Overlapping Rules and Logic Variables in Functional Logic Programs - PowerPoint PPT Presentation

ICLP 2006 Overlapping Rules and Logic Variables in Functional Logic Programs Michael Hanus Christian-Albrechts-Universit¨ at Kiel (joint work with Sergio Antoy, Portland State University)

F UNCTIONAL L OGIC L ANGUAGES Approach to amalgamate ideas of declarative programming • efficient execution principles of functional languages (determinism, laziness) • flexibility of logic languages (constraints, built-in search) • avoid non-declarative features of Prolog (arithmetic, I/O, cut) • combine best of both worlds in a single model (higher-order functions, declarative I/O, concurrent constraints) • Advantages: ➜ optimal evaluation strategies [JACM’00,ALP’97] ➜ new design patterns [FLOPS’02] ➜ better abstractions for application programming (GUI programming [PADL ’00], web programming [PADL ’01, PPDP’06]) CAU Kiel Michael Hanus 2

F UNCTIONAL L OGIC L ANGUAGES Approach to amalgamate ideas of declarative programming • efficient execution principles of functional languages (determinism, laziness) • flexibility of logic languages (constraints, built-in search) • avoid non-declarative features of Prolog (arithmetic, I/O, cut) • combine best of both worlds in a single model (higher-order functions, declarative I/O, concurrent constraints) • Advantages: ➜ optimal evaluation strategies [JACM’00,ALP’97] ➜ new design patterns [FLOPS’02] ➜ better abstractions for application programming (GUI programming [PADL ’00], web programming [PADL ’01, PPDP’06]) Minimal kernel language for FLP? CAU Kiel Michael Hanus 2

F UNCTIONAL L ANGUAGES ( E . G ., H ASKELL ) CAU Kiel Michael Hanus 3

F UNCTIONAL L ANGUAGES ( E . G ., H ASKELL ) ✓ ✏ [] ++ ys = ys (x:xs) ++ ys = x : xs ++ ys ✒ ✑ CAU Kiel Michael Hanus 3

F UNCTIONAL L ANGUAGES ( E . G ., H ASKELL ) + logic variables (expressive power) ✓ ✏ [] ++ ys = ys (x:xs) ++ ys = x : xs ++ ys ✒ ✑ last xs | ys ++ [x] =:= xs = x where x,ys free CAU Kiel Michael Hanus 3

F UNCTIONAL L ANGUAGES ( E . G ., H ASKELL ) + logic variables (expressive power) ✓ ✏ [] ++ ys = ys (x:xs) ++ ys = x : xs ++ ys ✒ ✑ last xs | ys ++ [x] =:= xs = x where x,ys free ★ + overlapping rules (demand-driven search) ✥ insert e [] = [e] insert e (x:xs) = e : x : xs -- overlapping insert e (x:xs) = x : insert e xs -- rules ✧ ✦ CAU Kiel Michael Hanus 3

F UNCTIONAL L ANGUAGES ( E . G ., H ASKELL ) + logic variables (expressive power) ✓ ✏ [] ++ ys = ys (x:xs) ++ ys = x : xs ++ ys ✒ ✑ last xs | ys ++ [x] =:= xs = x where x,ys free + overlapping rules (demand-driven search) ★ ✥ insert e [] = [e] insert e (x:xs) = e : x : xs -- overlapping insert e (x:xs) = x : insert e xs -- rules perm [] = [] ✧ ✦ perm (x:xs) = insert x (perm xs) perm [1,2,3] [1,2,3] | [1,3,2] | [2,1,3] | ... ❀ = functional logic languages Main result of this paper: only one of these extensions is sufficient! CAU Kiel Michael Hanus 3

T ERM R EWRITING S YSTEMS (TRS) Formal model for functional logic languages: CAU Kiel Michael Hanus 4

T ERM R EWRITING S YSTEMS (TRS) Formal model for functional logic languages: Datatypes ( ≈ admissible values): enumerate all data constructors ✓ ✏ data Bool = True | False data Nat = O | S Nat ✒ ✑ data List = [] | Nat : List CAU Kiel Michael Hanus 4

T ERM R EWRITING S YSTEMS (TRS) Formal model for functional logic languages: Datatypes ( ≈ admissible values): enumerate all data constructors ✓ ✏ data Bool = True | False data Nat = O | S Nat ✒ ✑ data List = [] | Nat : List Rewrite rules : define operations on values f ( t 1 , . . . , t n ) → r patterns defined expression operation (linear data terms) ☛ ✟ → y → False add(O,y) positive(O) ✡ ✠ add(S(x),y) → S(add(x,y)) positive(S(x)) → True Extra variable: rule variable not occurring in left-hand side CAU Kiel Michael Hanus 4

C LASSES OF T ERM R EWRITING S YSTEMS CAU Kiel Michael Hanus 5

C LASSES OF T ERM R EWRITING S YSTEMS Inductively sequential term rewriting systems: ➜ no overlapping left-hand sides in rules ✓ ✏ ➜ operations inductively defi ned on datatypes → True cond(True,x) → x leq(O,x) → False leq(S(x),O) ✒ ✑ leq(S(x),S(y)) → leq(x,y) seven → S(S(S(S(S(S(S(O))))))) CAU Kiel Michael Hanus 5

C LASSES OF T ERM R EWRITING S YSTEMS Inductively sequential term rewriting systems: ➜ no overlapping left-hand sides in rules ✓ ✏ ➜ operations inductively defi ned on datatypes → True cond(True,x) → x leq(O,x) → False leq(S(x),O) ✒ ✑ leq(S(x),S(y)) → leq(x,y) seven → S(S(S(S(S(S(S(O))))))) ✞ ☎ ISX : inductively sequential TRS with extra variables ✝ ✆ smallnum → cond(leq(x,seven),x) CAU Kiel Michael Hanus 5

C LASSES OF T ERM R EWRITING S YSTEMS Inductively sequential term rewriting systems: ➜ no overlapping left-hand sides in rules ✓ ✏ ➜ operations inductively defi ned on datatypes → True cond(True,x) → x leq(O,x) → False leq(S(x),O) ✒ ✑ leq(S(x),S(y)) → leq(x,y) seven → S(S(S(S(S(S(S(O))))))) ✞ ☎ ISX : inductively sequential TRS with extra variables ✝ ✆ smallnum → cond(leq(x,seven),x) OIS : overlapping inductively sequential TRS: ✞ ☎ allow rules with multiple right-hand sides: l → r 1 ? · · · ? r k ✝ ✆ parent(x) → mother(x) ? father(x) CAU Kiel Michael Hanus 5

C LASSES OF T ERM R EWRITING S YSTEMS Inductively sequential term rewriting systems: ➜ no overlapping left-hand sides in rules ✓ ✏ ➜ operations inductively defi ned on datatypes → True cond(True,x) → x leq(O,x) → False leq(S(x),O) ✒ ✑ leq(S(x),S(y)) → leq(x,y) seven → S(S(S(S(S(S(S(O))))))) ✞ ☎ ISX : inductively sequential TRS with extra variables ✝ ✆ smallnum → cond(leq(x,seven),x) OIS : overlapping inductively sequential TRS: ✞ ☎ allow rules with multiple right-hand sides: l → r 1 ? · · · ? r k ✝ ✆ parent(x) → mother(x) ? father(x) Main result: OIS with rewriting ⇐ ⇒ ISX with narrowing CAU Kiel Michael Hanus 5

E VALUATION : R EWRITING VS . N ARROWING Functional evaluation: (lazy) rewriting → → add(S(O),S(O)) S(add(O,S(O))) S(S(O)) CAU Kiel Michael Hanus 6

E VALUATION : R EWRITING VS . N ARROWING Functional evaluation: (lazy) rewriting → → add(S(O),S(O)) S(add(O,S(O))) S(S(O)) Functional logic evaluation: narrowing: guess values + rewriting { x �→ S(S(O)) } add(x,S(O)) =:= S(S(S(O))) ❀ CAU Kiel Michael Hanus 6

E VALUATION : R EWRITING VS . N ARROWING Functional evaluation: (lazy) rewriting → → add(S(O),S(O)) S(add(O,S(O))) S(S(O)) Functional logic evaluation: narrowing: guess values + rewriting { x �→ S(S(O)) } add(x,S(O)) =:= S(S(S(O))) ❀ Needed narrowing: demand-driven variable instantiation and rewriting leq(x,add(O,S(O))) True ❀ { x �→ O } Sound, complete, optimal evaluation strategy [JACM’00] CAU Kiel Michael Hanus 6

E LIMINATING O VERLAPPING R ULES Transformation OE : OIS − → ISX: Replace overlapping rule f ( t n ) → r 1 ? · · · ? r k by: (where V ar ( t n ) = { x 1 , . . . , x l } , y fresh, f ′ new) f ( t n ) → f ′ ( y, x l ) f ′ ( I 1 , x l ) → r 1 . . . f ′ ( I k , x l ) → r k data Ix = I 1 | · · · | I k (index type, e.g., natural numbers) CAU Kiel Michael Hanus 7

E LIMINATING O VERLAPPING R ULES Transformation OE : OIS − → ISX: Replace overlapping rule f ( t n ) → r 1 ? · · · ? r k by: (where V ar ( t n ) = { x 1 , . . . , x l } , y fresh, f ′ new) f ( t n ) → f ′ ( y, x l ) f ′ ( I 1 , x l ) → r 1 . . . f ′ ( I k , x l ) → r k data Ix = I 1 | · · · | I k (index type, e.g., natural numbers) Example: parent(x) → mother(x) ? father(x) OE �→ data Iparent = I0 | I1 parent(x) → parent’(y,x) parent’(IO,x) → mother(x) parent’(I1,x) → father(x) CAU Kiel Michael Hanus 7

E LIMINATING O VERLAPPING R ULES : R ESULTS Proposition: R ∈ OIS ⇒ OE ( R ) ∈ ISX CAU Kiel Michael Hanus 8

E LIMINATING O VERLAPPING R ULES : R ESULTS Proposition: R ∈ OIS ⇒ OE ( R ) ∈ ISX Correctness w.r.t. results computed by needed narrowing: Theorem: R ∈ OIS , R ′ = OE ( R ) , t, s terms of R : ❀ σ ′ s w.r.t. R ′ ⇒ ∃ t NN * NN * ❀ σ s w.r.t. R with σ = V ar ( t ) σ ′ Soundness: t ❀ σ ′ s w.r.t. R ′ with σ = V ar ( t ) σ ′ NN * NN * Completeness: t ❀ σ s w.r.t. R ⇒ ∃ t CAU Kiel Michael Hanus 8

E LIMINATING O VERLAPPING R ULES : R ESULTS Proposition: R ∈ OIS ⇒ OE ( R ) ∈ ISX Correctness w.r.t. results computed by needed narrowing: Theorem: R ∈ OIS , R ′ = OE ( R ) , t, s terms of R : ❀ σ ′ s w.r.t. R ′ ⇒ ∃ t NN * NN * ❀ σ s w.r.t. R with σ = V ar ( t ) σ ′ Soundness: t ❀ σ ′ s w.r.t. R ′ with σ = V ar ( t ) σ ′ NN * NN * Completeness: t ❀ σ s w.r.t. R ⇒ ∃ t ⇒ Implementations need not implement overlapping rules (done in several implementations but without formal justification) CAU Kiel Michael Hanus 8

E LIMINATING L OGIC V ARIABLES → OIS − Transformation XE : ISX − (extra variable elimination) OIS − : overlapping inductively sequential without extra variables Evaluation in OIS − : rewriting (not narrowing) Thus: programs transformed by XE ❀ implementation without handling of logic variables and substitutions CAU Kiel Michael Hanus 9