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Tree Proof Generator
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Introduction

This is the source code for https://www.umsu.de/trees, a website that implements a tableau prover for classical propositional and predicate logic, as well as some modal logics. The prover is written in Javascript and runs entirely in the browser.

Development tools

Run locally

The index.php file that serves the prover contains a few lines of php for turning debugging features on/off. To run the page locally, use

php -S localhost:1234 router.php

then go to localhost:1234.

Debugging information

Add ?debug=1 to the URL to get a detailed log of what the script is doing.

Tests

A few tests can be called in the browser at tests.html.

There’s also testdrive.php, which registers how fast the prover is at processing a sample of (valid and invalid) input formulas. This is useful for checking if a change leads to a significant speedup or slowdown or even to inaccurate classifications as valid or invalid.

Overview

Textbook tableau rules are inefficient for automated proving. Behind the scenes, we therefore use somewhat different rules. Once a closed tableau is found, we translate it back into the textbook rules to display on the page.

Free variables, unification, and run-time skolemization

A major problem for automated proving is to figure out which term to use when instantiating “gamma” formulas (of type ∀xA or ¬∃xA). Blindly trying all possibilities is inefficient. A useful trick is to use “free-variable” tableaux in which gamma formulas are initially expanded with dummy variables.

For example, in a free-variable tableau, we would expand ∀xFx as Fy, with y free. Whenever a new formula is added, we then check if some substitution of terms for the free variables (= some unification) would close the branch. For example, if the branch contains ¬Fc, we can substitute c for y and close the branch. (Even in a free-variable tableau, gamma nodes may need to be expanded more than once.)

A problem now arises for the expansion of “delta” nodes (∃xA or ¬∀xA). Suppose we’ve expanded Gamma(x) to Gamma(y). When expanding Delta(x), we want the instance term to be new, no matter what we later substitute for y. To secure this, we use run-time skolemization: we use f(y,…) as the new term for the delta expansion, where f is a new function symbol and y,… are all free variables on the branch.

(It would actually suffice to use only the free variables in Delta(x), but that makes conversion into textbook sentence tableaux hard.)

Negation normal form

A common trick in automated reasoning with tableaux is to convert the starting formulas into some normal form that is easier to work with. Clausal normal form is a popular choice, but again it makes back-conversion into textbook tableaux difficult. We instead use negation normal form. That is, the boolean operators are reduced to conjunction, disjunction, and negation, and negation is only applied to boolean atoms.

On reflection, I am not sure how much this helps to speed things up. It slightly simplifies the behind-the-scenes proof search, because we don’t need rules for expanding biconditionals etc., but it adds a layer of complexity to the code, due to the back-and-forth translation between behind-the-scenes formulas and displayed formulas, where we need e.g. the biconditional expansion rule after all.

Translation of modal formulas

Modal input formulas are converted (by the “standard translation”) into formulas of (two-sorted) predicate logic, so that the behind-the-scenes prover only works with predicate logic formulas.

Unfortunately, there are a few complications.

For one, the standard tableau rules applied to translated modal formulas easily create tableaux that are hard to convert back into textbook modal tableaux. For example, □A is translated into ∀x(¬wRx v Ax); by the standard free-variable rule for gamma expansions, this would be expanded to ¬wRy ∨ Ay, which in turn is expanded into two sub-branches. By the textbook rules, however, □A at w can only be expanded if we already have another node of form wR*. Moreover, expanding □A does not involve any kind of branching.

We could require that the expansion of ¬wRy ∨ Ay is only allowed if the ¬wRy subbranch immediately closes because it can be unified with some wR* node earlier on the branch. That would make the resulting tableau relatively easy to convert back into a textbook tableau. But then we might as well use a special rule for expanding ∀x(¬wRx v Ax), which mimics the standard rule for □A: expand directly to A*, but only if wR* is already on the branch.

This is the rule I currently use. Its downside is that it doesn’t make use of free variables at all: ∀x(¬wRx v Ax) is instantiated with a term * that already occurs in a wR* node; so we immediately unify the instance variable with an existing term. (Having no free variables for worlds, we also don’t need skolem terms for worlds in delta expansions.)

A second complication arises from constraints on the accessibility relation. In textbook tableaux, these correspond to special rules for expanding tableaux. We could instead translate the constraints into quantified formulas and add them to the starting formulas for the tableau construction. But again that would often lead to tableaux that don’t look like textbook tableaux; for example, expanding the transitivity node would lead to a two-fold branching. So the behind-the-scenes prover now has special accessibility rules mimicking the textbook rules.

All this makes it doubtful that much is gained by translating modal formulas into predicate-logic formulas. As in the case of negation normal form, the translation adds a layer of complexity to the code that should earn its keep.

On the other hand, the complications do not arise for the simplified tableau rules of S5, where we can ignore the accessibility clauses. Here we use the full free-variables machinery. The translation also simplifies the “model finder” (see below).

[Here’s a starting point for making use of free variables: if ∀x(wRx → px) and any wR* occurs on a branch, append px. If the branch also contains ¬pv, it can be closed. When closing the branch, we must take into account not only that it contains px and ¬pv, but also (i) that it contains wRv, and (ii) that px was expanded from w. So before we unify px and ¬pv, we would need to check that v is a “legitimate value” of x. Legitimate values are any names * for which wR* is on the branch. To find the legitimate values, we could note that px is derived from ∀x(wRx → px), which figures ‘w’ in the crucial place, meaning that we need substitutions whose value satisfies wR*.)

As it stands, this is too simple. Suppose we have a node ∀y(xRy → py), from an expansion of □□p at w, with free x. We also have wRv, vRu, and ¬pu. We want to expand ∀y(xRy → py) to py and unify y with u. So the rule for expanding box formulas ∀y(xRy → py) must allow expanding to y whenever some ^R* is on the tree, where ^ is a legitimate value of x. This needs more thought and/or research.]

The Modelfinder

Often there are simple countermodels that are hard to find through the tableau method. We therefore run a separate algorithm to find smallest countermodels.

In outline, this works as follows.

  1. We transform the formulas for which we want to find a model into clausal normal form, which gives us literal “constraints” that we’re trying to satisfy. For example, Fa ∧ Fb is split into two constraints, Fa and Fb; ∀x∃yRxy is turned into Rxf(x); Fa ∨ Fb is turned into the disjunctive constraint [Fa, Fb].
  2. Now we start with a domain of size 1, namely {0}. If no countermodel is found, we increase the domain to {0,1}, and so on. The interpretation of terms and predicates is initially empty. For each domain choice, we do the following:
  3. We replace free variables in the constraints by numbers. So for domain [0,1], Fx would replaced by two constraints, F0 and F1. [Numerals in formulas are never interpreted as terms, so there can be no clash.]
  4. We process the list of constraints from left to right. At each step, we look at one literal (disjunct) in one clause, with the aim of making it true. If it can’t be made true, we remove it from the list. If it can be made true, we simplify the remaining clauses by substituting all occurrences of newly interpreted terms by their values (e.g. turning Ac into A0), removing constraints for which any literal (disjunct) is known to be true, and removing constraints that known to be false.

Models for originally modal formulas have two domains, W and D. The elements of W are also natural numbers starting with 0. Accessibility conditions like reflexivity are added to the formulas for which we want to find a model.

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