Resource Combinatory Algebras

ion of variables A useful combinator: • linear identity: I ≡ S(1)K[ ] satisfies Ix̄ = { x0 if |x̄| = 1 0 otherwise Abstraction on monomials (i) λ∗x.t ≡ Kt if degx(t) = 0 (ii) λ∗x.x ≡ Iion on monomials (i) λ∗x.t ≡ Kt if degx(t) = 0 (ii) λ∗x.x ≡ I (iii) λ∗x.t0[t1, . . . , tk] ≡ Sn̄[λ∗x.t0][λ∗x.t1, . . . , λx.tk] with n̄ = (degx(t0), . . . , degx(tk)) and ∃i. degx(ti) 6= 0. (iv) ... and on polynomials: λ∗x. ∨n i=1 ti = ∨n i=1 λ ∗x.ti. Simulation of linear substitution Theorem 3 For any monomial t and polynomials s1, . . . , sn (λx.t)[s1, . . . , sn] = t〈x := s1〉 · · · 〈x := sn〉{x := 0} • axioms of bag-applicative structures + axiom schemes for the combinators Sn̄,K + rules of equational calculus = Resource Combinatory Logic (RCL) • weak extensionality fails in RCL: RCL ` t = s 6⇒ RCL ` λ∗x.t = λ∗x.s • equivalently, the implication t = s ⇒ λ∗x.t = λ∗x.s fails in some RCA: for example I[x] = x, but not S(0,1)[KI][I] = I in the free RCA over one indeterminate x