From: Richard Fikes (firstname.lastname@example.org)
During a recent joint committee telecon, I offered in support of the work on a rules language to provide pointers to ontologies that contain FOL axioms that would be difficult (or perhaps impossible) to express in DAML+OIL. Such examples are not difficult to find. It occurred to me that there are some (I think) easy to understand examples in the lecture slides for my KR course. So, I am attaching to this message Powerpoint files containing slides describing three such ontologies. All comments welcome. Richard Units and Measures The first file describes a portion of a "Units and Measures" ontology containing classes Physical-Dimension, Physical-Quantity, Unit-Of-Measure, Magnitude, Length-Unit-Of-Measure, etc. The notation used on the slides for class definitions is a simple monotonic frame language (OKBC) in which slot names are indented and template slots (i.e., slots that are inherited to instances) are denoted by an "*" preceding the name. So, for example, slide 3 says that Physical-Dimension is a class each of whose instances has exactly one Standard-Unit that is an instance of Unit-Of-Measure. (The frame language is described in the OKBC spec at http://www.ai.sri.com/~okbc/spec.html, but you shouldn't need to read any of the spec in order to understand the slides.) The ontology contains definitions of multiple logical functions that consist of complex axioms, including Unit* (an associative commutative mapping of all pairs of units to units), Unit^ (a mapping of reals to units that has the algebraic properties of exponentiation), Quantity-Magnitude (i.e., the magnitude of a physical quantity in a given unit of measure), The-Quantity (i.e., the physical quantity with a given magnitude in a given unit of measure), and an if-and-only-if definition for physical quantities being equal. The functions and classes are used to define objects, such as Length-Dimension, Meter, Kilometer, and Meter/Second. The definitions involve axioms such as: (= Meter/Second (Unit* Meter (Unit^ Second –1))) (=> (and (Instance-Of ?q1 Physical-Quantity) (Quantity-Dimension ?q1 Length-Dimension)) (= (Quantity-Magnitude ?q1 Kilometer) (/ (Quantity-Magnitude ?q1 Meter) 1000))) [Where slots (aka properties) are considered to be binary relations.] Time The second file describes a time ontology based on the paper at http://www.ksl.stanford.edu/KSL_Abstracts/KSL-00-01.html. There are many complex axioms in that ontology. For example there are axioms for computing the value of slot Year-Of, Hour-Of, Minute-Of, etc. of a time point based on its location on the time line (see slide 7), relations Before, After, and Equal for time points are defined in terms of Location-Of and for time points described at different granularities, constraints are expressed on the start and end time for intervals, the start and end time of an interval are defined as greatest lower bounds and least upper bounds on the points in an interval, axioms state that the time line is dense and that "infinite-past" and "infinite-future" are time points, etc. Electronic Circuits The third file describes a toy ontology that is the first example I present in the class. In that ontology, axioms are used to specify the relationship between the inputs and the output of an AndGate, OrGate, XorGate, and NotGate.
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