(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Reminder (cont.) Similarly a language L = ( L n ) n ❃ 0 ✷ Π ✦ if there exists a language L ✵ = ( L ✵ n ) n ❃ 0 ✷ P such that x ✷ L n ♠ ( Q 1 y 1 ✷ k m 1 )( Q 2 y 2 ✷ k m 2 ) ✁ ✁ ✁ ( Q ✦ y ✦ ✷ k m ✦ ) ( y 1 ❀ ✿ ✿ ✿ ❀ y ✦ ❀ x ) ✷ L ✵ m + n where m ( n ) = m 1 ( n ) + ✁ ✁ ✁ + m ✦ ( n ) = n O ( 1 ) and for 1 ✔ i ✔ ✦ , Q i ✷ ❢✾ ❀ ✽❣ , and Q j ✻ = Q j + 1 ❀ 1 ✔ j ❁ ✦ , Q 1 = ✽ . Notice that P = Σ 0 = Π 0 ❀ NP = Σ 1 ❀ coNP = Π 1 ✿ Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Reminder (cont.) Similarly a language L = ( L n ) n ❃ 0 ✷ Π ✦ if there exists a language L ✵ = ( L ✵ n ) n ❃ 0 ✷ P such that x ✷ L n ♠ ( Q 1 y 1 ✷ k m 1 )( Q 2 y 2 ✷ k m 2 ) ✁ ✁ ✁ ( Q ✦ y ✦ ✷ k m ✦ ) ( y 1 ❀ ✿ ✿ ✿ ❀ y ✦ ❀ x ) ✷ L ✵ m + n where m ( n ) = m 1 ( n ) + ✁ ✁ ✁ + m ✦ ( n ) = n O ( 1 ) and for 1 ✔ i ✔ ✦ , Q i ✷ ❢✾ ❀ ✽❣ , and Q j ✻ = Q j + 1 ❀ 1 ✔ j ❁ ✦ , Q 1 = ✽ . Notice that P = Σ 0 = Π 0 ❀ NP = Σ 1 ❀ coNP = Π 1 ✿ Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The polynomial time hierarchy Also, notice the inclusions Σ i ✚ Π i + 1 ❀ Σ i ✚ Σ i + 1 Π i ✚ Σ i + 1 ❀ Π i ✚ Π i + 1 The polynomial time hierarchy is defined to be ❬ ❬ ❬ def = (Σ ✦ ❬ Π ✦ ) = Σ ✦ = Π ✦ ✿ PH ✦ ✕ 0 ✦ ✕ 0 ✦ ✕ 0 Central problem of CS is to prove that PH is a proper hierarchy (as is widely believed), and in particular to prove P ✻ = NP . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The polynomial time hierarchy Also, notice the inclusions Σ i ✚ Π i + 1 ❀ Σ i ✚ Σ i + 1 Π i ✚ Σ i + 1 ❀ Π i ✚ Π i + 1 The polynomial time hierarchy is defined to be ❬ ❬ ❬ def = (Σ ✦ ❬ Π ✦ ) = Σ ✦ = Π ✦ ✿ PH ✦ ✕ 0 ✦ ✕ 0 ✦ ✕ 0 Central problem of CS is to prove that PH is a proper hierarchy (as is widely believed), and in particular to prove P ✻ = NP . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The polynomial time hierarchy Also, notice the inclusions Σ i ✚ Π i + 1 ❀ Σ i ✚ Σ i + 1 Π i ✚ Σ i + 1 ❀ Π i ✚ Π i + 1 The polynomial time hierarchy is defined to be ❬ ❬ ❬ def = (Σ ✦ ❬ Π ✦ ) = Σ ✦ = Π ✦ ✿ PH ✦ ✕ 0 ✦ ✕ 0 ✦ ✕ 0 Central problem of CS is to prove that PH is a proper hierarchy (as is widely believed), and in particular to prove P ✻ = NP . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The Class # P In order to develop an “algebraic” version of complexity theory Valiant introduced certain complexity classes of functions ; A sequence of functions ( f n : k n ✦ N ) n ❃ 0 is said to be in the class # P if there exists L = ( L n ) n ❃ 0 ✷ P such that for x ✷ k n f n ( x ) = card ( L m + n ❀ x ) ❀ m = n O ( 1 ) ❀ where L m + n ❀ x is the fibre ✙ � 1 ( x ) ❭ L m + n ❀ and ✙ : k m + n ✦ k n the projection map on the last n co-ordinates. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The Class # P In order to develop an “algebraic” version of complexity theory Valiant introduced certain complexity classes of functions ; A sequence of functions ( f n : k n ✦ N ) n ❃ 0 is said to be in the class # P if there exists L = ( L n ) n ❃ 0 ✷ P such that for x ✷ k n f n ( x ) = card ( L m + n ❀ x ) ❀ m = n O ( 1 ) ❀ where L m + n ❀ x is the fibre ✙ � 1 ( x ) ❭ L m + n ❀ and ✙ : k m + n ✦ k n the projection map on the last n co-ordinates. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The Class # P In order to develop an “algebraic” version of complexity theory Valiant introduced certain complexity classes of functions ; A sequence of functions ( f n : k n ✦ N ) n ❃ 0 is said to be in the class # P if there exists L = ( L n ) n ❃ 0 ✷ P such that for x ✷ k n f n ( x ) = card ( L m + n ❀ x ) ❀ m = n O ( 1 ) ❀ where L m + n ❀ x is the fibre ✙ � 1 ( x ) ❭ L m + n ❀ and ✙ : k m + n ✦ k n the projection map on the last n co-ordinates. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The Class # P In order to develop an “algebraic” version of complexity theory Valiant introduced certain complexity classes of functions ; A sequence of functions ( f n : k n ✦ N ) n ❃ 0 is said to be in the class # P if there exists L = ( L n ) n ❃ 0 ✷ P such that for x ✷ k n f n ( x ) = card ( L m + n ❀ x ) ❀ m = n O ( 1 ) ❀ where L m + n ❀ x is the fibre ✙ � 1 ( x ) ❭ L m + n ❀ and ✙ : k m + n ✦ k n the projection map on the last n co-ordinates. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The Class # P In order to develop an “algebraic” version of complexity theory Valiant introduced certain complexity classes of functions ; A sequence of functions ( f n : k n ✦ N ) n ❃ 0 is said to be in the class # P if there exists L = ( L n ) n ❃ 0 ✷ P such that for x ✷ k n f n ( x ) = card ( L m + n ❀ x ) ❀ m = n O ( 1 ) ❀ where L m + n ❀ x is the fibre ✙ � 1 ( x ) ❭ L m + n ❀ and ✙ : k m + n ✦ k n the projection map on the last n co-ordinates. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Toda’s Theorem Toda’s theorem is a seminal result in discrete complexity theory and gives the following inclusion. Theorem (Toda (1989)) PH ✚ P # P “illustrates the power of counting” Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Toda’s Theorem Toda’s theorem is a seminal result in discrete complexity theory and gives the following inclusion. Theorem (Toda (1989)) PH ✚ P # P “illustrates the power of counting” Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Toda’s Theorem Toda’s theorem is a seminal result in discrete complexity theory and gives the following inclusion. Theorem (Toda (1989)) PH ✚ P # P “illustrates the power of counting” Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Toda’s Theorem Toda’s theorem is a seminal result in discrete complexity theory and gives the following inclusion. Theorem (Toda (1989)) PH ✚ P # P “illustrates the power of counting” Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Blum-Shub-Smale model Generalized TM where k is allowed to be any ring (we restrict ourselves to the cases k = C or R ). Setting k = Z ❂ 2 Z (or any finite field) recovers the classical complexity classes. A B-S-S machine accepts for every n a subset S n ✚ k n . In case k = C , each S n is a constructible subset of C n , 1 in case k = R , each S n is a semi-algebraic subset of R n . 2 Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Blum-Shub-Smale model Generalized TM where k is allowed to be any ring (we restrict ourselves to the cases k = C or R ). Setting k = Z ❂ 2 Z (or any finite field) recovers the classical complexity classes. A B-S-S machine accepts for every n a subset S n ✚ k n . In case k = C , each S n is a constructible subset of C n , 1 in case k = R , each S n is a semi-algebraic subset of R n . 2 Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Blum-Shub-Smale model Generalized TM where k is allowed to be any ring (we restrict ourselves to the cases k = C or R ). Setting k = Z ❂ 2 Z (or any finite field) recovers the classical complexity classes. A B-S-S machine accepts for every n a subset S n ✚ k n . In case k = C , each S n is a constructible subset of C n , 1 in case k = R , each S n is a semi-algebraic subset of R n . 2 Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Blum-Shub-Smale model Generalized TM where k is allowed to be any ring (we restrict ourselves to the cases k = C or R ). Setting k = Z ❂ 2 Z (or any finite field) recovers the classical complexity classes. A B-S-S machine accepts for every n a subset S n ✚ k n . In case k = C , each S n is a constructible subset of C n , 1 in case k = R , each S n is a semi-algebraic subset of R n . 2 Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Blum-Shub-Smale model Generalized TM where k is allowed to be any ring (we restrict ourselves to the cases k = C or R ). Setting k = Z ❂ 2 Z (or any finite field) recovers the classical complexity classes. A B-S-S machine accepts for every n a subset S n ✚ k n . In case k = C , each S n is a constructible subset of C n , 1 in case k = R , each S n is a semi-algebraic subset of R n . 2 Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Blum-Shub-Smale model Generalized TM where k is allowed to be any ring (we restrict ourselves to the cases k = C or R ). Setting k = Z ❂ 2 Z (or any finite field) recovers the classical complexity classes. A B-S-S machine accepts for every n a subset S n ✚ k n . In case k = C , each S n is a constructible subset of C n , 1 in case k = R , each S n is a semi-algebraic subset of R n . 2 Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Complexity Classes Complexity classes P k ❀ NP k ❀ coNP k and more generally PH k are defined as before (for k = C ❀ R ). B-S-S developed a theory of NP -completeness. In case, k = C the problem of determining if a system of n + 1 polynomial equations in n variables has a common zero in C n is NP C -complete. In case, k = R the problem of determining if a quartic polynomial in n variables has a common zero in R n is NP R -complete. It is unknown if P C = NP C (respectively, P R = NP R ) just as in the discrete case. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Complexity Classes Complexity classes P k ❀ NP k ❀ coNP k and more generally PH k are defined as before (for k = C ❀ R ). B-S-S developed a theory of NP -completeness. In case, k = C the problem of determining if a system of n + 1 polynomial equations in n variables has a common zero in C n is NP C -complete. In case, k = R the problem of determining if a quartic polynomial in n variables has a common zero in R n is NP R -complete. It is unknown if P C = NP C (respectively, P R = NP R ) just as in the discrete case. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Complexity Classes Complexity classes P k ❀ NP k ❀ coNP k and more generally PH k are defined as before (for k = C ❀ R ). B-S-S developed a theory of NP -completeness. In case, k = C the problem of determining if a system of n + 1 polynomial equations in n variables has a common zero in C n is NP C -complete. In case, k = R the problem of determining if a quartic polynomial in n variables has a common zero in R n is NP R -complete. It is unknown if P C = NP C (respectively, P R = NP R ) just as in the discrete case. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Complexity Classes Complexity classes P k ❀ NP k ❀ coNP k and more generally PH k are defined as before (for k = C ❀ R ). B-S-S developed a theory of NP -completeness. In case, k = C the problem of determining if a system of n + 1 polynomial equations in n variables has a common zero in C n is NP C -complete. In case, k = R the problem of determining if a quartic polynomial in n variables has a common zero in R n is NP R -complete. It is unknown if P C = NP C (respectively, P R = NP R ) just as in the discrete case. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Complexity Classes Complexity classes P k ❀ NP k ❀ coNP k and more generally PH k are defined as before (for k = C ❀ R ). B-S-S developed a theory of NP -completeness. In case, k = C the problem of determining if a system of n + 1 polynomial equations in n variables has a common zero in C n is NP C -complete. In case, k = R the problem of determining if a quartic polynomial in n variables has a common zero in R n is NP R -complete. It is unknown if P C = NP C (respectively, P R = NP R ) just as in the discrete case. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Semi-algebraic sets From now we assume k = R , and restrict ourselves to real machines in the sense of B-S-S. Such a machine accepts a sequence ( S n ✚ R n ) n ❃ 0 where each S n is a semi-algebraic subset of R n . A semi-algebraic set, S ✚ R n , is a subset of R n defined by a Boolean formula whose atoms are polynomial equalities and inequalities. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Semi-algebraic sets From now we assume k = R , and restrict ourselves to real machines in the sense of B-S-S. Such a machine accepts a sequence ( S n ✚ R n ) n ❃ 0 where each S n is a semi-algebraic subset of R n . A semi-algebraic set, S ✚ R n , is a subset of R n defined by a Boolean formula whose atoms are polynomial equalities and inequalities. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Semi-algebraic sets From now we assume k = R , and restrict ourselves to real machines in the sense of B-S-S. Such a machine accepts a sequence ( S n ✚ R n ) n ❃ 0 where each S n is a semi-algebraic subset of R n . A semi-algebraic set, S ✚ R n , is a subset of R n defined by a Boolean formula whose atoms are polynomial equalities and inequalities. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Two classes of problems The most important algorithmic problems studied in this area fall into two broad sub-classes: the problem of quantifier elimination, and its special cases 1 such as deciding a sentence in the first order theory of reals, or deciding emptiness of semi-algebraic sets. the problem of computing topological invariants of 2 semi-algebraic sets, such as the number of connected components, Euler-Poincaré characteristic, and more generally all the Betti numbers of semi-algebraic sets. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Two classes of problems The most important algorithmic problems studied in this area fall into two broad sub-classes: the problem of quantifier elimination, and its special cases 1 such as deciding a sentence in the first order theory of reals, or deciding emptiness of semi-algebraic sets. the problem of computing topological invariants of 2 semi-algebraic sets, such as the number of connected components, Euler-Poincaré characteristic, and more generally all the Betti numbers of semi-algebraic sets. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Two classes of problems The most important algorithmic problems studied in this area fall into two broad sub-classes: the problem of quantifier elimination, and its special cases 1 such as deciding a sentence in the first order theory of reals, or deciding emptiness of semi-algebraic sets. the problem of computing topological invariants of 2 semi-algebraic sets, such as the number of connected components, Euler-Poincaré characteristic, and more generally all the Betti numbers of semi-algebraic sets. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Analogy with Toda’s Theorem The classes PH and # P appearing in the two sides of the inclusion in Toda’s Theorem can be identified with the two broad classes of problems in algorithmic semi-algebraic geometry; the class PH with the problem of deciding sentences with a fixed number of quantifier alternations; the class # P with the problem of computing topological invariants of semi-algebraic sets, namely their Betti numbers, which generalizes the notion of cardinality for finite sets; it is thus quite natural to seek a real analogue of Toda’s theorem. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Analogy with Toda’s Theorem The classes PH and # P appearing in the two sides of the inclusion in Toda’s Theorem can be identified with the two broad classes of problems in algorithmic semi-algebraic geometry; the class PH with the problem of deciding sentences with a fixed number of quantifier alternations; the class # P with the problem of computing topological invariants of semi-algebraic sets, namely their Betti numbers, which generalizes the notion of cardinality for finite sets; it is thus quite natural to seek a real analogue of Toda’s theorem. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Analogy with Toda’s Theorem The classes PH and # P appearing in the two sides of the inclusion in Toda’s Theorem can be identified with the two broad classes of problems in algorithmic semi-algebraic geometry; the class PH with the problem of deciding sentences with a fixed number of quantifier alternations; the class # P with the problem of computing topological invariants of semi-algebraic sets, namely their Betti numbers, which generalizes the notion of cardinality for finite sets; it is thus quite natural to seek a real analogue of Toda’s theorem. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Analogy with Toda’s Theorem The classes PH and # P appearing in the two sides of the inclusion in Toda’s Theorem can be identified with the two broad classes of problems in algorithmic semi-algebraic geometry; the class PH with the problem of deciding sentences with a fixed number of quantifier alternations; the class # P with the problem of computing topological invariants of semi-algebraic sets, namely their Betti numbers, which generalizes the notion of cardinality for finite sets; it is thus quite natural to seek a real analogue of Toda’s theorem. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Real Analogue of # P In order to define real analogues of counting complexity classes of discrete complexity theory, it is necessary to identify the proper notion of “counting” in the context of semi-algebraic geometry. Counting complexity classes over the reals have been defined previously by Meer (2000) and studied extensively by other authors Burgisser, Cucker et al (2006). These authors used a straightforward generalization to semi-algebraic sets of counting in the case of finite sets; namely f ( S ) = card ( S ) ❀ if card ( S ) ❁ ✶ ; = ✶ otherwise. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Real Analogue of # P In order to define real analogues of counting complexity classes of discrete complexity theory, it is necessary to identify the proper notion of “counting” in the context of semi-algebraic geometry. Counting complexity classes over the reals have been defined previously by Meer (2000) and studied extensively by other authors Burgisser, Cucker et al (2006). These authors used a straightforward generalization to semi-algebraic sets of counting in the case of finite sets; namely f ( S ) = card ( S ) ❀ if card ( S ) ❁ ✶ ; = ✶ otherwise. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof An alternative definition In our view this is not fully satisfactory, since the count gives no information when the semi-algebraic set is infinite, and most interesting semi-algebraic sets are infinite . If one thinks of “counting” a semi-algebraic set S ✚ R k as computing certain discrete invariants, then a natural mathematical candidate is its sequence of Betti numbers, b 0 ( S ) ❀ ✿ ✿ ✿ ❀ b k � 1 ( S ) , or more succinctly the Poincaré polynomial of S , namely ❳ def b i ( S ) T i ✿ P S ( T ) = i ✕ 0 In case card ( S ) ❁ ✶ , we have that b 0 ( S ) = P S ( 0 ) = card ( S ) . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof An alternative definition In our view this is not fully satisfactory, since the count gives no information when the semi-algebraic set is infinite, and most interesting semi-algebraic sets are infinite . If one thinks of “counting” a semi-algebraic set S ✚ R k as computing certain discrete invariants, then a natural mathematical candidate is its sequence of Betti numbers, b 0 ( S ) ❀ ✿ ✿ ✿ ❀ b k � 1 ( S ) , or more succinctly the Poincaré polynomial of S , namely ❳ def b i ( S ) T i ✿ P S ( T ) = i ✕ 0 In case card ( S ) ❁ ✶ , we have that b 0 ( S ) = P S ( 0 ) = card ( S ) . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof An alternative definition In our view this is not fully satisfactory, since the count gives no information when the semi-algebraic set is infinite, and most interesting semi-algebraic sets are infinite . If one thinks of “counting” a semi-algebraic set S ✚ R k as computing certain discrete invariants, then a natural mathematical candidate is its sequence of Betti numbers, b 0 ( S ) ❀ ✿ ✿ ✿ ❀ b k � 1 ( S ) , or more succinctly the Poincaré polynomial of S , namely ❳ def b i ( S ) T i ✿ P S ( T ) = i ✕ 0 In case card ( S ) ❁ ✶ , we have that b 0 ( S ) = P S ( 0 ) = card ( S ) . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof An alternative definition In our view this is not fully satisfactory, since the count gives no information when the semi-algebraic set is infinite, and most interesting semi-algebraic sets are infinite . If one thinks of “counting” a semi-algebraic set S ✚ R k as computing certain discrete invariants, then a natural mathematical candidate is its sequence of Betti numbers, b 0 ( S ) ❀ ✿ ✿ ✿ ❀ b k � 1 ( S ) , or more succinctly the Poincaré polynomial of S , namely ❳ def b i ( S ) T i ✿ P S ( T ) = i ✕ 0 In case card ( S ) ❁ ✶ , we have that b 0 ( S ) = P S ( 0 ) = card ( S ) . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Definition of # P ② R We call a sequence of functions ( f n : R n ✦ Z [ T ]) n ❃ 0 to be in class # P ② R if there exists ( S n ✚ R n ) n ❃ 0 ✷ P R such that for x ✷ R n f n ( x ) = P S m + n ❀ x ❀ m = n O ( 1 ) ❀ where S m + n ❀ x = S m + n ❭ ✙ � 1 ( x ) and ✙ : R m + n ✦ R n is the projection on the last n coordinates. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Definition of # P ② R We call a sequence of functions ( f n : R n ✦ Z [ T ]) n ❃ 0 to be in class # P ② R if there exists ( S n ✚ R n ) n ❃ 0 ✷ P R such that for x ✷ R n f n ( x ) = P S m + n ❀ x ❀ m = n O ( 1 ) ❀ where S m + n ❀ x = S m + n ❭ ✙ � 1 ( x ) and ✙ : R m + n ✦ R n is the projection on the last n coordinates. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Counting and Betti numbers The connection between counting points of varieties and their Betti numbers is more direct over fields of positive characteristic via the zeta function. The zeta function of a variety defined over F p is the exponential generating function of the sequence whose n -th term is the number of points in the variety over F p n . The zeta function depends on the Betti numbers of the variety with respect to a certain ( ❵ -adic) co-homology theory. Thus, the problems of “counting” varieties and computing their Betti numbers, are connected at a deeper level, and thus our definition of # P ② R is not entirely ad hoc. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Counting and Betti numbers The connection between counting points of varieties and their Betti numbers is more direct over fields of positive characteristic via the zeta function. The zeta function of a variety defined over F p is the exponential generating function of the sequence whose n -th term is the number of points in the variety over F p n . The zeta function depends on the Betti numbers of the variety with respect to a certain ( ❵ -adic) co-homology theory. Thus, the problems of “counting” varieties and computing their Betti numbers, are connected at a deeper level, and thus our definition of # P ② R is not entirely ad hoc. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Counting and Betti numbers The connection between counting points of varieties and their Betti numbers is more direct over fields of positive characteristic via the zeta function. The zeta function of a variety defined over F p is the exponential generating function of the sequence whose n -th term is the number of points in the variety over F p n . The zeta function depends on the Betti numbers of the variety with respect to a certain ( ❵ -adic) co-homology theory. Thus, the problems of “counting” varieties and computing their Betti numbers, are connected at a deeper level, and thus our definition of # P ② R is not entirely ad hoc. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Counting and Betti numbers The connection between counting points of varieties and their Betti numbers is more direct over fields of positive characteristic via the zeta function. The zeta function of a variety defined over F p is the exponential generating function of the sequence whose n -th term is the number of points in the variety over F p n . The zeta function depends on the Betti numbers of the variety with respect to a certain ( ❵ -adic) co-homology theory. Thus, the problems of “counting” varieties and computing their Betti numbers, are connected at a deeper level, and thus our definition of # P ② R is not entirely ad hoc. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Real analogue of Toda’s theorem It is now natural to formulate the following conjecture. Conjecture PH R ✚ P # P ② R For technical reasons we are unable to prove this without a further compactness hypothesis on the left hand-side. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Real analogue of Toda’s theorem It is now natural to formulate the following conjecture. Conjecture PH R ✚ P # P ② R For technical reasons we are unable to prove this without a further compactness hypothesis on the left hand-side. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Real analogue of Toda’s theorem It is now natural to formulate the following conjecture. Conjecture PH R ✚ P # P ② R For technical reasons we are unable to prove this without a further compactness hypothesis on the left hand-side. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The compact fragment of real polynomial hierarchy We say that a sequence of semi-algebraic sets ( S n ✚ S n ) n ❃ 0 ✷ Σ c R ❀✦ if there exists another sequence ( S ✵ n ) n ❃ 0 ✷ P R such that each S ✵ n is compact and x ✷ S n if and only if ( Q 1 y 1 ✷ S m 1 )( Q 2 y 2 ✷ S m 2 ) ✿ ✿ ✿ ( Q ✦ y ✦ ✷ S m ✦ ) ( y 1 ❀ ✿ ✿ ✿ ❀ y ✦ ❀ x ) ✷ S ✵ m + n where m ( n ) = m 1 ( n ) + ✁ ✁ ✁ + m ✦ ( n ) = n O ( 1 ) and for 1 ✔ i ✔ ✦ , Q i ✷ ❢✾ ❀ ✽❣ , and Q j ✻ = Q j + 1 ❀ 1 ✔ j ❁ ✦ , Q 1 = ✾ . The compact class Π c R ❀✦ is defined analogously. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The compact fragment of real polynomial hierarchy We say that a sequence of semi-algebraic sets ( S n ✚ S n ) n ❃ 0 ✷ Σ c R ❀✦ if there exists another sequence ( S ✵ n ) n ❃ 0 ✷ P R such that each S ✵ n is compact and x ✷ S n if and only if ( Q 1 y 1 ✷ S m 1 )( Q 2 y 2 ✷ S m 2 ) ✿ ✿ ✿ ( Q ✦ y ✦ ✷ S m ✦ ) ( y 1 ❀ ✿ ✿ ✿ ❀ y ✦ ❀ x ) ✷ S ✵ m + n where m ( n ) = m 1 ( n ) + ✁ ✁ ✁ + m ✦ ( n ) = n O ( 1 ) and for 1 ✔ i ✔ ✦ , Q i ✷ ❢✾ ❀ ✽❣ , and Q j ✻ = Q j + 1 ❀ 1 ✔ j ❁ ✦ , Q 1 = ✾ . The compact class Π c R ❀✦ is defined analogously. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The compact fragment of real polynomial hierarchy We say that a sequence of semi-algebraic sets ( S n ✚ S n ) n ❃ 0 ✷ Σ c R ❀✦ if there exists another sequence ( S ✵ n ) n ❃ 0 ✷ P R such that each S ✵ n is compact and x ✷ S n if and only if ( Q 1 y 1 ✷ S m 1 )( Q 2 y 2 ✷ S m 2 ) ✿ ✿ ✿ ( Q ✦ y ✦ ✷ S m ✦ ) ( y 1 ❀ ✿ ✿ ✿ ❀ y ✦ ❀ x ) ✷ S ✵ m + n where m ( n ) = m 1 ( n ) + ✁ ✁ ✁ + m ✦ ( n ) = n O ( 1 ) and for 1 ✔ i ✔ ✦ , Q i ✷ ❢✾ ❀ ✽❣ , and Q j ✻ = Q j + 1 ❀ 1 ✔ j ❁ ✦ , Q 1 = ✾ . The compact class Π c R ❀✦ is defined analogously. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The compact fragment of real polynomial hierarchy We say that a sequence of semi-algebraic sets ( S n ✚ S n ) n ❃ 0 ✷ Σ c R ❀✦ if there exists another sequence ( S ✵ n ) n ❃ 0 ✷ P R such that each S ✵ n is compact and x ✷ S n if and only if ( Q 1 y 1 ✷ S m 1 )( Q 2 y 2 ✷ S m 2 ) ✿ ✿ ✿ ( Q ✦ y ✦ ✷ S m ✦ ) ( y 1 ❀ ✿ ✿ ✿ ❀ y ✦ ❀ x ) ✷ S ✵ m + n where m ( n ) = m 1 ( n ) + ✁ ✁ ✁ + m ✦ ( n ) = n O ( 1 ) and for 1 ✔ i ✔ ✦ , Q i ✷ ❢✾ ❀ ✽❣ , and Q j ✻ = Q j + 1 ❀ 1 ✔ j ❁ ✦ , Q 1 = ✾ . The compact class Π c R ❀✦ is defined analogously. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The compact real polynomial hierarchy (cont.) We define ❬ ❬ ❬ PH c def (Σ c R ❀✦ ❬ Π c Σ c c = R ❀✦ ) = R ❀✦ = R ❀✦ ✿ R ✦ ✕ 0 ✦ ✕ 0 ✦ ✕ 0 Notice that the semi-algebraic sets belonging to any language in PH c R are all semi-algebraic compact (in fact closed semi-algebraic subsets of spheres). Also, notice the inclusion PH c R ✚ PH R ✿ Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof The compact real polynomial hierarchy (cont.) We define ❬ ❬ ❬ PH c def (Σ c R ❀✦ ❬ Π c Σ c c = R ❀✦ ) = R ❀✦ = R ❀✦ ✿ R ✦ ✕ 0 ✦ ✕ 0 ✦ ✕ 0 Notice that the semi-algebraic sets belonging to any language in PH c R are all semi-algebraic compact (in fact closed semi-algebraic subsets of spheres). Also, notice the inclusion PH c R ✚ PH R ✿ Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Main theorem Theorem (B-Zell,2008) # P ② PH c R ✚ P R ✿ R Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Remark about the compactness assumption Even though the restriction to compact semi-algebraic sets might appear to be only a technicality at first glance, this is actually an important restriction. For instance, it is a long-standing open question in real complexity theory whether there exists an NP R -complete problem which belongs to the class Σ c 1 (the compact version of the class NP R i.e. where the certificates are constrained to come from a compact set). Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Algorithmic Semi-algebraic Geometry Real Analogue of Toda’s Theorem Proof Remark about the compactness assumption Even though the restriction to compact semi-algebraic sets might appear to be only a technicality at first glance, this is actually an important restriction. For instance, it is a long-standing open question in real complexity theory whether there exists an NP R -complete problem which belongs to the class Σ c 1 (the compact version of the class NP R i.e. where the certificates are constrained to come from a compact set). Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Outline (Discrete) Polynomial Hierarchy 1 Blum-Shub-Smale Models of Computation 2 Algorithmic Semi-algebraic Geometry 3 Real Analogue of Toda’s Theorem 4 Proof 5 Outline Details Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Summary of the Main Idea Our main tool is a topological construction which given a semi-algebraic set S ✚ R m + n , p ✕ 0, and ✙ Y : R m + n ✦ R n denoting the projection along (say) the Y -co-ordinates, constructs efficiently a semi-algebraic set, D p Y ( S ) , such that b i ( ✙ Y ( S )) = b i ( D p Y ( S )) ❀ 0 ✔ i ❁ p ✿ Notice that even if there exists an efficient (i.e. polynomial time) algorithm for checking membership in S , the same need not be true for the image ✙ Y ( S ) . A second topological ingredient is Alexander-Lefshetz duality which relates the Betti numbers of a compact subset K of the sphere S n with those of S n � K . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Summary of the Main Idea Our main tool is a topological construction which given a semi-algebraic set S ✚ R m + n , p ✕ 0, and ✙ Y : R m + n ✦ R n denoting the projection along (say) the Y -co-ordinates, constructs efficiently a semi-algebraic set, D p Y ( S ) , such that b i ( ✙ Y ( S )) = b i ( D p Y ( S )) ❀ 0 ✔ i ❁ p ✿ Notice that even if there exists an efficient (i.e. polynomial time) algorithm for checking membership in S , the same need not be true for the image ✙ Y ( S ) . A second topological ingredient is Alexander-Lefshetz duality which relates the Betti numbers of a compact subset K of the sphere S n with those of S n � K . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Summary of the Main Idea Our main tool is a topological construction which given a semi-algebraic set S ✚ R m + n , p ✕ 0, and ✙ Y : R m + n ✦ R n denoting the projection along (say) the Y -co-ordinates, constructs efficiently a semi-algebraic set, D p Y ( S ) , such that b i ( ✙ Y ( S )) = b i ( D p Y ( S )) ❀ 0 ✔ i ❁ p ✿ Notice that even if there exists an efficient (i.e. polynomial time) algorithm for checking membership in S , the same need not be true for the image ✙ Y ( S ) . A second topological ingredient is Alexander-Lefshetz duality which relates the Betti numbers of a compact subset K of the sphere S n with those of S n � K . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof The case Σ c R ❀ 1 Consider a closed semi-algebraic set S ✚ S k ✂ S ❵ be defined by a quantifier free formula ✣ ( Y ❀ X ) and let ✙ Y : S k ✂ S ❵ ✦ S k be the projection map along the Y coordinates. Then the formula Φ( X ) = ✾ Y ✣ ( X ❀ Y ) is satisfied by x ✷ S k if and only if b 0 ( S x ) ✻ = 0, where S x = S ❭ ✙ � 1 Y ( x ) . Thus, the problem of deciding the truth of Φ( x ) is reduced to computing a Betti number (the 0-th) of the fiber of S over x . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof The case Σ c R ❀ 1 Consider a closed semi-algebraic set S ✚ S k ✂ S ❵ be defined by a quantifier free formula ✣ ( Y ❀ X ) and let ✙ Y : S k ✂ S ❵ ✦ S k be the projection map along the Y coordinates. Then the formula Φ( X ) = ✾ Y ✣ ( X ❀ Y ) is satisfied by x ✷ S k if and only if b 0 ( S x ) ✻ = 0, where S x = S ❭ ✙ � 1 Y ( x ) . Thus, the problem of deciding the truth of Φ( x ) is reduced to computing a Betti number (the 0-th) of the fiber of S over x . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof The case Π c R ❀ 1 Using the same notation as before we have that the formula Ψ( X ) = ✽ Y ✣ ( X ❀ Y ) is satisfied by x ✷ S k if and only if b 0 ( S ❵ ♥ S x ) = 0 which is equivalent to b ❵ ( S x ) = 1 (by Alexander duality). Notice, that as before the problem of deciding the truth of Ψ( x ) is reduced to computing a Betti number (the ❵ -th) of the fiber of S over x . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof The case Π c R ❀ 1 Using the same notation as before we have that the formula Ψ( X ) = ✽ Y ✣ ( X ❀ Y ) is satisfied by x ✷ S k if and only if b 0 ( S ❵ ♥ S x ) = 0 which is equivalent to b ❵ ( S x ) = 1 (by Alexander duality). Notice, that as before the problem of deciding the truth of Ψ( x ) is reduced to computing a Betti number (the ❵ -th) of the fiber of S over x . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Slightly more non-trivial case: Π c R ❀ 2 Let S ✚ S k ✂ S ❵ ✂ S m be a closed semi-algebraic set defined by a quantifier-free formula ✣ ( X ❀ Y ❀ Z ) and let ✙ Z : S k ✂ S ❵ ✂ S m ✦ S k ✂ S ❵ be the projection map along the Z variables, and ✙ Y : S k ✂ S ❵ ✦ S k be the projection map along the Y variables as before. Consider the formula Φ( X ) = ✽ Y ✾ Z ✣ ( X ❀ Y ❀ Z ) . For x ✷ S k , Φ( x ) is true if and only if ✙ Z ( S ) x = S ❵ , which is equivalent to b ❵ ( D ❵ + 1 ( S ) x ) = 1. Z Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Slightly more non-trivial case: Π c R ❀ 2 Let S ✚ S k ✂ S ❵ ✂ S m be a closed semi-algebraic set defined by a quantifier-free formula ✣ ( X ❀ Y ❀ Z ) and let ✙ Z : S k ✂ S ❵ ✂ S m ✦ S k ✂ S ❵ be the projection map along the Z variables, and ✙ Y : S k ✂ S ❵ ✦ S k be the projection map along the Y variables as before. Consider the formula Φ( X ) = ✽ Y ✾ Z ✣ ( X ❀ Y ❀ Z ) . For x ✷ S k , Φ( x ) is true if and only if ✙ Z ( S ) x = S ❵ , which is equivalent to b ❵ ( D ❵ + 1 ( S ) x ) = 1. Z Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Slightly more non-trivial case: Π c R ❀ 2 Let S ✚ S k ✂ S ❵ ✂ S m be a closed semi-algebraic set defined by a quantifier-free formula ✣ ( X ❀ Y ❀ Z ) and let ✙ Z : S k ✂ S ❵ ✂ S m ✦ S k ✂ S ❵ be the projection map along the Z variables, and ✙ Y : S k ✂ S ❵ ✦ S k be the projection map along the Y variables as before. Consider the formula Φ( X ) = ✽ Y ✾ Z ✣ ( X ❀ Y ❀ Z ) . For x ✷ S k , Φ( x ) is true if and only if ✙ Z ( S ) x = S ❵ , which is equivalent to b ❵ ( D ❵ + 1 ( S ) x ) = 1. Z Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof The case : Π c R ❀ 2 (cont.) Thus for any x ✷ S k , the truth or falsity of Φ( x ) is determined by a certain Betti number of the fiber D ❵ + 1 ( S ) x Z over x of a certain semi-algebraic set D ❵ + 1 ( S ) which can Z be constructed efficiently in terms of the set S . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof In general ... The idea behind the proof of the main theorem is a recursive application of the above argument in case when the number of quantifier alternations is larger (but still bounded by some constant) while keeping track of the growth in the sizes of the intermediate formulas and also the number of quantified variables. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Key Proposition Suppose there exists a real Turing machine M , and a sequence of formulas Φ n ( X 0 ❀ ✿ ✿ ✿ ❀ X n ❀ Y 0 ❀ ✿ ✿ ✿ ❀ Y m � 1 ) := ( Q 1 Z 1 ✷ S k 1 ) ✁ ✁ ✁ ( Q ✦ Z ✦ ✷ S k ✦ ) ✣ n ( X ❀ Y ❀ Z 1 ❀ ✿ ✿ ✿ ❀ Z ✦ ) ❀ having free variables ( X ❀ Y ) = ( X 0 ❀ ✿ ✿ ✿ ❀ X n ❀ Y 0 ❀ ✿ ✿ ✿ ❀ Y m � 1 ) , with Q 1 ❀ ✿ ✿ ✿ ❀ Q ✦ ✷ ❢✾ ❀ ✽❣ ❀ Q i ✻ = Q i + 1 ❀ where ✣ n a quantifier-free formula defining a closed (respectively open) semi-algebraic subset of S n , and such that M tests membership in the semi-algebraic sets defined by ✣ n in polynomial time. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Key Proposition (cont.) Then, there exists a polynomial time real Turing machine M ✵ which recognizes the semi-algebraic sets defined by a sequence of quantifier-free first order formulas (Θ n ( X ❀ V 0 ❀ ✿ ✿ ✿ ❀ V N )) n ❃ 0 such that for each x ✷ S n , where Θ n ( x ❀ V ) describes a closed (respectively open) semi-algebraic subset T n ✚ S N , with N = n O ( 1 ) , and polynomial-time computable maps F n : Z [ T ] ✔ N ✦ Z [ T ] ✔ m such that P ❘ (Φ n ( x ❀ Y )) = F n ( P ❘ (Θ n ( x ❀ V )) ) ✿ Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Outline (Discrete) Polynomial Hierarchy 1 Blum-Shub-Smale Models of Computation 2 Algorithmic Semi-algebraic Geometry 3 Real Analogue of Toda’s Theorem 4 Proof 5 Outline Details Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Topological Join The join J ( X ❀ Y ) of two topological spaces X and Y is defined by = X ✂ Y ✂ ∆ 1 ❂ ✘ ❀ def J ( X ❀ Y ) where ( x ❀ y ❀ t 0 ❀ t 1 ) ✘ ( x ✵ ❀ y ✵ ❀ t 0 ❀ t 1 ) if t 0 = 1 ❀ x = x ✵ or t 1 = 1 ❀ y = y ✵ . Intuitively, J ( X ❀ Y ) is obtained by joining each point of X with each point of Y by a unit interval. Example: J ( S m ❀ S n ) ✘ = S m + n + 1 ✿ Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Topological Join The join J ( X ❀ Y ) of two topological spaces X and Y is defined by = X ✂ Y ✂ ∆ 1 ❂ ✘ ❀ def J ( X ❀ Y ) where ( x ❀ y ❀ t 0 ❀ t 1 ) ✘ ( x ✵ ❀ y ✵ ❀ t 0 ❀ t 1 ) if t 0 = 1 ❀ x = x ✵ or t 1 = 1 ❀ y = y ✵ . Intuitively, J ( X ❀ Y ) is obtained by joining each point of X with each point of Y by a unit interval. Example: J ( S m ❀ S n ) ✘ = S m + n + 1 ✿ Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Topological Join The join J ( X ❀ Y ) of two topological spaces X and Y is defined by = X ✂ Y ✂ ∆ 1 ❂ ✘ ❀ def J ( X ❀ Y ) where ( x ❀ y ❀ t 0 ❀ t 1 ) ✘ ( x ✵ ❀ y ✵ ❀ t 0 ❀ t 1 ) if t 0 = 1 ❀ x = x ✵ or t 1 = 1 ❀ y = y ✵ . Intuitively, J ( X ❀ Y ) is obtained by joining each point of X with each point of Y by a unit interval. Example: J ( S m ❀ S n ) ✘ = S m + n + 1 ✿ Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Iterated joins For p ✕ 0, the ( p + 1 ) -fold join J p ( X ) of X is J p ( X ) def ✂ ∆ p ❂ ✘ ❀ = X ✂ ✁ ✁ ✁ ✂ X ⑤ ④③ ⑥ ( p + 1 ) times where ( x 0 ❀ ✿ ✿ ✿ ❀ x p ❀ t 0 ❀ ✿ ✿ ✿ ❀ t p ) ✘ ( x ✵ 0 ❀ ✿ ✿ ✿ ❀ x ✵ p ❀ t 0 ❀ ✿ ✿ ✿ ❀ t p ) i . It is easy to see that , J p ( S 0 ) , of if for each i with t i ✻ = 0, x i = x ✵ the zero dimensional sphere is homeomorphic to S p . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Iterated joins For p ✕ 0, the ( p + 1 ) -fold join J p ( X ) of X is J p ( X ) def ✂ ∆ p ❂ ✘ ❀ = X ✂ ✁ ✁ ✁ ✂ X ⑤ ④③ ⑥ ( p + 1 ) times where ( x 0 ❀ ✿ ✿ ✿ ❀ x p ❀ t 0 ❀ ✿ ✿ ✿ ❀ t p ) ✘ ( x ✵ 0 ❀ ✿ ✿ ✿ ❀ x ✵ p ❀ t 0 ❀ ✿ ✿ ✿ ❀ t p ) i . It is easy to see that , J p ( S 0 ) , of if for each i with t i ✻ = 0, x i = x ✵ the zero dimensional sphere is homeomorphic to S p . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof p -equivalence We call a map f : A ✦ B between two topological spaces to be a p -equivalence if the induced homomorphism f ✄ : H i ( A ) ✦ H i ( B ) is an isomorphism for all 0 ✔ i ❁ p , and an epimorphism for = S p is p -equivalent to a point. In i = p . Observe that J p ( S 0 ) ✘ fact, this holds much more generally and we have that Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof p -equivalence We call a map f : A ✦ B between two topological spaces to be a p -equivalence if the induced homomorphism f ✄ : H i ( A ) ✦ H i ( B ) is an isomorphism for all 0 ✔ i ❁ p , and an epimorphism for = S p is p -equivalent to a point. In i = p . Observe that J p ( S 0 ) ✘ fact, this holds much more generally and we have that Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Connectivity Property of Join Spaces Theorem Let X be a compact semi-algebraic set (in fact any reasonable top space). Then, the ( p + 1 ) -fold join J p ( X ) is p-equivalent to a point. Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
(Discrete) Polynomial Hierarchy Blum-Shub-Smale Models of Computation Outline Algorithmic Semi-algebraic Geometry Details Real Analogue of Toda’s Theorem Proof Topological join over a map Let f : A ✦ B be a map between topological spaces A and B . For p ✕ 0 the ( p + 1 ) -fold join J p f ( A ) of A over f is J p def ✂ ∆ p ❂ ✘ ❀ f ( A ) = A ✂ B ✁ ✁ ✁ ✂ B A ⑤ ④③ ⑥ ( p + 1 ) times where ( x 0 ❀ ✿ ✿ ✿ ❀ x p ❀ t 0 ❀ ✿ ✿ ✿ ❀ t p ) ✘ ( x ✵ 0 ❀ ✿ ✿ ✿ ❀ x ✵ p ❀ t 0 ❀ ✿ ✿ ✿ ❀ t p ) if for each i with t i ✻ = 0, x i = x ✵ i . Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’
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