Policy Existence Problem Overview

• Policy Existence Problem is defined as determining whether a valid policy, certificate, or equilibrium exists under given model constraints across diverse domains.
• The topic covers formulations in access control, offline policy learning, and reinforcement learning, each with distinct criteria and computational guarantees.
• Methodologies include statistical validations, constructive synthesis, and combinatorial optimizations to certify policy existence under varying objectives and adversarial settings.

Searching arXiv for recent and foundational papers on “Policy Existence Problem” and closely related formulations. The policy existence problem denotes a family of formally distinct questions about whether some policy, policy witness, policy-compatible exchange, policy property, or equilibrium policy exists under a given model. In the surveyed literature, it appears as the existence of a valid authorization relation in access control (Bergé et al., 2016), the existence of an improving policy in offline policy learning (Bastani et al., 3 Jul 2026), the existence of a true off-policy policy-gradient object for a meaningful objective (Imani et al., 2018), the existence of a bounded-memory policy satisfying a PCTL* specification (Baumgartner et al., 2017), the existence of an equilibrium policy in quantified answer set programs (Diéguez et al., 7 Jan 2026), and the existence of a robust property that holds across all future extensions of a partial policy (Gheorghiu, 13 Mar 2026). The shared theme is existential certification under constraints, but the quantified object, the semantics of existence, and the computational guarantees differ substantially across domains.

1. Core meanings of “policy existence”

The literature uses “policy existence” for several non-equivalent objects. In some settings, the question is whether there exists a policy itself; in others, whether there exists a certificate that some policy class contains a desirable element; in still others, whether a current partial policy already forces a property in every completion.

Domain	Existence object	Representative criterion
Authorization and workflow	Valid authorization relation $A \subseteq U \times R$	Authorized, complete, and eligible (Bergé et al., 2016)
Offline policy learning	Boolean answer to whether $\Pi_P^{\text{imp}} \neq \varnothing$	Validator $\mathcal{V}_{\text{exist}}$ (Bastani et al., 3 Jul 2026)
Robust policy verification	Robust property $\phi$	$\Vdash_P \phi$ under all extensions (Gheorghiu, 13 Mar 2026)
Temporal/logic synthesis	Witness policy satisfying a specification	Satisfiable tableau constraints or extracted equilibrium policy (Baumgartner et al., 2017, Diéguez et al., 7 Jan 2026)
Resource exchange	Agreement-labeled exchange or rational policy	CEL provability or constructive rational-policy existence (Ceragioli et al., 2024)
Strategic competition	Pure-strategy Nash equilibrium	No profitable unilateral deviation (Lin et al., 27 Dec 2025)

A closely related but distinct line of work studies the existence of a representative compressed subset of policies. There the target is not an optimal or improving policy, but a size-$K$ compressed cover of the policy space satisfying

$$\max_{\theta\in\Theta}\min_{k\in[K]} D(d_\theta \,\|\, d_{\theta_k})\le \sigma,$$

together with high-probability certification that estimated representative occupancy distributions are close to their true counterparts (Molaei et al., 2024). That formulation is explicitly about statistical recoverability and certification of a compressed representative policy set, not a direct performance-existence theorem for control.

2. Statistical and reinforcement-learning formulations

In offline policy learning, the policy existence problem is formalized as a distinct objective alongside the optimal policy problem and the improving policy problem. The output space is Boolean,

$\mathcal{O}_{\text{exist}}=\mathbb{B},$

with validator

$$\mathcal{V}_{\text{exist}}(o,P)=\mathbf{1}(o=\mathbf{1}(\Pi_P^{\text{imp}\neq\varnothing})).$$

Thus the task is to determine, with abstention allowed, whether there exists any improving policy in the class. Within the reduction framework of the paper, the policy existence problem reduces to the improving policy problem, which in turn reduces to the optimal policy problem, so each problem is at least as easy as the next one in sample complexity (Bastani et al., 3 Jul 2026).

The same paper gives partial evidence that this hierarchy is strict. It states that the gap between the optimal policy and improving policy problems is strict, and proves that a sublinear polynomial gap exists between the improving policy and policy existence problems under natural conditions on improving policy learning algorithms. On a constructed family $P_k$, Algorithm 3 for policy existence satisfies

$\Pi_P^{\text{imp}} \neq \varnothing$0

for sufficiently large $\Pi_P^{\text{imp}} \neq \varnothing$1, while any valid monotone algorithm for the improving policy problem must satisfy

$\Pi_P^{\text{imp}} \neq \varnothing$2

which yields an $\Pi_P^{\text{imp}} \neq \varnothing$3 lower bound because $\Pi_P^{\text{imp}} \neq \varnothing$4 (Bastani et al., 3 Jul 2026). This shows that aggregated evidence may suffice to decide existence even when it does not suffice to identify which specific types should be treated.

A more general statistical formulation appears in the theory of policy transforms under incomplete and partially identified models. There the central object is the robust maximin benchmark

$\Pi_P^{\text{imp}} \neq \varnothing$5

with lower envelope

$\Pi_P^{\text{imp}} \neq \varnothing$6

The paper does not generally prove existence of an exact argmax policy; instead it proves sufficient conditions for existence of a sample-dependent decision rule that is PAMPAC-learnable, meaning that for any target error/confidence pair $\Pi_P^{\text{imp}} \neq \varnothing$7, sufficiently large $\Pi_P^{\text{imp}} \neq \varnothing$8 guarantees a rule $\Pi_P^{\text{imp}} \neq \varnothing$9 whose worst-case value is within $\mathcal{V}_{\text{exist}}$0 of the maximin benchmark with probability at least $\mathcal{V}_{\text{exist}}$1 (Russell, 2020). Ex post, it gives high-probability guarantees for an $\mathcal{V}_{\text{exist}}$2-maximin empirical rule and for policies in an empirical $\mathcal{V}_{\text{exist}}$3-level set.

Reinforcement learning supplies a different theorem-level existence result. For the off-policy objective

$\mathcal{V}_{\text{exist}}$4

the off-policy policy-gradient problem had been elusive because the outer weighting $\mathcal{V}_{\text{exist}}$5 and the Bellman recursion under $\mathcal{V}_{\text{exist}}$6 are mismatched. The paper proves that a true off-policy policy-gradient theorem does exist for general differentiable policy parameterizations, but only with emphatic weighting

$\mathcal{V}_{\text{exist}}$7

yielding

$\mathcal{V}_{\text{exist}}$8

This is an existence result for a valid gradient object, not for a policy itself. The paper further reports counterexamples in which OffPAC and DPG converge to the wrong solution whereas ACE finds the optimal solution (Imani et al., 2018).

Policy compression gives yet another statistical-existence formulation. The target is the existence of $\mathcal{V}_{\text{exist}}$9 representative policies $\phi$0 such that

$\phi$1

The paper is explicit that it does not prove that a near-optimal reward-maximizing policy exists in the compressed set; instead it studies when the representative occupancy distributions can be learned or certified from data with high confidence, in known-model and unknown-model settings, using total variation and order-2 Rényi divergence (Molaei et al., 2024).

3. Authorization, workflow, completion, and repair

In access control, the authorization policy existence problem is the existence of a valid authorization relation. Given users $\phi$2, resources $\phi$3, a base authorization relation $\phi$4, and constraints $\phi$5, an authorization relation $\phi$6 is authorized if $\phi$7, complete if $\phi$8 for every $\phi$9, eligible if it satisfies every constraint in $\Vdash_P \phi$0, and valid if it is authorized, complete, and eligible. D-APEP asks whether there exists a valid authorization relation; O-APEP asks for a best valid authorization relation under an objective. The paper shows that APEP generalizes WSP, with the global cardinality constraint $\Vdash_P \phi$1 recovering the one-user-per-resource setting, and proves fixed-parameter tractability for broad classes of user-independent bounded constraints parameterized by $\Vdash_P \phi$2 (Bergé et al., 2016).

The same line of work also develops a more practically permissive optimization version. Valued APEP replaces all-or-nothing validity by a weighted objective

$\Vdash_P \phi$3

where

$\Vdash_P \phi$4

The task is to return a complete, optimal authorization relation of minimum weight. The main generic tractability result is that Valued APEP is fixed-parameter tractable when all weighted constraints are user-independent and the constraint family is $\Vdash_P \phi$5-wbounded; user-independence alone and $\Vdash_P \phi$6-wboundedness alone are insufficient. The algorithmic core is the user profile

$\Vdash_P \phi$7

together with profile enumeration and a minimum-cost perfect matching construction (Crampton et al., 2021).

A closely related completion-and-repair formulation appears in XML write-access control. There policies are pairs $\Vdash_P \phi$8 over valid update access types. A policy is consistent iff no forbidden atomic update can be simulated by a sequence of allowed updates. The paper proves that consistency is decidable in PTIME, that a partial policy is quasiconsistent iff it has a consistent total extension, and that

$\Vdash_P \phi$9

is necessary and sufficient for existence of a consistent total extension, where $K$0 is the closure operator capturing privileges forced by simulation. When this condition holds, the unique least-privilege consistent total extension is

$K$1

The paper also studies repair by deleting privileges and proves that the total-repair and partial-repair problems are NP-complete (0708.2076).

4. Universal existence, robust forcing, and monitoring distinctions

A distinct formulation asks not whether some completion exists, but whether every completion satisfies a property. Robust property verification formalizes this with the support judgment $K$2, defined semantically over all extensions $K$3. The central result is monotonicity: $K$4 Thus a robustly verified property persists under policy extension. The paper is explicit that this addresses the universal side of the policy existence problem: whether the current partial policy already commits the system to $K$5 in every future coherent elaboration. It does not solve the existential completion problem $K$6 (Gheorghiu, 13 Mar 2026).

That framework is proof-theoretically executable. It introduces a translation $K$7 from robust formulas to a second-order logic-programming language and proves soundness and completeness: $K$8 As a result, universal quantification over all future policy extensions is reduced to finitary proof search. This turns a universal existence problem for robust properties into a decidable verification problem within the paper’s fragment (Gheorghiu, 13 Mar 2026).

History-based transaction monitoring provides an instructive contrast. There the main decision problem is model checking a fixed policy $K$9 against a finite history $$\max_{\theta\in\Theta}\min_{k\in[K]} D(d_\theta \,\|\, d_{\theta_k})\le \sigma,$$0, namely whether $$\max_{\theta\in\Theta}\min_{k\in[K]} D(d_\theta \,\|\, d_{\theta_k})\le \sigma,$$1. The paper does not define a standalone policy existence problem in the sense of synthesizing a formula with a desired semantics. It does, however, study two partial-observability variants: potential satisfiability, where there exists a completion of unknown parameters making the fixed policy true, and adherence, where all completions make it true. Model checking for $$\max_{\theta\in\Theta}\min_{k\in[K]} D(d_\theta \,\|\, d_{\theta_k})\le \sigma,$$2 is PSPACE-complete, and potential satisfiability and adherence are decidable under a restriction to linear arithmetic (0903.2904). This sharply separates existence of a policy from existence of a completion for a fixed policy.

5. Synthesis, extraction, and constructive witnesses

In MDP policy synthesis with temporal logic constraints, the existence problem is explicit. Given a finite MDP

$$\max_{\theta\in\Theta}\min_{k\in[K]} D(d_\theta \,\|\, d_{\theta_k})\le \sigma,$$3

a partially specified finite-memory policy

$$\max_{\theta\in\Theta}\min_{k\in[K]} D(d_\theta \,\|\, d_{\theta_k})\le \sigma,$$4

and a PCTL* state formula $$\max_{\theta\in\Theta}\min_{k\in[K]} D(d_\theta \,\|\, d_{\theta_k})\le \sigma,$$5, the question is whether there exists an action function $$\max_{\theta\in\Theta}\min_{k\in[K]} D(d_\theta \,\|\, d_{\theta_k})\le \sigma,$$6 such that

$$\max_{\theta\in\Theta}\min_{k\in[K]} D(d_\theta \,\|\, d_{\theta_k})\le \sigma,$$7

Because unrestricted history-dependent synthesis is undecidable, the paper fixes the finite-memory skeleton $$\max_{\theta\in\Theta}\min_{k\in[K]} D(d_\theta \,\|\, d_{\theta_k})\le \sigma,$$8 in advance and synthesizes only $$\max_{\theta\in\Theta}\min_{k\in[K]} D(d_\theta \,\|\, d_{\theta_k})\le \sigma,$$9. Its tableau procedure generates a system of arithmetic constraints, generally nonlinear because of products in the $\mathcal{O}_{\text{exist}}=\mathbb{B},$0-rule. The main existence equivalence is that a satisfying bounded-memory policy exists iff the tableau-generated constraint system has a solution, with soundness, completeness, and termination proved for the method (Baumgartner et al., 2017).

Quantified answer set programs provide a more explicitly strategic formulation. A policy is a strategy tree over the quantifier prefix: existential quantifiers choose one Boolean value, while universal quantifiers branch over both Boolean values. Under the paper’s equilibrium-logic semantics, a policy exists for $\mathcal{O}_{\text{exist}}=\mathbb{B},$1 iff there exists an equilibrium policy, equivalently an equilibrium configuration $\mathcal{O}_{\text{exist}}=\mathbb{B},$2. The extraction algorithm $\mathcal{O}_{\text{exist}}=\mathbb{B},$3 computes the set of equilibrium policies exactly; for the initial call,

$\mathcal{O}_{\text{exist}}=\mathbb{B},$4

and $\mathcal{O}_{\text{exist}}=\mathbb{B},$5 is the set of all equilibrium policies of $\mathcal{O}_{\text{exist}}=\mathbb{B},$6. Thus policy existence is decided by whether $\mathcal{O}_{\text{exist}}=\mathbb{B},$7 (Diéguez et al., 7 Jan 2026).

Resource-exchange systems add a logic in which proofs witness agreements. In an exchange environment, states are resource allocations and transitions are labeled by exchanges. Coalition policies consist of rules $\mathcal{O}_{\text{exist}}=\mathbb{B},$8, meaning that a coalition is willing to perform $\mathcal{O}_{\text{exist}}=\mathbb{B},$9 in return for $$\mathcal{V}_{\text{exist}}(o,P)=\mathbf{1}(o=\mathbf{1}(\Pi_P^{\text{imp}\neq\varnothing})).$$0. The paper compiles states, exchanges, and policies into CEL formulas and proves that

$$\mathcal{V}_{\text{exist}}(o,P)=\mathbf{1}(o=\mathbf{1}(\Pi_P^{\text{imp}\neq\varnothing})).$$1

is valid iff there exists an agreement $$\mathcal{V}_{\text{exist}}(o,P)=\mathbf{1}(o=\mathbf{1}(\Pi_P^{\text{imp}\neq\varnothing})).$$2 such that

$$\mathcal{V}_{\text{exist}}(o,P)=\mathbf{1}(o=\mathbf{1}(\Pi_P^{\text{imp}\neq\varnothing})).$$3

With the Cut rule added, the same correspondence holds for exchange environments with debts. The paper also proves that, given valuation functions of all agents in a coalition $$\mathcal{V}_{\text{exist}}(o,P)=\mathbf{1}(o=\mathbf{1}(\Pi_P^{\text{imp}\neq\varnothing})).$$4, there exists a rational policy for $$\mathcal{V}_{\text{exist}}(o,P)=\mathbf{1}(o=\mathbf{1}(\Pi_P^{\text{imp}\neq\varnothing})).$$5 (Ceragioli et al., 2024).

6. Equilibria, geometric nonemptiness, and decidability frontiers

In two-party policy competition, the existence problem becomes equilibrium existence. Each party chooses a policy vector from

$$\mathcal{V}_{\text{exist}}(o,P)=\mathbf{1}(o=\mathbf{1}(\Pi_P^{\text{imp}\neq\varnothing})).$$6

voter utility is an inner product, and the winning probability is affine isotonic: $$\mathcal{V}_{\text{exist}}(o,P)=\mathbf{1}(o=\mathbf{1}(\Pi_P^{\text{imp}\neq\varnothing})).$$7 Payoff is expected utility to a party’s supporters. Under these assumptions, the paper proves existence of a pure-strategy Nash equilibrium in both one- and multi-dimensional settings. The one-dimensional case yields a closed-form equilibrium; the multi-dimensional case is handled by reduction to the span of $$\mathcal{V}_{\text{exist}}(o,P)=\mathbf{1}(o=\mathbf{1}(\Pi_P^{\text{imp}\neq\varnothing})).$$8 and $$\mathcal{V}_{\text{exist}}(o,P)=\mathbf{1}(o=\mathbf{1}(\Pi_P^{\text{imp}\neq\varnothing})).$$9, best-response structure, and fixed-point reasoning. The paper also constructs a counterexample showing the game’s non-monotonicity, studies decentralized projected gradient ascent empirically, and gives a grid-based algorithm for computing an $P_k$0-approximate PSNE in time polynomial in the input size and $P_k$1 (Lin et al., 27 Dec 2025).

For ML-rich policy languages, the existence problem often becomes nonemptiness of rule regions or overlap of rule predicates. The paper on probabilistic ML predicates identifies a three-level decidability hierarchy. For crisp predicates, conflict detection is decidable via SAT or linear integer arithmetic. For embedding predicates, co-firing reduces to spherical cap intersection, with pairwise overlap characterized by

$P_k$2

For classifier predicates, calibration-conflict detection is undecidable without knowledge of the input distribution $P_k$3. The same paper gives a constructive nonexistence result for overlap within a signal group: under Voronoi normalization with

$P_k$4

and threshold $P_k$5, at most one signal in the group fires for any input $P_k$6 as $P_k$7 (Liu et al., 18 Mar 2026).

Taken together, these results show that “policy existence” has at least five technically distinct readings: existence of a feasible policy object, existence of a true optimization derivative or value functional, existence of a robust property under all completions, existence of a proof or extracted strategy witnessing realizability, and existence of an equilibrium or nonempty admissible region. A plausible implication is that progress on one formulation does not automatically transfer to the others: a framework may certify existence without construction, construct without robustness, or prove universal support without solving existential synthesis. The surveyed literature therefore treats the policy existence problem not as a single theorem, but as a family of existence, certification, and realizability problems indexed by semantics, objective, and adversarial structure.