Nominal Completion for Rewrite Systems with Binders Maribel Fern´ andez King’s College London July 2012 Joint work with Albert Rubio M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Summary Motivations Nominal Rewriting Closed nominal rules Confluence and Termination Completion Future work M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
First-order languages vs. languages with binders First-order data structures: numbers, lists, trees, etc. available in most programming languages. Few languages provide data structures with binding, needed in type checkers, program analysers, compilers, etc. M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Binding operators: Examples Some concrete examples of binding operators (informally): • Operational semantics: let a = N in M − → (fun a . M ) N M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Binding operators: Examples Some concrete examples of binding operators (informally): • Operational semantics: let a = N in M − → (fun a . M ) N • π -calculus: P | ν a . Q → ν a . ( P | Q ) ( a �∈ fv( P )) M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Binding operators: Examples Some concrete examples of binding operators (informally): • Operational semantics: let a = N in M − → (fun a . M ) N • π -calculus: P | ν a . Q → ν a . ( P | Q ) ( a �∈ fv( P )) • Logic equivalences: P and ( ∀ x . Q ) ⇔ ∀ x ( P and Q ) ( x �∈ fv( P )) M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Binders - α -equivalence Binding operators are defined modulo renaming of bound variables, i.e., α -equivalence. Example: In ∀ x . P the variable x can be renamed. • α -conversion is implicit, but ∀ x . P � = α ∀ y . P — x may occur in P . M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Binders - α -equivalence Binding operators are defined modulo renaming of bound variables, i.e., α -equivalence. Example: In ∀ x . P the variable x can be renamed. • α -conversion is implicit, but ∀ x . P � = α ∀ y . P — x may occur in P . • ∀ x . P = α ∀ y . P { x �→ y } Substitution of a bound name by a term has to avoid capture of other bound names: y fresh? M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Binders - α -equivalence Binding operators are defined modulo renaming of bound variables, i.e., α -equivalence. Example: In ∀ x . P the variable x can be renamed. • α -conversion is implicit, but ∀ x . P � = α ∀ y . P — x may occur in P . • ∀ x . P = α ∀ y . P { x �→ y } Substitution of a bound name by a term has to avoid capture of other bound names: y fresh? • Formal definition. There are several alternatives. M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Formally: Alternatives: • Encode in a first-order rewrite system e.g. use De Bruijn indices, encode alpha using operators such as “lift” and “shift”; explicit substitutions. (-) α -equivalence has to be “implemented” (+) simple notion of substitution, efficient first-order matching M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Formally: Alternatives: • Encode in a first-order rewrite system e.g. use De Bruijn indices, encode alpha using operators such as “lift” and “shift”; explicit substitutions. (-) α -equivalence has to be “implemented” (+) simple notion of substitution, efficient first-order matching • Higher-order rewrite systems (CRS, HRS, ERS, etc.), e.g. app ( lam ([ a ] Z ( a )) , Z ′ ) → Z ( Z ′ ) (+) Binders, functions, implicit α -equivalence (-) Substitution as a meta-operation, using β (-) Unification is undecidable in general M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Alternative: Nominal Approach [Pitts, Gabbay, Urban, ...] Inspired by nominal set theory (Fraenkel-Mostowski). Key ideas: Freshness conditions a # t , name swapping ( a b ) · t . Example: app ( lam ([ a ] Z ) , Z ′ ) subst ([ a ] Z , Z ′ ) → a # M ⊢ ( λ ([ a ] app ( M , a )) → M • Terms with binders • Built-in α -equivalence • Efficient matching and unification algorithms • Simple notion of substitution (“first-order”), non-capturing substitution has to be specified using rewrite rules. • Dependencies of terms on names are implicit. M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Nominal Syntax [Urban,Pitts,Gabbay: TCS’04] • Function symbols: f , g . . . Variables: M , N , X , Y , . . . Atoms: a , b , . . . Swappings: ( a b ) Def. ( a b ) a = b , ( a b ) b = a , ( a b ) c = c Permutations: lists of swappings, denoted π ( Id empty). • Nominal Terms: s , t ::= a | π · X | [ a ] t | f t | ( t 1 , . . . , t n ) Id · X written as X . • Example (ML): var ( a ), app ( t , t ′ ), lam ([ a ] t ), let ( t , [ a ] t ′ ), letrec [ f ]([ a ] t , t ′ ), subst ([ a ] t , t ′ ) Syntactic sugar: a , ( tt ′ ), λ a . t , let a = t in t ′ , letrec fa = t in t ′ , t [ a �→ t ′ ] M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
α -equivalence We use freshness to avoid name capture. a # X means a �∈ fv( X ) when X is instantiated. ds ( π, π ′ )# X a ≈ α a π · X ≈ α π ′ · X s 1 ≈ α t 1 · · · s n ≈ α t n s ≈ α t ( s 1 , . . . , s n ) ≈ α ( t 1 , . . . , t n ) fs ≈ α ft s ≈ α t a # t s ≈ α ( a b ) · t [ a ] s ≈ α [ a ] t [ a ] s ≈ α [ b ] t where ds ( π, π ′ ) = { n | π ( n ) � = π ′ ( n ) } • a # X , b # X ⊢ ( a b ) · X ≈ α X M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
α -equivalence We use freshness to avoid name capture. a # X means a �∈ fv( X ) when X is instantiated. ds ( π, π ′ )# X a ≈ α a π · X ≈ α π ′ · X s 1 ≈ α t 1 · · · s n ≈ α t n s ≈ α t ( s 1 , . . . , s n ) ≈ α ( t 1 , . . . , t n ) fs ≈ α ft s ≈ α t a # t s ≈ α ( a b ) · t [ a ] s ≈ α [ a ] t [ a ] s ≈ α [ b ] t where ds ( π, π ′ ) = { n | π ( n ) � = π ′ ( n ) } • a # X , b # X ⊢ ( a b ) · X ≈ α X • b # X ⊢ λ [ a ] X ≈ α λ [ b ]( a b ) · X M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Freshness Also defined by induction: π − 1 ( a )# X a #[ a ] s a # b a # π · X a # s 1 · · · a # s n a # s a # s a #( s 1 , . . . , s n ) a # fs a #[ b ] s M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Nominal Rewriting Rewriting on nominal terms. Applications: • equational reasoning on data structures with binding; • algebraic specifications of operators and data structures; • operational semantics of programs; • compilers, program transformations, etc. M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Nominal Rewriting Nominal Rewriting Rules: ∆ ⊢ l → r V ( r ) ∪ V (∆) ⊆ V ( l ) Example: prenex normal forms in first-order logic ⊢ P ∧ ∀ [ a ] Q → ∀ [ a ]( P ∧ Q ) a # P a # P ⊢ ( ∀ [ a ] Q ) ∧ P → ∀ [ a ]( Q ∧ P ) ⊢ P ∨ ∀ [ a ] Q → ∀ [ a ]( P ∨ Q ) a # P a # P ⊢ ( ∀ [ a ] Q ) ∨ P → ∀ [ a ]( Q ∨ P ) a # P ⊢ P ∧ ∃ [ a ] Q → ∃ [ a ]( P ∧ Q ) a # P ⊢ ( ∃ [ a ] Q ) ∧ P → ∃ [ a ]( Q ∧ P ) a # P ⊢ P ∨ ∃ [ a ] Q → ∃ [ a ]( P ∨ Q ) a # P ⊢ ( ∃ [ a ] Q ) ∨ P → ∃ [ a ]( Q ∨ P ) ⊢ ¬ ( ∃ [ a ] Q ) → ∀ [ a ] ¬ Q ⊢ ¬ ( ∀ [ a ] Q ) → ∃ [ a ] ¬ Q Reduction relation generated by (equivariant) nominal matching. M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Nominal matching problem l ≈ α t where V ( l ) ∩ V ( t ) = ∅ has solution (∆ , θ ) if ∆ ⊢ l θ ≈ α t • Nominal matching is decidable [Urban, Pitts, Gabbay 2003] • A solvable problem Pr has a unique most general solution: (Γ , θ ) such that Γ ⊢ Pr θ . • Efficient algorithms: linear in time and space [Calves-F.2008] BTW, nominal unification is quadratic [Calves-F.2010,Levy-Villaret2010]. • But for general NRSs nominal matching is not sufficient: we need Equivariant Nominal Matching — NP [Cheney2004]. • If rules are closed , then nominal matching is sufficient. M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Closed Rules Intuitively: no free names. R ≡ ∇ ⊢ l → r is closed when ( ∇ ′ ⊢ ( l ′ , r ′ )) ? ≈ ( ∇ , A ( R ′ )# V ( R ) ⊢ ( l , r )) has a solution σ (where R ′ is a freshened copy of R ). Given R ≡ ∇ ⊢ l → r and ∆ ⊢ s we write ∆ , A ( R ′ )# V (∆ , s ) ⊢ s R ′ R ∆ ⊢ s → c t when → t and call this closed rewriting . M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Examples The following rules are not closed: g ( a ) → a [ a ] X → X Why? M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
Examples The following rule is closed: a # X ⊢ [ a ] X → X Why? M. Fern´ andez Nominal Completion for Rewrite Systems with Binders
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