FOTRINV: Qℝ-Complete Real Theory Fragment

• FOTRINV is a restricted fragment of the first-order theory of reals defined solely by addition and inversion constraints with alternating quantifiers over fixed compact intervals.
• It is proven Qℝ-complete through polynomial-time reductions utilizing structural methods like closed-constraints and compactification lemmas, establishing tight complexity bounds.
• FOTRINV serves as a canonical kernel for continuous adversarial games and geometric realizability, underpinning fundamental lower bounds in real computation and decision problems.

FOTRINV refers to a restrictive but expressive fragment of the first-order theory of the reals (FOTR). It is defined by formulas involving only addition and inversion constraints, with alternating quantifiers ranging over fixed compact intervals. FOTRINV has been proven to be $\text{Q}\mathbb{R}$-complete, and thus precisely characterizes the computational complexity of a broad class of continuous games and geometric realizability problems where choices are over the continuum. This role as a canonical $\text{Q}\mathbb{R}$-complete problem makes FOTRINV both a subject of intrinsic interest in mathematical logic and complexity theory, and a key tool in establishing lower bounds for decision problems in real computation and continuous combinatorial games (Meijer et al., 2 Dec 2025).

1. Formal Definition

A FOTRINV sentence is constructed in the following prenex form: $$\exists x_1 \in I_\exists\, \forall x_2 \in I_\forall\, \exists x_3 \in I_\exists\, \cdots\, Q_n x_n \in I_{Q_n}:\ \Phi(x_1,\ldots,x_n)$$ where

• $I_\exists = [\frac{1}{2},2]$ for existentially quantified variables,
• $I_\forall = [\frac{3}{4}, 1]$ for universally quantified variables,
• $Q_i \in \{\exists, \forall\}$ (with quantifiers alternating),
• $\Phi$ is a conjunction of atomic equalities of the following restricted forms:
  • $x = 1$
  • $x + y = z$
  • $x \cdot y = 1$

The decision problem FOTRINV asks, given such a sentence, whether it is true over the real numbers with the specified quantifier structure and constraints (Meijer et al., 2 Dec 2025).

2. Complexity Class $\text{Q}\mathbb{R}$ and FOTRINV-Completeness

The complexity class $\text{Q}\mathbb{R}$ is defined as the class of decision problems many-one (polynomial time) reducible to FOTR—the general first-order theory of the reals. Equivalently, a language $L$ is in $\text{Q}\mathbb{R}$ if there is a real-RAM (or BSS) machine $M$ running in time polynomial in the input $|w|$ and real parameters $x_1, y_1, ..., x_k, y_k$ such that

$$w \in L \iff \exists x_1 \in \mathbb{R}^n\, \forall y_1 \in \mathbb{R}^n\, \cdots\, \exists x_k \in \mathbb{R}^n\, \forall y_k \in \mathbb{R}^n \ [M(w, x_1, y_1, ..., x_k, y_k) = 1]$$

The class $\text{Q}\mathbb{R}$ contains the existential theory of the reals (ETR) and is contained in EXPTIME; Fischer–Rabin and Berman provided unconditional exponential-time lower bounds (Meijer et al., 2 Dec 2025).

FOTRINV is shown to be $\text{Q}\mathbb{R}$-complete:

• Membership: Any FOTRINV sentence is an FOTR sentence, hence is in $\text{Q}\mathbb{R}$.
• Hardness: Any FOTR sentence can be transformed, in polynomial time, into an equivalent FOTRINV sentence via:
  • Elimination of inequalities and unbounded domains using standard algebraic geometry (Basu–Pollack–Roy quantifier elimination and the closed-constraints lemma).
  • Replacement of general multiplication by inversion and defined scaling (Abrahamsen–Adamaszek–Miltzow gadgets), restricted to compact intervals.
  • Range reduction to the prescribed compact intervals $[\frac{1}{2},2]$ and $[\frac{3}{4},1]$ using affine rescaling and additional constraints.

These reductions preserve logical equivalence, and the polynomial-time nature ensures that FOTRINV is $\text{Q}\mathbb{R}$-hard, thereby establishing completeness (Meijer et al., 2 Dec 2025).

3. Structural Reductions and the Compactification Lemma

The proof of FOTRINV's completeness centers around a sequence of structural reductions:

• Closed-Constraints Lemma: Every Boolean combination of polynomial constraints with inequalities can be rewritten as a single polynomial equation $f=0$ with auxiliary variables.
• Monotone-Limit Swap: Under appropriate monotonicity and closure assumptions, quantifiers and limiting processes can be interchanged and ultimately eliminated in favor of large but finite bounds.
• Limit-Realization Lemma: For prenex FOTR formulas, one can find explicit (though doubly/triply exponential) numeric bounds to replace all limit quantifiers by bounded quantifiers, preserving truth.
• Reduction to Addition and Inversion: General multiplicative constraints $x y = z$ are simulated by systems involving only $x y = 1$ (inversion) and addition, coupled with scaling and variable substitution, again relying on variables confined to compact intervals.

These results enable the complete compactification of arbitrary FOTR sentences to the FOTRINV form, a key step in making the problem an effective starting point for reductions (Meijer et al., 2 Dec 2025).

4. FOTRINV as a Kernel for Continuous Games and Geometric Realizability

FOTRINV acts as a canonical $\text{Q}\mathbb{R}$-complete problem for reductions to numerous continuous adversarial games and geometric decision problems. For a range of "devil's games"—where two players alternately pick real values subject to continuous constraints—the alternation of quantifiers in FOTRINV directly mirrors the sequence of moves:

• Packing Game: Players alternately place pieces, with the spatial constraints encoding FOTRINV instances involving only addition and inversion among compact-real coordinates.
• Planar Extension Game: Vertices and edges are added by alternating players; the extension constraints are enforced by geometric gadgets that simulate addition and inversion constraints.
• Order-Type Game: Players alternately place points in a planar order, where order-type constraints can be realized combinatorially through von Staudt constructions enforcing $x + y = z$ and $x y = 1$.

In every case, reductions from FOTRINV show that these games are $\text{Q}\mathbb{R}$-complete, establishing their intrinsic algorithmic intractability and aligning their complexity with the general first-order theory of the reals (Meijer et al., 2 Dec 2025).

5. Complexity-Theoretic Consequences

The $\text{Q}\mathbb{R}$-completeness of FOTRINV has several foundational consequences:

• Intractability: Unconditional exponential-time lower bounds (Berman [1980]) carry over to FOTRINV, as well as to all continuous adversarial games and decision problems reducible from it.
• Hierarchy Characterization: Each level $\Sigma_k^R, \Pi_k^R$ of the real-RAM (or BSS) polynomial hierarchy can be characterized using bounded-interval FOTRINV formulas with $k$ quantifier alternations.
• Reduction Framework: FOTRINV serves as a kernel for showing $\text{Q}\mathbb{R}$-hardness of numerous geometric and combinatorial problems in the continuous setting, analogous to the role of SAT or ETR in classical and real complexity.

This suggests that significant gaps remain between the complexity of discrete and continuous games, and that FOTRINV encodes a complexity-theoretic threshold for continuous problems with quantifier alternation over the reals (Meijer et al., 2 Dec 2025).

6. Comparison with Other Inverse and Real-Theoretic Problems

FOTRINV differs fundamentally from continuous inverse problems such as those arising in PDE-constrained optimization (e.g., inverse optical tomography (Katzourakis, 2018)) in both structure and complexity. Whereas PDE inverse problems often involve functional minimization and are analyzed using variational and PDE theory, FOTRINV deals with finite-dimensional sentences constrained to addition and inversion, and occupies a sharply delineated position in the theory of real computation and logic.

A plausible implication is that, while both classes of problems may share real-valued variables and constraints, the presence of quantifier alternation and the specific algebraic structure of FOTRINV leads to distinctly higher complexity and different algorithmic barriers compared to variational PDE settings, where optimality conditions and approximations can be employed (Katzourakis, 2018).

7. Summary Table: Defining Features of FOTRINV

Feature	Description	Constraint/Form
Quantifier Structure	Alternating $\exists$, $\forall$	Fixed: $\exists \forall \exists ...$
Variable Ranges	Compact intervals	$[\frac{1}{2},2]$ and $[\frac{3}{4},1]$
Atomic Formulas	Addition, inversion, constants	$x=1$, $x+y=z$, $x\cdot y=1$
Completeness	$\text{Q}\mathbb{R}$-complete	Under polynomial-time reductions
Reduction Kernel	For continuous games & geometric problems	Packing, Planar Extension, Order-Type

FOTRINV thus serves as a central object in real complexity theory, underpinning the analysis of continuous games, real computation, and the expressiveness of the first-order theory of the reals under quantifier alternation and bounded domains (Meijer et al., 2 Dec 2025).