Locality of Counting Logics

Transcription

1 Locality of Counting Logics Dietrich Kuske Technische Universität Ilmenau 1/22

2 Example 1 Can a first-order formula distinguish the following two graphs? vs. no, but why? Hanf (1965): every single-centered sphere appears equally often in both graphs 2/22

3 Example 2 Can a first-order formula distinguish the following two graphs with distinguished vertex? (N,S,4) = (N,S,5) = vs. Can we do with smaller formula / formula of lower quantifier rank? Fagin, Stockmeyer, Vardi (1995): give bounds on quantifier rank of distinguishing formula 3/22

4 Example 3 Given a node-labeled graph G and a first-order formula ϕ, how difficult is it to check whether G = ϕ holds? e.g. ϕ = x,y: red(x) blue(y) E(x,y) ϕ = (x i ) 1 i n : color i (x i ) 1 i n 1 i<j n E(x i,x j ) space 2 log( G ), time G 2 space ϕ log( G ), time G ϕ if G has degree d (Seese 1996): first compute condition on sphere counts from ϕ and d, then evaluate this condition on G i.e., time f( ϕ,d) G ( fixed parameter tractable ) 4/22

5 A = (A,(R i ) 1 i k ): structure over purely relational signature σ Gaifman-graph G(A) = (A,E) with (a,b) E (...,a,...,b,...) R i R 1 i for some i distance d A (a,b) and degree to be understood in G(A) (possibly ) ball B r A (a 1,...,a n ) = {b A d A (a i,b) r for some i} (a 1,...,a n A, r N) sphere S r A (a 1,...,a n ) = (A B r A (a 1,...,a n),a 1,...,a n ) is σ-structure with n constants ( centers ) for sphere τ of radius r with n centers: # τ (A) = {a A n : τ = S r A(a)} is the number of realizations of τ in A 5/22

6 A = τ = # τ (A) =? 10 Theorem (Hanf, 1965) Let A and B be locally finite such that # τ (A) = # τ (B) for any 1-centered sphere τ (of any radius). Then A and B are indistinuishable by first-order formulas. Consequence (N,S) (N,S) (Z,S) 6/22

7 Theorem (Fagin, Stockmeyer, Vardi 1995) The following holds for any quantifier rank q and any degree bound d: Let A and B be locally finite structures of degree d such that # τ (A) = # τ (B)min(# τ (A),d 2q ) = min(# τ (B),d 2q ) for all 1-centered spheres τ. of radius 2 q. of degree d. Then A and B cannot be distinguished by first-order formulas. of quantifier-rank q. Consequence (N,S,2 q ) and (N,S,2 q +1) cannot be distinguished by formulas of small quantifier rank. 7/22

8 Theorem (Fagin, Stockmeyer, Vardi 1995) The following holds for any quantifier rank q and any degree bound d: Let A and B be locally finite structures of degree d such that # τ (A) = # τ (B)min(# τ (A),d 2q ) = min(# τ (B),d 2q ) for all 1-centered spheres τ. of radius 2 q. of degree d. Then A and B cannot be distinguished by first-order formulas. of quantifier-rank q. Note τ n-centered sphere of radius r 2 q and degree d = τ n d 2q elements = there is a formula sph τ (x) with A = sph τ (a) iff S r A (a) = τ f.a. A and a A n. 7/22

9 Hanf normal form Consequence (Fagin, Stockmeyer, Vardi 1995) For any d 0 and any formula ϕ(x) FO, there exists a d-equivalent formula ψ(x) in Hanf normal form, i.e., a Boolean combination of formulas sph ρ (x) with ρ a sphere with x centres, k y sph τ (y) with k 1 and τ a 1-centered sphere. Theorem (Bollig, K 2012) ψ is computable from d and ϕ in time exp(d 2 ϕ ) = 3-exp( ϕ +loglogd) (and not faster). 8/22

10 Consequences (1) there is no formula distinguishing, for all n 1, the graphs C n C n and C 2n, in particular connectivity is not expressible (2) evaluation of FO-formula ϕ on finite structure A: in general in time A ϕ if A has degree d: 3-exp(ϕ,d) A, i.e., fixed parameter linear with elementary parameter dependence Remark. fixed parameter polynomial (=fpt) is know for any class of nowhere dense structures (3) evaluation of FO-queries on dynamic databases of degree d with constant delay 9/22

11 From now on, all structures are finite (and usually symmetric graphs). 10/22

12 A counting extension of first-order logic FO(Q) = first-order logic with unary counting quantifiers Qy ϕ for Q Q P(N) (Hella, Nurmonen et al.) (A,β) = Qy ϕ {a A (A,β) = ϕ(a)} Q examples: 1. Q = N 1 A = Qx ϕ(x) iff {a A A = ϕ(a)} 1 iff A = x ϕ 2. 1 mod 2 = 2N+1 A = 1 mod 2 x ϕ(x) iff {a A A = ϕ(a)} odd which is not expressible in FO 11/22

13 Hanf normal form Theorem (Heimberg, K, Schweikardt 2016) Let Q be a class of ultimately periodic counting quantifiers. For any d 0 and any formula ϕ(x) FO(Q), there exists a d-equivalent formula ψ(x) in Hanf normal form, i.e., a Boolean combination of formulas sph ρ (x) with ρ a sphere with x centres, k y sph τ (y) with k 1 and τ a sphere with 1 centre, and (Q+k)y sph τ (y) with τ a sphere with 1 centre. Such a ψ is computable in time 3-exp( ϕ +loglogd) and not faster. 12/22

14 Hanf normal form Theorem (K, Schweikardt 2017) Let Q be a class of///////////// ultimately////////// periodic counting quantifiers. For any d 0 and any formula ϕ(x) FO(Q), there exists a d-equivalent formula ψ(x) in Hanf normal form, i.e., a Boolean combination of formulas sph ρ (x) with ρ a sphere with x centres, k y sph τ (y) with k 1 and τ a sphere with 1 centre, and τ T sph τ(y) with T asetofsphereswith1centre. Such a ψ is computable in time 4-exp( ϕ +loglogd) and (for many classes Q) not faster. 12/22

15 Further counting extensions of first-order logic FO(Cnt) = FO+Count (Immerman, Libkin) = FO+C (Grohe) FO+predicates on witness counts (given by formulas of bounded arithmetic) 2-sorted structures: usual σ-structure A together with (N, +, ) 2-sorted logic: FO[σ] + quantification of number variables κ,λ,... (extends over {0,1,..., A }) + equality of polynomials over number variables + counting operator λ #y ϕ (A,β) = (λ #y ϕ) β(λ) {a A (A,β) = ϕ(a)} 13/22

16 FO(unary) = FO+unary generalized quantifiers Q K y (ϕ 1,...,ϕ k ) for K a class of structures with k unary relations FO+predicates on witness counts (from a fixed set P of predicates) (Hella, Nurmonen, Libkin et al.) FO[σ] + counting expressions ψ = P(#y ϕ 1,#y ϕ 2...,#y ϕ n ) for P P n-ary (A,β) = ψ ( ϕ A 1, ϕ 2 A,..., ϕ A n ) P 14/22

17 New counting logic FOCN(P) FO+predicates on counts of witnessing tuples (given by formulas of bounded arithmetic and, in addition, from a fixed set P of predicates) 2-sorted structures: usual σ-structure A Z with functions + and, and numerical predicates from P 2-sorted logic: FO[σ] + quantification over number variables (extends over {0,1,..., A }) + atomic formulas P(t 1,...,t k ) with P P of arity k and t 1,...,t k counting terms + counting terms: #(y 1,...,y n )ϕ for structure variables y i, integers, number variables κ,λ,..., + and. 15/22

18 For I = (A,β): [#(y 1,...,y n )ϕ] I = {(a 1,...,a n ) A n I = ϕ(a 1,...,a n )} Examples = number of witnessing tuples for ϕ (1) with P = {1,2,...}: P (#(y)ϕ) y ϕ (2) LE(#(y)ϕ,#(z)ψ) with LE = {(m,n) m n} expresses Rescher quantifier (3) all previously mentioned counting logics can be encoded into FOCN(P) (for suitable P) 16/22

19 (4) Prime(#(x)x = x +#(x,y)e(x,y)) expresses: V + E is a prime (5) EQ(#(x,y)E(x,y),#(x,y)E(x,y)) expresses: there are as many red edges as there are blue edges Luosto 2000: not expressible in FO(unary). (6) κpoweroftwo(#(y) EQ(κ,#(z)E(y,z) )) }{{} out-degree of y } {{ } out-degree of y is κ expresses: there is some out-degree κ whose number of realisations is a power of 2 17/22

20 Theorem (K, Schweikardt 2017) For any set of numerical predicates P, any formula ϕ(x) from FOCN(P), and any degree bound d, there exists a d-equivalent formula ψ(x) in Hanf normal form., i.e., a Boolean combination of formulas sph ρ (x) with ρ a sphere with x centres, k y sph τ (y) with k 1 and τ a sphere with 1 centre, and oc-type conditions: built by,, κ from atomic oc-type conditions atomic oc-type conditions: P(t 1,...,t k ) with P P of arity k and simple counting terms t i simple counting terms: built from #(y)sph τ (y) and κ using +,, and integers 18/22

21 Example. ϕ = P ( #(x,y): red(x) blue(y) E(x,y) ) ψ ( = P #(x)sph τ (x) with τ T τ T #(y)sph τ (y) τ T k τ #(x)sph τ (x) T set of all 1-centered spheres of radius 0 with red center, T set of all 1-centered spheres of radius 0 with blue center, T set of all d-bounded 1-centered spheres of radius 1 with red center, and k τ number of blue nodes in τ. ) 19/22

22 Example. ϕ(x) = P ( #(y)α(x,y) ) where α is a Boolean combination of atomic oc-type conditions (hence independent from y) from S and formulas sph τ (x,y) with τ of radius r ϕ θ θ P(#(y)α H (x,y)) }{{} H S θ H θ S\H =:β H where α H is a Boolean combination of formulas sph τ (x,y). β H d sph σ (x) P #(y)sph τ (y) c σ +l σ τ T σ σ T T is set of all d-bounded 1-centered spheres of radius 3r +1 T σ is set of d-bounded 1-centered spheres of radius r s.t. S r σ(x) τ = α H c σ = {y σ d σ (x,y) 2r and S r σ(x) S r σ(y) = α H } l σ = {y σ d σ (x,y) 2r and S r σ(x,y) = α H } 20/22

23 Theorem (algorithmic version) Fix countable P. Computable in time 5-exp( ϕ +loglogd): input: degree bound d 0 and FOCN(P)-formula ϕ(x) output: d-equivalent formula ψ(x) in Hanf normal form with nqr(ψ) nqr(ϕ) (number quantifier rank) and locality radius of ψ < (2bw(ϕ)+1) br(ϕ) 21/22

24 Consequences (1) FOCN(P) is Hanf-local, hence, e.g., connectivity is not expressible (2) FO-formulas ϕ Σ n have Hanf locality radius ϕ n Does this hold for infinite structures of bounded degree? (3) evaluation of FOCN(P)-formulas on structures of degree d with oracles for P in time 5-exp(ϕ,d)+3-exp(ϕ,d) A +5-exp(ϕ,d) A nqr(ϕ) hence fixed parameter linear with elementary parameter dependence for numberfree fragment FOC(P) of FOCN(P) Grohe, Schweikardt: holds for natural fragment of FOC(P) on nowhere dense classes of structures (4) evaluation of FOC(P)-queries on dynamic databases of degree d with constant delay 22/22