mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Logic bot, pt. 2

 2015-03-15 
A few months ago, I set @mathslogicbot going on the long task of tweeting all the tautologies (containing 140 characters or less) in propositional calculus with the symbols \(\neg\) (not), \(\rightarrow\) (implies), \(\leftrightarrow\) (if and only if), \(\wedge\) (and) and \(\vee\) (or). My first post on logic bot contains a full explanation of propositional calculus, formulae and tautologies.

An alternative method

Since writing the original post, I have written an alternative script to generate all the tautologies. In this new method, I run through all possible strings of length 1 made with character in the logical language, then strings of length 2, 3 and so on. The script then checks if they are valid formulae and, if so, if they are tautologies.
In the new script, only formulae where the first appearances of variables are in alphabetical order are considered. This means that duplicate tautologies are removed. For example, \((b\rightarrow(b\wedge a))\) will now be counted as it is the same as \((a\rightarrow(a\wedge b))\).
You can view or download this alternative code on github. All the terms of the sequence that I have calculated so far can be viewed here and the tautologies for these terms are here.

Sequence

One advantage of this method is that it generates the tautologies sorted by the number of symbols they contain, meaning we can generate the sequence whose \(n\)th term is the number of tautologies of length \(n\).
The first ten terms of this sequence are
$$0, 0, 0, 0, 2, 2, 12, 6, 57, 88$$
as there are no tautologies of length less than 5; and, for example two tautologies of length 6 (\((\neg a\vee a)\) and \((a\vee \neg a)\)).
This sequence is listed as A256120 on OEIS.

Properties

There are a few properties of this sequence that can easily be shown. Throughout this section I will use \(a_n\) to represent the \(n\)th term of the sequence.
Firstly, \(a_{n+2}\geq a_n\). This can be explained as follows: let \(A\) be a tautology of length \(n\). \(\neg\neg A\) will be of length \(n+2\) and is logically equivalent to \(A\).
Another property is \(a_{n+4}\geq 2a_n\): given a tautology \(A\) of length \(n\), both \((a\vee A)\) and \((A\vee a)\) will be tautologies of length \(n+4\). Similar properties could be shown for \(\rightarrow\), \(\leftrightarrow\) and \(\wedge\).
Given properties like this, one might predict that the sequence will be increasing (\(a_{n+1}\geq a_n\)). However this is not true as \(a_7\) is 12 and \(a_8\) is only 6. It would be interesting to know at how many points in the sequence there is a term that is less than the previous one. Given the properties above it is reasonable to conjecture that this is the only one.
Edit: The sequence has been published on OEIS!

Similar posts

Logical contradictions
Logic bot
How OEISbot works
Raspberry Pi weather station

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "rebmun" backwards in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Feb 2020

PhD thesis, chapter ∞
PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

dates royal baby talking maths in public ucl captain scarlet binary gaussian elimination games video games logic dataset folding tube maps chalkdust magazine london underground triangles pac-man error bars news curvature oeis raspberry pi probability python statistics weather station phd geometry chebyshev london countdown bempp latex javascript dragon curves matrix of minors european cup boundary element methods wave scattering misleading statistics frobel stickers big internet math-off realhats php palindromes polynomials pythagoras noughts and crosses sorting hannah fry game of life football chess preconditioning bubble bobble graph theory ternary simultaneous equations platonic solids people maths hexapawn cambridge inverse matrices sound hats tennis final fantasy light estimation a gamut of games asteroids craft reuleaux polygons martin gardner go programming twitter world cup matrix multiplication pizza cutting map projections gerry anderson data advent calendar data visualisation inline code fractals golden spiral mathslogicbot wool radio 4 books manchester rhombicuboctahedron interpolation trigonometry braiding propositional calculus puzzles plastic ratio reddit weak imposition matrices christmas coins nine men's morris rugby determinants mathsteroids finite element method accuracy speed computational complexity flexagons numerical analysis machine learning manchester science festival matt parker cross stitch sport mathsjam folding paper signorini conditions bodmas matrix of cofactors tmip approximation arithmetic golden ratio electromagnetic field christmas card menace game show probability national lottery draughts the aperiodical sobolev spaces royal institution harriss spiral

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020