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 ` ' 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 :' _ ', for all formulas ' 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 (9w)(x + S(w) = y). recursive functions Let (9!y)'(x; y) abbreviate Conclusion Future directions Concluding (9y)'(x; y) ^ (8y)(8z)('(x; y) ^ '(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 ) (9y)(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 '(x; y) satisfying Statements Proof k+1 Total (a) for all m; n 2 , if f(m) = n, then ` '(m; n) recursive N functions (b) for all m 2 k, ` (9!y)'(m; y) Conclusion N Future directions f is strongly representable in T as a total function if there Concluding remarks exists a formula '(x; y) satisfying (a) and Appendix (b)' ` (9!y)'(x; y) Representability of partial functions Representing Recursive Definition Functions k Yan Steimle For f : N 99K N and '(x; y) consider the conditions k+1 Definitions (P1) for all m; n 2 N , f(m) ' n iff ` '(m; n) k Main (P2) for all m 2 N , ` '(m; y) ^ '(m; z) ) y = z theorems Statements (P3) ` '(x; y) ^ '(x; y) ) y = z Proof Total recursive (P4) ` (9!y)'(x; y) functions Conclusion Future k directions For f : N 99K N, if there exists '(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 ' in T and any variable x occurring free Total recursive functions in ', n Conclusion if ` (9x)', then 9 n 2 N such that ` ' 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 2 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 2 N such that Conclusion Future directions f(m) ' U(µn(Tk(e; m; n) = 0)) Concluding remarks k Appendix for all m 2 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 (y; z) and '(x; y), respectively, then Appendix (9y)('(x; y) ^ (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) ^ (8u)(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 (x) with exactly k free variables such that, for theorems k Statements all m 2 N , Proof Total recursive functions E(m) iff ` (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 2 N such that Main theorems k Statements f(m) ' U(µn(Tk(e; m; n) = 0)) 8m 2 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) 8m 2 N : Appendix As U is primitive recursive, there exists a formula '(z; y) that strongly represents U as a total function. ' 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) ^ (9z)(σ(x; z) ^ '(z; y)): Total recursive functions Indeed, Conclusion Future directions (P3) for θ follows from (P3) for σ and '. 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 (x) in T with exactly k free variables such Main k theorems that, for all m 2 N , Statements Proof Total recursive E(m) iff ` (m) functions F (m) iff ` : (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 2 N be distinct. By Lemma 3, we obtain a Definitions −1 formula σ(x) that exactly separates f (fn0g) and Main −1 theorems f (fn1g).