Overview

Allen's algebra is industry standard for reasoning about temporal interval relationships. Good overview of Allen's algebra here: http://www.ics.uci.edu/~alspaugh/cls/shr/allen.html For practical purposes it is convenient to use less complicated sub-algebras (e.g. SQL:2011 approach).

Assumptions about a and b where SQL:2008 equivalent is provided:

• inclusive-exclusive model, i.e. [from, to)
• from-to interval is not empty, i.e. from < to - follows natural time flow and not empty

Allen's Relations

Relation	Name	Allen	Signature	SQL:2011	SQL:2008
	before	<	<<<<	PRECEDES	a.to < b.from
	meets	m	<=<<	IMMEDIATELY PRECEDES	a.to = b.from
	overlaps	o	<><<	OVERLAPS	a.from < b.from AND b.from < a.to AND a.to < b.to
	finished-by	fi	<><=	CONTAINS	a.from < b.from AND a.to = b.to
	contains	di	<>>>	CONTAINS	a.from < b.from AND a.to > b.to
	starts	s	=><<	OVERLAPS	a.from = b.from AND a.to < b.to
	equals	=	=><=	EQUALS	a.from = b.from AND a.to = b.to
	started-by	si	=<<>	CONTAINS	a.from = b.from AND a.to > b.to
	during	d	>><<	OVERLAPS	a.from > b.from AND a.to < b.to
	finishes	f	>><=	OVERLAPS	a.from > b.from AND a.to = b.to
	overlapped-by	oi	>><>	OVERLAPS	b.from < a.from AND a.from < b.to AND b.to < a.to
	met-by	mi	>>=>	IMMEDIATELY SUCCEEDS	a.from = b.to
	after	>	>>>>	SUCCEEDS	a.from > b.to

Signature = relation between endpoints a.from-b.from / a.to-b.from / a.from-b.to / a.to-b.to (Kovarik)

source:

• Asplaugh T. Allen's Interval Algebra http://www.ics.uci.edu/~alspaugh/cls/shr/allen.html
• Jonhston T. 2014 Bitemporal Data: Theory and Practice. Morgan-Kaufmann. (ISBN: 978-0124080676)
• Kulkarni K. et al. 2012 Temporal features in SQL:2011 SIGMOD Record
• Kovarik V. et al. 2000 An interval-based temporal algebra based on binary encoding of point relations IJIS

SQL:2011 Relations

Predicate	Allens relation	Description	SQL:2008	Set Properties
CONTAINS	(fi, di, =, si)	first interval includes every point of the second	a.from <= b.from AND a.to >= b.to	reflexive, symmetric
OVERLAPS	(o, fi, di, s, =, si, d, f, oi)	at least one time point in common	a.to > b.from AND a.from < b.to	reflexive, antisymmetric
EQUALS	(=)	intervals equal	a.from = b.from AND a.to = b.to	reflexive, symmetric, transitive
PRECEDES	(<, m)	first interval is before the second	a.to <= b.from	irreflexive, asymmetric, transitive
SUCCEEDS	(>, mi)	first interval is after the second	a.from >= b.to	irreflexive, asymmetric, transitive
IMMEDIATELY PRECEDES	(m)	first interval is before and adjacent to the second	a.to = b.from	irreflexive, asymmetric
IMMEDIATELY SUCCEEDS	(mi)	first interval is after and adjacent to the second	a.from = b.to	irreflexive, asymmetric