The intensional content of Rices Theorem Andrea Asperti Department - PowerPoint PPT Presentation

The intensional content of Rice’s Theorem The intensional content of Rice’s Theorem Andrea Asperti Department of Computer Science, University of Bologna Mura Anteo Zamboni 7, 40127, Bologna, ITALY asperti@cs.unibo.it

The intensional content of Rice’s Theorem Content 1 Rice’s Theorem 2 Blum’s Abstract Complexity 3 Similarity and Complexity Cliques 4 Rice-Shapiro’s Theorem Monotonicity compactness 5 Corollaries 6 Kleene’s Fixed Point Theorem 7 Conclusions Main results Future works and applications

The intensional content of Rice’s Theorem Content 1 Rice’s Theorem 2 Blum’s Abstract Complexity 3 Similarity and Complexity Cliques 4 Rice-Shapiro’s Theorem Monotonicity compactness 5 Corollaries 6 Kleene’s Fixed Point Theorem 7 Conclusions Main results Future works and applications

The intensional content of Rice’s Theorem Content 1 Rice’s Theorem 2 Blum’s Abstract Complexity 3 Similarity and Complexity Cliques 4 Rice-Shapiro’s Theorem Monotonicity compactness 5 Corollaries 6 Kleene’s Fixed Point Theorem 7 Conclusions Main results Future works and applications

The intensional content of Rice’s Theorem Content 1 Rice’s Theorem 2 Blum’s Abstract Complexity 3 Similarity and Complexity Cliques 4 Rice-Shapiro’s Theorem Monotonicity compactness 5 Corollaries 6 Kleene’s Fixed Point Theorem 7 Conclusions Main results Future works and applications

The intensional content of Rice’s Theorem Content 1 Rice’s Theorem 2 Blum’s Abstract Complexity 3 Similarity and Complexity Cliques 4 Rice-Shapiro’s Theorem Monotonicity compactness 5 Corollaries 6 Kleene’s Fixed Point Theorem 7 Conclusions Main results Future works and applications

The intensional content of Rice’s Theorem Content 1 Rice’s Theorem 2 Blum’s Abstract Complexity 3 Similarity and Complexity Cliques 4 Rice-Shapiro’s Theorem Monotonicity compactness 5 Corollaries 6 Kleene’s Fixed Point Theorem 7 Conclusions Main results Future works and applications

The intensional content of Rice’s Theorem Content 1 Rice’s Theorem 2 Blum’s Abstract Complexity 3 Similarity and Complexity Cliques 4 Rice-Shapiro’s Theorem Monotonicity compactness 5 Corollaries 6 Kleene’s Fixed Point Theorem 7 Conclusions Main results Future works and applications

The intensional content of Rice’s Theorem Rice’s Theorem Outline 1 Rice’s Theorem 2 Blum’s Abstract Complexity 3 Similarity and Complexity Cliques 4 Rice-Shapiro’s Theorem Monotonicity compactness 5 Corollaries 6 Kleene’s Fixed Point Theorem 7 Conclusions Main results Future works and applications

The intensional content of Rice’s Theorem Rice’s Theorem Rice’s Theorem Rice 1953 An estensional property of programs is decidable if and only if it is trivial. estensional = closed w.r.t. estensional equivalence

The intensional content of Rice’s Theorem Rice’s Theorem Rice’s Yin Yang ∀ x , φ m ( x ) ↑

The intensional content of Rice’s Theorem Rice’s Theorem the function h Let K = dom ( φ k ), and define φ h ( x ) ( y ) = φ k ( x ); φ a ( y ) Clearly, if φ m is the everywhere divergent function, � if x ∈ K φ a φ h ( x ) ≈ if x �∈ K φ m Does h preserve any other property, in addition to extensional equivalence? Yes, complexity! Next: investigates the complexity assumptions needed to formalize such result.

The intensional content of Rice’s Theorem Rice’s Theorem the function h Let K = dom ( φ k ), and define φ h ( x ) ( y ) = φ k ( x ); φ a ( y ) Clearly, if φ m is the everywhere divergent function, � if x ∈ K φ a φ h ( x ) ≈ if x �∈ K φ m Does h preserve any other property, in addition to extensional equivalence? Yes, complexity! Next: investigates the complexity assumptions needed to formalize such result.

The intensional content of Rice’s Theorem Rice’s Theorem the function h Let K = dom ( φ k ), and define φ h ( x ) ( y ) = φ k ( x ); φ a ( y ) Clearly, if φ m is the everywhere divergent function, � if x ∈ K φ a φ h ( x ) ≈ if x �∈ K φ m Does h preserve any other property, in addition to extensional equivalence? Yes, complexity! Next: investigates the complexity assumptions needed to formalize such result.

The intensional content of Rice’s Theorem Rice’s Theorem the function h Let K = dom ( φ k ), and define φ h ( x ) ( y ) = φ k ( x ); φ a ( y ) Clearly, if φ m is the everywhere divergent function, � if x ∈ K φ a φ h ( x ) ≈ if x �∈ K φ m Does h preserve any other property, in addition to extensional equivalence? Yes, complexity! Next: investigates the complexity assumptions needed to formalize such result.

The intensional content of Rice’s Theorem Rice’s Theorem the function h Let K = dom ( φ k ), and define φ h ( x ) ( y ) = φ k ( x ); φ a ( y ) Clearly, if φ m is the everywhere divergent function, � if x ∈ K φ a φ h ( x ) ≈ if x �∈ K φ m Does h preserve any other property, in addition to extensional equivalence? Yes, complexity! Next: investigates the complexity assumptions needed to formalize such result.

The intensional content of Rice’s Theorem Blum’s Abstract Complexity Outline 1 Rice’s Theorem 2 Blum’s Abstract Complexity 3 Similarity and Complexity Cliques 4 Rice-Shapiro’s Theorem Monotonicity compactness 5 Corollaries 6 Kleene’s Fixed Point Theorem 7 Conclusions Main results Future works and applications

The intensional content of Rice’s Theorem Blum’s Abstract Complexity Blum’s Abstract Complexity A pair � φ, Φ � is a computational complexity measure if φ is a principal effective enumeration of partial recursive functions and Φ satisfies Blum’s axioms (Blum 1967): n ) ↓↔ Φ i ( � n ) ↓ ( a ) φ i ( � ( b ) the predicate Φ i ( � n ) = m is decidable

The intensional content of Rice’s Theorem Similarity and Complexity Cliques Outline 1 Rice’s Theorem 2 Blum’s Abstract Complexity 3 Similarity and Complexity Cliques 4 Rice-Shapiro’s Theorem Monotonicity compactness 5 Corollaries 6 Kleene’s Fixed Point Theorem 7 Conclusions Main results Future works and applications

The intensional content of Rice’s Theorem Similarity and Complexity Cliques Big O notation Big O remind: 1 f ∈ O ( g ) if and only if there exist n and c such that for any m ≥ n , if g ( m ) ↓ then f ( m ) ≤ cg ( m ); 2 f ∈ Θ( g ) if and only if f ∈ O ( g ) and g ∈ O ( f ).

The intensional content of Rice’s Theorem Similarity and Complexity Cliques Similarity and Complexity Clique Definition Two programs i and j are similar (write i ≈ j ) if and only if φ j ∼ = φ i ∧ Φ j ∈ Θ(Φ i ) Similarity is an equivalence relation. Definition Let � φ, Φ � be an abstract complexity measure. A set P of natural numbers is a Complexity Clique , if and only if for all i and j i ∈ P ∧ j ≈ i → j ∈ P

The intensional content of Rice’s Theorem Similarity and Complexity Cliques Examples of Complexity Cliques 1 ∅ and ω ; 2 for any index i , [ i ] ≈ ; 3 for any index i , { j | Φ j ∈ O (Φ i ) } . 4 all programs with polynomial (exponential, . . . ) complexity. Warning : not every Complexity Class is a Complexity Cliques. Complexity Cliques are closed w.r.t to union, intersection, and complementation.

The intensional content of Rice’s Theorem Similarity and Complexity Cliques Complexity Assumptions: s-m-n Definition A pair � φ, Φ � has the s-m-n property if for all m and n there exists a recursive function s n m such that, for any i and all x 1 , . . . , x m m ( i , x 1 ,..., x m ) ∼ ( a ) φ s n = λ y 1 , . . . , y n .φ i ( x 1 , . . . , x m , y 1 , . . . , y n ) ( b ) Φ s n m ( i , x 1 ,..., x m ) ∈ Θ( λ y 1 , . . . , y n . Φ i ( x 1 , . . . , x m , y 1 , . . . , y n ))

The intensional content of Rice’s Theorem Similarity and Complexity Cliques Complexity Assumptions: s-m-n Definition A pair � φ, Φ � has the s-m-n property if for all m and n there exists a recursive function s n m such that, for any i and all x 1 , . . . , x m m ( i , x 1 ,..., x m ) ∼ ( a ) φ s n = λ y 1 , . . . , y n .φ i ( x 1 , . . . , x m , y 1 , . . . , y n ) ( b ) Φ s n m ( i , x 1 ,..., x m ) ∈ Θ( λ y 1 , . . . , y n . Φ i ( x 1 , . . . , x m , y 1 , . . . , y n ))

The intensional content of Rice’s Theorem Similarity and Complexity Cliques Complexity Assumptions: composition Definition A pair � φ, Φ � has the composition property if there exists a total computable function h such that ( a ) φ h ( i , j ) ∼ = φ i ◦ φ j ( b ) Φ h ( i , j ) ∈ Θ( max { Φ i ◦ φ j , Φ j } ) we only ask that there exists a way of composing functions with the above complexity.

The intensional content of Rice’s Theorem Similarity and Complexity Cliques Complexity Assumptions: composition Definition A pair � φ, Φ � has the composition property if there exists a total computable function h such that ( a ) φ h ( i , j ) ∼ = φ i ◦ φ j ( b ) Φ h ( i , j ) ∈ Θ( max { Φ i ◦ φ j , Φ j } ) we only ask that there exists a way of composing functions with the above complexity.