mirror of
https://github.com/Brandon-Rozek/website.git
synced 2024-11-25 09:36:31 -05:00
36 lines
3.1 KiB
Markdown
36 lines
3.1 KiB
Markdown
---
|
|
title: "Caching Theorem Prover Results"
|
|
date: 2024-03-02T11:15:04-05:00
|
|
draft: false
|
|
tags: ["Logic"]
|
|
math: true
|
|
medium_enabled: false
|
|
---
|
|
|
|
[Spectra](https://github.com/rairlab/spectra) is an automated planner that uses theorem proving to decide which actions to try out and determine when the goal is reached. Given an execution of Spectra on a problem within a domain, it's current architecture involves making many calls of similar structure to a theorem prover.
|
|
|
|
For example we know that $\\{ P, P \rightarrow Q \\} \vdash Q$ holds. In classical first order logic, it is also the case that $\\{ P, P \rightarrow Q, R\\} \vdash Q$ holds.
|
|
|
|
That is to say, adding additional formulae does not affect the result. In fact in classical first order logic, adding contradictory formulae also does not affect the provability of a given statement. This is due to the principle of explosion which states that from a contradiction we can derive anything.
|
|
|
|
What can we say about the other direction? How do we know when something does not hold? Let's say we know the following statement: $\\{ R, Z, P \rightarrow Q \\} \not\vdash Q$. What can we say about $\\{ Z, P \rightarrow Q \\} \not\vdash Q$?
|
|
|
|
Intuitively, if we cannot show a formula $Q$ given some set of information. Then in classical first order logic, it goes to show that we cannot show $Q$ with less information.
|
|
|
|
Let's discuss how this works in practice. Let $P$ be the set of assumptions which prove a corresponding goal. That is, given $(\Gamma_i, G_i) \in P, \Gamma_i \vdash G_i$. On the other hand, let $N$ be the set of assumptions which don't prove a corresponding goal, i.e. $(\Gamma_i, G_i) \in N, \Gamma_i \not\vdash G_i$.
|
|
|
|
Given a problem $\Gamma \vdash G$. We perform two caching checks before calling the full automated theorem prover.
|
|
|
|
1) If there exists a $(\Gamma_i, G_i) \in P$ such that $G = G_i$ and $\Gamma_i \subseteq \Gamma$, then we know that $\Gamma \vdash G$ holds.
|
|
2) On the other hand, if there exists a $(\Gamma_i, G_i) \in N$ such that $G = G_i$ and $\Gamma \subseteq \Gamma_i$, then we know that $\Gamma \not\vdash G$.
|
|
|
|
If either of those two conditions don't get matched, then we can call the full automated theorem prover to determine the result and cache it in either $P$ or $N$.
|
|
|
|
This caching technique might not work well in your application. It works well in Spectra since we're trying to prove a small set of goals and action preconditions and the assumptions consist of state variables which don't vary by much between steps.
|
|
|
|
Though keep in mind that this technique will give you flawed answers if the corresponding logic is non-monotonic. Classical first-order and propositional logic is monotonic, however, so this caching technique is safe to use in those settings.
|
|
|
|
In non-monotonic or defeasible logic, you could have a statement $B^2(\neg P)$ which defeats another statement $B^1(P)$. We can read the last example as "I have a strong belief that P does not holds and a weak belief that P holds". This depending on the defeasible logic, can change whether of not given $B^\sigma(P \rightarrow Q)$, if $B^{\sigma_i}(Q)$ holds.
|
|
|
|
|
|
|