 2 ) true,..., eq(s n, t n ) true add eq(x, x) true true fresh boolean constant, eq fresh binary predicate s R t iff eq(s, t) R true (in many-sorted setting) Syntactical properties of CTRSs R (via R u ) R u = {l r l r = c R} R left-linear, right-linear, linear, (non-)collapsing, (non-)duplicating, non-overlapping, orthogonal, overlaying,..., if u is left-linear,..., respectively advantage of these syntactic definitions: easily decidable! disadvantage (partially): semantically misleading (Ex.: f (x, y) a = Bernhard x Gramlich y)! Conditional Rewriting ISR 2009, July 22-26,

7 Basics in Conditional Rewriting Classification of CTRSs according to extra variables VAR(s) set of all variables occurring in s, l r = c R type 1: VAR(l) VAR(r) VAR(c) (no extra variables) type 2: VAR(l) VAR(r) (extra variables at most in cond.) type 3: VAR(l) VAR(c) VAR(r) (extra variables in right-hand sides must appear in conditions) type 4: no restrictions. R n-ctrs if all its rules are of type n type 2 involves existential search in conditions type 3 involves existential search in conditions and rhs s and may cause infinite branching type 4 very hard to understand / handle computationally (almost no results known for systems of this type) Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

8 Basics in Conditional Rewriting Induced equality for CES E Inference system for conditional equational logic (CEL): Given CES E, CEL defines conditional equational derivability = E : Reflexivity: Symmetry: Transitivity: Congruence: Application: t = t s = t t = s s = t, t = u s = u s 1 = t 1,..., s n = t n f (s,..., s n ) = f (t 1,..., t n ) s 1 σ = t 1 σ,..., s n σ = t n σ lσ = rσ if l = r = s 1 = t 1,..., s n = t n E Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

9 Basics in Conditional Rewriting Induced rewrite relation for CTRS R for R consisting of rules of the form l r = s 1 t 1,..., s n t n with {,, }: R 0 = R k+1 = {lσ rσ l r = c R, s i σ Rk t i σ for all s i t i in c}. = R = i 0 R i. If s R t with some concrete derivation s R n t, the latter has level n. The depth of a reduction s t is the minimal (level) n with s R n t. Relationships between different induced rewrite relations R, R, R, = R Logicality: R, = R (hence R, = R )? Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

10 Basics in Conditional Rewriting Examples For R = {a b = c d; e c; e d}: e R1 c, e R1 d a R2 b, a Rk b for every k 2 a R1 b, depth(a b) = 2 For R = {a b = c d; e c; e d}: c, d in normal form w.r.t. every R k a Rk b for no k, hence a R b For R = {f (x) a = f (x) x; b f (b)}: b R f (b) R a, since f (b) R b due to f (b) R b f (a) R a iff f (a) R a iff f (a) R a (since a NF), hence f (a) R a Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

11 Major Problems and Difficulties effectiveness in testing conditions conditions are recursively(!) evaluated entails in general a possibly non-terminating search process consequence: basic notions like one-step reduction, one-step reducibility, being a normal form etc. are undecidable in general! no modular decomposition of steps (non-locality) s R t using rule ρ does not imply s {ρ} t, because verifiation of conditions may need more rules, possibly the whole system! major obstacle in extending modularity results form TRSs to CTRSs Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

12 Major Problems and Difficulties confluence: variable overlaps may be critical! TRS:lσ rσ, σ σ = lσ rσ for l r R CTRS:lσ rσ, σ σ = lσ rσ for l r = c R? In general NO! s i σ t i σ, σ σ, hence: s i σ s i σ, t i σ t i σ but: s i σ t i σ? termination: may be non-effective! R = {a b = a c is terminating (has empty rewrite relation R ) but trying to apply the rule leads to a cycle Hence, from a practical point of view, R should not only be terminating, but also the evaluation of conditions (leading to stronger notions of termination) Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

13 Major Problems and Difficulties realization / implementation of conditional rewriting how to realize / implement rewriting with conditional rewrite rules? strategies for sequentialization / backtracking? how to avoid complicated rewrite machine architecture? why CTRSs instead of transformed unconditional TRSs? conceptually more adequate, intuitive any encoding / transformation entails a loss of structural information! appropriate encodings / transformations into TRSs are not easy! Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

14 More tomorrow! Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

15 Lecture 2 Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

16 Termination operational termination problems effective computation (reduction, normalization) example: a b = a c confluence criteria for TRSs do not directly generalize to CTRSs (see later) idea for recovery: require well-founded decrease not only from l to r, but also from l to u, u condition term in c, l r = c R yields stronger notions of termination Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

17 Termination strengthened approaches [Kaplan 84-87, Dershowitz et al ] R simplifying if there exists simplification ordering (rewrite ordering with subterm property) s.t. l r, s i, t i for all l r = s 1 t 1,..., s n t n R R reductive if there exists reduction ordering s.t. l r, s i, t i for all l r = s 1 t 1,..., s n t n R R decreasing if there exists well-founded term ordering s.t. R lσ st s 1 σ, t 1 σ,..., s n σ, t n σ for all l r = s 1 t 1,..., s n t n R ( st = ( ) + ) Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

18 Termination strengthened relationships simplifying = reductive = decreasing = terminating all implications are proper: f (f (x)) a = f (g(f (x))) b reductive, but not simplifying f (b) f (a); a c = b d decreasing, but not reductive a b = a c terminating, but not decreasing Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

19 Termination strengthened remarks oriented + normal CTRSs: for decreasingness e.g.: lσ st s 1 σ,..., s n σ instead of lσ st s 1 σ, t 1 σ,..., s n σ, t n σ decreasingness depends on interpretation of = in conditions ( or ) simplifyingness, reductivity and decreasingness can be refined: check conditions sequentially e.g. from left to right, instead of simultaneously) 2-CTRSs with extra variables in conditions cannot be simplifying,... 3-CTRSs with extra variables in right-hand sides: sequentialization becomes essential to make sense (see later)! Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

20 Confluence criteria for TRSs without termination orthogonality implies confluence (CR) with termination Critical Pair Theorem: local confluence (WCR) joinability of CPs (JCP) Newman s Lemma: SN = (WCR CR) hence: SN = (JCP CR) critical pairs (peaks), CPs l 1 σ[r 2 σ] p l 1 σ[l 2 σ] p ɛ r 1 σ where l i r i R, i = 1, 2, variable disjoint l1 p F (no overlap into variables) σ mgu of l1 p and l 2 Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

21 Confluence conditional critical pairs (peaks), CCPs l 1 σ[r 2 σ] p l 1 σ[l 2 σ] p ɛ r 1 σ with l 1 r 1 = c 1 R, σ at ɛ and l 2 r 2 = c 2 R, σ at p l 1 p F σ mgu of l1 p and l 2 yielding CCP l 1 σ[r 2 σ] = r 1 σ = c 1 σ, c 2 σ joinability: s = t = c CCP(R) joinable if σ : sσ t σ = cσ (in)feasibility: s = t = c feasible if there exists σ with cσ even for terminating systems: (in)feasibility and joinability of CCPs undecidable R non-overlapping CTRS (via R u ) syntactic criterion for non-overlappingness R has only infeasible CCPs semantic criterion for non-overlappingness Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

22 Confluence of (Join 2-)CTRSs criteria without termination orthogonality? No! b f (b); f (x) a = f (x) x is orthogonal (left-linear and non-overlapping) b f (b) a, due to f (b) b due to f (b) b, hence: f (a) f (b) a, but f (a) a: f (a) a f (a) a f (a) a... No! sufficient criterion orthogonality + normality example: b f (b); f (x) a = f (x) c proof idea: show that m commutes over n, via induction on m+n (here: m = Rm, m = Rm ) Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

23 Confluence of (Join 2-)CTRSs criteria with termination for Newman s Lemma we need: WCR does WCR JCP hold for CTRSs? No! counterexample R: (1) h(x) k(b) = k(x) h(b) (2) k(a) h(a) (3) a b CCPs: k(b) (3) k(a) (2) h(a), due to k(a) h(b) via k(a) h(a) h(b) joinable via: k(b) h(a) however, the variable overlap h(b) h(a) k(b) is NOT joinable anymore! properties of R: decreasing, normal, overlaying, shallow joinable Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

24 Confluence of (Join 2-)CTRSs strengthened notions of confluence for CTRSs shallow joinable CCP t s u = c: σ with cσ : tσ m sσ n uσ implies tσ n v m uσ for some v (notation: n = Rn ) shallow confluence: m n n m level confluence: n n n n shallow confluence = level confluence = confluence Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

25 Confluence of (Join 2-)CTRSs confluence criteria with termination [Dershowitz et al ] A terminating (join 2-) CTRS with joinable CCPs is confluent if it is (a) decreasing; or (b) left-linear, normal, shallow joinable; or (c) overlaying proof ideas (a) by well-founded induction (as for TRSs), but using the stronger decreasing order (b) by proving shallow confluence (c) by using + as induction order and exploiting the overlay property Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

26 Confluence of (Join 2-)CTRSs beyond overlay systems consider R given by a b f (a, a) b f (b, x) f (x, x) = f (x, x) x f (x, b) f (x, x) = f (x, x) x CCPs (the non-trivial ones) joinable: f (b, a) f (a, a) b: f (b, a) f (a, a) b f (a, b) f (a, a) b: f (a, b) f (a, a) b but: f (b, b) f (a, a) a not joinable anymore shared parallel critical peaks play a crucial role this can be exploited for generalizing the overlay confluence criterion (c) beyond overlay systems [Gramlich/Wirth 96] Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

27 Exercises A Confluence of CTRS Test the following join (1-) CTRSs for confluence (via constructing conditional critical pairs and applying known confluence criteria): (a) (b) (c) { } h(g(x), y) h(g(x), g(y)) = h(x, x) a R = h(a, g(y)) y R = R = f (x) k(x) = h(x) x h(a) a g(f (x)) b g(k(a)) b g(k(h(x))) g(k(x)) ins(x, c(y, )) c(x, c(y, l)) = x y true ins(x, c(y, )) c(y, ins(x, l)) = x y false 0 y true s(x) 0 false s(x) s(y) false Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

28 Lecture 3 Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

29 Lecture 4 Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,

30 Selected literature on conditional rewriting Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26,