## Knowledge Representation and Classical Logic (2007)

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Citations: | 11 - 4 self |

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@MISC{Lifschitz07knowledgerepresentation,

author = {Vladimir Lifschitz and Leora Morgenstern and David Plaisted},

title = {Knowledge Representation and Classical Logic},

year = {2007}

}

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### Abstract

Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields. The language of classical logic that is most widely used in the theory of knowledge representation is the language of first-order (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of first-order logic; recent

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Citation Context ...d its inferential power. Note [115] that it is not a particular syntax for which logicists argue; it is the notion of a formal, declarative semantics and methods of inference that are important. (See =-=[96, 65, 240, 40]-=- for examples of how AI logicism is used.) Anti-logicists have argued that the program, outside of textbook examples, is undesirable and infeasible. To paraphrase McDermott [186], You Don’t Want To Do... |

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Citation Context ...world in order to make simple inferences about moving blocks in a simple blocks-world domain. In the late 1960s and early 1970s, SRI’s Shakey project [201] attempted to use the planning system STRIPS =-=[86]-=- for robot motion planning; automated theorem proving was used to determine applicability of operators and differences between states [239]. The difficulties posed by the intractability of theorem pro... |

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Citation Context ...cist AI 5 —that knowledge is best represented using formal logic— has been debated as long as the field of knowledge representation has existed. Among logicist AI’s strong advocates are John McCarthy =-=[176, 181]-=-, Patrick Hayes [115, 117, 114], and Robert Moore [192], critics of the logicist approach have included Yehoshua Bar-Hillel [22], Marvin Minsky [191], Drew McDermott[186], and Rodney Brooks [43]. (McD... |

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Citation Context ...[176, 181], Patrick Hayes [115, 117, 114], and Robert Moore [192], critics of the logicist approach have included Yehoshua Bar-Hillel [22], Marvin Minsky [191], Drew McDermott[186], and Rodney Brooks =-=[43]-=-. (McDermott can be counted in both the logicist and anti-logicist camps, having advocated for and contributed to logicist AI [184, 187, 185] before losing faith in the enterprise.) The crux of the de... |

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Citation Context ... At the time Minsky wrote his critique, early work on nonmonotonicity had already begun. Several years later, most of the major formal approaches to nonmonotonic reasoning had already been mapped out =-=[178, 230, 187]-=-. This validated both the logicist AI approach, since it demonstrated that formal systems could be used for default reasoning, and the anti-logicists, who had from the first argued that first-order lo... |

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Citation Context ... in particular, is important for both expert and commonsense reasoning. Diagnosis is a form of abduction; understanding natural language requires abduction as well [125]. Some philosophers of science =-=[222, 120, 121]-=- have suggested that abduction can be grounded in deduction. The idea is to hypothesize or guess an explanation for a particular phenomenon, and then try to justify this guess using deduction. A well-... |

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Citation Context ...y kind of respectable performance from machines on theorem proving problems. Some of the earliest automatic theorem proving methods, such as those of Gilmore [101], Prawitz [224], and Davis and Putnam=-=[70]-=- were based on Herbrand’s theorem, which gives an enumeration process for testing if a theorem of first-order logic is true. Davis and Putnam used Skolem functions and conjunctive normal form clauses,... |

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Citation Context ...icularly effective for proofs involving equations and rewriting. The area of rewriting was largely originated by the work of Knuth and Bendix [148]. For a discussion of term-rewriting techniques, see =-=[77, 12, 78, 205, 263]-=-. Syntax of equational systems A term u is said to be a subterm of t if u is t or if t is f(t1, ..., tn) and u is a subterm of ti for some i. An equation is an expression of the form s = t where s and... |

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Citation Context ... ∧ xn = yn)) (1.7) for all function constants f of arity > 0. These formulas entail t1 �= t2 for any distinct variable-free terms t1, t2. The set of equality axioms that was introduced by Keith Clark =-=[58]-=- and is often used in the theory of logic programming includes, in addition to (1.6) and (1.7), the axioms t �= x, where t is a term containing x as a proper subterm. Domain Closure Consider the first... |

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Citation Context ...s. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution =-=[236]-=-. Propositional logic is, of course, the most important subset of first-order logic; recent surge of interest in representing knowledge by propositional formulas is related to the creation of fast sat... |

942 | Symbolic Boolean manipulation with ordered binary decision diagrams
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Citation Context ...general solution. Nevertheless, there are propositional provers that are surprisingly efficient, and becoming increasingly more so; see Chapter 2 of this handbook for details. Binary decision diagrams=-=[44]-=- are a particular form of propositional formulas for which efficient provers exist. BDD’s are used in hardware verification, and initiated a tremendous surge of interest by industry in formal verifica... |

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Citation Context ...icist AI’s strong advocates are John McCarthy [176, 181], Patrick Hayes [115, 117, 114], and Robert Moore [192], critics of the logicist approach have included Yehoshua Bar-Hillel [22], Marvin Minsky =-=[191]-=-, Drew McDermott[186], and Rodney Brooks [43]. (McDermott can be counted in both the logicist and anti-logicist camps, having advocated for and contributed to logicist AI [184, 187, 185] before losing... |

865 |
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Citation Context ... At the time Minsky wrote his critique, early work on nonmonotonicity had already begun. Several years later, most of the major formal approaches to nonmonotonic reasoning had already been mapped out =-=[178, 230, 187]-=-. This validated both the logicist AI approach, since it demonstrated that formal systems could be used for default reasoning, and the anti-logicists, who had from the first argued that first-order lo... |

795 |
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Citation Context ...tioned in the previous section, these award winning theorem provers have been used for many verification applications.s1. Knowledge Representation and Classical Logic 57 • The Isabelle theorem prover =-=[210, 203]-=- can handle higher-order logics and temporal logics . Isabelle is thus especially well-suited in cases where program specifications are written in temporal or dynamic logic (as is frequently the case)... |

779 | Rewrite Systems - Dershowitz, Jouannaud - 1990 |

749 | Cyc: A Large-Scale Investment in Knowledge Infrastructure
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Citation Context ... detail in Chapter 20, Bos and Markert [36] have used Vampire (as well as the Paradox model finder) to determine whether a hypothesis is entailed by some text. The Cyc artificial intelligence project =-=[161, 160, 174]-=- is another widespread application of non-interactive automated theorem proving. The ultimate goal of Cyc is the development of a comprehensive, encyclopedic knowledge base of commonsense facts, along... |

731 | The Description Logic Handbook: Theory, Implementation and Applications, chapter Basic Description Logics - Baader, Nutt - 2003 |

648 | L.J.: Symbolic model checking: 10 20 states and beyond. Inf Comput 98(2
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Citation Context ...for which efficient provers exist is that of temporal logic, the logic of time (see Chapter 12 of this handbook). This has applications to concurrency. The model-checking approach of Clarke and others=-=[49]-=- has proven to be particularly efficient in this area, and has also stimulated considerable interest by industry. Other logical systems for which provers have been developed are the theory of equation... |

614 | Interpretation as abduction
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Citation Context ...Prolog fails as a general-purpose theorem prover, namely, its unsound unification algorithm, its incomplete search strategy, and its incomplete inference system. PTTP was used in SRI’s TACITUS system =-=[124, 127]-=-, a message understanding system for reports on equipment failure, naval operations, and terrorist activities. PTTP was used specifically to furnish minimal-cost abductive explanations . It is frequen... |

597 |
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Citation Context ...ent, but it does not produce a CNF equivalent to the input formula; it only gives us a conservative extension. Natural Deduction in Propositional Logic Natural deduction , invented by Gerhard Gentzen =-=[97]-=-, formalizes the process of introducing and discharging assumptions , common in informal mathematical proofs. In the natural deduction system for propositional system described below, derivable object... |

591 |
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Citation Context ... of countless non-monotonic logics and almost as many logics of knowledge and belief, and none of the work shows any obvious application to actual knowledge-representation problems. — Eugene Charniak =-=[55]-=- During the last fifteen years, statistical learning techniques have become increasingly popular within AI, particularly for applications such as natural language processing for which classic knowledg... |

573 |
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Citation Context ...ient provers exist. BDD’s are used in hardware verification, and initiated a tremendous surge of interest by industry in formal verification techniques. Also, the Davis-PutnamLogemann-Loveland method =-=[69]-=- for propositional logic is heavily used in industry for hardware verification. Another restricted logic for which efficient provers exist is that of temporal logic, the logic of time (see Chapter 12 ... |

571 | PVS: A Prototype Verification System
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Citation Context ...overs in prominence today, including Otter [183], the provers of Boyer, Moore, and Kaufmann [144, 143], Andrew’s matings prover [3], the HOL prover [103], Isabelle [210], Mizar [267], NuPrl [63], PVS =-=[208]-=-, and many more. Many of these require substantial human guidance to find proofs. The Omega system[247] is a higher order logic proof development system that attempts to overcome some of the shortcomi... |

548 |
A Computational Logic
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Citation Context ...ve the even case directly without induction and the odd case by induction, using the usual ordering on natural numbers.s54 1. Knowledge Representation and Classical Logic The Boyer-Moore prover NqTHM =-=[39, 37]-=- has mathematical induction techniques built in, and many difficult proofs have been done on it, generally with substantial human guidance. For example, correctness of AMD Athlon’s elementary floating... |

518 |
Simple word problems in universal algebras
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Citation Context ...ore and more powerful. The Waldmeister system [93] is particularly effective for proofs involving equations and rewriting. The area of rewriting was largely originated by the work of Knuth and Bendix =-=[148]-=-. For a discussion of term-rewriting techniques, see [77, 12, 78, 205, 263]. Syntax of equational systems A term u is said to be a subterm of t if u is t or if t is f(t1, ..., tn) and u is a subterm o... |

508 | Applications of circumscription to formalizing common-sense reasoning
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Citation Context ...o solve the frame problem, for example, one needs to formalize the principle of inertia —that properties tend to persist over time. However, a naive formalization of this principle along the lines of =-=[179]-=- leads to the multiple extension problem; a phenomenon in which the theory supports several models, some of which are unintuitive. Hanks and McDermott [113] demonstrated a particular example of this, ... |

488 |
The Description Logic Handbook
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Citation Context ... effort to make general theorem proving more efficient (this is discussed at length in Section 1.3); special-purpose reasoning techniques have been developed (e.g., by the description logic community =-=[11]-=- as well as by Cyc (see Section 1.4.2) to determine subsumption and disjointness of classes; and logic programming techniques (for both Prolog (see Section 1.4.4) and answer set programming (see Chapt... |

487 | Inductive logic programming: theory and methods - Muggleton, Raedt - 1994 |

469 | Termination of rewriting
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Citation Context ..., 1. This operation can be repeated any number of times, still yielding a smaller multiset; in fact, the relation ≫ can be defined in this way as the smallest transitive relation having this property =-=[76]-=-. One can show that if > is well founded, so is ≫. For a comparison with other definitions of multiset ordering, see [133]. We now give some examples of termination orderings. The simplest kind of ter... |

464 | Knowledge in Action: Logical Foundations for Specifying & Implementing Dynamical Systems - Reiter |

433 |
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Citation Context ...ly very inefficient. There are a number of classical proof systems for first-order logic: Hilbert-style systems, Gentzen-style systems, natural deduction systems, semantic tableau systems, and others =-=[88]-=-. Since these generally have not found much application to automated deduction, except for semantic tableau systems, they are not discussed here. Typically they specify inference rules of the form A1,... |

433 | Isabelle: A Generic Theorem Prover
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Citation Context ...and belief. There are a number of provers in prominence today, including Otter [183], the provers of Boyer, Moore, and Kaufmann [144, 143], Andrew’s matings prover [3], the HOL prover [103], Isabelle =-=[210]-=-, Mizar [267], NuPrl [63], PVS [208], and many more. Many of these require substantial human guidance to find proofs. The Omega system[247] is a higher order logic proof development system that attemp... |

418 | Simplification by cooperating decision procedures
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(Show Context)
Citation Context ...m for resolution, and for automated deduction in general, soon wore off. This reaction led, for example, to the development of specialized decision procedures for proving theorems in certain theories =-=[196, 197]-=- and the development of expert systems. However, resolution and similar approaches continued to be developed. Data structures were developed permitting the resolution operation to be implemented much ... |

385 |
Logic for Problem-Solving
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Citation Context ...ning (i.e., performing subsumption and classification, and performing consistency checks) over the ontologies. 1.4.4 Prolog In terms of its use in working applications, the logic programming paradigm =-=[151]-=- represents an important success in automated theorem proving. Its main advantage is its efficiency; this makes it suitable for real-world applications. The most popular language for logic programming... |

371 | Gröbner bases: an algorithmic method in polynomial ideal theory. Multidimensional systems theory, Progress, directions and open problems - Buchberger - 1985 |

349 | Programs with Common Sense
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(Show Context)
Citation Context ...nowledge representation is the language of first-order (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper =-=[176]-=-, and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of first-order logic; recent surge of interest in representing ... |

347 | An efficient unification algorithm
- Martelli, Montanari
(Show Context)
Citation Context ...ented by Robinson. This algorithm is worst-case exponential time, but often efficient in practice. Algorithms that are more efficient (and even linear time) on large terms have been devised since then=-=[172, 209]-=-. If s and t are two terms and α is a most general unifier of s and t, then sα can be of size exponential in the sizes of s and t, so constructing sα is inherently exponential unless the proper encodi... |

333 |
The second naive physics manifesto
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Citation Context ...lying issues. At the time that Minsky wrote his paper, his critique on this point was not universally appreciated by proponents of AI logicism. The sense one gets from reading the papers of Pat Hayes =-=[116, 117, 114]-=- 6 , for example, is one of confidence and optimism. Hayes decried the paucity of existing domain formalizations, but at the time seemed to believe that creating the formalizations could be done as lo... |

285 |
ComputerAided Reasoning: An Approach
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(Show Context)
Citation Context ... to a given problem set. Some other provers dealt with higher-order logic, such as the TPS prover of Andrews and others [4, 5] and the interactive NqTHM and ACL2 provers of Boyer, Moore, and Kaufmann =-=[144, 143]-=- for proofs by mathematical induction. Today, a variety of approaches including formal methods and theorem proving seem to be accepted as part of the standard AI tool kit. Despite early difficulties, ... |

284 |
Symbolic Logic and Mechanical Theorem Proving
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(Show Context)
Citation Context ... were developed in the early days of theorem proving. We present a few of them, and mention a number of others. For a discussion of resolution and its refinements, and theorem proving in general, see =-=[54, 168, 46, 278, 88, 159]-=-. It is hard to know which refinements will help on any given example, but experience with a theorem prover can help to give one a better idea of which refinements to try. In general, none of these re... |

279 | A temporal logic for reasoning about processes and plans, Cognitive Science 6
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Citation Context ...lel [22], Marvin Minsky [191], Drew McDermott[186], and Rodney Brooks [43]. (McDermott can be counted in both the logicist and anti-logicist camps, having advocated for and contributed to logicist AI =-=[184, 187, 185]-=- before losing faith in the enterprise.) The crux of the debate is simply this: Logicists believe that first-order logic, along with its modifications, is a language particularly well suited to captur... |

272 | Orderings for term-rewriting systems
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(Show Context)
Citation Context ...atisfies the replacement property. Definition 1.3.15 A quasi-ordering ≥ satisfies the full invariance property (see section 1.3.5) if s > t implies sΘ > tΘ for all s, t, Θ. Theorem 1.3.16 (Dershowitz =-=[75]-=-) For terms over a finite set of function symbols, all quasi-simplification orderings have strict parts which are well founded. Proof. Using Kruskal’s tree theorem [152]. Theorem 1.3.17 Suppose R is a... |

271 |
The intractability of resolution
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Citation Context ...ulas in the propositional calculus because the total number of resolutions performed by a propositional resolution prover in the search for a proof is typically much larger than for DPLL. Also, Haken =-=[110]-=- showed that there are unsatisfiable sets S of propositional clauses for which the length of the shortest resolution refutation is exponential in the size (number of clauses) in S. Despite these ineff... |

262 |
Non-monotonic Logic and Temporal Projection
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(Show Context)
Citation Context ...zation of this principle along the lines of [179] leads to the multiple extension problem; a phenomenon in which the theory supports several models, some of which are unintuitive. Hanks and McDermott =-=[113]-=- demonstrated a particular example of this, the Yale shooting problem. They wrote up a simple nonmonotonic theory containing some general facts about actions (that loading a gun causes the gun to be l... |

247 |
Representations of Commonsense Knowledge
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(Show Context)
Citation Context ...d its inferential power. Note [115] that it is not a particular syntax for which logicists argue; it is the notion of a formal, declarative semantics and methods of inference that are important. (See =-=[96, 65, 240, 40]-=- for examples of how AI logicism is used.) Anti-logicists have argued that the program, outside of textbook examples, is undesirable and infeasible. To paraphrase McDermott [186], You Don’t Want To Do... |

238 | Unification theory
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(Show Context)
Citation Context ...t general E-unifier is an E-unifier that is as general as possible in a certain technical sense relative to the theory E. Many unification algorithms for various sets of equations have been developed =-=[246, 9]-=-. For some theories, there may be at most one most general E-unifier, and for others, there may be more than one, or even infinitely many, most general E-unifiers. An important special case, already m... |

238 |
Non-monotonic logic
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(Show Context)
Citation Context ...lel [22], Marvin Minsky [191], Drew McDermott[186], and Rodney Brooks [43]. (McDermott can be counted in both the logicist and anti-logicist camps, having advocated for and contributed to logicist AI =-=[184, 187, 185]-=- before losing faith in the enterprise.) The crux of the debate is simply this: Logicists believe that first-order logic, along with its modifications, is a language particularly well suited to captur... |

238 | Logical foundations of artificial intelligence - Genesereth, Nilsson - 1987 |

237 | Application of theorem proving to problem solving
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Citation Context ...ioms and an automated theorem prover to reason with those axioms. The first implementation of this vision came in the late 1960s with Cordell Green’s question-answering system QA3 and planning system =-=[105, 106]-=-, Given a set of facts and a question, Green’s question-answering system worked by resolving the (negated) question against the set of facts. Green’s planning system used resolution theorem proving on... |

231 |
Shakey the robot
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(Show Context)
Citation Context ...g on a set of axioms representing facts about the world in order to make simple inferences about moving blocks in a simple blocks-world domain. In the late 1960s and early 1970s, SRI’s Shakey project =-=[201]-=- attempted to use the planning system STRIPS [86] for robot motion planning; automated theorem proving was used to determine applicability of operators and differences between states [239]. The diffic... |