Complexity Classes and Theories for the Comparator Circuit Value Problem Dai Tri Man Lˆ e Joint work with Stephen Cook and Yuli Ye University of Toronto Canada Prague Fall Logic School 2011 1 / 30
Stephen Cook (’68) Yuli Ye 2 / 30
Bounded Reverse Mathematics [Cook-Nguyen ’10] Motivation Classify theorems according to the computational complexity of concepts needed to prove them. Program in Chapter 9 Introduce a general method for associating 1 a canonical minimal theory VC for “nice” complexity classes C AC 0 ⊆ C ⊆ P Given a theorem τ , try to find the smallest 2 complexity class C such that VC ⊢ τ 3 / 30
Outline of the talk The complexity classes for the Comparator Circuit Value Problem 1 Define a theory for CC ∗ 2 Natural complete problems: stable marriage and lex-first maximal 3 matching Conclusion and open problems 4 4 / 30
The complexity classes for the Comparator Circuit Value Problem 1 Define a theory for CC ∗ 2 Natural complete problems: stable marriage and lex-first maximal 3 matching Conclusion and open problems 4 5 / 30
Comparator Circuits Comparator gate a x • min( a , b ) Originally invented for sorting, e.g., ◮ Ajtai-Koml´ y os-Szemer´ edi (AKS) max( a , b ) b � O (log n )-depth sorting networks (’83) abek (’11) in VNC 1 ◮ Formalized by Jeˇ r´ ∗ . 6 / 30
Comparator Circuits Comparator gate a x • min( a , b ) Originally invented for sorting, e.g., ◮ Ajtai-Koml´ y os-Szemer´ edi (AKS) max( a , b ) b � O (log n )-depth sorting networks (’83) abek (’11) in VNC 1 Boolean comparator gate ◮ Formalized by Jeˇ r´ ∗ . p p ∧ q x • Can also be seen as boolean circuits. q y p ∨ q � 6 / 30
Comparator Circuits Comparator gate a x • min( a , b ) Originally invented for sorting, e.g., ◮ Ajtai-Koml´ y os-Szemer´ edi (AKS) max( a , b ) b � O (log n )-depth sorting networks (’83) abek (’11) in VNC 1 Boolean comparator gate ◮ Formalized by Jeˇ r´ ∗ . p p ∧ q x • Can also be seen as boolean circuits. q y p ∨ q � Example w 0 1 • 0 • 0 0 w 1 • 1 0 1 � w 2 1 1 w 3 • 0 1 0 � w 4 0 1 1 � w 5 0 0 0 � 6 / 30
Comparator Circuit Value ( Ccv ) Problem (decision) 1 w 0 • • Given a comparator circuit with specified 1 w 1 � • 1 w 2 Boolean inputs, determine the output ? 0 w 3 � • value of a designated wire. 0 w 4 � 0 w 5 � Complexity classes CC Subr = � � decision problems log-space many-one-reducible to Ccv 1 ◮ [Subramanian ’90], [Mayr-Subramanian ’92] 7 / 30
Comparator Circuit Value ( Ccv ) Problem (decision) 1 w 0 • • Given a comparator circuit with specified 1 w 1 � • 1 w 2 Boolean inputs, determine the output ? 0 w 3 � • value of a designated wire. 0 w 4 � 0 w 5 � Complexity classes CC Subr = � � decision problems log-space many-one-reducible to Ccv 1 ◮ [Subramanian ’90], [Mayr-Subramanian ’92] decision problems AC 0 many-one-reducible to Ccv � � CC = 2 Complete problems: stable marriage, lex-first maximal matching. . . ◮ CC ∗ = decision problems AC 0 oracle-reducible to Ccv � � 3 ◮ Needed when developing a Cook-Nguyen style theory for CC ◮ The function class FCC ∗ is closed under compostion 7 / 30
Comparator Circuit Value ( Ccv ) Problem (decision) 1 w 0 • • Given a comparator circuit with specified 1 w 1 � • 1 w 2 Boolean inputs, determine the output ? 0 w 3 � • value of a designated wire. 0 w 4 � 0 w 5 � Complexity classes CC Subr = � � decision problems log-space many-one-reducible to Ccv 1 ◮ [Subramanian ’90], [Mayr-Subramanian ’92] decision problems AC 0 many-one-reducible to Ccv � � CC = 2 Complete problems: stable marriage, lex-first maximal matching. . . ◮ CC ∗ = decision problems AC 0 oracle-reducible to Ccv � � 3 ◮ Needed when developing a Cook-Nguyen style theory for CC ◮ The function class FCC ∗ is closed under compostion 7 / 30
Comparator Circuit Value ( Ccv ) Problem (decision) 1 w 0 • • Given a comparator circuit with specified 1 w 1 � • 1 w 2 Boolean inputs, determine the output ? 0 w 3 � • value of a designated wire. 0 w 4 � 0 w 5 � Complexity classes CC Subr = � � decision problems log-space many-one-reducible to Ccv 1 ◮ [Subramanian ’90], [Mayr-Subramanian ’92] decision problems AC 0 many-one-reducible to Ccv � � CC = 2 Complete problems: stable marriage, lex-first maximal matching. . . ◮ CC ∗ = decision problems AC 0 oracle-reducible to Ccv � � 3 ◮ Needed when developing a Cook-Nguyen style theory for CC ◮ The function class FCC ∗ is closed under compostion NC 1 ⊆ NL ⊆ CC ⊆ CC Subr ⊆ CC ∗ ⊆ P 7 / 30
The complexity classes for the Comparator Circuit Value Problem 1 Define a theory for CC ∗ 2 Natural complete problems: stable marriage and lex-first maximal 3 matching Conclusion and open problems 4 8 / 30
Two-sorted language L 2 A (Zambella ’96) Vocabulary L 2 � � A = 0 , 1 , + , · , | | ; ∈ , ≤ , = 1 , = 2 Standard model N 2 = � N , finite subsets of N � 0 , 1 , + , · , ≤ , = have usual meaning over N | X | = length of X Set membership y ∈ X “number” variables x , y , z , . . . (range over N ) “string” variables X , Y , Z , . . . (range over finite subsets of N ) Number terms are built from x , y , z , . . . , 0 , 1 , + , · and | X | , | Y | , | Z | ,. . . The only string terms are variable X , Y , Z , . . . 9 / 30
Two-sorted language L 2 A (Zambella ’96) Vocabulary L 2 � � A = 0 , 1 , + , · , | | ; ∈ , ≤ , = 1 , = 2 Note Standard model N 2 = � N , finite subsets of N � The natural inputs for Turing machines 0 , 1 , + , · , ≤ , = have usual meaning over N and circuits are | X | = length of X finite strings. Set membership y ∈ X “number” variables x , y , z , . . . (range over N ) “string” variables X , Y , Z , . . . (range over finite subsets of N ) Number terms are built from x , y , z , . . . , 0 , 1 , + , · and | X | , | Y | , | Z | ,. . . The only string terms are variable X , Y , Z , . . . 9 / 30
Two-sorted language L 2 A (Zambella ’96) Vocabulary L 2 � � A = 0 , 1 , + , · , | | ; ∈ , ≤ , = 1 , = 2 Note Standard model N 2 = � N , finite subsets of N � The natural inputs for Turing machines 0 , 1 , + , · , ≤ , = have usual meaning over N and circuits are | X | = length of X finite strings. Set membership y ∈ X “number” variables x , y , z , . . . (range over N ) “string” variables X , Y , Z , . . . (range over finite subsets of N ) Number terms are built from x , y , z , . . . , 0 , 1 , + , · and | X | , | Y | , | Z | ,. . . The only string terms are variable X , Y , Z , . . . Definition ( Σ B 0 formula) All the number quantifiers are bounded. 1 No string quantifiers (free string variables are allowed) 2 9 / 30
Two-sorted complexity classes x , � A two-sorted complexity class consists of relations R ( � X ), where x are number arguments (in unary) and � X are string arguments � Definition (Two-sorted AC 0 ) X ) is in AC 0 iff some alternating Turing machine accepts x , � A relation R ( � R in time O (log n ) with a constant number of alternations. Σ B 0 -Representation Theorem [Zambella ’96, Cook-Nguyen] X ) is in AC 0 iff it is represented by a Σ B x , � x , � R ( � 0 -formula ϕ ( � X ). Useful consequences Don’t need to work with uniform circuit families or alternating Turing 1 machines when defining AC 0 functions or relations. Useful when working with AC 0 -reductions 2 10 / 30
The theory V 0 for AC 0 reasoning The theory V 0 2 -BASIC axioms : essentially the axioms of Robinson arithmetic plus 1 ◮ the defining axioms for ≤ and the string length function | | ◮ the axiom of extensionality for finite sets (bit strings). Σ B 0 -COMP (Comprehension): for every Σ B 0 -formula ϕ ( z ) without X , 2 � � ∃ X ≤ y ∀ z < y X ( z ) ↔ ϕ ( z ) Theorem Σ B 0 -IND: for ϕ ∈ Σ B 1 0 � �� � ϕ (0) ∧ ∀ x ϕ ( x ) → ϕ ( x + 1) → ∀ x ϕ ( x ) The provably total functions in V 0 are precisely FAC 0 . 2 Note: Theories, developed using Cook-Nguyen method, extend V 0 . 11 / 30
The 2-BASIC axioms B1. x + 1 � = 0 B8. x ≤ x + y B2. x + 1 = y + 1 → x = y B9. 0 ≤ x B3. x + 0 = x B10. x ≤ y ∨ y ≤ x B4. x + ( y + 1) = ( x + y ) + 1 B11. x ≤ y ↔ x < y + 1 B5. x · 0 = 0 B12. x � = 0 → ∃ y ≤ x ( y + 1 = x ) B6. x · ( y + 1) = ( x · y ) + x L1. X ( y ) → y < | X | B7. ( x ≤ y ∧ y ≤ x ) → x = y L2. y + 1 = | X | → X ( y ) � �� � SE. | X | = | Y | ∧ ∀ i < | X | X ( i ) = Y ( i ) → X = Y 12 / 30
The theory VCC ∗ for CC ∗ Comparator Circuit Value ( Ccv ) Problem (decision) 1 w 0 • • Given a comparator circuit with 1 w 1 � • specified Boolean inputs 1 w 2 ? 0 w 3 � • Determine the output value of a 0 w 4 � designated wire. 0 w 5 � Recall that CC ∗ = decision problems AC 0 oracle-reducible to Ccv � � The two-sorted theory VCC ∗ [using the Cook-Nguyen method] VCC ∗ has vocabulary L 2 A Axiom of VCC ∗ = Axiom of V 0 + one additional axiom asserting the existence of a solution to the Ccv problem. 13 / 30
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