Derive Right-Branching Proper Interval Algebra
NOTE: This algebra derivation restricts intervals to be proper intervals, similar to Allen’s time interval algebra. So, time points are not permitted. However, this algebra adds a new point relation, incomparable, denoted by, r~, which expresses the relationship between two points on different time branches. The transitivity/composition table of the right-branching point algebra is defined so that time branches to the right (i.e., into the future). See the section, below, titled, “Right-Branching Point Algebra”.
References
“Maintaining Knowledge about Temporal Intervals” by J.F. Allen - Allen’s original paper
Allen’s Interval Algebra or here - summarizes Allen’s algebra of proper time intervals
“Intervals, Points, and Branching Time” by A.J. Reich - basis for the extensions here to Allen’s algebra
W3C Time Ontology in OWL - temporal vocabulary used here is based on the W3C vocabulary of time
bitsets Python package - used to implement Algebra relation sets and operations
NetworkX Python package - used to represent directed graph of constraints
Python format string syntax - used in Algebra summary method
Spatial Ontology - I’m still looking for a standard spatial vocabulary; maybe start here
Qualitative Spatial Relations (QSR) Library - an alternative library to the one defined here
Dependencies
import os
import qualreas as qr
import numpy as np
import sys
sys.setrecursionlimit(10000)
path = os.path.join(os.getenv('PYPROJ'), 'qualreas')
Deriving the RB Proper Interval Algebra from RB Point Algebra
Right-Branching Point Algebra
pt_alg = qr.Algebra(os.path.join(path, "Algebras/Right_Branching_Point_Algebra.json"))
pt_alg.summary()
Algebra Name: Right_Branching_Point_Algebra
Description: Right-Branching Point Algebra
Equality Rels: =
Relations:
NAME (SYMBOL) CONVERSE (ABBREV) REFLEXIVE SYMMETRIC TRANSITIVE DOMAIN RANGE
LessThan ( <) GreaterThan ( >) False False True Pt Pt
Equals ( =) Equals ( =) True True True Pt Pt
GreaterThan ( >) LessThan ( <) False False True Pt Pt
Incomparable ( r~) Incomparable ( r~) False True False Pt Pt
Domain & Range Abbreviations:
Pt = Point
PInt = Proper Interval
qr.print_point_algebra_composition_table(pt_alg)
Right_Branching_Point_Algebra Elements: <, =, >, r~ ============================== rel1 ; rel2 = composition ============================== < < < < = < < > <|=|> < r~ <|r~ ------------------------------ = < < = = = = > > = r~ r~ ------------------------------ > < <|=|>|r~ > = > > > > > r~ r~ ------------------------------ r~ < r~ r~ = r~ r~ > >|r~ r~ r~ <|=|>|r~ ------------------------------
Derive Right-Branching Proper Interval Algebra as a Dictionary
The definition of less than, below, either restricts intervals to be proper (‘<’) or allows intervals to be degenerate (‘=|<’) (i.e., integrates points and intervals).
#less_than_rel = '=|<'
less_than_rel = '<'
rb_proper_alg_name="Derived_Right_Branching_Proper_Interval_Algebra"
rb_proper_alg_desc="Extended right-branching proper interval algebra derived from point relations"
%time test_rb_proper_alg_dict = qr.derive_algebra(pt_alg, less_than_rel, name=rb_proper_alg_name, description=rb_proper_alg_desc)
19 consistent networks
CPU times: user 2.18 s, sys: 436 ms, total: 2.62 s
Wall time: 2.04 s
test_rb_proper_alg_dict
{'Name': 'Derived_Right_Branching_Proper_Interval_Algebra',
'Description': 'Extended right-branching proper interval algebra derived from point relations',
'Relations': {'B': {'Name': 'Before',
'Converse': 'BI',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': True},
'BI': {'Name': 'After',
'Converse': 'B',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': True},
'D': {'Name': 'During',
'Converse': 'DI',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': True},
'DI': {'Name': 'Contains',
'Converse': 'D',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': True},
'E': {'Name': 'Equals',
'Converse': 'E',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': True,
'Symmetric': True,
'Transitive': True},
'F': {'Name': 'Finishes',
'Converse': 'FI',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': True},
'FI': {'Name': 'Finished-by',
'Converse': 'F',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': True},
'M': {'Name': 'Meets',
'Converse': 'MI',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': False},
'MI': {'Name': 'Met-By',
'Converse': 'M',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': False},
'O': {'Name': 'Overlaps',
'Converse': 'OI',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': False},
'OI': {'Name': 'Overlapped-By',
'Converse': 'O',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': False},
'RB': {'Name': 'Right-Before',
'Converse': 'RBI',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': True},
'RBI': {'Name': 'Right-After',
'Converse': 'RB',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': True},
'RO': {'Name': 'Right-Overlaps',
'Converse': 'ROI',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': False},
'ROI': {'Name': 'Right-Overlapped-By',
'Converse': 'RO',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': False},
'RS': {'Name': 'Right-Starts',
'Converse': 'RS',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': True,
'Transitive': False},
'R~': {'Name': 'Right-Incomparable',
'Converse': 'R~',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': True,
'Transitive': False},
'S': {'Name': 'Starts',
'Converse': 'SI',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': True},
'SI': {'Name': 'Started-By',
'Converse': 'S',
'Domain': ['ProperInterval'],
'Range': ['ProperInterval'],
'Reflexive': False,
'Symmetric': False,
'Transitive': True}},
'TransTable': {'B': {'B': 'B',
'BI': 'B|BI|D|DI|E|F|FI|M|MI|O|OI|S|SI',
'D': 'B|D|M|O|S',
'DI': 'B',
'E': 'B',
'F': 'B|D|M|O|S',
'FI': 'B',
'M': 'B',
'MI': 'B|D|M|O|S',
'O': 'B',
'OI': 'B|D|M|O|S',
'RB': 'B',
'RBI': 'B|D|M|O|RBI|RO|ROI|RS|S',
'RO': 'B',
'ROI': 'B|D|M|O|S',
'RS': 'B',
'R~': 'B|RB|R~',
'S': 'B',
'SI': 'B'},
'BI': {'B': 'B|BI|D|DI|E|F|FI|M|MI|O|OI|RB|RBI|RO|ROI|RS|R~|S|SI',
'BI': 'BI',
'D': 'BI|D|F|MI|OI|RBI|ROI',
'DI': 'BI',
'E': 'BI',
'F': 'BI',
'FI': 'BI',
'M': 'BI|D|F|MI|OI|RBI|ROI',
'MI': 'BI',
'O': 'BI|D|F|MI|OI|RBI|ROI',
'OI': 'BI',
'RB': 'R~',
'RBI': 'RBI',
'RO': 'RBI',
'ROI': 'RBI',
'RS': 'RBI',
'R~': 'R~',
'S': 'BI|D|F|MI|OI|RBI|ROI',
'SI': 'BI'},
'D': {'B': 'B',
'BI': 'BI',
'D': 'D',
'DI': 'B|BI|D|DI|E|F|FI|M|MI|O|OI|S|SI',
'E': 'D',
'F': 'D',
'FI': 'B|D|M|O|S',
'M': 'B',
'MI': 'BI',
'O': 'B|D|M|O|S',
'OI': 'BI|D|F|MI|OI',
'RB': 'B|RB|R~',
'RBI': 'RBI',
'RO': 'B|D|M|O|RBI|RO|ROI|RS|S',
'ROI': 'D|RBI|ROI',
'RS': 'D|RBI|ROI',
'R~': 'R~',
'S': 'D',
'SI': 'BI|D|F|MI|OI'},
'DI': {'B': 'B|DI|FI|M|O|RB|RO',
'BI': 'BI|DI|MI|OI|SI',
'D': 'D|DI|E|F|FI|O|OI|RO|ROI|RS|S|SI',
'DI': 'DI',
'E': 'DI',
'F': 'DI|OI|SI',
'FI': 'DI',
'M': 'DI|FI|O|RO',
'MI': 'DI|OI|SI',
'O': 'DI|FI|O|RO',
'OI': 'DI|OI|SI',
'RB': 'RB',
'RBI': 'RBI|RO|ROI|RS',
'RO': 'RO',
'ROI': 'RO|ROI|RS',
'RS': 'RO',
'R~': 'RB|R~',
'S': 'DI|FI|O|RO',
'SI': 'DI'},
'E': {'B': 'B',
'BI': 'BI',
'D': 'D',
'DI': 'DI',
'E': 'E',
'F': 'F',
'FI': 'FI',
'M': 'M',
'MI': 'MI',
'O': 'O',
'OI': 'OI',
'RB': 'RB',
'RBI': 'RBI',
'RO': 'RO',
'ROI': 'ROI',
'RS': 'RS',
'R~': 'R~',
'S': 'S',
'SI': 'SI'},
'F': {'B': 'B',
'BI': 'BI',
'D': 'D',
'DI': 'BI|DI|MI|OI|SI',
'E': 'F',
'F': 'F',
'FI': 'E|F|FI',
'M': 'M',
'MI': 'BI',
'O': 'D|O|S',
'OI': 'BI|MI|OI',
'RB': 'RB|R~',
'RBI': 'RBI',
'RO': 'RBI|RO|ROI|RS',
'ROI': 'RBI|ROI',
'RS': 'RBI|ROI',
'R~': 'R~',
'S': 'D',
'SI': 'BI|MI|OI'},
'FI': {'B': 'B',
'BI': 'BI|DI|MI|OI|SI',
'D': 'D|O|S',
'DI': 'DI',
'E': 'FI',
'F': 'E|F|FI',
'FI': 'FI',
'M': 'M',
'MI': 'DI|OI|SI',
'O': 'O',
'OI': 'DI|OI|SI',
'RB': 'RB',
'RBI': 'RBI|RO|ROI|RS',
'RO': 'RO',
'ROI': 'RO|ROI|RS',
'RS': 'RO',
'R~': 'RB|R~',
'S': 'O',
'SI': 'DI'},
'M': {'B': 'B',
'BI': 'BI|DI|MI|OI|SI',
'D': 'D|O|S',
'DI': 'B',
'E': 'M',
'F': 'D|O|S',
'FI': 'B',
'M': 'B',
'MI': 'E|F|FI',
'O': 'B',
'OI': 'D|O|S',
'RB': 'B',
'RBI': 'RBI|RO|ROI|RS',
'RO': 'B',
'ROI': 'D|O|S',
'RS': 'M',
'R~': 'RB|R~',
'S': 'M',
'SI': 'M'},
'MI': {'B': 'B|DI|FI|M|O|RB|RO',
'BI': 'BI',
'D': 'D|F|OI|ROI',
'DI': 'BI',
'E': 'MI',
'F': 'MI',
'FI': 'MI',
'M': 'E|RS|S|SI',
'MI': 'BI',
'O': 'D|F|OI|ROI',
'OI': 'BI',
'RB': 'R~',
'RBI': 'RBI',
'RO': 'RBI',
'ROI': 'RBI',
'RS': 'RBI',
'R~': 'R~',
'S': 'D|F|OI|ROI',
'SI': 'BI'},
'O': {'B': 'B',
'BI': 'BI|DI|MI|OI|SI',
'D': 'D|O|S',
'DI': 'B|DI|FI|M|O',
'E': 'O',
'F': 'D|O|S',
'FI': 'B|M|O',
'M': 'B',
'MI': 'DI|OI|SI',
'O': 'B|M|O',
'OI': 'D|DI|E|F|FI|O|OI|S|SI',
'RB': 'B|RB',
'RBI': 'RBI|RO|ROI|RS',
'RO': 'B|M|O|RO',
'ROI': 'D|O|RO|ROI|RS|S',
'RS': 'O|RO',
'R~': 'RB|R~',
'S': 'O',
'SI': 'DI|FI|O'},
'OI': {'B': 'B|DI|FI|M|O|RB|RO',
'BI': 'BI',
'D': 'D|F|OI|ROI',
'DI': 'BI|DI|MI|OI|SI',
'E': 'OI',
'F': 'OI',
'FI': 'DI|OI|SI',
'M': 'DI|FI|O|RO',
'MI': 'BI',
'O': 'D|DI|E|F|FI|O|OI|RO|ROI|RS|S|SI',
'OI': 'BI|MI|OI',
'RB': 'RB|R~',
'RBI': 'RBI',
'RO': 'RBI|RO|ROI|RS',
'ROI': 'RBI|ROI',
'RS': 'RBI|ROI',
'R~': 'R~',
'S': 'D|F|OI|ROI',
'SI': 'BI|MI|OI'},
'RB': {'B': 'RB',
'BI': 'BI|DI|MI|OI|RB|RO|ROI|RS|SI',
'D': 'RB|RO|ROI|RS',
'DI': 'RB',
'E': 'RB',
'F': 'RB|RO|ROI|RS',
'FI': 'RB',
'M': 'RB',
'MI': 'RB|RO|ROI|RS',
'O': 'RB',
'OI': 'RB|RO|ROI|RS',
'RB': 'RB',
'RBI': 'D|DI|E|F|FI|O|OI|RB|RBI|RO|ROI|RS|S|SI',
'RO': 'RB',
'ROI': 'RB|RO|ROI|RS',
'RS': 'RB',
'R~': 'B|DI|FI|M|O|RB|RO|R~',
'S': 'RB',
'SI': 'RB'},
'RBI': {'B': 'R~',
'BI': 'BI',
'D': 'RBI',
'DI': 'BI|RBI|R~',
'E': 'RBI',
'F': 'RBI',
'FI': 'RBI|R~',
'M': 'R~',
'MI': 'BI',
'O': 'RBI|R~',
'OI': 'BI|RBI',
'RB': 'B|BI|D|DI|E|F|FI|M|MI|O|OI|RB|RBI|RO|ROI|RS|R~|S|SI',
'RBI': 'RBI',
'RO': 'BI|D|F|MI|OI|RBI|ROI|R~',
'ROI': 'BI|D|F|MI|OI|RBI|ROI',
'RS': 'BI|D|F|MI|OI|RBI|ROI',
'R~': 'R~',
'S': 'RBI',
'SI': 'BI|RBI'},
'RO': {'B': 'RB',
'BI': 'BI|DI|MI|OI|SI',
'D': 'RO|ROI|RS',
'DI': 'DI|RB|RO',
'E': 'RO',
'F': 'RO|ROI|RS',
'FI': 'RB|RO',
'M': 'RB',
'MI': 'DI|OI|SI',
'O': 'RB|RO',
'OI': 'DI|OI|RO|ROI|RS|SI',
'RB': 'B|DI|FI|M|O|RB|RO',
'RBI': 'RBI|RO|ROI|RS',
'RO': 'DI|FI|O|RB|RO',
'ROI': 'D|DI|E|F|FI|O|OI|RO|ROI|RS|S|SI',
'RS': 'DI|FI|O|RO',
'R~': 'RB|R~',
'S': 'RO',
'SI': 'DI|RO'},
'ROI': {'B': 'RB',
'BI': 'BI',
'D': 'ROI',
'DI': 'BI|DI|MI|OI|RB|RO|ROI|RS|SI',
'E': 'ROI',
'F': 'ROI',
'FI': 'RB|RO|ROI|RS',
'M': 'RB',
'MI': 'BI',
'O': 'RB|RO|ROI|RS',
'OI': 'BI|MI|OI|ROI',
'RB': 'B|DI|FI|M|O|RB|RO|R~',
'RBI': 'RBI',
'RO': 'D|DI|E|F|FI|O|OI|RB|RBI|RO|ROI|RS|S|SI',
'ROI': 'D|F|OI|RBI|ROI',
'RS': 'D|F|OI|RBI|ROI',
'R~': 'R~',
'S': 'ROI',
'SI': 'BI|MI|OI|ROI'},
'RS': {'B': 'RB',
'BI': 'BI',
'D': 'ROI',
'DI': 'DI|RB|RO',
'E': 'RS',
'F': 'ROI',
'FI': 'RB|RO',
'M': 'RB',
'MI': 'MI',
'O': 'RB|RO',
'OI': 'OI|ROI',
'RB': 'B|DI|FI|M|O|RB|RO',
'RBI': 'RBI',
'RO': 'DI|FI|O|RB|RO',
'ROI': 'D|F|OI|ROI',
'RS': 'E|RS|S|SI',
'R~': 'R~',
'S': 'RS',
'SI': 'RS|SI'},
'R~': {'B': 'R~',
'BI': 'BI|RBI|R~',
'D': 'RBI|R~',
'DI': 'R~',
'E': 'R~',
'F': 'RBI|R~',
'FI': 'R~',
'M': 'R~',
'MI': 'RBI|R~',
'O': 'R~',
'OI': 'RBI|R~',
'RB': 'R~',
'RBI': 'BI|D|F|MI|OI|RBI|ROI|R~',
'RO': 'R~',
'ROI': 'RBI|R~',
'RS': 'R~',
'R~': 'B|BI|D|DI|E|F|FI|M|MI|O|OI|RB|RBI|RO|ROI|RS|R~|S|SI',
'S': 'R~',
'SI': 'R~'},
'S': {'B': 'B',
'BI': 'BI',
'D': 'D',
'DI': 'B|DI|FI|M|O',
'E': 'S',
'F': 'D',
'FI': 'B|M|O',
'M': 'B',
'MI': 'MI',
'O': 'B|M|O',
'OI': 'D|F|OI',
'RB': 'B|RB',
'RBI': 'RBI',
'RO': 'B|M|O|RO',
'ROI': 'D|ROI',
'RS': 'RS|S',
'R~': 'R~',
'S': 'S',
'SI': 'E|S|SI'},
'SI': {'B': 'B|DI|FI|M|O|RB|RO',
'BI': 'BI',
'D': 'D|F|OI|ROI',
'DI': 'DI',
'E': 'SI',
'F': 'OI',
'FI': 'DI',
'M': 'DI|FI|O|RO',
'MI': 'MI',
'O': 'DI|FI|O|RO',
'OI': 'OI',
'RB': 'RB',
'RBI': 'RBI',
'RO': 'RO',
'ROI': 'ROI',
'RS': 'RS',
'R~': 'R~',
'S': 'E|RS|S|SI',
'SI': 'SI'}}}
Save Right-Branching Proper Interval Algebra Dictionary to JSON File
test_rb_proper_json_path = os.path.join(path, "Algebras/test_derived_right_branching_proper_interval_algebra.json")
test_rb_proper_json_path
'/Users/alfredreich/Documents/Python/github/myrepos/qualreas/Algebras/test_derived_right_branching_proper_interval_algebra.json'
qr.algebra_to_json_file(test_rb_proper_alg_dict, test_rb_proper_json_path)
Instantiate a Right-Branching Proper Interval Algebra Object from JSON File
test_rb_proper_alg = qr.Algebra(test_rb_proper_json_path)
test_rb_proper_alg
<qualreas.Algebra at 0x7fab72b158e0>
test_rb_proper_alg.summary()
Algebra Name: Derived_Right_Branching_Proper_Interval_Algebra
Description: Extended right-branching proper interval algebra derived from point relations
Equality Rels: E
Relations:
NAME (SYMBOL) CONVERSE (ABBREV) REFLEXIVE SYMMETRIC TRANSITIVE DOMAIN RANGE
Before ( B) After ( BI) False False True PInt PInt
After ( BI) Before ( B) False False True PInt PInt
During ( D) Contains ( DI) False False True PInt PInt
Contains ( DI) During ( D) False False True PInt PInt
Equals ( E) Equals ( E) True True True PInt PInt
Finishes ( F) Finished-by ( FI) False False True PInt PInt
Finished-by ( FI) Finishes ( F) False False True PInt PInt
Meets ( M) Met-By ( MI) False False False PInt PInt
Met-By ( MI) Meets ( M) False False False PInt PInt
Overlaps ( O) Overlapped-By ( OI) False False False PInt PInt
Overlapped-By ( OI) Overlaps ( O) False False False PInt PInt
Right-Before ( RB) Right-After (RBI) False False True PInt PInt
Right-After (RBI) Right-Before ( RB) False False True PInt PInt
Right-Overlaps ( RO) Right-Overlapped-By (ROI) False False False PInt PInt
Right-Overlapped-By (ROI) Right-Overlaps ( RO) False False False PInt PInt
Right-Starts ( RS) Right-Starts ( RS) False True False PInt PInt
Right-Incomparable ( R~) Right-Incomparable ( R~) False True False PInt PInt
Starts ( S) Started-By ( SI) False False True PInt PInt
Started-By ( SI) Starts ( S) False False True PInt PInt
Domain & Range Abbreviations:
Pt = Point
PInt = Proper Interval
test_rb_proper_alg.check_composition_identity()
True
test_rb_proper_alg.is_associative()
TEST SUMMARY: 6859 OK, 0 Skipped, 0 Failed (6859 Total)
True