Viewing -terms through Maps Masahiko Sato Kyoto University Randy - PowerPoint PPT Presentation

Viewing λ -terms through Maps Viewing λ -terms through Maps Masahiko Sato Kyoto University Randy Pollack Harvard University Helmut Schwichtenberg University of Munich Takafumi Sakurai Chiba University Version of April 28, 2013

Viewing λ -terms through Maps Outline Motivation The Intuition Maps Lambda terms with maps Syntax and well formedness Hole filling Use of parameters Λ : Raw λ -terms Working with Λ The βη -calculus Conclusion

Viewing λ -terms through Maps Motivation Motivation: Formal representation of binding with natural reasoning ◮ Concrete: inductively definable in (say) Coq and HOL. ◮ Eliminates nominal: needs extensionality and quotients. ◮ Canonical: α -equivalence is identity. ◮ Eliminates McKinna and Pollack representation (from 1993). ◮ Reasoning: structural. ◮ Eliminates pure de Bruijn. ◮ Reasoning without equivariance, name swapping, special derived induction principles, etc. ◮ Eliminates locally nameless and Sato canonical representations. ??? The representation of this talk makes some progress.

Viewing λ -terms through Maps The Intuition Intuition: Maps for binding ◮ Map s generalize the notion of occurrance . ◮ Maps are binary trees over 0 and 1 . ◮ Example: ◮ Occurrences of x in ( xz )( yz ) represented by map ( 10 )( 00 ) . ◮ Occurrences of z in ( xz )( yz ) represented by map ( 01 )( 01 ) . λ -term S = λ xyz . ( xz )( yz ) is represented ◮ ( 10 00 ) \ ( 00 10 ) \ ( 01 01 ) \ ( ✷✷ ✷✷ ) (We drop some parentheses for readability.) ◮ Bound positions represented only by constant ✷ (called box ).

Viewing λ -terms through Maps The Intuition Open terms ✷ may occur unbound. ◮ ✷ is a distinguished constant. ◮ ◮ We accept ✷ as a term. 1 \ ✷ represents λ z . z . ◮ 0 \ ✷ represents λ x . z . ◮ Unbound box is available for binding or substitution, ◮ 1 \ 0 \ ✷ represents λ z . λ x . z . ◮ ◮ Free variables may occur in terms, ◮ the informal term λ z . ( xz ) is written as ( 0 1 ) \ ( x ✷ ) . ◮ There are no bound names or de Bruijn indices.

Viewing λ -terms through Maps The Intuition Well-formedness conditions needed ◮ Free variables cannot be bound: ◮ maps can only bind ✷ , 0 \ x is a term, 1 \ x is not a term. ◮ ◮ We will show how to bind names. ◮ Want canonical representation: one representative per λ -term. 0 \ 1 \ ✷ is our notation for λ x . λ x . x (which equals λ y . λ x . x ) ◮ 1 \ 1 \ ✷ is not a term. ◮ ◮ Substitution: Consider the term ( 0 1 ) \ ( ✷ ✷ ) ; ◮ position ( 1 0 ) (the red ✷ ) is free, ◮ substitute ( ✷ ✷ ) in that position, ◮ get ( 0 1 ) \ (( ✷ ✷ ) ✷ ) which is not a term because 0 is not a position in ( ✷ ✷ ) . ◮ The solution: identify maps 0 and ( 0 0 ) .

Viewing λ -terms through Maps The Intuition Compare with other notations ◮ Abstraction by names (raw terms or nominal terms): ◮ Binding information shared between binding occurrences and bound occurrences (shared names). ◮ Substitution may require α -conversion of the base term. ◮ Abstraction by indexes (de Bruijn): ◮ Binding information only at bound occurrences (indexes). ◮ At binding point, only λ to mark structure. ◮ Substitution may require de Bruijn lifting of the implanted term. ◮ Abstraction by maps: ◮ Binding information only at binding occurrences (maps). ◮ At bound points, only ✷ to mark structure. ◮ No adjustment required for substitution.

Viewing λ -terms through Maps The Intuition Formalization ◮ Everything that follows is formalized in Isabelle/HOL. ◮ The apparent quotients and partial functions are coded in HOL without any actual quotienting of datatypes or “domain predicates” of functions. ◮ Correctness of the map representation is proved w.r.t. Nominal Isabelle. ◮ Independently, correctness of the map representation is proved w.r.t. de Bruijn nameless terms in Minlog. ◮ However our favorite form of the map approach is not representable in HOL or easily representable in Coq: ◮ Requires induction-recursion or induction-induction.

Viewing λ -terms through Maps Maps Maps, M , defined inductively ◮ Maps are binary trees over 0 and 1 , with the identification ( 0 0 ) = 0 . ◮ Can formalize this inductively without quotienting using an auxiliary type M + not containing 0 : m + ∈ M + n + ∈ M + 1 ∈ M + inl ( m + ) ∈ M + inr ( n + ) ∈ M + m + ∈ M + n + ∈ M + cons ( m + , n + ) ∈ M + ◮ Extend M + with 0 to get M m + ∈ M + m + ∈ M 0 ∈ M

Viewing λ -terms through Maps Maps Map application ◮ For “cons” on M we define:  0 if m = n = 0 ,   inl ( m ) if m � = 0 and n = 0 ,  mapp ( m , n ) := inr ( n ) if m = 0 and n � = 0 ,   cons ( m , n ) if m � = 0 and n � = 0 .  (Eliding explicit inclusion of M + in M .) ◮ Write ( m n ) for mapp ( m , n ) , ( m 1 m 2 m 3 ) for (( m 1 m 2 ) m 3 ) , etc. mapp is injective. ◮

Viewing λ -terms through Maps Maps Orthoganality on maps ◮ A symmetric orthogonality relation ⊥ : m ′ ⊥ n ′ m ⊥ n mm ′ ⊥ nn ′ m ⊥ 0 0 ⊥ n m ⊥ n means: ◮ m and n have the same shape ◮ m and n bind different positions in that shape. ◮ 0 has every shape and binds no positions. ◮

Viewing λ -terms through Maps Lambda terms with maps Syntax and well formedness Lambda terms as a subtype ◮ Symbolic expressions ( S ) are raw syntax: S ∈ S T ∈ S m ∈ M S ∈ S x ∈ S ✷ ∈ S ( S T ) ∈ S m \ S ∈ S ◮ Well formedness ( m | S ; m divides S ): m | S n | T 0 | x 0 | ✷ 1 | ✷ mn | ST m | T n | T m ⊥ n m | ( n \ T ) m | S means ◮ “ S is well-formed and m is a position of unbound boxes in S ”. m | S = ⇒ 0 | S . ◮ 0 | S means “ S is well formed”. ◮

Viewing λ -terms through Maps Lambda terms with maps Syntax and well formedness Aside: Syntax and well-formedness simultaneously L is a type. ◮ M ∈ L N ∈ L x ∈ L ✷ ∈ L ( M N ) ∈ L m ∈ M M ∈ L m | M m \ M ∈ L ◮ Divides is a relation | ⊆ M × L . m | M n | N 0 | x 0 | ✷ 1 | ✷ ( m n ) | ( M N ) m | N n | N m ⊥ n m | ( n \ N ) ◮ Not simultaneous inductive definition due to L in the type of | . ◮ Need induction-induction or induction-recursion to formalize.

Viewing λ -terms through Maps Lambda terms with maps Hole filling Hole filling ◮ Define the partial operation M m [ P ] : L × M × L → L : ✷ 1 [ P ] := P . ✷ 0 [ P ] := ✷ . x 0 [ P ] := x . ( M N ) ( m n ) [ P ] := ( M m [ P ] N n [ P ]) if m | M and n | N . ( n \ N ) m [ P ] := n \ ( N m [ P ]) if m | ( n \ N ) . ◮ Only defined if m | M ( m is a position of unbound holes in M ). ◮ Hole filling is a homomorphism, even going under binders. ◮ Hole filling respects well-formedness: m | M ∧ 0 | N = ⇒ 0 | M m [ N ] . ◮ Why is the last equation well-formed?

Viewing λ -terms through Maps Lambda terms with maps Use of parameters Parameters: map, skeleton, abstraction ◮ map , M x , computes the map of all the occurrences of x in M . ◮ skel , M x , replaces all occurrences of x in M by ✷ . map : X × L → M skel : X × L → L � 1 � ✷ if x = y , if x = y , y x := y x := 0 if x � = y . y if x � = y . ✷ x := ✷ . ✷ x := 0 . ( M N ) x := ( M x N x ) . ( M N ) x := ( M x N x ) . ( m \ M ) x := m \ M x . ( m \ M ) x := M x . ◮ With map and skel can define abstraction of a name from a term. lam ( x , M ) := M x \ M x lam ( x , M ) does not contain x . ◮

Viewing λ -terms through Maps Lambda terms with maps Use of parameters Substitution defined by hole filling subst : L × X × L → L M { x \ P } := ( M x ) M x [ P ] . ◮ Some provable equations of substitution � P if x = y , y { x \ P } = y if x � = y . ✷ { x \ P } = ✷ . ( M N ) { x \ P } = ( M { x \ P } N { x \ P } ) if 0 | M and 0 | N . ( m \ M ) { x \ P } = ( m \ M { x \ P } ) if m | M . ◮ These equations eliminate substitution on concrete terms. ◮ Substitution is a homomorphism. ◮ There are no name-freshness conditions on these equations.

Viewing λ -terms through Maps Lambda terms with maps Use of parameters Substitution lemma of λ -calculus: better proof If x � = y and x ♯ P , then M { x \ N }{ y \ P } = M { y \ P }{ x \ N { y \ P }} . ◮ In named representations (including locally nameless and nominal) this proof requires choosing a fresh name. ◮ When M = λ z . M ′ we must assume z ♯ ( x , y , N , P ) ◮ By equivariance, strengthened induction principle, . . . Our proof ◮ By induction on (well-formedness of) M . ◮ Each case completely solved by equational reasoning. ◮ Using the equations of substitution and the IH. ◮ No need for fresh names to appy the equations of substitution.

Viewing λ -terms through Maps Λ : Raw λ -terms Datatype Λ of raw λ -syntax K ∈ Λ L ∈ Λ K ∈ Λ x ∈ Λ ✷ ∈ Λ ( K L ) ∈ Λ lam ( x , K ) ∈ Λ ◮ Define map ( K x ) and skel ( K x ) on Λ K x computes the map of occurrences of x in K . ◮ K x replaces every x in K with ✷ . ◮ map : X × λ → M skel : X × λ → λ � � 1 if x = y , if x = y , ✷ y x := y x := 0 if x � = y . y if x � = y . ✷ x := ✷ . ✷ x := 0 . ( K L ) x := ( K x L x ) . ( K L ) x := ( K x L x ) . � 0 � lam ( y , K ) if x = y , if x = y , lam ( y , K ) x := lam ( y , K ) x := lam ( y , K x ) K x if x � = y . if x � = y .