Descriptive Complexity Neil Immerman College of Computer and Information Sciences University of Massachusetts, Amherst Amherst, MA, USA people.cs.umass.edu/˜immerman Neil Immerman Descriptive Complexity P = 1 [ DTIME[nk ] k=1 P is a good mathematical wrapper for “truly feasible”. co-r.e. complete r.e. complete co-r.e. r.e. Recursive co-NP complete NP complete co-NP NP NP co-NP ∩ P complete P “truly feasible” “truly feasible” is FO(CFL) the informal set of problems we can FO(REGULAR) solve exactly on all reasonably sized FO instances. Neil Immerman Descriptive Complexity P is a good mathematical wrapper for “truly feasible”. co-r.e. complete r.e. complete co-r.e. r.e. P = Recursive 1 [ DTIME[nk ] k=1 co-NP complete NP complete co-NP NP NP co-NP ∩ P complete P “truly feasible” “truly feasible” is FO(CFL) the informal set of problems we can FO(REGULAR) solve exactly on all reasonably sized FO instances. Neil Immerman Descriptive Complexity co-r.e. complete r.e. complete co-r.e. r.e. P = Recursive 1 [ DTIME[nk ] k=1 P is a good co-NP complete NP complete co-NP NP mathematical NP co-NP ∩ wrapper for “truly P complete P feasible”. “truly feasible” “truly feasible” is FO(CFL) the informal set of problems we can FO(REGULAR) solve exactly on all reasonably sized FO instances. Neil Immerman Descriptive Complexity NTIME[t(n)]: a mathematical fiction 0 0 0 0 input w, w = n 0 j j 0 0 0 N accepts w 0 1 0 t(n) if at least s 0 2 0 one of the 2t(n) 0 0 0 paths accepts. 0 0 0 0 0 0 t(n) 0 Neil Immerman Descriptive Complexity b b b b 1 2 3 ··· t(n) Many optimization problems we want to solve are NP complete. SAT, TSP, 3-COLOR, CLIQUE, . As descison problems, all NP complete problems are isomorphic. co-r.e. complete r.e. complete NP = co-r.e. r.e. 1 Recursive [ NTIME[nk ] k=1 co-NP complete NP complete SAT SAT co-NP NP NP co-NP ∩ P complete P “truly feasible” FO(CFL) FO(REGULAR) FO Neil Immerman Descriptive Complexity As descison problems, all NP complete problems are isomorphic. co-r.e. complete r.e. complete NP = co-r.e. r.e. 1 Recursive [ NTIME[nk ] k=1 Many optimization problems we want co-NP complete NP complete to solve are NP SAT SAT co-NP NP NP co-NP complete. ∩ P complete P SAT, TSP, 3-COLOR, “truly CLIQUE, . feasible” FO(CFL) FO(REGULAR) FO Neil Immerman Descriptive Complexity co-r.e. complete r.e. complete NP = co-r.e. r.e. 1 Recursive [ NTIME[nk ] k=1 Many optimization problems we want co-NP complete NP complete to solve are NP SAT SAT co-NP NP NP co-NP complete. ∩ P complete P SAT, TSP, 3-COLOR, “truly CLIQUE, . feasible” FO(CFL) As descison problems, all NP FO(REGULAR) complete problems are isomorphic. FO Neil Immerman Descriptive Complexity co-r.e. complete r.e. complete NP = co-r.e. r.e. 1 Recursive [ NTIME[nk ] k=1 Many optimization PSPACE problems we want co-NP complete NP complete to solve are NP SAT SAT co-NP NP NP co-NP complete. ∩ P complete P SAT, TSP, 3-COLOR, “truly CLIQUE, . feasible” FO(CFL) As descison problems, all NP FO(REGULAR) complete problems are isomorphic. FO Neil Immerman Descriptive Complexity co-r.e. complete r.e. complete NP = co-r.e. r.e. 1 Recursive [ NTIME[nk ] k=1 EXPTIME Many optimization PSPACE problems we want co-NP complete NP complete to solve are NP SAT SAT co-NP NP NP co-NP complete. ∩ P complete P SAT, TSP, 3-COLOR, “truly CLIQUE, . feasible” FO(CFL) As descison problems, all NP FO(REGULAR) complete problems are isomorphic. FO Neil Immerman Descriptive Complexity S ··· ··· Restrict attention to the complexity of computing individual bits of the output, i.e., decision problems. How hard is it to check if input has property S ? How rich a language do we need to express property S? There is a constructive isomorphism between these two approaches. Descriptive Complexity Query Answer Computation q1 q2 qn 7! 7! a1 a2 ai am ··· ··· ··· Neil Immerman Descriptive Complexity S ··· ··· How hard is it to check if input has property S ? How rich a language do we need to express property S? There is a constructive isomorphism between these two approaches. Descriptive Complexity Query Answer Computation q1 q2 qn 7! 7! a1 a2 ai am ··· ··· ··· Restrict attention to the complexity of computing individual bits of the output, i.e., decision problems. Neil Immerman Descriptive Complexity How rich a language do we need to express property S? There is a constructive isomorphism between these two approaches. Descriptive Complexity Query Answer Computation q1 q2 qn 7! 7! a1 a2 ai am ··· ··· S ··· ··· ··· Restrict attention to the complexity of computing individual bits of the output, i.e., decision problems. How hard is it to check if input has property S ? Neil Immerman Descriptive Complexity There is a constructive isomorphism between these two approaches. Descriptive Complexity Query Answer Computation q1 q2 qn 7! 7! a1 a2 ai am ··· ··· S ··· ··· ··· Restrict attention to the complexity of computing individual bits of the output, i.e., decision problems. How hard is it to check if input has property S ? How rich a language do we need to express property S? Neil Immerman Descriptive Complexity Descriptive Complexity Query Answer Computation q1 q2 qn 7! 7! a1 a2 ai am ··· ··· S ··· ··· ··· Restrict attention to the complexity of computing individual bits of the output, i.e., decision problems. How hard is it to check if input has property S ? How rich a language do we need to express property S? There is a constructive isomorphism between these two approaches. Neil Immerman Descriptive Complexity Think of the Input as a Finite Logical Structure Graph G = ( v ;:::; vn ; ; E; s; t) f 1 g ≤ ¨* H ¨¨ H ¨ r r HHj HH ¨¨* ¨¨* t s H ¨ ¨ r Hj¨H r ¨ r H 2 H Σg = (E ; s; t) r Hj r r Binary String w = ( p ;:::; p ; ; S) A f 1 8g ≤ S = p ; p5; p7; p f 2 8g 1 Σs = (S ) w = 01001011 Neil Immerman Descriptive Complexity In this setting, with the structure of interest being the finite input, FO is a weak, low-level complexity class. First-Order Logic input symbols: from Σ variables: x; y; z;::: boolean connectives: ; ; ^ _ : quantifiers: ; 8 9 numeric symbols: =; ; +; ; min; max ≤ × α x y(E(x; y)) (Σg) ≡8 9 2L β x y(x y S(x)) (Σs) ≡9 8 ≤ ^ 2L β S(min) (Σs) ≡ 2L Neil Immerman Descriptive Complexity First-Order Logic input symbols: from Σ variables: x; y; z;::: boolean connectives: ; ; ^ _ : quantifiers: ; 8 9 numeric symbols: =; ; +; ; min; max ≤ × α x y(E(x; y)) (Σg) ≡8 9 2L β x y(x y S(x)) (Σs) ≡9 8 ≤ ^ 2L β S(min) (Σs) ≡ 2L In this setting, with the structure of interest being the finite input, FO is a weak, low-level complexity class. Neil Immerman Descriptive Complexity Fagin’s Theorem: NP = SO 9 s d t b a c f g e Second-Order Logic: FO plus Relation Variables Φ R1 G1 B1 x y ((R(x) G(x) B(x)) (E(x; y) 3color ≡ 9 8 _ _ ^ ! ( (R(x) R(y)) (G(x) G(y)) (B(x) B(y))))) : ^ ^ : ^ ^ : ^ s d t b a c f g e Neil Immerman Descriptive Complexity s d t b a c f g e Second-Order Logic: FO plus Relation Variables Fagin’s Theorem: NP = SO 9 Φ R1 G1 B1 x y ((R(x) G(x) B(x)) (E(x; y) 3color ≡ 9 8 _ _ ^ ! ( (R(x) R(y)) (G(x) G(y)) (B(x) B(y))))) : ^ ^ : ^ ^ : ^ s d t b a c f g e Neil Immerman Descriptive Complexity co-r.e. complete r.e. complete co-r.e. r.e. Recursive EXPTIME PSPACE SO co-NP complete PTIME Hierarchy NP complete SAT SAT SO SO co-NP ∀ ∃ NP NP co-NP ∩ P complete P “truly feasible” FO(CFL) FO(REGULAR) FO Neil Immerman Descriptive Complexity C(i) ( j > i) A(j) B(j) ≡ 9 ^ ^ ( k:j > k > i)(A(k) B(k)) 8 _ Q+(i) A(i) B(i) C(i) ≡ ⊕ ⊕ Addition is First-Order Q+ : STRUC[Σ ] STRUC[Σs] AB ! A a1 a2 ... an−1 an B + b1 b2 ... bn−1 bn S s1 s2 ... sn−1 sn Neil Immerman Descriptive Complexity Q+(i) A(i) B(i) C(i) ≡ ⊕ ⊕ Addition is First-Order Q+ : STRUC[Σ ] STRUC[Σs] AB ! A a1 a2 ... an−1 an B + b1 b2 ... bn−1 bn S s1 s2 ... sn−1 sn C(i) ( j > i) A(j) B(j) ≡ 9 ^ ^ ( k:j > k > i)(A(k) B(k)) 8 _ Neil Immerman Descriptive Complexity Addition is First-Order Q+ : STRUC[Σ ] STRUC[Σs] AB ! A a1 a2 ... an−1 an B + b1 b2 ... bn−1 bn S s1 s2 ... sn−1 sn C(i) ( j > i) A(j) B(j) ≡ 9 ^ ^ ( k:j > k > i)(A(k) B(k)) 8 _ Q+(i) A(i) B(i) C(i) ≡ ⊕ ⊕ Neil Immerman Descriptive Complexity Quantifiers are Parallel Assume array A[x]: x = 1;:::; r in memory. x(A(x)) write(1); proc p : if (A[i] = 0) then write(0) 8 ≡ i 01 Parallel Machines: CRAM[t(n)] = CRCW-PRAM-TIME[t(n)]-HARD[nO(1)] P P r 1 P 2 Global Memory P3 P 4 P 5 Neil Immerman Descriptive Complexity x(A(x)) write(1); proc p : if (A[i] = 0) then write(0) 8 ≡ i 01 Parallel Machines: Quantifiers are Parallel CRAM[t(n)] = CRCW-PRAM-TIME[t(n)]-HARD[nO(1)] Assume array A[x]: x = 1;:::; r in memory.