---
id: 20250818132022
title: Temporal Logic
type: permanent
created: 2025-08-18T17:20:22Z
modified: 2025-08-21T16:42:25Z
tags: []
---

# Temporal Logic

## What is Temporal Logic?

An extension of **predicate logic** that allows reasoning
about how statements evolve over **time**.  
- Used heavily in **formal verification**, **model
checking**, and **control systems safety**.  
- Reasoning is over **infinite traces** (sequences of
states).  

---

## Core Temporal Operators

| Symbol | LaTeX | Name | Meaning |
|--------|-------|------|---------|
| X φ | `X \varphi` | Next | φ holds in the **next** state |
| F φ | `\Diamond \varphi` | Eventually | φ will hold at some time in the **future** |
| G φ | `\Box \varphi` | Globally (Always) | φ holds at **all future times** |
| φ U ψ | `\varphi \, U \, \psi` | Until | φ must hold until ψ holds |

---

## Example Specifications

1. **Safety:**  
   *“The reactor temperature never exceeds 600°C.”*  
   $G\,(temp < 600)$

2. **Liveness:**  
   *“The pump will eventually turn on.”*  

    $F(PumpOn)$

3. **Response:**  
   *“Whenever a request occurs, it will eventually be granted.”*  
   $G\,(request \rightarrow F\,grant)$

4. **Stability:**  
   *“If the valve opens, it stays open until pressure is safe.”*  
    $G\,(ValveOpen\rightarrow (ValveOpen\;U\;PressureSafe))$

## Why It Matters
- **Formal Methods:** Express constraints on **system behaviors over time**.  
- **Model Checking:** Tools like SPIN, NuSMV, Spot verify LTL specs.  
- **Engineering Applications:**  
  - Ensure control systems obey safety invariants.  
  - Specify liveness goals for controllers (e.g., “eventually stabilize temperature”).  
  - Used in **TLA+**, runtime monitoring, and shield synthesis.

## Related Notes
[[strix]] - a reactive synthesis tool using temporal logic
specs