The Invariant Problem for Binary String Structures and the Parallel Complexity Theory of Queries

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1 The Invariant Problem for Binary String Structures and the Parallel Complexity Theory of Queries Steven Lindell Haverford College Haverford, PA Introduction A fundamental problem in finite model theory: The logical complexity of determining if two structures are isomorphic, and of computing canonical invariants for structures. Provide query-theoretic formulations of these problems a. Are two structures isomorphic? b. Finding an invariant for each isomorphism type. Study these problems on binary string structures. 2 1

2 The Isomorphism Problem Let K be a class of finite structures in the common language of predicate symbols L ={R 1,... R k }. The isomorphism problem for K is the Boolean query ISO K on pairs of structures from K with a common domain such that for all A, R 1,... R k and A, R 1 ',... R k ' in K: A, R 1,... R k, R 1 ',...R k ' ISO K A, R 1,... R k A,R 1 ',... R k ' 3 Graph Isomorphism A single binary (edge) relation, L ={E}. V, E, E' ISO V, E V,E' 4 2

3 Binary String Isomorphism Definition: A binary string structure is a tuple A,<, U where A is a finite set totally ordered by <, and U is a unary relation on A (which represents the location of ones in a binary string) A = a 1 < a 2 < a 3 < a 4 (black dots indicate true values of U). A,<, U,<',U' ISO A,<, U A,<',U' 5 The Isomorphism Query for Binary Strings is not First-Order Theorem: The isomorphism problem for binary strings, as defined above, is not first-order definable. Proof: a 2 k +1 b 2 k +1 a 2 k +1 a 2 k middle b 2 k b 2 k -1 a 1 b 1 A B A and B are k-fraisséequivalent. 6 3

4 The Importance of Arithmetic Allow our binary string structures to include the logical equivalent of random-access addressing. Add arithmetic relations: addition (+), multiplication ( ), and binary exponentiation (2^) on the domain {0,1,..., n-1} ordered by < (all operations are modulo n). Equivalent expressive power can be achieved with just a total linear ordering, <, and a single binary predicate, bit(i, j) satisfying bit(i, j) the jth digit of the binary expansion of i is 1 (idiv2^j)mod2=1 7 Binary String Isomorphism Revisited Definition: A binary string structure is a tuple A,<, bit, U where A is a finite set, < is a total order, bit gives the binary expansion of any element with respect to < as explained above, and U is an arbitrary unary relation (representing the location of ones in a binary string). A, <, bit, U, <', bit', U' ISO A, <, bit, U A, <', bit', U' A,<, U A,<',U' This ISO is first-order definable! 8 4

5 First-Order Axiomatizability of Bit Fact: The binary string structures are first-order definable. Proof: To say that < is a strict total linear order is easy: (x < x) (x y) [(x < y) (y<x)] [(x < y) (y<z)] (x<z) The axioms for bit are not much harder: bit(0, x) x +1=y X+1=Y where X ={z bit(x, z)}, 1= {0}, Y = {z bit(y, z)}. 9 Graph Invariants There is not a single graph invariant. Rather, a binary query INV is said to solve the invariant problem for graphs if when given an arbitrarily ordered graph G = V,<, E, INV G satisfies V, E V,INV G and for all total orderings <' of V V,<, INV V,<, E V,<',INV V,<',E In other words, the graph V,<, INV is an ordered copy of the unordered graph V, E. Also, INV satisfies the further property of isomorphism invariance simultaneously for both of its relations E and <. As a result, V,<, INV V,<, E is always isomorphic to V, E (ignoring <) and independent of <. 10 5

6 Canonical Representations Note that the invariant problem is at least as hard to compute as the isomorphism problem since two structures are isomorphic if and only if their invariants for arbitrary orderings are identical. I.e. V, E₁ V, E₂ V,<₁, INV V,<₁, E₁ V,<₂, INV V,<₂, E₂ for any <₁, <₂ For definiteness, we shall specify a particular canonical representation. Namely, we will define the binary query invariant of a graph is to be the smallest binary relation that solves the invariant problem. 11 Graph Comparison A Boolean query which totally orders isomorphism classes of graphs of a given size with respect to canonical representatives: V, E, E' COM canon( V, E ) < canon( V, E' ) On isomorphism classes: ISO is to equality (=) as COM is to less than (<). Proposition: COM is polynomial-time equivalent to INV. Proof: V, E, E' COM INV V,<, E < INV V,<, E'. Use binary search. 12 6

7 A First-Order Solution to the Invariant Problem for Binary Strings Definition: The invariant query for the class of binary string structures in the language {<, bit, U} is the unary query INV on structures of the form A = A,<, bit, U,<',bit' defined as satisfying: A,<, bit, U A,<',bit', INV A Theorem: The query INV for binary string structures is a firstorder definable query. Proof: Idea is to find a first-order formula which witnesses the order-preserving isomorphism, h, between A,< and A,<'. From h we could then define: INV(x') ( x)(h(x)= x') U(x). 13 Binary Encodings of Finite Structures To discuss connections with complexity theory, it will be necessary to encode finite structures as real binary strings so that we can consider machine computations of queries. Example: G = c Encoding: a < (G)=

8 Sequential Complexity Classes Definition: A set of binary languages C is a sequential complexity class if L C {w f(w) L} C for all logarithmic-space computable functions f:{0,1}* {0,1}*. Examples: L = SPACE(logn) P = TIME(n O(1) ) 15 Parallel Complexity Classes Definition: A set of binary languages C is a parallel complexity class if L C {w f(w) L} C for all constant-time computable functions f:{0,1}* {0,1}*. Examples: CP = TIME(O(1)) LP = TIME(O(logn)) 16 8

9 Computational Complexity of Queries Definition: Let q be a query of arity l for the predicate symbols R 1,..., R k, let C be a complexity class, and let (A, c 1,..., c l ) denote the expansion of the structure A = A, R 1,..., R k by the constants c 1,..., c l. Then q is said to be computable in the complexity class C if L q = {a < (A, c 1,..., c l ) where < orders A, and A q(c 1,..., c l )} C. Let Q(C) = {q L q C} denote the set of queries which are C-computable. 17 Correspondence Results Letting IND(<) denote the queries syntactically expressible in an inductive fixed-point logic on ordered structures, it has been established [Immerman, Vardi] that: Q(P) = IND(<) In a similar fashion, letting DTC(<) denote the queries syntactically expressible in a deterministic transitive closure logic on ordered structures yields [Immerman]: Q(L) = DTC(<). Immermanhas very good surveys of these and many other correspondence results. 18 9

10 Constant-Time Computability The constant-time computable queries, Q(CP), are particularly interesting because of their close connection with first-order logic. Let FO(<, bit) denote the set of first-order expressible queries on finite structures that include the < and bit predicates (internally). The following result is due to Immerman: Theorem : Q(CP) = FO(<, bit) Indicates that parallel computation plays an important role in the computational complexity of queries expressed by logical formulas 19 Complexity Theory There is a tight relationship between the stratification of sequential complexity classes and their correspondingly induced query classes [Chandra& Harel]: C₁ C₂ Q(C₁) Q(C₂) Theorem: The invariant problem for binary strings with bit is first-order solvable. This allows us to extend the result: Corollary: Let C₁ and C₂ be parallel complexity classes. Then C₁ C₂ Q(C₁) Q(C₂) 20 10

11 Acknowledgments Sheila Greibach Yiannis Moschovakis Ashok Chandra Neil Immerman David Barrington Sam Buss 21 11

DATABASE THEORY. Lecture 11: Introduction to Datalog. TU Dresden, 12th June Markus Krötzsch Knowledge-Based Systems

DATABASE THEORY. Lecture 11: Introduction to Datalog. TU Dresden, 12th June Markus Krötzsch Knowledge-Based Systems DATABASE THEORY Lecture 11: Introduction to Datalog Markus Krötzsch Knowledge-Based Systems TU Dresden, 12th June 2018 Announcement All lectures and the exercise on 19 June 2018 will be in room APB 1004