A Machine-Oriented Logic Based on the Resolution Principle
TL;DR: The paper concludes with a discussion of several principles which are applicable to the design of efficient proof-procedures employing resolution as the basle logical process.
Abstract: :tb.~tract. Theorem-proving on the computer, using procedures based on the fund~mental theorem of Herbrand concerning the first-order predicate etdeulus, is examined with ~ view towards improving the efticieney and widening the range of practical applicability of these procedures. A elose analysis of the process of substitution (of terms for variables), and the process of t ruth-funct ional analysis of the results of such substitutions, reveals that both processes can be combined into a single new process (called resolution), i terating which is vastty more ef[ieient than the older cyclic procedures consisting of substitution stages alternating with truth-functional analysis stages. The theory of the resolution process is presented in the form of a system of first<~rder logic with .just one inference principle (the resolution principle). The completeness of the system is proved; the simplest proof-procedure based oil the system is then the direct implementation of the proof of completeness. Howew~r, this procedure is quite inefficient, ~nd the paper concludes with a discussion of several principles (called search principles) which are applicable to the design of efficient proof-procedures employing resolution as the basle logical process.
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TL;DR: This paper proposes a logic for default reasoning, develops a complete proof theory and shows how to interface it with a top down resolution theorem prover, and provides criteria under which the revision of derived beliefs must be effected.
4,146 citations
Cites methods from "A Machine-Oriented Logic Based on t..."
...Then the Skolemizedform of w is obtained as follows (Robinson (1965)): replace each existentially quantified variable y of w by ¢(xl, • •....
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TL;DR: Experiments with a real-world database and knowledge base in a university domain illustrate the promise of this approach to combining first-order logic and probabilistic graphical models in a single representation.
Abstract: We propose a simple approach to combining first-order logic and probabilistic graphical models in a single representation. A Markov logic network (MLN) is a first-order knowledge base with a weight attached to each formula (or clause). Together with a set of constants representing objects in the domain, it specifies a ground Markov network containing one feature for each possible grounding of a first-order formula in the KB, with the corresponding weight. Inference in MLNs is performed by MCMC over the minimal subset of the ground network required for answering the query. Weights are efficiently learned from relational databases by iteratively optimizing a pseudo-likelihood measure. Optionally, additional clauses are learned using inductive logic programming techniques. Experiments with a real-world database and knowledge base in a university domain illustrate the promise of this approach.
2,916 citations
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31 Oct 1995
TL;DR: A theory of the nature of knowledge is proposed, namely, that there is another computer system level immediately above the symbol (or program) level and knowledge itself is the processing medium at this level and the principle of rationality plays a central role.
Abstract: : This paper focuses on a basic substantive problem: the nature of knowledge and representation. There are ample indications that artificial intelligence is in need of substantial work in this area. The paper proposes a theory of the nature of knowledge, namely, that there is another computer system level immediately above the symbol (or program) level. The nature of computer system levels is reviewed, the new level proposed, and its definition is treated in detail. Knowledge itself is the processing medium at this level and the principle of rationality plays a central role. Some consequences of the existence of the knowledge level and some relations to other fields are discussed.
2,512 citations
Cites background from "A Machine-Oriented Logic Based on t..."
...Early work in theorem proving programs for quantified logics culminated in 1965 with Alan Robinson's development of a machine-oriented formulation of first-order logic called Resolution (Robinson, 1965)....
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Book•
01 Jan 1972TL;DR: A computer system for understanding English that contains a parser, a recognition grammar of English, programs for semantic analysis, and a general problem solving system based on the belief that in modeling language understanding, it must deal in an integrated way with all of the aspects of language—syntax, semantics, and inference.
Abstract: This paper describes a computer system for understanding English. The system answers questions, executes commands, and accepts information in an interactive English dialog. It is based on the belief that in modeling language understanding, we must deal in an integrated way with all of the aspects of language—syntax, semantics, and inference. The system contains a parser, a recognition grammar of English, programs for semantic analysis, and a general problem solving system. We assume that a computer cannot deal reasonably with language unless it can understand the subject it is discussing. Therefore, the program is given a detailed model of a particular domain. In addition, the system has a simple model of its own mentality. It can remember and discuss its plans and actions as well as carrying them out. It enters into a dialog with a person, responding to English sentences with actions and English replies, asking for clarification when its heuristic programs cannot understand a sentence through the use of syntactic, semantic, contextual, and physical knowledge. Knowledge in the system is represented in the form of procedures, rather than tables of rules or lists of patterns. By developing special procedural representations for syntax, semantics, and inference, we gain flexibility and power. Since each piece of knowledge can be a procedure, it can call directly on any other piece of knowledge in the system.
2,441 citations
Cites background from "A Machine-Oriented Logic Based on t..."
...…than the full predicate calculus (Bar-Hillel, 1964; Coles, 1968; Darlington, 1964), but the big boost to theorem proving research was the development of the Robinson resolution algorithm (Robinson, 1965)) a very simple “complete uniform proof procedure” for the first-order predicate calculus....
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Book•
01 Aug 1993
TL;DR: SOAR, an implemented proposal for a foundation for a system capable of general intelligent behavior, is presented and its organizational principles, the system as currently implemented, and demonstrations of its capabilities are described.
Abstract: The ultimate goal of work in cognitive architecture is to provide the foundation for a system capable of general intelligent behavior. That is, the goal is to provide the underlying structure that would enable a system to perform the full range of cognitive tasks, employ the full range of problem solving methods and representations appropriate for the tasks, and learn about all aspects of the tasks and its performance on them. In this article we present SOAR, an implemented proposal for such an architecture. We describe its organizational principles, the system as currently implemented, and demonstrations of its capabilities.
2,429 citations
Cites methods from "A Machine-Oriented Logic Based on t..."
...In each case the performance and knowledge of an existing system has been adopted as a target in order to learn as much as possible by comparison: Dypar [6], Version Spaces [44] and Resolution [60]....
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References
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TL;DR: In the present paper, a uniform proof procedure for quantification theory is given which is feasible for use with some rather complicated formulas and which does not ordinarily lead to exponentiation.
Abstract: The hope that mathematical methods employed in the investigation of formal logic would lead to purely computational methods for obtaining mathematical theorems goes back to Leibniz and has been revived by Peano around the turn of the century and by Hilbert's school in the 1920's. Hilbert, noting that all of classical mathematics could be formalized within quantification theory, declared that the problem of finding an algorithm for determining whether or not a given formula of quantification theory is valid was the central problem of mathematical logic. And indeed, at one time it seemed as if investigations of this “decision” problem were on the verge of success. However, it was shown by Church and by Turing that such an algorithm can not exist. This result led to considerable pessimism regarding the possibility of using modern digital computers in deciding significant mathematical questions. However, recently there has been a revival of interest in the whole question. Specifically, it has been realized that while no decision procedure exists for quantification theory there are many proof procedures available—that is, uniform procedures which will ultimately locate a proof for any formula of quantification theory which is valid but which will usually involve seeking “forever” in the case of a formula which is not valid—and that some of these proof procedures could well turn out to be feasible for use with modern computing machinery.Hao Wang [9] and P. C. Gilmore [3] have each produced working programs which employ proof procedures in quantification theory. Gilmore's program employs a form of a basic theorem of mathematical logic due to Herbrand, and Wang's makes use of a formulation of quantification theory related to those studied by Gentzen. However, both programs encounter decisive difficulties with any but the simplest formulas of quantification theory, in connection with methods of doing propositional calculus. Wang's program, because of its use of Gentzen-like methods, involves exponentiation on the total number of truth-functional connectives, whereas Gilmore's program, using normal forms, involves exponentiation on the number of clauses present. Both methods are superior in many cases to truth table methods which involve exponentiation on the total number of variables present, and represent important initial contributions, but both run into difficulty with some fairly simple examples.In the present paper, a uniform proof procedure for quantification theory is given which is feasible for use with some rather complicated formulas and which does not ordinarily lead to exponentiation. The superiority of the present procedure over those previously available is indicated in part by the fact that a formula on which Gilmore's routine for the IBM 704 causes the machine to computer for 21 minutes without obtaining a result was worked successfully by hand computation using the present method in 30 minutes. Cf. §6, below.It should be mentioned that, before it can be hoped to employ proof procedures for quantification theory in obtaining proofs of theorems belonging to “genuine” mathematics, finite axiomatizations, which are “short,” must be obtained for various branches of mathematics. This last question will not be pursued further here; cf., however, Davis and Putnam [2], where one solution to this problem is given for ele
2,743 citations
TL;DR: It is shown that the general case of the Entscheidungsproblem is unsolvable in any system of symbolic logic which is adequate to a certain portion of arithmetic and is ω-consistent.
Abstract: In a recent paper the author has proposed a definition of the commonly used term “effectively calculable” and has shown on the basis of this definition that the general case of the Entscheidungsproblem is unsolvable in any system of symbolic logic which is adequate to a certain portion of arithmetic and is ω-consistent. The purpose of the present note is to outline an extension of this result to the engere Funktionenkalkul of Hilbert and Ackermann.In the author's cited paper it is pointed out that there can be associated recursively with every well-formed formula a recursive enumeration of the formulas into which it is convertible. This means the existence of a recursively defined function a of two positive integers such that, if y is the Godel representation of a well-formed formula Y then a(x, y) is the Godel representation of the xth formula in the enumeration of the formulas into which Y is convertible.Consider the system L of symbolic logic which arises from the engere Funktionenkalkul by adding to it: as additional undefined symbols, a symbol 1 for the number 1 (regarded as an individual), a symbol = for the propositional function = (equality of individuals), a symbol s for the arithmetic function x+1, a symbol a for the arithmetic function a described in the preceding paragraph, and symbols b1, b2, …, bk for the auxiliary arithmetic functions which are employed in the recursive definition of a; and as additional axioms, the recursion equations for the functions a, b1, b2, …, bk (expressed with free individual variables, the class of individuals being taken as identical with the class of positive integers), and two axioms of equality, x = x, and x = y →[F(x)→F(y)].
633 citations
TL;DR: A program is described which can provide a computer with quick logical facility for syllogisms and moderately more complicated sentences and realizes a method for proving that a sentence of quantification theory is logically true.
Abstract: A program is described which can provide a computer with quick logical facility for syllogisms and moderately more complicated sentences. The program realizes a method for proving that a sentence of quantification theory is logically true. The program, furthermore, provides a decision procedure over a subclass of the sentences of quantification theory. The subclass of sentences for which the program provides a decision procedure includes all syllogisms. Full justification of the method is given.
A program for the IBM 704 Data Processing Machine is outlined which realizes the method. Production runs of the program indicate that for a class of moderately complicated sentences the program can produce proofs in intervals ranging up to two minutes.
147 citations
TL;DR: The paper discusses the "combinatorial explosion" difficulties encountered by computer programs embodying proof-construction procedures, and a program developed at Argonne National Laboratory is described in which these difficulties are somewhat alleviated in two ways.
Abstract: There are excellent explanations in the literature of the formulation, in quantification theory, of problems in which a proof is to be found, ff one exists, for a given conclusion from a set of given premises. In particular, in [1] and [3] it is shown how to transform the original problem into a standard form which contains no quantifiers and which consists of a conjunction of disjunctions, each disjunct being an open atomic sentence-form or the negation of one. We assume familiarity with these methods of formulation and preliminary transformation, and provide just those definitions of our working terminology which will be required for the immediate purposes of the paper. The paper discusses the \"combinatorial explosion\" difficulties encountered by computer programs embodying proof-construction procedures. A program developed at Argonne National Laboratory is described in which these difficulties are somewhat alleviated in two ways. The first way, which although very useful in practice is less intellectually satisfying than the second way, consists essentially in incorporating the mathematician-user of the program into the searchqoop. Several examples of proofs obtained by this means are exhibited and discussed, one interesting feature of them being that they are \"reasonably nontrivial\" mathematical exercises. The second way involves a complete proof procedure which seems to be new to the literature but which has not yet been programmed and tested. Following [1] and [3], then, consider logical expressions
81 citations
"A Machine-Oriented Logic Based on t..." refers background or methods or result in this paper
...[2] and [ 5 ]) that in order to detemfine whether a finite set S of sentences of first-order logic is satisfiable, it is sufficient to assume that each sentence in S is in prenex form with no existential quantifiers in the prefix; moreover the matrix of each sentence in S carl be assumed to be a disjunction of formulas each of which is either au atomic formula or the negation of an atomic formula....
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...Is was ~oted in aa earlier paper [ 5 ] that one can express Herbrand's Theorem i~ the following form:...
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...An interesting heuristic remark is that, for every finite set S of clauses which is unsatisfiable and which has a refutation one could possibly construct, there is at least one reasonably small finite subset of the Herbrand universe of S such that P(S) is unsatisfiable and such that P is minimal in the sense that Q(S) is satisfiable for each proper subset Q of P. Such a P was called a proof set for S in [ 5 ]....
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...Accordingly we introduce the following definitions (following in part the nomenclature of [2] and [ 5 ]): 2.1 Variables....
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...The major combinatorial obstacle to efficiency for level-satm'ation procedures is the enormous rate of growth of the finite sets Hi and Hi(S) as 3' increases, at least for most interesting sets S. These growth rates were analyzed in some det~til in [ 5 ], and some examples were there given of some quite simple unsatisfiable S for which the earliest unsatisfiable Hi(N) is so large as to be absolutely beyond the limits of feasibility....
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TL;DR: Algorithms by which decision procedures for the functional calculus can be applied mechanically and it is proved that the method provides a solution to the decision problem in the sense that the given expression is a theorem if and only if M* is tautologous.
Abstract: AbstracL This paper develops algorithms by which decision procedures for the functional calculus can be applied mechanically. Given any expression in Skolem normal form with prefix (3y~)(3y2) . . . (3y,~,)(z~)(z~) . . . (z,~) and matrix M, the algorithms lead to the construction of a nu~trix M* such that M D M* is valid and such that the expressions (3yl)(3y~) . . . (3y,~)(zl)(z~) . . . (z,~).Mand (3y l ) (3y2) \" \" (3y,,,)(zl)(z~) . . (z,~).M*areinterprovable. This procedure is thus a semi-decision procedure for the general Skolem case. For two special cases of this prefix, it is further proved that the method provides a solution to the decision problem in the sense that the given expression is a theorem if and only if M* is tautologous. These cases are ~1) a matrix M in which every elementary part contains at least one of the individual variables z~ , z~, .. • , z,~ or contains only one individual variable or contains both Yl and y2 and no other individual variables; and (2) a matrix M in which every elementary part contains at least one of the individual variables z~ , z2, ., z,, or contains only one individual variable or contains all of y~, y2, ' \" , ym •
21 citations