Derive Extended Interval Algebra
NOTE: From a derivation point-of-view, what distinquishes this algebra from Allen’s algebra it the definition of less than used to define intervals. In particular, this derivation uses ‘=|<’ rather than ‘<’, which allows intervals to be degenerate (i.e., equal a point). See the section, below, titled, “Derive the Extended Interval Algebra as a Dictionary”.
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 Extended Interval Algebra from Extended Point Algebra
Extended Point Algebra
pt_alg = qr.Algebra(os.path.join(path, "Algebras/Linear_Point_Algebra.json"))
pt_alg.summary()
Algebra Name: Linear_Point_Algebra
Description: Linear 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
Domain & Range Abbreviations:
Pt = Point
PInt = Proper Interval
qr.print_point_algebra_composition_table(pt_alg)
Linear_Point_Algebra Elements: <, =, > ============================== rel1 ; rel2 = composition ============================== < < < < = < < > <|=|> ------------------------------ = < < = = = = > > ------------------------------ > < <|=|> > = > > > > ------------------------------
Derive the Extended 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 = '<'
ext_alg_name="Derived_Extended_Interval_Algebra"
ext_alg_desc="Extended linear interval algebra derived from point relations"
verbose = True
%time test_ext_alg_dict = qr.derive_algebra(pt_alg, less_than_rel, name=ext_alg_name, description=ext_alg_desc, verbose=verbose)
========================== <,<,<,< B (['Point', 'ProperInterval'], ['Point', 'ProperInterval']) [['=' '<|=' '<' '<'] ['=|>' '=' '<' '<'] ['>' '>' '=' '<|='] ['>' '>' '=|>' '=']] ========================== <,<,=,< M (['ProperInterval'], ['ProperInterval']) [['=' '<' '<' '<'] ['>' '=' '=' '<'] ['>' '=' '=' '<'] ['>' '>' '>' '=']] ========================== <,<,=,= PFI (['ProperInterval'], ['Point']) [['=' '<' '<' '<'] ['>' '=' '=' '='] ['>' '=' '=' '='] ['>' '=' '=' '=']] ========================== <,<,>,< O (['ProperInterval'], ['ProperInterval']) [['=' '<' '<' '<'] ['>' '=' '>' '<'] ['>' '<' '=' '<'] ['>' '>' '>' '=']] ========================== <,<,>,= FI (['ProperInterval'], ['ProperInterval']) [['=' '<' '<' '<'] ['>' '=' '>' '='] ['>' '<' '=' '<'] ['>' '=' '>' '=']] ========================== <,<,>,> DI (['ProperInterval'], ['Point', 'ProperInterval']) [['=' '<' '<' '<'] ['>' '=' '>' '>'] ['>' '<' '=' '<|='] ['>' '<' '=|>' '=']] ========================== =,<,=,< PS (['Point'], ['ProperInterval']) [['=' '=' '=' '<'] ['=' '=' '=' '<'] ['=' '=' '=' '<'] ['>' '>' '>' '=']] ========================== =,=,=,= PE (['Point'], ['Point']) [['=' '=' '=' '='] ['=' '=' '=' '='] ['=' '=' '=' '='] ['=' '=' '=' '=']] ========================== =,<,>,< S (['ProperInterval'], ['ProperInterval']) [['=' '<' '=' '<'] ['>' '=' '>' '<'] ['=' '<' '=' '<'] ['>' '>' '>' '=']] ========================== =,<,>,= E (['ProperInterval'], ['ProperInterval']) [['=' '<' '=' '<'] ['>' '=' '>' '='] ['=' '<' '=' '<'] ['>' '=' '>' '=']] ========================== =,<,>,> SI (['ProperInterval'], ['ProperInterval']) [['=' '<' '=' '<'] ['>' '=' '>' '>'] ['=' '<' '=' '<'] ['>' '<' '>' '=']] ========================== =,=,>,> PSI (['ProperInterval'], ['Point']) [['=' '<' '=' '='] ['>' '=' '>' '>'] ['=' '<' '=' '='] ['=' '<' '=' '=']] ========================== >,<,>,< D (['Point', 'ProperInterval'], ['ProperInterval']) [['=' '<|=' '>' '<'] ['=|>' '=' '>' '<'] ['<' '<' '=' '<'] ['>' '>' '>' '=']] ========================== >,<,>,= F (['ProperInterval'], ['ProperInterval']) [['=' '<' '>' '<'] ['>' '=' '>' '='] ['<' '<' '=' '<'] ['>' '=' '>' '=']] ========================== >,<,>,> OI (['ProperInterval'], ['ProperInterval']) [['=' '<' '>' '<'] ['>' '=' '>' '>'] ['<' '<' '=' '<'] ['>' '<' '>' '=']] ========================== >,=,>,= PF (['Point'], ['ProperInterval']) [['=' '=' '>' '='] ['=' '=' '>' '='] ['<' '<' '=' '<'] ['=' '=' '>' '=']] ========================== >,=,>,> MI (['ProperInterval'], ['ProperInterval']) [['=' '<' '>' '='] ['>' '=' '>' '>'] ['<' '<' '=' '<'] ['=' '<' '>' '=']] ========================== >,>,>,> BI (['Point', 'ProperInterval'], ['Point', 'ProperInterval']) [['=' '<|=' '>' '>'] ['=|>' '=' '>' '>'] ['<' '<' '=' '<|='] ['<' '<' '=|>' '=']] 18 consistent networks CPU times: user 11 s, sys: 644 ms, total: 11.7 s Wall time: 10.8 s
test_ext_alg_dict
{'Name': 'Derived_Extended_Interval_Algebra', 'Description': 'Extended linear interval algebra derived from point relations', 'Relations': {'B': {'Name': 'Before', 'Converse': 'BI', 'Domain': ['Point', 'ProperInterval'], 'Range': ['Point', 'ProperInterval'], 'Reflexive': False, 'Symmetric': False, 'Transitive': True}, 'BI': {'Name': 'After', 'Converse': 'B', 'Domain': ['Point', 'ProperInterval'], 'Range': ['Point', 'ProperInterval'], 'Reflexive': False, 'Symmetric': False, 'Transitive': True}, 'D': {'Name': 'During', 'Converse': 'DI', 'Domain': ['Point', 'ProperInterval'], 'Range': ['ProperInterval'], 'Reflexive': False, 'Symmetric': False, 'Transitive': True}, 'DI': {'Name': 'Contains', 'Converse': 'D', 'Domain': ['ProperInterval'], 'Range': ['Point', '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}, 'PE': {'Name': 'Point-Equals', 'Converse': 'PE', 'Domain': ['Point'], 'Range': ['Point'], 'Reflexive': True, 'Symmetric': True, 'Transitive': True}, 'PF': {'Name': 'Point-Finishes', 'Converse': 'PFI', 'Domain': ['Point'], 'Range': ['ProperInterval'], 'Reflexive': False, 'Symmetric': False, 'Transitive': False}, 'PFI': {'Name': 'Point-Finished-By', 'Converse': 'PF', 'Domain': ['ProperInterval'], 'Range': ['Point'], 'Reflexive': False, 'Symmetric': False, 'Transitive': False}, 'PS': {'Name': 'Point-Starts', 'Converse': 'PSI', 'Domain': ['Point'], 'Range': ['ProperInterval'], 'Reflexive': False, 'Symmetric': False, 'Transitive': False}, 'PSI': {'Name': 'Point-Started-By', 'Converse': 'PS', 'Domain': ['ProperInterval'], 'Range': ['Point'], 'Reflexive': False, 'Symmetric': False, '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|PE|PF|PFI|PS|PSI|S|SI', 'D': 'B|D|M|O|PS|S', 'DI': 'B', 'E': 'B', 'F': 'B|D|M|O|PS|S', 'FI': 'B', 'M': 'B', 'MI': 'B|D|M|O|PS|S', 'O': 'B', 'OI': 'B|D|M|O|PS|S', 'PE': 'B', 'PF': 'B|D|M|O|PS|S', 'PFI': 'B', 'PS': 'B', 'PSI': 'B', 'S': 'B', 'SI': 'B'}, 'BI': {'B': 'B|BI|D|DI|E|F|FI|M|MI|O|OI|PE|PF|PFI|PS|PSI|S|SI', 'BI': 'BI', 'D': 'BI|D|F|MI|OI|PF', 'DI': 'BI', 'E': 'BI', 'F': 'BI', 'FI': 'BI', 'M': 'BI|D|F|MI|OI|PF', 'MI': 'BI', 'O': 'BI|D|F|MI|OI|PF', 'OI': 'BI', 'PE': 'BI', 'PF': 'BI', 'PFI': 'BI', 'PS': 'BI|D|F|MI|OI|PF', 'PSI': 'BI', 'S': 'BI|D|F|MI|OI|PF', 'SI': 'BI'}, 'D': {'B': 'B', 'BI': 'BI', 'D': 'D', 'DI': 'B|BI|D|DI|E|F|FI|M|MI|O|OI|PE|PF|PFI|PS|PSI|S|SI', 'E': 'D', 'F': 'D', 'FI': 'B|D|M|O|PS|S', 'M': 'B', 'MI': 'BI', 'O': 'B|D|M|O|PS|S', 'OI': 'BI|D|F|MI|OI|PF', 'PE': '', 'PF': '', 'PFI': 'B', 'PS': '', 'PSI': 'BI', 'S': 'D', 'SI': 'BI|D|F|MI|OI|PF'}, 'DI': {'B': 'B|DI|FI|M|O|PFI', 'BI': 'BI|DI|MI|OI|PSI|SI', 'D': 'D|DI|E|F|FI|O|OI|S|SI', 'DI': 'DI', 'E': 'DI', 'F': 'DI|OI|SI', 'FI': 'DI', 'M': 'DI|FI|O', 'MI': 'DI|OI|SI', 'O': 'DI|FI|O', 'OI': 'DI|OI|SI', 'PE': 'DI', 'PF': 'DI|OI|SI', 'PFI': 'DI', 'PS': 'DI|FI|O', 'PSI': 'DI', 'S': 'DI|FI|O', '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', 'PE': '', 'PF': '', 'PFI': 'PFI', 'PS': '', 'PSI': 'PSI', 'S': 'S', 'SI': 'SI'}, 'F': {'B': 'B', 'BI': 'BI', 'D': 'D', 'DI': 'BI|DI|MI|OI|PSI|SI', 'E': 'F', 'F': 'F', 'FI': 'E|F|FI', 'M': 'M', 'MI': 'BI', 'O': 'D|O|S', 'OI': 'BI|MI|OI', 'PE': '', 'PF': '', 'PFI': 'PFI', 'PS': '', 'PSI': 'BI', 'S': 'D', 'SI': 'BI|MI|OI'}, 'FI': {'B': 'B', 'BI': 'BI|DI|MI|OI|PSI|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', 'PE': '', 'PF': '', 'PFI': 'PFI', 'PS': '', 'PSI': 'DI', 'S': 'O', 'SI': 'DI'}, 'M': {'B': 'B', 'BI': 'BI|DI|MI|OI|PSI|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', 'PE': '', 'PF': '', 'PFI': 'B', 'PS': '', 'PSI': 'PFI', 'S': 'M', 'SI': 'M'}, 'MI': {'B': 'B|DI|FI|M|O|PFI', 'BI': 'BI', 'D': 'D|F|OI', 'DI': 'BI', 'E': 'MI', 'F': 'MI', 'FI': 'MI', 'M': 'E|S|SI', 'MI': 'BI', 'O': 'D|F|OI', 'OI': 'BI', 'PE': '', 'PF': '', 'PFI': 'PSI', 'PS': '', 'PSI': 'BI', 'S': 'D|F|OI', 'SI': 'BI'}, 'O': {'B': 'B', 'BI': 'BI|DI|MI|OI|PSI|SI', 'D': 'D|O|S', 'DI': 'B|DI|FI|M|O|PFI', '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', 'PE': '', 'PF': '', 'PFI': 'B', 'PS': '', 'PSI': 'DI', 'S': 'O', 'SI': 'DI|FI|O'}, 'OI': {'B': 'B|DI|FI|M|O|PFI', 'BI': 'BI', 'D': 'D|F|OI', 'DI': 'BI|DI|MI|OI|PSI|SI', 'E': 'OI', 'F': 'OI', 'FI': 'DI|OI|SI', 'M': 'DI|FI|O', 'MI': 'BI', 'O': 'D|DI|E|F|FI|O|OI|S|SI', 'OI': 'BI|MI|OI', 'PE': '', 'PF': '', 'PFI': 'DI', 'PS': '', 'PSI': 'BI', 'S': 'D|F|OI', 'SI': 'BI|MI|OI'}, 'PE': {'B': 'B', 'BI': 'BI', 'D': 'D', 'DI': '', 'E': '', 'F': '', 'FI': '', 'M': '', 'MI': '', 'O': '', 'OI': '', 'PE': 'PE', 'PF': 'PF', 'PFI': '', 'PS': 'PS', 'PSI': '', 'S': '', 'SI': ''}, 'PF': {'B': 'B', 'BI': 'BI', 'D': 'D', 'DI': 'BI', 'E': 'PF', 'F': 'PF', 'FI': 'PF', 'M': 'PS', 'MI': 'BI', 'O': 'D', 'OI': 'BI', 'PE': '', 'PF': '', 'PFI': 'PE', 'PS': '', 'PSI': 'BI', 'S': 'D', 'SI': 'BI'}, 'PFI': {'B': 'B', 'BI': 'BI|DI|MI|OI|PSI|SI', 'D': 'D|O|S', 'DI': '', 'E': '', 'F': '', 'FI': '', 'M': '', 'MI': '', 'O': '', 'OI': '', 'PE': 'PFI', 'PF': 'E|F|FI', 'PFI': '', 'PS': 'M', 'PSI': '', 'S': '', 'SI': ''}, 'PS': {'B': 'B', 'BI': 'BI', 'D': 'D', 'DI': 'B', 'E': 'PS', 'F': 'D', 'FI': 'B', 'M': 'B', 'MI': 'PF', 'O': 'B', 'OI': 'D', 'PE': '', 'PF': '', 'PFI': 'B', 'PS': '', 'PSI': 'PE', 'S': 'PS', 'SI': 'PS'}, 'PSI': {'B': 'B|DI|FI|M|O|PFI', 'BI': 'BI', 'D': 'D|F|OI', 'DI': '', 'E': '', 'F': '', 'FI': '', 'M': '', 'MI': '', 'O': '', 'OI': '', 'PE': 'PSI', 'PF': 'MI', 'PFI': '', 'PS': 'E|S|SI', 'PSI': '', 'S': '', 'SI': ''}, 'S': {'B': 'B', 'BI': 'BI', 'D': 'D', 'DI': 'B|DI|FI|M|O|PFI', 'E': 'S', 'F': 'D', 'FI': 'B|M|O', 'M': 'B', 'MI': 'MI', 'O': 'B|M|O', 'OI': 'D|F|OI', 'PE': '', 'PF': '', 'PFI': 'B', 'PS': '', 'PSI': 'PSI', 'S': 'S', 'SI': 'E|S|SI'}, 'SI': {'B': 'B|DI|FI|M|O|PFI', 'BI': 'BI', 'D': 'D|F|OI', 'DI': 'DI', 'E': 'SI', 'F': 'OI', 'FI': 'DI', 'M': 'DI|FI|O', 'MI': 'MI', 'O': 'DI|FI|O', 'OI': 'OI', 'PE': '', 'PF': '', 'PFI': 'DI', 'PS': '', 'PSI': 'PSI', 'S': 'E|S|SI', 'SI': 'SI'}}}
Save Algebra Dictionary to JSON File
test_ext_json_path = os.path.join(path, "Algebras/test_derived_extended_interval_algebra.json")
test_ext_json_path
'/Users/alfredreich/Documents/Python/github/myrepos/qualreas/Algebras/test_derived_extended_interval_algebra.json'
qr.algebra_to_json_file(test_ext_alg_dict, test_ext_json_path)
Instantiate an Algebra Object from JSON File
test_ext_alg = qr.Algebra(test_ext_json_path)
test_ext_alg
<qualreas.Algebra at 0x7fa84a716430>
test_ext_alg.summary()
Algebra Name: Derived_Extended_Interval_Algebra Description: Extended linear interval algebra derived from point relations Equality Rels: E|PE Relations: NAME (SYMBOL) CONVERSE (ABBREV) REFLEXIVE SYMMETRIC TRANSITIVE DOMAIN RANGE Before ( B) After ( BI) False False True Pt|PInt Pt|PInt After ( BI) Before ( B) False False True Pt|PInt Pt|PInt During ( D) Contains ( DI) False False True Pt|PInt PInt Contains ( DI) During ( D) False False True PInt Pt|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 Point-Equals ( PE) Point-Equals ( PE) True True True Pt Pt Point-Finishes ( PF) Point-Finished-By (PFI) False False False Pt PInt Point-Finished-By (PFI) Point-Finishes ( PF) False False False PInt Pt Point-Starts ( PS) Point-Started-By (PSI) False False False Pt PInt Point-Started-By (PSI) Point-Starts ( PS) False False False PInt Pt 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_ext_alg.check_composition_identity()
True
test_ext_alg.is_associative()
TEST SUMMARY: 3609 OK, 2223 Skipped, 0 Failed (5832 Total)
True
Load Original Extended Interval Algebra
ext_alg = qr.Algebra(os.path.join(path, "Algebras/Extended_Linear_Interval_Algebra.json"))
ext_alg
<qualreas.Algebra at 0x7fa8788ced90>
ext_alg.summary()
Algebra Name: Extended_Linear_Interval_Algebra Description: Extension of Allen's algebra to include points and intervals Equality Rels: E|PE Relations: NAME (SYMBOL) CONVERSE (ABBREV) REFLEXIVE SYMMETRIC TRANSITIVE DOMAIN RANGE Before ( B) After ( BI) False False True Pt|PInt Pt|PInt After ( BI) Before ( B) False False True Pt|PInt Pt|PInt During ( D) Contains ( DI) False False True Pt|PInt PInt Contains ( DI) During ( D) False False True PInt Pt|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 Point-Equals ( PE) Point-Equals ( PE) True True True Pt Pt Point-Finishes ( PF) Point-Finished-By (PFI) False False False Pt PInt Point-Finished-By (PFI) Point-Finishes ( PF) False False False PInt Pt Point-Starts ( PS) Point-Started-By (PSI) False False False Pt PInt Point-Started-By (PSI) Point-Starts ( PS) False False False PInt Pt 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
Compare Derived Extended Interval Algebra with Original
print(f"Same as original algebra? {ext_alg.equivalent_algebra(test_ext_alg)}")
Same as original algebra? True