1. Steven Lindell Scott Weinstein
Traversal-invariant elementary definability and logarithmic-space computation
Steven Lindell Scott Weinstein
11/29/2018
Workshop on Logic and Computational Complexity

2. Elementary (first-order) definability on graphs
Graphs are finite relational structures with a single binary relation.
First-order sentences/formulas define elementary (Boolean) queries, i.e. global properties/relations over the class of all graphs.
A graph 〈V; E〉 is simple (symmetric and irreflexive) if and only if:
For every x, y, z in V, [E(x, y) → E(y, x)] ∧ ¬E(z, z).
A binary string is a graph whose edge relation lies in between a strict linear order < and a non-strict linear order ≤ of its vertices.
E.g would be: 1 2 3 4
11/29/2018
Workshop on Logic and Computational Complexity

3. Order invariant definability
Motivation: all inputs to computers are arbitrarily linearly ordered
Are there properties of (directed) graphs which can be defined in the presence of a linear order, independent of that linear order?
Example [Yuri Gurevich]: Formulate Boolean algebras in the language of subsets, and ask if there is a set of atoms (minimal elements) containing the first atom in the linear order, alternate atoms thereafter, and not the last atom.
This order invariant property defines an even number of atoms.
By compactness this cannot be elementary because all infinite atomic Boolean algebras are elementarily equivalent [Tarski].
11/29/2018
Workshop on Logic and Computational Complexity

4. Connectivity & Acyclicity (with order) [Yuri]
Corollary: Neither connectivity nor acyclicity are order invariant.
Proof: Given any finite linear order, define an edge between every other adjacent element, as well as the first and last elements.
This always results in a chain, plus a cycle iff the length is odd.
If either connectivity or acyclicity were elementary, then we could define even length linear orders, which is impossible because the theory of infinite discrete linear orders with endpoints is complete. 1 2 3 4 5 6 1 2 3 4 5 6 7
11/29/2018
Workshop on Logic and Computational Complexity

5. Graph traversals (for simple graphs)
Definition: A traversal of a connected simple graph is a linear ordering of its vertices in which every initial segment is connected.
Example: Take a simple path (a − b − c − d) ordered as shown:
OK: NOT:
Note: I.e. every node (except the first) has a preceding neighbor.
A traversal of an arbitrary simple graph is a linear ordering of traversals of its connected components (i.e. their direct sum).
Fact: Every simple graph admits many traversals (e.g. using DFS). a b c d a c b d
Motivate traversals By saying that elements must when possible be connected to previously input data.
11/29/2018
Workshop on Logic and Computational Complexity

6. Traversal invariant definability (idea)
Idea: A graph property P is traversal invariant if it can be defined by a first-order sentence θ on ordered graphs (G, <) in which < is a traversal of G. I.e. θ ⊧ (G, <) ⇔ G ∈ P independent of the traversal <.
Examples: these properties are elementary with any traversal:
Connected: every element except the first has a preceding neighbor.
Acyclic: no element has two preceding neighbors.
The non-Boolean query transitive-closure is elementary as well:
Reachability: there is a path between two nodes s and t iff every node between them (except the first one) has a preceding neighbor.
11/29/2018
Workshop on Logic and Computational Complexity

7. Breadth-first traversals
Definition: A graph traversal (G, <) is breadth-first if the earliest preceding neighbor function p is monotone (p(x) = x if undefined). Example: Take a simple cycle (a − b − c − d − a) ordered as shown: OK: NOT: Definition: A formula on ordered graphs is breadth-first invariant if it determines a query on a breadth-first traversal of the underlying simple graph independent of any particular breadth-first traversal. Example: Are two nodes x and y equidistant from a fixed node c? a b d c a b c d
11/29/2018
Workshop on Logic and Computational Complexity

8. Logarithmic-space computability (L and NL)
Fact: Equivalent to a multi-head finite automaton with bidirectional read-only input tapes and a unidirectional write-only output tape.
Capturing L and NL in logic requires a successor relation, so we assume that all of our graphs come with an arbitrary successor.
Example: An empty graph has an even number of vertices if and only if the square S² of its successor relation S, together with an edge between the two endpoints, is connected. S S S …
S²: 1 2 3 4 1 2 3 4 5
11/29/2018
Workshop on Logic and Computational Complexity

9. Traversal invariance − full version
Close the notion of traversal invariance under elementary interpretations, to be the composition of the following:
S → π(S) = G → (G, <) → σ(G, <)
A (finite) structure S, in some signature with successor;
A first-order translation π which results in a simple graph G;
A (breadth-first) traversal < of G; followed by
A traversal invariant sentence σ over ordered simple graphs.
11/29/2018
Workshop on Logic and Computational Complexity

10. Workshop on Logic and Computational Complexity
Results
Theorem: A problem on (finite) structures with successor is computable in (nondeterministic) log-space iff it is elementarily definable in a (breadth-first) traversal invariant manner. I.e.,
L = FO(successor) + simple graph traversals
NL = FO(successor) + breadth-first traversals
The proofs of these rely on essential internal ideas from the arguments within the celebrated results of SL = L and co-NL = NL.
11/29/2018
Workshop on Logic and Computational Complexity