The Representability of Partial Recursive Yan Steimle Functions in - PowerPoint PPT Presentation

Representing Recursive Functions The Representability of Partial Recursive Yan Steimle Functions in Arithmetical Theories and Definitions Main Categories theorems Statements Proof Total recursive functions Yan Steimle Conclusion Future directions Department of Mathematics and Statistics Concluding 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 calculus theorems Statements Logical axioms: Proof Total recursive For all theories, decidability of equality (DE): functions Conclusion x � = y ∨ x = y Future directions 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 L M be the first-order language with 0, S , · , +. Main Let S n (0) be the n th numeral, denoted n . theorems Statements Proof Let x < y abbreviate ( ∃ w )( x + S ( w ) = y ). Total recursive functions Let ( ∃ ! y ) ϕ ( x , y ) abbreviate Conclusion Future directions ( ∃ y ) ϕ ( x , y ) ∧ ( ∀ y )( ∀ z )( ϕ ( x , y ) ∧ ϕ ( x , z ) ⇒ y = z ) . Concluding remarks Appendix

The arithmetical theory M Representing Recursive Definition Functions M is a theory over L M with the nonlogical axioms Yan Steimle (M1) S ( x ) � = 0 Definitions (M2) S ( x ) = S ( y ) ⇒ x = y Main theorems (M3) x + 0 = x Statements Proof Total (M4) x + S ( y ) = S ( x + y ) recursive functions (M5) x · 0 = 0 Conclusion Future directions (M6) x · S ( y ) = ( x · y ) + x Concluding remarks (M7) x � = 0 ⇒ ( ∃ y )( x = S ( y )) Appendix (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 Primitive recursive: basic functions; closed under Main subsitution (S) and primitive recursion (PR) theorems Statements Proof Total recursive: basic functions; closed under (S), (PR), Total recursive and total µ functions Conclusion Partial recursive: basic functions; closed under (S), Future directions (PR), and partial µ Concluding remarks Appendix

Representability of total functions Representing Recursive Functions Yan Steimle Definition Definitions A function f : N k → N is numeralwise representable in T as Main theorems a total function if there exists a formula ϕ ( x , y ) satisfying Statements Proof (a) for all m , n ∈ N k +1 , if f ( m ) = n , then ⊢ ϕ ( m , n ) Total recursive functions (b) for all m ∈ N k , ⊢ ( ∃ ! y ) ϕ ( m , y ) Conclusion f is strongly representable in T as a total function if there Future directions Concluding exists a formula ϕ ( x , y ) satisfying (a) and remarks Appendix (b)’ ⊢ ( ∃ ! y ) ϕ ( x , y )

Representability of partial functions Representing Definition Recursive Functions For f : N k ��� N and ϕ ( x , y ) consider the conditions Yan Steimle (P1) for all m , n ∈ N k +1 , f ( m ) ≃ n iff ⊢ ϕ ( m , n ) Definitions (P2) for all m ∈ N k , ⊢ ϕ ( m , y ) ∧ ϕ ( m , z ) ⇒ y = z Main theorems (P3) ⊢ ϕ ( x , y ) ∧ ϕ ( x , y ) ⇒ y = z Statements Proof Total (P4) ⊢ ( ∃ ! y ) ϕ ( x , y ) recursive functions Conclusion For f : N k ��� N , if there exists ϕ ( x , y ) in T such that Future directions Concluding (P1) and (P2) hold, f is numeralwise representable in T remarks 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 Theorem (I) Definitions Let T be any arithmetical theory. All partial recursive Main theorems functions are type-one representable in T . Statements Proof Total recursive functions Theorem (II) Conclusion Let T be a classical arithmetical theory. All partial Future directions recursive functions are strongly representable in T as partial Concluding remarks functions. Appendix

Consequences of the Existence Property (EP) Representing Recursive Functions Yan Steimle Definitions Main The Existence Property (EP) theorems Statements For every formula ϕ in T and any variable x occurring free Proof Total recursive in ϕ , functions � n � if ⊢ ( ∃ x ) ϕ , then ∃ n ∈ N such that ⊢ ϕ . Conclusion x Future directions Concluding remarks Appendix

The Kleene normal form theorem (alternate version) Representing Recursive Functions Yan Steimle Theorem (Kleene normal form) Definitions For each k ∈ N , k > 0 , there exist primitive recursive Main functions U : N → N and T k : N k +2 → N such that, for any theorems Statements partial recursive function f : N k ��� N , there exists a number Proof Total recursive e ∈ N such that functions Conclusion Future f ( m ) ≃ U ( µ n ( T k ( e, m , n ) = 0)) directions Concluding remarks for all m ∈ N k . Appendix

The strong representability of primitive recursive functions in arithmetical theories Representing Theorem Recursive Functions Let T be any arithmetical theory. All primitive recursive Yan Steimle functions are strongly representable in T as total functions. Definitions Proof. Main theorems It suffices to express the basic functions and the recursion Statements Proof schemes (S) and (PR) by formulas in T . For example: Total recursive functions y = S ( x ) strongly represents the successor function. Conclusion Future If g, h : N → N are primitive recursive and strongly directions Concluding representable by ψ ( y, z ) and ϕ ( x, y ), respectively, then remarks Appendix ( ∃ y )( ϕ ( x, y ) ∧ ψ ( y, z )) strongly represents f = g ( h ) : N → N .

Representing functions obtained by partial minimisation Representing Lemma (1) Recursive Functions Let g : N k +1 → N ( k ≥ 0) be a total function that is Yan Steimle numeralwise representable in T as a total function, and let f : N k ��� N be obtained from g by partial µ . Then, f is Definitions Main type-one representable in T . theorems Statements Proof Proof. Total recursive functions g is numeralwise representable in T by σ ( x , y, z ) and f is Conclusion defined by Future directions Concluding f ( m ) ≃ µ n ( g ( m , n ) = 0) . remarks 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 Definition (Weak representability) Yan Steimle A relation E ⊆ N k is weakly representable in T if there exists Definitions a formula ψ ( x ) with exactly k free variables such that, for Main theorems all m ∈ N k , Statements Proof Total E ( m ) iff ⊢ ψ ( m ). recursive functions Conclusion Future Lemma (2) directions Concluding remarks All k -ary r.e. relations on N ( k ≥ 0) are weakly Appendix representable in T . (long technical proof)

Proof of Theorem (I) Representing Recursive Functions Yan Steimle Theorem (I) Definitions Let T be any arithmetical theory. All partial recursive Main functions are type-one representable in T . theorems Statements Proof Total recursive Proof. functions Let f : N k ��� N be a partial recursive function. Conclusion Future directions Concluding remarks k = 0: If f is the constant n in N , take the formula n = y . If Appendix f is completely undefined, take the formula y = y ∧ 0 � = 0.

Proof of Theorem (I) Representing Recursive Proof (continued). Functions Yan Steimle k ≥ 1: By the Kleene normal form theorem, we obtain U : N → N , T k : N k +2 → N , and e ∈ N such that Definitions Main theorems ∀ m ∈ N k . f ( m ) ≃ U ( µ n ( T k ( e, m , n ) = 0)) Statements Proof Total As T k is primitive recursive, by Lemma 1 there exists a recursive functions formula σ ( x , z ) that type-one represents the partial function Conclusion Future given by directions Concluding ∀ m ∈ N k . µ n ( T k ( e, m , n ) = 0) remarks 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 Proof (continued). Functions By Lemma 2, there exists a formula η ( x ) that weakly Yan Steimle represents the r.e. domain D f of f . Then, f is type-one Definitions representable in T by the formula θ ( x , y ) defined by Main theorems Statements η ( x ) ∧ ( ∃ z )( σ ( x , z ) ∧ ϕ ( z, y )) . Proof Total recursive functions Indeed, Conclusion Future (P3) for θ follows from (P3) for σ and ϕ . directions Concluding remarks For (P1), since η weakly represents D f , 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 .