# P3134: Decidability of the real exponential field

- ID: `P3134`
- Reference: `real-exponential-field-decidability`
- Page: https://theoremdb.org/statements/P3134
- Record maturity: Reviewed problem with recorded work

## Problem

Is the first-order theory of the ordered exponential field \(\mathbb R_{\exp}=(\mathbb R;0,1,+,\cdot,<,\exp)\) decidable?

### Context

Known frontier: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability.

Open boundary: An unconditional decision procedure or undecidability proof remains unknown.

### Problem setup

- **Definition (first-order theory).** All sentences in the displayed language true in the real exponential field.
- **Definition (decidable).** Membership in that set of true sentences is algorithmically decidable.
- **Remark.** Tarski proved decidability for real closed fields without exp. Adding the total exponential function preserves o-minimality, yet no unconditional decision procedure is known.

### What counts as a solution

- Give a terminating algorithm deciding every sentence of R_exp.
- Or prove undecidability by an effective reduction.

## Status

OPEN as checked on 2026-08-01. Strongest checked neighboring result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Exact unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (Current status and exact unresolved remainder).** OPEN as checked on 2026-08-01. Strongest checked neighboring result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Exact unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability.

The exact unresolved remainder is: An unconditional decision procedure or undecidability proof remains unknown.

A complete resolution must meet the following acceptance conditions:
- Give a terminating algorithm deciding every sentence of R_exp.
- Or prove undecidability by an effective reduction.

### Background and intake notes

- Original intake status: OPEN as checked on 2026-08-01. Strongest checked neighboring result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Exact unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown.
- The release review checked 2 structured sources on 2026-08-01.
- Equivalent-formulation queries: decidability real exponential field open 2026; Tarski exponential function problem current status Schanuel
- Strongest checked neighboring result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability.
- Exact unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown.

### Other known results

- **Claim 2** (supported): R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): An unconditional decision procedure or undecidability proof remains unknown.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `real-exponential-field-decidability`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>A. Macintyre and A. Wilkie, On the decidability of the real exponential field, Kreiseliana (1996). bibliographic record for pages 441-467. bibliographic record for pages 441-467 https://ris.uni-paderborn.de/record/18304
   - Also cited at A. Macintyre and A. Wilkie, On the decidability of the real exponential field, Kreiseliana (1996). bibliographic record for pages 441-467
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Proves decidability assuming a form of Schanuel's conjecture.
   - Source used to assess the problem's recorded status.
   - For Decidability of the real exponential field: This is the dated publication status for the canonical target Decidability of the real exponential field.
   - Source named by the research packet.
2. <a id="reference-2"></a>L. van den Dries, Some results about exponential fields, survey (2024). Tarski exponential problem discussion https://www.numdam.org/item/10.24033/msmf.315.pdf
   - journal_article; secondary source; checked 2026-08-01
   - Source use: original_summary
   - Describes the decision problem as open and reviews current model theory.
   - Source used to assess the problem's recorded status.
   - For Decidability of the real exponential field: Describes the decision problem as open and reviews current model theory.