From ed2edb82b455728bae8823d667e46510ae8d4ff9 Mon Sep 17 00:00:00 2001
From: Brandon Rozek <brozek@brandonrozek.com>
Date: Sat, 2 Mar 2024 11:49:19 -0500
Subject: [PATCH] New post

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+---
+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.
+
+
+