Definability of Recursive Predicates in the Induced Subgraph Order

Abstract

Consider the set of all finite simple graphs $\mathcal {G}$ ordered by the induced subgraph order $\le _i$. Building on previous work by Wires [14] and Jezek and Mckenzie [5,6,7,8], we show that every recursive predicate over graphs is definable in the first order theory of ($\mathcal {G},\le _i, P_3$) where $P_3$ is the path on 3 vertices.

Appendix: Proof Sketch of Theorem 2

Theorem Statement: Every recursive predicate R on numbers is definable in first order arithmetic.

Proof

(sketch). For simplicity we look at the case of only unary predicates, assume $R \subseteq \mathbb {N}$. Let $M=(Q,\delta ,s,F)$ be a turing machine over the alphabet $\{0,1\}$. First, consider strings over the alphabet $\varSigma = (0,1,\#,s,q_1,...,q_n)$ where $Q=\{s,q_1,...,q_n\}$. They can be encoded as binary strings by using some encoding e.g. 0 is mapped to 01, 1 to 001, $\#$ to 0001, s to 00001, $q_i$ to $0^{i+4}1$. Given any input x, we can encode the run of the Turing machine as a number y, which we will think of a string over the extended alphabet $\varSigma$ (ignoring the 1 in the most significant digit). $y=c_1\#c_2\#...\#c_m$ where each $c_i$ is a string containing exactly one state symbol and remaining 0’s and 1’s. The placement of the head of the machine is given by the position just after the state symbol. $c_1$ is sx i.e. the starting state s concatenated with the input x, $c_m$ is a configuration containing a final state and the relationship between any two consecutive configurations is restricted based on the transition function $\delta$. All of this can be written as a formula $\phi _R(x)$ which essentially states “there exists a number y such that the binary encoding of the number represents an accepting run of the machine on x”, making crucial use of the bit predicate and exponentiation. For details on definability in arithmetic, please see Kaye [9].   $\square$