A note on the expressive power of linear orders

This article shows that there exist two particular linear orders such that first-order logic with these two linear orders has the same expressive power as first-order logic with the Bit-predicate FO(Bit). As a corollary we obtain that there also exists a built-in permutation such that first-order logic with a linear order and this permutation is as expressive as FO(Bit).

💡 Research Summary

The paper investigates the expressive power of first‑order logic (FO) when equipped with built‑in linear orders, and shows that two carefully constructed linear orders suffice to capture the full power of FO with the Bit‑predicate (FO(Bit)). The authors begin by recalling known equivalences: FO with addition and multiplication, FO with various numeric predicates (e.g., Squares, Exp), and FO with the Bit‑predicate all characterize exactly the class of languages decidable in DLOGTIME‑uniform AC⁰. Contrary to the intuition that arithmetic operations require explicit numeric relations, the authors demonstrate that arithmetic can be encoded solely through order relations.

The core construction maps the natural numbers ℕ onto an infinite lower‑right triangular matrix. Column i contains i + 1 consecutive numbers, while each row is infinite. For each element x, the column index c(x), the row index r(x), and the bottom element of its column q(x) are defined. The standard order “<” corresponds to a column‑major traversal (left‑to‑right, bottom‑to‑top), whereas a second order “≺₀” corresponds to a row‑major traversal (bottom‑to‑top, left‑to‑right). Two unary predicates C and Q are introduced: C encodes, in each column i, the binary representation of i + 1 using the rows as bit positions; Q encodes, in each column i, the binary representation of q_i + 1 (where q_i = i(i + 1)/2). Thus, the presence of an element in C (or Q) at a given row directly tells whether a particular bit of the associated number is 1.

The main theorem (Theorem 3.1) states that FO(<, ≺₀, C, Q) has exactly the same expressive power as FO(Bit). The proof proceeds in two directions. First, using known FO(Bit) formulas for the basic arithmetic functions c(x), r(x), and q(x), the authors define FO(Bit) formulas that capture the relations “x < y”, “x ≺₀ y”, “C(x)”, and “Q(x)”. Second, they construct, within FO(<, ≺₀, C, Q), a formula that reproduces the Bit‑predicate. This construction relies on a suite of auxiliary formulas: ϕ_same‑col and ϕ_same‑row to test column/row equality, ϕ_q to locate the bottom element of a column, ϕ_rc to relate a row number to a column number, and formulas that identify diagonal elements and the last full column. Using these, they define formulas ϕ_q,Bit,r and ϕ_r,Bit,r that test whether a given row‑indexed bit of q(x) or r(x) is 1. To handle the addition of the binary numbers q(x) and r(x) (which yields the original number x), they introduce ϕ_q+r,carry,r to detect carry bits, and then combine the cases with and without carry to obtain ϕ_Bit,r(x, z). Finally, they define ϕ_r(x, y) to express that the row number of x equals y, and assemble the full Bit‑predicate as ϕ_Bit(x, y) ≡ ∃u(ϕ_r(u, y) ∧ ϕ_Bit,r(x, u)). This completes the equivalence proof.

In Section 4 the authors show that the two orders alone, FO(<, ≺₀), are strictly weaker than FO(Bit). They employ the Crane Beach property: a logic has this property if every language definable in it that admits a neutral letter is also definable in FO(<). FO(Bit) lacks the Crane Beach property, whereas FO(<, ≺₀) possesses it (proved via Ehrenfeucht‑Fraïssé games). Hence FO(<, ≺₀) cannot capture FO(Bit).

Section 5 eliminates the auxiliary unary predicates C and Q by embedding their information directly into a single linear order ≺, and further shows that this order can be represented as a built‑in permutation π. Consequently, FO(<, π) (or FO(<, ≺)) already has the full power of FO(Bit).

The paper concludes by emphasizing that two linear orders (or a linear order plus a permutation) suffice to simulate arbitrary arithmetic predicates, thereby providing a new perspective on the minimal structural resources needed for expressive completeness in finite model theory. It also suggests future work on characterising other fragments of FO with limited built‑in relations and exploring the impact on circuit‑complexity characterisations.