Representing Recursive Functions Yan Steimle The Representability of Partial Recursive Definitions Functions in Arithmetical Theories and Main theorems Categories Statements Proof Total recursive functions Conclusion Yan Steimle Future directions Concluding Department of Mathematics and Statistics remarks University of Ottawa Appendix Foundational Methods in Computer Science May 30 – June 2, 2018 First-order theories
Representing Recursive Functions
Yan Steimle With equality Definitions Γ ` ϕ satisfying the rules for intuitionistic sequent Main theorems calculus Statements Proof Logical axioms: Total recursive functions For all theories, decidability of equality (DE): Conclusion Future directions x 6= y ∨ x = y Concluding remarks To obtain classical theories, the excluded middle (EM): Appendix ¬ϕ ∨ ϕ, for all formulas ϕ The arithmetical theory M
Representing Recursive Functions
Yan Steimle
Definitions Let LM be the first-order language with 0, S, ·, +. Main theorems Let Sn(0) be the nth numeral, denoted n. Statements Proof Total Let x < y abbreviate (∃w)(x + S(w) = y). recursive functions Let (∃!y)ϕ(x, y) abbreviate Conclusion Future directions Concluding (∃y)ϕ(x, y) ∧ (∀y)(∀z)(ϕ(x, y) ∧ ϕ(x, z) ⇒ y = z). remarks Appendix The arithmetical theory M
Representing Recursive Functions Definition
Yan Steimle M is a theory over LM with the nonlogical axioms
Definitions (M1) S(x) 6= 0
Main theorems (M2) S(x) = S(y) ⇒ x = y Statements Proof (M3) x + 0 = x Total recursive functions (M4) x + S(y) = S(x + y) Conclusion (M5) x · 0 = 0 Future directions Concluding (M6) x · S(y) = (x · y) + x remarks Appendix (M7) x 6= 0 ⇒ (∃y)(x = S(y)) (M8) x < y ∨ x = y ∨ y < x
We consider an arbitrary arithmetical theory T , i.e. a consistent r.e. extension of M. Recursive functions, brief overview
Representing Recursive Functions
Yan Steimle
Definitions
Main Primitive recursive: basic functions; closed under theorems subsitution (S) and primitive recursion (PR) Statements Proof Total Total recursive: basic functions; closed under (S), (PR), recursive functions and total µ Conclusion Future Partial recursive: basic functions; closed under (S), directions Concluding (PR), and partial µ remarks Appendix Representability of total functions
Representing Recursive Functions
Yan Steimle Definition Definitions k Main A function f : N → N is numeralwise representable in T as theorems a total function if there exists a formula ϕ(x, y) satisfying Statements Proof k+1 Total (a) for all m, n ∈ , if f(m) = n, then ` ϕ(m, n) recursive N functions (b) for all m ∈ k, ` (∃!y)ϕ(m, y) Conclusion N Future directions f is strongly representable in T as a total function if there Concluding remarks exists a formula ϕ(x, y) satisfying (a) and Appendix (b)’ ` (∃!y)ϕ(x, y) Representability of partial functions
Representing Recursive Definition Functions k Yan Steimle For f : N 99K N and ϕ(x, y) consider the conditions k+1 Definitions (P1) for all m, n ∈ N , f(m) ' n iff ` ϕ(m, n) k Main (P2) for all m ∈ N , ` ϕ(m, y) ∧ ϕ(m, z) ⇒ y = z theorems Statements (P3) ` ϕ(x, y) ∧ ϕ(x, y) ⇒ y = z Proof Total recursive (P4) ` (∃!y)ϕ(x, y) functions Conclusion Future k directions For f : N 99K N, if there exists ϕ(x, y) in T such that Concluding remarks (P1) and (P2) hold, f is numeralwise representable in T Appendix as a partial function (P1) and (P3) hold, f is type-one representable in T (P1) and (P4) hold, f is strongly representable in T as a partial function Representability theorems for partial recursive functions
Representing Recursive Functions
Yan Steimle
Definitions Theorem (I) Main Let T be any arithmetical theory. All partial recursive theorems Statements functions are type-one representable in T . Proof Total recursive functions Theorem (II) Conclusion Future directions Let T be a classical arithmetical theory. All partial Concluding remarks recursive functions are strongly representable in T as partial Appendix functions. Consequences of the Existence Property (EP)
Representing Recursive Functions
Yan Steimle
Definitions
Main theorems The Existence Property (EP) Statements Proof For every formula ϕ in T and any variable x occurring free Total recursive functions in ϕ, n Conclusion if ` (∃x)ϕ, then ∃ n ∈ N such that ` ϕ x . Future directions Concluding remarks Appendix The Kleene normal form theorem (alternate version)
Representing Recursive Functions
Yan Steimle Theorem (Kleene normal form) Definitions Main For each k ∈ N, k > 0, there exist primitive recursive theorems k+2 Statements functions U : N → N and Tk : N → N such that, for any Proof k Total partial recursive function f : N 99K N, there exists a number recursive functions e ∈ N such that Conclusion Future directions f(m) ' U(µn(Tk(e, m, n) = 0)) Concluding remarks k Appendix for all m ∈ N . The strong representability of primitive recursive functions in arithmetical theories
Representing Recursive Theorem Functions Let T be any arithmetical theory. All primitive recursive Yan Steimle functions are strongly representable in T as total functions. Definitions Main Proof. theorems Statements It suffices to express the basic functions and the recursion Proof Total recursive schemes (S) and (PR) by formulas in T . For example: functions Conclusion y = S(x) strongly represents the successor function. Future directions If g, h : → are primitive recursive and strongly Concluding N N remarks representable by ψ(y, z) and ϕ(x, y), respectively, then Appendix (∃y)(ϕ(x, y) ∧ ψ(y, z))
strongly represents f = g(h): N → N. Representing functions obtained by partial minimisation
Representing Recursive Lemma (1) Functions k+1 Yan Steimle Let g : N → N (k ≥ 0) be a total function that is numeralwise representable in T as a total function, and let Definitions k f : N 99K N be obtained from g by partial µ. Then, f is Main theorems type-one representable in T . Statements Proof Total recursive Proof. functions Conclusion g is numeralwise representable in T by σ(x, y, z) and f is Future directions defined by Concluding remarks f(m) ' µn(g(m, n) = 0). Appendix Thus, f is type-one representable in T by the formula
σ(x, y, 0) ∧ (∀u)(u < y ⇒ ¬σ(x, u, 0)). Weak representability of r.e. relations
Representing Recursive Functions
Yan Steimle Definition (Weak representability) k Definitions A relation E ⊆ N is weakly representable in T if there exists Main a formula ψ(x) with exactly k free variables such that, for theorems k Statements all m ∈ N , Proof Total recursive functions E(m) iff ` ψ(m). Conclusion Future directions Lemma (2) Concluding remarks Appendix All k-ary r.e. relations on N (k ≥ 0) are weakly representable in T .
(long technical proof) Proof of Theorem (I)
Representing Recursive Functions
Yan Steimle Theorem (I) Definitions
Main Let T be any arithmetical theory. All partial recursive theorems functions are type-one representable in T . Statements Proof Total recursive functions Proof. Conclusion k Future Let f : N 99K N be a partial recursive function. directions Concluding remarks k = 0: If f is the constant n in , take the formula n = y. If Appendix N f is completely undefined, take the formula y = y ∧ 0 6= 0. Proof of Theorem (I)
Representing Recursive Functions Proof (continued). Yan Steimle k ≥ 1: By the Kleene normal form theorem, we obtain k+2 Definitions U : N → N, Tk : N → N, and e ∈ N such that Main theorems k Statements f(m) ' U(µn(Tk(e, m, n) = 0)) ∀m ∈ N . Proof Total recursive functions As Tk is primitive recursive, by Lemma 1 there exists a Conclusion formula σ(x, z) that type-one represents the partial function Future directions given by Concluding remarks k µn(Tk(e, m, n) = 0) ∀m ∈ N . Appendix As U is primitive recursive, there exists a formula ϕ(z, y) that strongly represents U as a total function. ϕ also type-one represents U. Proof of Theorem (I)
Representing Recursive Functions Proof (continued). Yan Steimle By Lemma 2, there exists a formula η(x) that weakly represents the r.e. domain D of f. Then, f is type-one Definitions f
Main representable in T by the formula θ(x, y) defined by theorems Statements Proof η(x) ∧ (∃z)(σ(x, z) ∧ ϕ(z, y)). Total recursive functions Indeed, Conclusion Future directions (P3) for θ follows from (P3) for σ and ϕ. Concluding remarks For (P1), since η weakly represents Df , we only have to Appendix consider inputs on which f is defined. Hence, we can show that ` θ(m, p) implies f(m) ' p by (P3) for θ and the fact that ` f(m) = p iff f(m) = p. Exact separability of r.e. relations
Representing Recursive Functions Definition (Exact separability) Yan Steimle k Two relations E,F ⊆ N are exactly separable in T if there Definitions exists a formula ψ(x) in T with exactly k free variables such Main k theorems that, for all m ∈ N , Statements Proof Total recursive E(m) iff ` ψ(m) functions F (m) iff ` ¬ψ(m) Conclusion Future directions Concluding remarks Lemma (3) Appendix Let T be a classical arithmetical theory. Any two disjoint k-ary r.e. relations on N (k ≥ 0) are exactly separable in T . (long technical proof) Proof of Theorem (II)
Representing Recursive Functions Theorem (II) Yan Steimle Let T be a classical arithmetical theory. All partial Definitions recursive functions are strongly representable in T as partial Main theorems functions. Statements Proof Total recursive functions Proof. Conclusion Let f : k be a partial recursive function. Future N 99K N directions Concluding remarks k = 0: If f is completely undefined, let G be a closed Appendix undecidable formula in T and take
(y = 0 ⇒ ¬G) ∧ (y 6= 0 ⇒ G) ∧ y < 2 Proof of Theorem (II)
Representing Recursive Functions Proof (continued). Yan Steimle k ≥ 1: Let n0, n1 ∈ N be distinct. By Lemma 3, we obtain a Definitions −1 formula σ(x) that exactly separates f ({n0}) and Main −1 theorems f ({n1}). By Theorem (I), we obtain a formula ϕ(x, y) Statements Proof that type-one represents f. Consider Total recursive functions ψ(x) ≡ (∃z)ϕ(x, z) ∧ ¬ϕ(x, n0) ∧ ¬ϕ(x, n1) Conclusion def Future directions θ(x, y) ≡ (ψ(x) ∧ ϕ(x, y)) ∨ (¬ψ(x) ∧ σ(x) ∧ y = n0) Concluding def remarks ∨(¬ψ(x) ∧ ¬σ(x) ∧ y = n1). Appendix By (EM), ` (¬ψ(x) ∧ ¬σ(x)) ∨ (¬ψ(x) ∧ σ(x)) ∨ ψ(x), from which (P4) follows. (P1) is obtained by cases. Representability of total recursive functions
Representing Recursive Functions
Yan Steimle
Definitions Corollary (of Theorem (I)) Main theorems Let T be an arithmetical theory. All total recursive functions Statements Proof are numeralwise representable in T as total functions. Total recursive functions Conclusion Corollary (of Theorem (II)) Future directions Let T be a classical arithmetical theory. All total recursive Concluding remarks functions are strongly representable in T as total functions. Appendix Classifying categories
Representing Recursive Functions Yan Steimle Given a theory T , we construct a classifying category
Definitions C (T ):
Main objects: formulas of T theorems morphisms: equivalence classes of provably functional Statements Proof relations between formulas Total recursive functions For a general theory T , C (T ) is regular. Conclusion If T is an intuitionistic arithmetical theory, we claim Future directions that in (T ): Concluding C remarks there is at least a weak NNO; Appendix numerals are standard; 1 is projective and indecomposable. Work in progress
Representing Recursive Functions Yan Steimle For an arithmetical theory T :
Definitions 1 Consider the formulas representing recursive functions Main in C (T ) (for all possible variations). What theorems Statements sub-categories do we obtain? Proof Total 2 recursive Construct a partial map category associated with C (T ) functions and show it is a Turing category. Conclusion Future 3 directions Ultimately, we want to consider partial recursive Concluding remarks functionals of higher type using a notion of λ-calculus Appendix with equalisers and a construction of the free CCC with equalisers. Acknowledgements
Representing Recursive Functions
Yan Steimle
Definitions
Main theorems Statements I would like to thank my supervisor, Professor Scott, the Proof Total conference organisers, the University of Ottawa, and recursive functions NSERC. Conclusion Future directions Concluding remarks Appendix References
Representing Recursive [LS] J. Lambek and P. Scott. Introduction to higher order Functions categorical logic. Vol. 7. Cambridge studies in advanced Yan Steimle mathematics. Cambridge University Press, 1986. Definitions
Main [RY] R. W. Ritchie and P. R. Young. ‘Strong theorems Representability of Partial Functions in Arithmetic Statements Proof Theories’. In: Information Sciences 1.2 (1969), pp. Total recursive functions 189–204. Conclusion Future [S] J. C. Shepherdson. ‘Representability of recursively directions Concluding enumerable sets in formal theories’. In: Archiv f¨ur remarks Appendix mathematische Logik und Grundlagenforschung 5.3 (1961), pp. 119–127. [Sh] J. R. Shoenfield. Mathematical Logic. Addison-Wesley Series in Logic. Addison-Wesley Publishing Company, 1967. Kleene Equality
Representing Recursive Functions
Yan Steimle To deal with partialness, we use Kleene Equality. If e1 and e2 are two expressions on N that may or may not be defined, Definitions then Main theorems Statements e1 ' e2 iff (e1, e2 are defined and equal) Proof Total OR (e , e are undefined). recursive 1 2 functions Conclusion For example, if f : k is a partial function and Future N 99K N directions m, n ∈ k+1, Concluding N remarks Appendix f(m) 6' n iff (f(m) is defined but not equal to n) OR (f(m) is undefined). Consequences of the Existence Property (EP)
Representing Recursive Functions
Yan Steimle If T is classical: Definitions G is a closed undecidable formula in T , f : k the Main N 99K N theorems completely undefined function. Statements Proof Total recursive ϕ(x, y) ≡ x = x ∧ (y = 0 ⇒ ¬G) ∧ (y 6= 0 ⇒ G) ∧ y < 2 functions def Conclusion Future directions strongly represents f in T as a partial function. Concluding remarks If T were to satisfy EP, then ` G or ` ¬G, a Appendix contradiction. Consequences of the Existence Property (EP)
Representing Recursive Functions If T is intuitionistic: Yan Steimle Let f : N 99K N be a partial function undefined at Definitions m ∈ N. Main theorems Suppose that there exists ϕ(x, y) satisfying (P1) and Statements Proof (P4). Total recursive functions By (P4), ` (∃y)ϕ(m, y). Conclusion Future By EP, there exists n ∈ N such that ` ϕ(m, n). directions Concluding remarks By (P1), f(m) ' n, and so f(m) is defined. Appendix Contradiction. So, strong representability of partial functions doesn’t make sense and Theorem (II) fails. A technical lemma
Representing Recursive Functions
Yan Steimle
Definitions Lemma Main k k+j theorems Let E1 ⊆ and E2 ⊆ (k, j ≥ 0) be r.e. relations. Statements N N Proof There exists a formula ϕ(x, u) in T with k + j free variables Total recursive k j functions such that, for all m ∈ N and p ∈ N , Conclusion if E (m) and ¬E (m, p), then ` ϕ(m, p) Future 1 2 directions Concluding if ¬E1(m) and E2(m, p), then 6` ϕ(m, p). remarks Appendix A technical lemma
Representing Recursive Proof (idea). Functions k Yan Steimle (Adapted from the case for j = 0 in [S]) Let E1 ⊆ N and k+j E2 ⊆ N (k, j ≥ 0) be r.e. relations. There exist primitive Definitions k+1 k+j+1 recursive relations F1 ⊆ N , F2 ⊆ N such that, for all Main k j theorems m ∈ N and p ∈ N , Statements Proof Total E1(m) iff ∃n ∈ s.t. F1(m, n) recursive N functions E2(m, p) iff ∃n ∈ N s.t. F2(m, p, n). Conclusion Future directions We obtain formulas ψ1(x, y) and ψ2(x, u, y) that Concluding remarks numeralwise represent F1 and F2, respectively, in T . Appendix Then, ϕ(x, u) given by
(∃y)(ψ1(x, y) ∧ (∀z)(z ≤ y ⇒ ¬ψ2(x, u, z))
is the required formula. Weak representability of r.e. relations (proof)
Representing Recursive Functions Proof (Lemma 2). Yan Steimle 0 k = 0: N = {∗} is weakly representable by 0 = 0 and ∅ is Definitions weakly representable by 0 6= 0.
Main theorems k ≥ 1: Let E ⊆ k, let x, y be k + 1 distinct fixed variables. Statements N Proof T has an associated G¨odelnumbering where ψ denotes Total p q recursive functions the G¨odelnumber of ψ and γn is the formula with G¨odel Conclusion number n. Then, we can construct a primitive recursive Future directions function g : k+1 → such that Concluding N N remarks ( h i Appendix γ m , n if γ exists g(m, n) = p n x y q n n otherwise Weak representability of r.e. relations (proof)
Representing Recursive Functions Proof (continued). Yan Steimle Since T is an r.e. theory, D ⊆ k+1 given by Definitions N Main h m n i theorems D(m, n) iff GTHM T (g(m, n)) iff ` γn x , y Statements Proof Total recursive is an r.e. relation. By the technical lemma, we obtain functions ϕ(x, y) in T such that, for all m, n ∈ k+1, Conclusion N Future directions h i Concluding m n remarks if E(m) and 6` γn x , y , then ` ϕ(m, n) Appendix h m n i if ¬E(m) and ` γn x , y , then 6` ϕ(m, n). Weak representability of r.e. relations (proof)
Representing Recursive Functions Yan Steimle Proof (continued). Definitions k Let p = ϕ(x, y) . Then, γp = ϕ and so, for all m ∈ N , Main p q theorems Statements if E(m) and 6` ϕ(m, p), then ` ϕ(m, p) Proof Total recursive if ¬E(m) and ` ϕ(m, p), then 6` ϕ(m, p). functions Conclusion k Future It follows that, for all m ∈ N , directions Concluding remarks E(m) iff ` ϕ(m, p), Appendix and so ϕ(x, p) weakly represents E in T .