A Canonical 1 Local Representation of Binding. α -equivalence is identity. Randy Pollack. Masahiko Sato. LFCS, University of Edinburgh

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1 A Canonical 1 Local Representation of Binding Randy Pollack LFCS, University of Edinburgh Masahiko Sato Graduate School of Informatics, Kyoto University Version of May 12, α -equivalence is identity

2 Isabelle theory files: Full paper (submitted): These are older than the work presented in this talk.

3 Outline Introduction: Local Representations Symbolic Expressions (sexpr) Syntax B-Algebras, Substitution and Equivariance Variable-Closed Sexprs A Canonical Representation Examples: β -reduction and typing

4 Introduction: Local Representations Outline Introduction: Local Representations Symbolic Expressions (sexpr) Syntax B-Algebras, Substitution and Equivariance Variable-Closed Sexprs A Canonical Representation Examples: β -reduction and typing

5 Introduction: Local Representations Local Representations Syntactically distinct classes for (locally) bound variables vs (globally bound) free parameters. The idea goes back to Frege, Gentzen and Prawitz. Different styles: Locally named: two species of names. McKinna/Pollack (1993) formalized Pure Type System metatheory. Not canonical representation. This talk: idea of Sato allows canonical representation. Locally nameless: names for parameters, de Bruijn indices for locally bound variables. Canonical representation. POPL 08 paper by Ademir, Chargueraud, Pierce, Pollack and Weirich.

6 Introduction: Local Representations Local Representations are Concrete Close to informal usage. Anything true can be proved. Relatively light infrastructure (compared to Twelf or nominal Isabelle). Can be used in intensional logics (e.g. Coq). Some technologies make local representations convenient: McKinna/Pollack style strengthened induction and inversion. Urban and Pollack, WMM 07, Strong Induction Principles in the Locally Nameless Representation of Binders. POPL 08 paper by Ademir, Chargueraud, Pierce, Pollack and Weirich.

7 Symbolic Expressions (sexpr) Outline Introduction: Local Representations Symbolic Expressions (sexpr) Syntax B-Algebras, Substitution and Equivariance Variable-Closed Sexprs A Canonical Representation Examples: β -reduction and typing

8 Symbolic Expressions (sexpr) Syntax Syntax Names: Countable set V of atoms used for local variables: x, y, z. Countable set X of atoms, used for global parameters: X, Y, Z. Only relation needed on V, X is decidable equality. Nominal Isabelle atom types are convenient. Symbolic Expressions ( S[X] ): The syntax of pure λ -terms, ranged over by M, N, P, Q : M ::= x X P Q [x]m Usual induction principles for this datatype. Name-carrying syntax. In general, may be other classes of variables, parameters and expressions e.g. types and terms in System F.

9 Symbolic Expressions (sexpr) Syntax Occurrences of (Global) Parameters Define X A means X does not occur syntactically in A. We use X A polymorphically for... X from any type of parameters A from types of structures: terms, contexts, judgements,... Each instance of is easily defined by structural recursion. In nominal Isabelle, our corresponds to nominal freshness (also written ). Nominal Isabelle provides polymorphic over classes of atoms and finitely supported structures for free.

10 Symbolic Expressions (sexpr) Syntax Occurrences of Local Variables (LV) Defined by structural recursion. Respects intended scoping of abstraction. LV(X) LV(x) LV(M N) LV([x]M) = {} = {x} = LV(M) LV(N) = LV(M) {x}

11 Symbolic Expressions (sexpr) B-Algebras, Substitution and Equivariance B-Algebras A B-algebra is a triple A, () : A A A, [ ] : V A A where A is a set containing V as a subset. A B-algebra homomorphism is a function h on B-algebras s.t.: 1. h(x) = x ( h fixes V ), 2. h(m N) = h(m) h(n), 3. h([x]m) = [x]h(m).

12 Symbolic Expressions (sexpr) B-Algebras, Substitution and Equivariance Free B-Algebras: Substitution Abstractly S[X] is a free B-algebra with free generating set X. S[X], () : S S S, [] : V S S Let B be a B-algebra; any ρ : X B can be uniquely extended to a B-algebra homomorphism [ρ] : S[X] B : 1. [ρ]x = ρ(x). 2. [ρ]x = x. 3. [ρ]m N = [ρ]m [ρ]n. 4. [ρ][x]m = [x][ρ]m. In particular, a finite map ρ : X S is a substitution. [ρ] : S S is an endomorphism

13 Symbolic Expressions (sexpr) B-Algebras, Substitution and Equivariance Substitution, Concretely If ρ : X i M i (and fixes the rest) then [ρ] is concretely defined by structural recursion: [M i /X i ]x = x [M i /X i ]Y = if X i = Y then M i else Y [M i /X i ]N 1 N 2 = ([M i /X i ]N 1 ) [M i /X i ]N 2 [M i /X i ]([x]n) = [x][m i /X i ]N Deterministic: no choosing arbitrary names. Thus has natural properties; e.g. [X/X]M = M. X M [P/X]M = M. Does not prevent capture, e.g. [x/x][x]x = [x]x. Will only be use in safe ways.

14 Symbolic Expressions (sexpr) B-Algebras, Substitution and Equivariance Equivariance G X group of finite permutations of X (with composition). For π G X, [π] : S S is a B-algebra automorphism. G X acts on B-algebra S[X] by group action [π]m. [πσ]m = [π][σ]m, [ ]M = M. Suppose that G X acts on two sets U, V. f : U V is equivariant if π u. f ([π]u) = [π]f (u). U or V might be a product (e.g. multi-argument f ). U or V might have trivial G X action (e.g. V or B, truth values). If R : U B is an equivariant relation π u. R([π]u) = [π]r(u) = R(u) then the relation is preserved by permutations of parameters. Can permute parameters in arguments about equivariant relations.

15 Symbolic Expressions (sexpr) B-Algebras, Substitution and Equivariance Not Substitution: a purely technical operation Used to fill a hole (free variable) created by going under a binder. Defined by structural recursion: [M/y]x = if y = x then M else x [M/y]X = X [M/y]([x]N) = [x](if y = x then N else [M/y]N) [M/y]N 1 N 2 = ([M/y]N 1 ) [M/y]N 2 Not a B-algebra homomorophism. E.g. doesn t fix V. Does not prevent capture, e.g. [x/y][x]y = [x]x.

16 Outline Introduction: Local Representations Symbolic Expressions (sexpr) Syntax B-Algebras, Substitution and Equivariance Variable-Closed Sexprs A Canonical Representation Examples: β -reduction and typing

17 Overview: Symbolic expressions vs λ -terms Sexprs do not faithfully represent λ -terms for two reasons. 1. Local variables may appear unbound in sexprs. X is an sexpr representing a λ -term with one (particular) global variable. x is an sexpr, but is not intended to represent any λ -term. The fix: select the set of sexprs with no unbound variables (variable closed, vclosed ). Substitution is capture-avoiding on vclosed. 2. Different sexprs in vclosed may represent the same λ -term. [x]x and [y]y The fix: select a canonical subset of vclosed.

18 Overview: Symbolic expressions vs λ -terms Sexprs do not faithfully represent λ -terms for two reasons. 1. Local variables may appear unbound in sexprs. X is an sexpr representing a λ -term with one (particular) global variable. x is an sexpr, but is not intended to represent any λ -term. The fix: select the set of sexprs with no unbound variables (variable closed, vclosed ). Substitution is capture-avoiding on vclosed. 2. Different sexprs in vclosed may represent the same λ -term. [x]x and [y]y The fix: select a canonical subset of vclosed.

19 Overview: Symbolic expressions vs λ -terms Sexprs do not faithfully represent λ -terms for two reasons. 1. Local variables may appear unbound in sexprs. X is an sexpr representing a λ -term with one (particular) global variable. x is an sexpr, but is not intended to represent any λ -term. The fix: select the set of sexprs with no unbound variables (variable closed, vclosed ). Substitution is capture-avoiding on vclosed. 2. Different sexprs in vclosed may represent the same λ -term. [x]x and [y]y The fix: select a canonical subset of vclosed.

20 Overview: Symbolic expressions vs λ -terms Sexprs do not faithfully represent λ -terms for two reasons. 1. Local variables may appear unbound in sexprs. X is an sexpr representing a λ -term with one (particular) global variable. x is an sexpr, but is not intended to represent any λ -term. The fix: select the set of sexprs with no unbound variables (variable closed, vclosed ). Substitution is capture-avoiding on vclosed. 2. Different sexprs in vclosed may represent the same λ -term. [x]x and [y]y The fix: select a canonical subset of vclosed.

21 Variable-Closed Sexprs Variable-Closed Sexprs A predicate meaning no free variables. vclosed X vclosed M vclosed N vclosed M N vclosed M vclosed [x][x/x]m Every parameter is vclosed and no variable is vclosed. Use vclosed induction instead of sexpr structural induction no case for unbound variables. Essential property: vclosed is closed under substitution: Trivial to prove. vclosed M vclosed N vclosed [M/X]N Remark: We could equivalently replace the last rule with vclosed [X/x]M vclosed [x]m

22 Variable-Closed Sexprs Variable-Closed Sexprs(2) Think of vclosed as a weak typing judgement. vclosed terms behave well for substitution, just as well-typed terms behave well for computation. vclosed M is provably equivalent to LV(M) = {}. Thus vclosed is intuitively correct. It is the induction principle for vclosed that we want. Remark: The vclosed representation has been used for a big formalisation of type theory [McKinna/Pollack, TLCA 93]. The technology of [McKinna/Pollack] is another story it is necessary for the present story too. Remember: vclosed representation not canonical.

23 A Canonical Representation A Canonical Representation Consider the vclosed rules: vclosed X vclosed M vclosed N vclosed M N vclosed M vclosed [x][x/x]m The variable x is not determined in the rule for abstraction. To define a canonical relation L, choose x deterministically: X : L M : L M N : L N : L M : L x = F X (M) [x][x/x]m : L where F : X S V is a height function. Still to do: specify F such that L is closed under substitution.

24 A Canonical Representation The Height Function Interpret V as N. H : X S N defined by structural recursion: H X (Y ) H X (x) H X (M N) H X ([x]m) = = 0 { 1 if X = Y 0 if X Y = max(h { X (M), H X (N)) = HX (M) if H X (M) = 0 or x = 0 or H X (M) > x x + 1 otherwise

25 A Canonical Representation Three Axioms About F : X S V (FE) Assume every F -like function is equivariant: (FP) F is preserved by substitution: X Y X Q F X (M) = F [π]x ([π]m). F X (M) = F X ([Q/Y ]M). (FF) F X (M) does not occur in binding position on any path from the root of M to any occurrence of X in M. F X (M) / E X (M) where E X (M) : X S (V set) is defined: E X (α) E X (M N) E X ([x]m) = {} if α is atomic = E{ X (M) E X (N) {} if X M = {x} E X (M) otherwise

26 A Canonical Representation Derived Properties L is equivariant: M : L [π]m : L Among equivariant functions F, (FP) and (FF) are independent properties. By simple examples. H satisfies all three axioms. Height lemma: F X (M) / LV(M) [Z/F X (M)][F X (M)/X]M = [Z/X]M. A common case is when X = Z

27 A Canonical Representation Equivalent Forms of L X : L M : L M N : L N : L M : L x = F X (M) [x][x/x]m : L ( ) ( ) can equivalently be written X M [X/x]M : L x = F X ([X/x]M) [x]m : L ( ) In ( ) X varies independently of M. Any sufficiently fresh X will do in the premises of ( ) so the following rule is also equivalent X. X M [X/x]M : L x = F X ([X/x]M) [x]m : L ( )

28 A Canonical Representation Strong Induction Principle for L As in [McKinna/Pollack, TLCA 93] and [Ayedemir et al., POPL 08], the following strong induction rule is admissible: (1) X. Φ(X) (2) M N. M : L Φ(M) N : L Φ(N) Φ(M N) (3) x M. ( X. X M x = F X ([X/x]M) [X/x]M : L Φ([X/x]M)) Φ([x]M) N. N : L Φ(N) This makes reasoning about L convenient. Nominal Isabelle can automatically infer a similar strong induction principle.

29 A Canonical Representation Some Properties of L Using the strong induction, can prove key theorem: L is closed by substitution M : L N : L [M/X]N : L There is a substitution preserving isomorphism between L and the nominal Isabelle representation of pure lambda terms. In extensional logics (HOL, NuPrl,... ) can make a type of L.

30 Examples: β -reduction and typing Example: β -reduction [x]p : L N : L ([x]p) N [N/x]P M 1 M 2 N : L M 1 N M 2 N (β) M : L N 1 N 2 M N 1 M N 2 M N x = F X (M) y = F X (N) [x][x/x]m [y][y/x]n Note change of bound names in rule (ξ) and side conditions on x and y. is well behaved, e.g. is equivariant. M N implies M : L and N : L. (ξ)

31 Examples: β -reduction and typing Example: Simple Type Assignment Let S, T range over simple types. A type context, Γ, is a set of pairs (X, T ) such that no two different pairs have the same first component. (X, T ) Γ Γ X : T Γ M : S T Γ (X, S) M : T Type assignment is equivariant. Γ M N : T Γ [x][x/x]m : S T x = F X (M) Γ M : S Γ M : T M : L. To prove weakening of we must derive a strengthened induction principle, as usual. Nominal Isabelle can do this automatically.

32 Examples: β -reduction and typing Conclusion We presented a canonical name-carrying representation of binding. Well formed terms are an inductively defined subset of a datatype. All definitions by structural recursion. More beautiful than [McKinna/Pollack, TLCA 93] ours is canonical. More beautiful than locally nameless [Ayedemir et al., POPL 08]... name carrying, no indexes. Light infrastructure. Formalisable in intensional constructive logic in a few days. Can use nominal Isabelle package for some free automation.