Elementary Greek Vocabulary

Elementary Greek Vocabulary

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Elementary Greek Vocabulary Coordinated with Mounce, Basics of Biblical Greek (2d ed.) and with two common sets of vocabulary cards (Gromacki and Mounce) by Rodney J. Decker, Th.D. (v. 2.0, 2006) Card Number Gromacki / Mounce Greek Word Freq/NT Gloss 04Ch. 26[ words] 9999 73 Abraham1 7 175 angel, messenger2 55 129 amen, truly, verily, “so let it be”3 80 550 man, mankind, person, people, human being, humankind4 105 80 apostle, envoy, messenger5 9999 61 Galilee6 184 50 writing, Scripture7 9999 59 David8 242 166 glory
  • husband109 280 114 assembly
  • scripture7 9999 59 david8 242 166 glory
  • after67 620 93 house68 620 114 house69 667 175 crowd
  • sight139 675 141 again140 767 93 foot
  • breath21 787 144 prophet22 805 68 sabbath
  • belief122 913 76 water123
  • messenger5 9999 61 galilee6 184 50 writing
  • nor120 705 413 father121 731 243 faith
  • live208 9999 43 judea209

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Science of Computational Logic
1| Working Material |
Ste en H olldobler
International Center for Computational Logic
Technische Universit at Dresden
D{01062 Dresden
sh@iccl.tu-dresden.de
January 16, 2012
1 The working material is incomplete and may contain errors. Any suggestions are greatly
appreciated.Contents
1 Description Logic 3
1.1 Terminologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Assertions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Subsumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Unsatis ability Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Equational Logic 11
2.1 Equational Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Paramodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Term Rewriting Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Con uence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.3 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Uni cation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 Uni cation under Equality . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.4 Multisets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Actions and Causality 41
3.1 Conjunctive Planning Problems . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Blocks World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.1 A Fluent Calculus Implementation . . . . . . . . . . . . . . . . . . . 44
3.2.2 SLDE-Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.3 Solving Conjunctive Planning Problems . . . . . . . . . . . . . . . . 46
3.2.4 Solving the Frame Problem . . . . . . . . . . . . . . . . . . . . . . . 46
iiiiv CONTENTS
3.2.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Deduction, Abduction, and Induction 51
4.1 Deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.1 Sorts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 Abduction in Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.2 Knowledge Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.3 Theory Revision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.4 Abduction and Model Generation . . . . . . . . . . . . . . . . . . . 60
4.2.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Non-Monotonic Reasoning 65Notation
In this book we will make the following notational conventions:
a constant
bt
C unary relation symbol denoting a concept
C set of concept formulas
D non-empty domain of an interpretaion
E set of equations
Edse expression containing an occurrence of the term s
Eds=te where an occurrence of the term s has been replaced by t
E equational system obtained from the term rewriting systemRR
E axioms of equality
" empty substitution
g function symbol
f symbol
F formula
F set of formulas
g function symbol
G formula
H formula
I interpretation
K a set of formulas often called knowledge base
l term; left-hand side of an equation or rewrite rule
L literal
p relation symbol
r term; the right-hand side of equations or rewrite rules
R binary relation symbol denoting a role
R term rewriting system
s term
t term
substitution
u term
U variable
V v
W variable
X v
Y variable
Z v
In addition, we will consider the following precedence hierarchy among connectives:
f8;9g:f^;_g!$:2 CONTENTSChapter 1
Description Logic
In the late 1960s and early 1970s, it was recognized that knowledge representation and
reasoning is at the heart of any intelligent system. Heavily in uenced by the work of
Quillian on so-called semantic networks [Qui68] and the work of Minsky on so-called
frames [Min75] simple graphs and structured objects were used to represent knowledge
and many algorithms were developed which manipulated these data structures. At rst
sight, these systems were quite attractive because they apparently admitted an intuitive
semantics, which was easy to understand. For example, a graph like the one shown in
Figure 1.1 seems to represent the following short story.
Dogs, cats and mice are mammals. Dogs dislike cats and, in particular, the
dog Rudi, which is a German shepherd, has bitten the cat Tom while Tom was
chasing the mouse Jerry.
Simple algorithms operating on this graph can be applied to conclude that, for example,
German shepherds are mammals, Rudi dislikes Tom, etc.
Shortly afterwards, however, it was recognized that systems based on these techniques
lack a formal semantics (see e.g. [Woo75]). What precisely is denoted by a link? What
precisely is denoted by a vertex? It was also observed that the algorithms which operated
on these data structures did not always yield the intended results. This led to a formal
reconstruction of semantic networks as well as frame systems within logic (see e.g. [Sch76,
Hay79]). At around the same time, Brachman developed the idea that formally de ned
concepts should be interrelated and organized in networks such that the structure of
these networks allows reasoning about possible conclusions [Bra78]. This line of research
led to the knowledge representation and reasoning system KlOne [BS85], which is the
ancester of a whole family of systems. Such systems have been used in a wide range
of practical applications including nancial management systems, computer con guration
systems, software information systems and database interfaces. KlOne has also led to a
thorough investigation of the semantics of the representations used in these systems and the
development of correct and complete algorithms for computing with these representations.
Today the eld is called description logic and this chapter gives an introduction into such
logics.
Description logics focus on descriptions of concepts and their interrelationships in cer-
tain domains. Based on so-called atomic concepts and relations between concepts, which
are traditionally called roles, more complex concepts are formed with the help of certain
34 CHAPTER 1. DESCRIPTION LOGIC
mammals
are areare
dogs cats mice
dislike
are
german shephards is a
is a
jerryrudi tom
has bitten was chasing
Figure 1.1: A simple semantic network with apparently obvious intended meaning.
operators. Furthermore, assertions about certain aspects of the world can be made. For
example, a certain individual may be an instance of a certain concept or two individuals
are connected via a certain role. The basic inference tasks provided by description logics
are subsumption and unsatis ability testing . Subsumption is used to check whether a cat-
egory is a subset of another category. As we shall see in the next paragraph, description
logics do not allow the speci cation of subsumption hierarchies explicitly but these hier-
archies depend on the de nitions of the concepts. The unsatis ability check allows the
determination of whether an individual belongs to a certain concept. A formal account of
these notions will be developed in the following sections.
1.1 Terminologies
We consider an alphabet with constant symbols, the variables X; Y; ::: , the connectives
:;^;_;!;$ , the quanti ers 8 and9 , and the special symbols (; ;; ) . For notational
C convenicence, C shall denote a unary relation symbols and R a binary relation symbol
R in the sequel. Informally, C denotes a concept whereas R denotes a role.
Terms are dened as usual, ie., the set of terms is the union of the set of constant
symbols and the set of variables. The set of role formulas consists of all strings of the
form R(X;Y ). The set of atomic concept formulas consists of all strings of the form
C(X). As we will see shortly, each concept formula contains precisely one free variable.
Hence, concept formulas will be denoted by F (X) and G(X), where X is the only
concept formula free variable occurring in F and G. The set of concept formulas is the smallest set C
satisfying the following conditions:
1. All atomic formulas are in C.
2. If F (X) is in C , so is :F (X).
3. If F (X) and G(X) are in C, so are F (X)^G(X) and F (X)_G(X).6
1.1. TERMINOLOGIES 5
4. if R(X;Y ) is a role formula and F (Y ) is in C, then (9Y )(R(X;Y )^F (Y )) and
(8X)(R(X;Y )!F (Y )) are in C as well.
The set of concept axioms consists of all strings of the form (8X)(C(X)! F (X)) or concept axioms
(8X)(C(X)$ F (X)). A terminology or T-box is a nite set K of concept axioms terminologyT
such that T-box
KT
1. each atomic concept C occurs at most once as left-hand side of an axiom and
12. the set does not contain any cycles.
The set of generalized concept axioms consists of all strings of the form (8X) (F (X)! gerneralized
concept axiomG(X)) or (8X) (F (X)$G(X)) .
An example of a T-box is shown in Table 1.1. Informally, the concepts woman and
man are not completely de ned but a necessary condition is stated, viz. that both are
persons. The remaining concepts are completely de ned. For example, a father is a
man who has a child which is a person. By inspection we observe that all axioms are
universally closed in a T-box. Hence, the universal quanti ers can be omitted. Likewise,
because each concept formula has precisely one free variable, this variable can be omitted as
well. Furthermore, the structure of remaining quanti ed formulas like ( 9Y ) (child(X;Y )^
parent(Y )) and (8Y ) (child(X;Y )!:man(Y )) is also quite regular, which allows for
further abbreviations like 9child : parent and 8child ::man , respectively. Alltogether,
Table 1.1 depicts the simple terminology also in abbreviated form, where the usage of the
symbols v; =; u and t instead of !; $; ^ and _ , respectively, is motivated by the
following semantics.
The semantics for terminologies is the usual semantics for rst order logic formulas.
However, the restricted form of concept formulas and concept axioms allows the represen-
tation of the semantics in a more convenient and intuitive form. Let I be an interpretation
with nite, non-empty domain D.
I I assigns to each constant a an element a of D.
I I assigns to each unary predicate symbol C a subset C D. This subset contains
Iprecisely the individuals from D which belong to C .
I I Let F and G be the subsets of D assigned to the concept formulas D(X) and
I I I IE(X), respectively. Then, I assigns DnF , F \G , and F [G to the concept1
formulas :F (X), F (X)^G(X), and F (X)_G(X), respectively.
I I I assigns to each binary relation symbol symbol R a set R DD . Let R (d)
0 Idenote the set of all d 2D obtained from R by selecting all tuples whose rst
argument is d and projecting this selection onto the second argument, i.e.,
I 0 0 IR (d) =fd 2Dj (d;d )2Rg:
Then, I assigns
I Ifd2DjR (d)\F =;g
1 0A concept C depends on the concept C wrt the T-box K i K contains a concept axiom of theT T
0form (8X)(C(X)!F (X)) or (8X)(C(X)$F (X)) such that C occurs in F . A T-box is said to
be cyclic i it contains a concept which recursively depends on itself.