1. NummSquared Coercion make it so! Samuel Howse poohbist.com samuelhowse@poohbist.com November 29, 2006 Copyright © 2006 Samuel Howse. All rights reserved.

2. Type theory Types can ensure termination and avoid paradoxes. But more complex – 2 fundamental concepts: –Type –Function Compile-time type checking is computable. –Benefit – catch some bugs early and automatically –Cost – additional constraints on programmer my research addresses by using runtime coercion instead of compile-time type checking Coq proof assistant –nice mix of computation and proof checking –practical tools, including extraction to OCaml –highly developed

3. Set theory Everything is a set. Well-founded set theory –Membership (ε) is well-founded, i.e. no infinite descending ε chains. –A set is formed “after” its elements. Some well-founded set theories: –Zermelo Fraenkel (ZF) –von Neumann-Bernays-Gödel (NBG) Usually no reduction (computation)

4. Well-founded functions Everything is a function. Well-founded –A function is formed “after” elements of its domain and range. [Jones] –Membership in field of function a well-founded relation Few foundations of this kind, and not popular von Neumann in 1925 –Others changed functions to sets – became NBG Jones’s Pure Functions in 1998 von Neumann and Jones – do not define reduction (computation) or proof

5. NummSquared Everything a function, no types, no side-effects, no global state, well-founded Reduction (computation) and proof Sound: the proposition of a valid proof is true Termination ensured without proofs by programmer Proofs as desired, but not required Classical logic Follows set theory as much as possible Simple variable-free syntax Reflection –NummSquared is its own macro language. –NummSquared is used to manipulate NummSquared proofs.

6. Domain membership When calling a function, how to ensure that the argument belongs to the function’s domain? Unrestricted domain – untyped lambda calculus – non-termination Type constraints on programmer Set theory: higher-order domain membership not computable – need proofs by programmer NummSquared: coercion

7. NummSquared coercion for domain membership NummSquared coercion: if it isn’t so, then make it so! Type conversion generalized to higher- order (function with function argument) Coerce a function to domain/codomain by applying pre-coercion/post-coercion. NummSquared coercion somewhat related to Howe’s restriction of untyped lambda terms. –Howe does not ensure termination.

8. Semantics: small function extensions p = left(p) p = right(p) zero one pair p p = null null zero = null null one = null one = zero zero leaf null null (base case) null = null rule f f x dom(f) small (no larger than a ZFC set) … null simple

9. Semantics: small function extensions Well-founded to ensure termination and avoid paradoxes Defined inductively Rule f with: –dom(f) a small set of small function extensions Small means no larger than a ZF set. –for x in dom(f), f is a small function extension Leaf: null, zero, one –null means absence of relevant information – does not mean 0, false, undefined or non-termination Pair p such that left(p) and right(p) are small function extensions A tree is a small function extension recursively containing only leaves and pairs.

10. Semantics: domain extensions For a rule f, dom(f) is not, in general, amenable to coercion. So dom(f) represented by domain extension (tag): –Same information as a type in type theory –Different purpose – not compile-time type checking, but runtime coercion –Domain extensions never appear directly in NummSquared programs, but are available as small function extensions. Domain extension: –Constant: Dom.Null, Dom.Nuro (Null or Zero), Dom.Leaf, Dom.Tree –Combination: dependent sum/product A with domExtFam(A) a domain extension family Domain extension family F with: –dom(F) a small set of small function extensions –for x in dom(F), F is a domain extension –domExt(F) is a domain extension

11. Semantics: domain of domain extensions dom(Dom.Null) = Null = {null}. dom(Dom.Nuro) = Nuro = {null, zero}. dom(Dom.Leaf) = Leaf = {null, zero, one}. dom(Tree) = Tree = set of all trees For dependent sum A, dom(A) = set of null and all pairs p such that: –left(p) in dom(domExtFam(A)) –right(p) in dom(domExtFam(A) ) For dependent product A, dom(A) = set of null and all rules f such that: –dom(f) = dom(domExtFam(A)) –for x in dom(f), f in dom(domExtFam(A) )

12. Semantics: Domain extension irrelevance Domain extension f is valid iff, for each domain extension family G contained, directly or indirectly, in f: –dom(domExt(G)) = dom(G) Domain extension irrelevance theorem: for valid domain extensions A and B, if dom(A) = dom(B), then A = B. –A domain extension contains no more information than the domain it represents.

13. Semantics: tagged small function extensions Tagging –Constrain rules by domain extensions –Somewhat analogous to runtime type information Defined inductively Tagged rule f with: –dom(f) a small set of small function extensions –for x in dom(f), f is a tagged small function extension –domExt(f) is a valid domain extension such that dom(domExt(f)) = dom(f) Leaf: as before Tagged pair p such that left(p) and right(p) are tagged small function extensions

14. Semantics: Untag and tagged For tagged small function extension f, untag(f) is obtained by removing domain extensions from f. Tag irrelevance theorem: for tagged small function extensions f and g, if untag(f) = untag(g), then f = g. For valid domain extension A, and f in dom(A), tagged(A, f) is the tagged small function extension such that untag(tagged(A, f)) = f. For valid domain extension family F, and x in dom(F), tagged(F, x) = tagged(domExt(F), x). For tagged small function extension f, and dom(f) element x, tagged(f, x) = tagged(domExt(f), x).

15. Semantics: coercion For valid domain extension A and tagged small function extension f, coercion to A of f, denoted by coer(A, f), is in dom(A). If A is a dependent sum, and f is a pair, coerce left(f) first, then right(f). –coer(A, f) is the pair p such that: left(p) = coer(domExt(domExtFam(A)), left(f)) right(p) = coer(domExtFam(A), right(f)) If A is a dependent product, and f is a rule, add pre- coercion and post-coercion to f: –coer(A, f) is the rule r such that: dom(r) = dom(domExtFam(A)) For x in dom(r), r = coer(domExtFam(A), f )

16. Semantics: result See NummSquared Explained for: –The precise recursive definition of coercion –Justification of that recursive definition by a well-founded relation Use coercion to define tagged small function extensions over all tagged small function extensions For tagged small function extensions f and x, f(x) = f Coercion and result are computable and always terminate.

17. Semantics: large function extensions Abstract over all tagged small function extensions f such that, for each tagged small function extension x, f(x) is a tagged small function extension A large function extension is also a proposition extension: –f is true iff, for each tagged small function extension x, f(x) = one

18. Semantics: recursion Terminating recursion principle based on structure of small function extensions For large function extensions start and step, ~r[start step] is the large function extension such that, for each tagged small function extension x: –if x = null, ~r[start step](x) = start(x) –otherwise, ~r[start step](x) = step({rDom, rRan, x}) where: rDom is the tagged rule such that domExt(rDom) = domExt(x) and, for y in dom(rDom), rDom = ~r[start step](tagged(rDom, y)) rRan is the tagged rule such that domExt(rRan) = domExt(x) and, for y in dom(rRan), rRan = ~r[start step](x(tagged(rDom, y)))

19. NummSquared syntax Variable-free Large functions –Constants –Combinations Proofs –Axioms –Inferences Reflection –Quoting and unquoting large functions –Quoting and unquoting proofs –The quoted representation is a tree. –Quoting is easy because NummSquared is variable-free. See NummSquared Explained for details.

20. The Future Thanks to Hugo Herbelin and Bruno Barras for suggesting some directions for the future. Current recursion principle is like ε or ordinal recursion, but well-founded recursion requires proof –Solution: In well-founded recursion, the proof is irrelevant for computation. Define/prove NummSquared within first order logic + NBG axioms First order logic axioms for NummSquared Model typed languages within NummSquared –NummSquared models (unlike set theory models) are computational

21. References My PhD thesis, NummSquared 2006a0 Explained –poohbist.com Other references are in the bibliography of NummSquared Explained. Short paper: NummSquared: a New Foundation for Formal Methods –poohbist.com These slides –poohbist.com