Weak Cardinality Theorems for First-Order Logic Till Tantau Fakult - PowerPoint PPT Presentation

Weak Cardinality Theorems for First-Order Logic Till Tantau Fakult¨ at f¨ ur Elektrotechnik und Informatik Technische Universit¨ at Berlin Fundamentals of Computation Theory 2003 logo

History Unification by Logic Applications Summary Outline History 1 Enumerability in Recursion and Automata Theory Known Weak Cardinality Theorem Why Do Cardinality Theorems Hold Only for Certain Models? Unification by First-Order Logic 2 Elementary Definitions Enumerability for First-Order Logic Weak Cardinality Theorems for First-Order Logic Applications 3 A Separability Result for First-Order Logic logo

History Unification by Logic Applications Summary Outline History 1 Enumerability in Recursion and Automata Theory Known Weak Cardinality Theorem Why Do Cardinality Theorems Hold Only for Certain Models? Unification by First-Order Logic 2 Elementary Definitions Enumerability for First-Order Logic Weak Cardinality Theorems for First-Order Logic Applications 3 A Separability Result for First-Order Logic logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Motivation of Enumerability Problem Many functions are not computable or not efficiently computable. Example #SAT : How many satisfying assignments does a formula have? logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Motivation of Enumerability Problem Many functions are not computable or not efficiently computable. Example in A For difficult languages A : Cardinality function # n A : ( w 1 , w 2 , w 3 , w 4 , w 5 ) How many input words are in A ? Characteristic function χ n A : # 5 χ 5 Which input words are in A ? A A 2 01001 logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Motivation of Enumerability Problem Many functions are not computable or not efficiently computable. Solutions Difficult functions can be computed using probabilistic algorithms, computed efficiently on average, approximated, or enumerated. logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Enumerators Output Sets of Possible Function Values input tapes w 1 . . . Definition (1987, 1989, 1994, 2001) w n An m -enumerator for a function f reads n input words w 1 , . . . , w n , 1 does a computation, 2 computerimage outputs at most m values, 3 one of which is f ( w 1 , . . . , w n ) . 4 output tape logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Enumerators Output Sets of Possible Function Values input tapes w 1 w 1 . . . . . . Definition (1987, 1989, 1994, 2001) w n w n An m -enumerator for a function f reads n input words w 1 , . . . , w n , 1 does a computation, 2 computerimage outputs at most m values, 3 one of which is f ( w 1 , . . . , w n ) . 4 output tape logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Enumerators Output Sets of Possible Function Values input tapes w 1 . . . Definition (1987, 1989, 1994, 2001) w n An m -enumerator for a function f reads n input words w 1 , . . . , w n , 1 does a computation, 2 computerimage outputs at most m values, 3 one of which is f ( w 1 , . . . , w n ) . 4 output tape logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Enumerators Output Sets of Possible Function Values input tapes w 1 . . . Definition (1987, 1989, 1994, 2001) w n An m -enumerator for a function f reads n input words w 1 , . . . , w n , 1 does a computation, 2 computerimage outputs at most m values, 3 one of which is f ( w 1 , . . . , w n ) . 4 output tape logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Enumerators Output Sets of Possible Function Values input tapes w 1 . . . Definition (1987, 1989, 1994, 2001) w n An m -enumerator for a function f reads n input words w 1 , . . . , w n , 1 does a computation, 2 computerworkingimage outputs at most m values, 3 one of which is f ( w 1 , . . . , w n ) . 4 output tape logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Enumerators Output Sets of Possible Function Values input tapes w 1 . . . Definition (1987, 1989, 1994, 2001) w n An m -enumerator for a function f reads n input words w 1 , . . . , w n , 1 does a computation, 2 computerimage outputs at most m values, 3 one of which is f ( w 1 , . . . , w n ) . 4 u 1 output tape logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Enumerators Output Sets of Possible Function Values input tapes w 1 . . . Definition (1987, 1989, 1994, 2001) w n An m -enumerator for a function f reads n input words w 1 , . . . , w n , 1 does a computation, 2 computerimage outputs at most m values, 3 one of which is f ( w 1 , . . . , w n ) . 4 u 1 u 2 output tape logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Enumerators Output Sets of Possible Function Values input tapes w 1 . . . Definition (1987, 1989, 1994, 2001) w n An m -enumerator for a function f reads n input words w 1 , . . . , w n , 1 does a computation, 2 computerimage outputs at most m values, 3 one of which is f ( w 1 , . . . , w n ) . 4 u 3 u 1 u 2 output tape logo

History Unification by Logic Applications Summary Enumerability in Recursion and Automata Theory Enumerators Output Sets of Possible Function Values input tapes w 1 . . . Definition (1987, 1989, 1994, 2001) w n An m -enumerator for a function f reads n input words w 1 , . . . , w n , 1 does a computation, 2 computerimage outputs at most m values, 3 one of which is f ( w 1 , . . . , w n ) . 4 u 3 u 1 u 2 output tape logo

History Unification by Logic Applications Summary Known Weak Cardinality Theorem How Well Can the Cardinality Function Be Enumerated? Observation For fixed n , the cardinality function # n A can be 1-enumerated by Turing machines only for recursive A , but can be ( n + 1 ) -enumerated for every language A . Question What about 2-, 3-, 4-, . . . , n -enumerability? logo

History Unification by Logic Applications Summary Known Weak Cardinality Theorem How Well Can the Cardinality Function Be Enumerated? Observation For fixed n , the cardinality function # n A can be 1-enumerated by Turing machines only for recursive A , but can be ( n + 1 ) -enumerated for every language A . Question What about 2-, 3-, 4-, . . . , n -enumerability? logo

History Unification by Logic Applications Summary Known Weak Cardinality Theorem How Well Can the Cardinality Function Be Enumerated by Turing Machines? Cardinality Theorem (Kummer, 1992) If # n A is n-enumerable by a Turing machine, then A is recursive. Weak Cardinality Theorems (1987, 1989, 1992) If χ n A is n-enumerable by a Turing machine, then A is 1 recursive. If # 2 A is 2 -enumerable by a Turing machine, then A is 2 recursive. If # n A is n-enumerable by a Turing machine that never 3 enumerates both 0 and n, then A is recursive. logo

History Unification by Logic Applications Summary Known Weak Cardinality Theorem How Well Can the Cardinality Function Be Enumerated by Turing Machines? Cardinality Theorem (Kummer, 1992) If # n A is n-enumerable by a Turing machine, then A is recursive. Weak Cardinality Theorems (1987, 1989, 1992) If χ n A is n-enumerable by a Turing machine, then A is 1 recursive. If # 2 A is 2 -enumerable by a Turing machine, then A is 2 recursive. If # n A is n-enumerable by a Turing machine that never 3 enumerates both 0 and n, then A is recursive. logo

History Unification by Logic Applications Summary Known Weak Cardinality Theorem How Well Can the Cardinality Function Be Enumerated by Turing Machines? Cardinality Theorem (Kummer, 1992) If # n A is n-enumerable by a Turing machine, then A is recursive. Weak Cardinality Theorems (1987, 1989, 1992) If χ n A is n-enumerable by a Turing machine, then A is 1 recursive. If # 2 A is 2 -enumerable by a Turing machine, then A is 2 recursive. If # n A is n-enumerable by a Turing machine that never 3 enumerates both 0 and n, then A is recursive. logo

History Unification by Logic Applications Summary Known Weak Cardinality Theorem How Well Can the Cardinality Function Be Enumerated by Turing Machines? Cardinality Theorem (Kummer, 1992) If # n A is n-enumerable by a Turing machine, then A is recursive. Weak Cardinality Theorems (1987, 1989, 1992) If χ n A is n-enumerable by a Turing machine, then A is 1 recursive. If # 2 A is 2 -enumerable by a Turing machine, then A is 2 recursive. If # n A is n-enumerable by a Turing machine that never 3 enumerates both 0 and n, then A is recursive. logo

History Unification by Logic Applications Summary Known Weak Cardinality Theorem How Well Can the Cardinality Function Be Enumerated by Finite Automata? Conjecture If # n A is n -enumerable by a finite automaton, then A is regular. Weak Cardinality Theorems (2001, 2002) If χ n A is n-enumerable by a finite automaton, then A is 1 regular. If # 2 A is 2 -enumerable by a finite automaton, then A is 2 regular. If # n A is n-enumerable by a finite automaton that never 3 enumerates both 0 and n, then A is regular. logo