Loveland,D., 1978, Automated Theorem Proving: A Logical Basis, North Holland.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....caching and lemmaizing, that have reduced by more than an order of magnitude the time required to find proofs of several problems and that have enabled the prover to prove theorems previously unobtainable by top down model elimination theorem provers. 1 Introduction Model Elimination (ME) [20, 22] is a complete inference procedure for the first order predicate calculus. It is the method underlying the Prolog Technology Theorem Prover (PTTP) 33, 34] the SETHEO prover [19] and several or parallel theorem provers [31, 8, 2] The use of model elimination, an input proof procedure, has ....

....extension and reduction inference rules and other ME terminology su#cient for an understanding of the remaining sections. We assume familiarity with terminology of resolution proof procedures, e.g. terms, atomic formulas (atoms) literals, clauses and unification. For a description of these, see [22], which also gives a complete description of the model elimination procedure. We use Prolog notation in which variables are represented by capital letters and functions, constants and predicates are represented by lowercase letters. Although the juxtaposition of vowels in this word may be ....

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D. W. Loveland. Automated Theorem Proving: A Logical Basis. North-Holland, 1978.

....is based. It also includes proofs of the theorems that establish the needed logical properties of various inference rules and strategies. Another good choice for a discussion of logic oriented toward automated reasoning is the book by D. Loveland entitled Automated Theorem Proving: A Logical Basis [Loveland, 1978]. Most recently, J. Kalman [2001] published a book entitled Automated Reasoning with Otter. This scholarly work covers in fine detail such topics as how to convey a problem to an automated reasoning program, how to find a proof by contradiction, and how to reason about equality. It is a unique ....

D. W. Loveland. Automated theorem proving: A logical basis. North-Holland, Amsterdam, 1978.

....1.2 Theorem Proving as Question Answering More specifically, our exploration of answers takes place in the context of automated reasoning. Resolution refutation, along with many variations and enhancement, is widely employed for automated reasoning tasks [Robinson, 1965, Chang and Lee, 1973, Loveland, 1978, Kowalski, 1979, McCune, 1994] Resolution plus factoring is refutation complete for first order predicate calculus (FOPC) Chang and Lee, 1973, Loveland, 1978] given a consistent knowledge base K and a proposition p, it can be determined if K [ f: pg is inconsistent. This in turn means that p ....

.... along with many variations and enhancement, is widely employed for automated reasoning tasks [Robinson, 1965, Chang and Lee, 1973, Loveland, 1978, Kowalski, 1979, McCune, 1994] Resolution plus factoring is refutation complete for first order predicate calculus (FOPC) Chang and Lee, 1973, Loveland, 1978] given a consistent knowledge base K and a proposition p, it can be determined if K [ f: pg is inconsistent. This in turn means that p follows from K . Automated theorem proving served as an early model for question answering in the field of AI [Green and Raphael, 1968, Green, 1969b, Luckham ....

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Donald W. Loveland. Automated Theorem Proving: A Logical Basis. Fundamental Studies in Computer Science. North-Holland, 1978.

....of the activated link is deleted. Nevertheless, Theorem 4 seems to lift for ordered clause set resolution with little modification. Of course, unit resolutions cannot be pursued indefinitely as in the ground case, but must be alternated with rounds of ordered resolution in some fair way (see [16]) However, subsumptions by units and of parents by unit resolvents can be employed, since all instances of these operations would have been done in the ground proof. ....

Loveland, D.W., Automated Theorem Proving: A Logical Basis, North-Holland, New York, 1978.

....nonmonotonic reasoning system to have a practical implementation [10] Since then, systems implementing versions of Reiter s default logic (e.g. 19] and Nute s defeasible logic (e.g. 9] have appeared. Implementations of THEORIST are usually based on Loveland s MESON reasoning procedure [8] and use the Prolog logic programming language. Modern Prolog systems offer high inference rates as well as the traditional benefits of inbuilt logical variables and unification. Lemmas partial results saved during a proof for later reuse provide opportunities for improving the efficiency of ....

....=12 is h, h2, h, 17] so we can store potential crucial literal informa tion for 12. Note that this inference is only valid if there are no model elimination steps during the derivation of 12. 4 Implementation THEORIST is typically implemented by extending Loveland s MESON theorem prover [8]. The first interpreters appeared in the mid to late 1980s and Poole and Goodwin s compiler appeared in 1987 [12] Though these systems bear similarities to Stickels Prolog Technology Theorem Prover [20] which is it self derived from the MESON system, they were developed independently. The ....

Donald W. Loveland. Automated Theorem Prov- ing: A Logical Basis, volume 6 of Fundamental Studies in Computer Science. North Holland, Amsterdam, The Netherlands, 1978.

....lemmata, cut rule. 1 Introduction Tableau and connection calculi offer interesting approaches to automated deduction which deviate from the uniform resolution paradigm. A convincing illustration of this fact is the relative success of proof procedures based on model elimination [Loveland, 1968, Loveland, 1978] as described in [Stickel, 1988] Letz et al. 1992] or [Astrachan and Loveland, 1991] Although, historically, introduced against the background of resolution as a procedure generating formulae, in its essence model elimination belongs to the family of analytic tableau and connection calculi. ....

....=m= Q(b) m= S(x ) 0. c 4 :2 P (a) m= Q(b) 0 S(x ) m= Q(x ) Figure 2: Subgoal matrix notation of a depth first construction of the tableau in Figure 1. 2.5. 2 Model Elimination Chains The model elimination calculus was introduced in [Loveland, 1968] and improved in [Loveland, 1969, Loveland, 1978] and [Astrachan and Loveland, 1991] Model elimination can be viewed as a refinement of the connection tableau calculus in which subgoals are selected in a depth first right most manner. This re interpretation of model elimination has various advantages concerning generality, elegance, and the ....

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D. W. Loveland. Automated Theorem Proving: a Logical Basis. North-Holland, 1978.

....clauses. Although input resolution is complete for Horn clauses, it is incomplete in general. However, the linear restriction of resolution, in which derived clauses can be resolved with their own ancestor clauses or with input clauses, is complete in general. The model elimination (ME) procedure [19, 20] can be viewed as a very convenient and efficient way to implement linear resolution. It is a complete inference system for non Horn as well as Horn sets of clauses. The SL resolution procedure [17] is similar; the principal difference is its need for an additional factoring operation. Prolog s ....

Loveland, D.W. Automated Theorem Proving: A Logical Basis. North-Holland, Amsterdam, the Netherlands, 1978.

.... by standard university textbooks on artificial intelligence, such as Nils Nilsson s [88] and George Luger and William Stubblefield s [78] More thoroghly it is described in Wos, Overbeek, Lusk and Boyle s book [126] A thoroughly theoretical but a little outdated account is given by Loveland [77]. The recent North Holland handbook [100] is an advanced treatment. 4.1 The limits of automated reasoning The ultimate goal of research in the field of automated reasoning is to find an efficient decision procedure for each interesting logical system. In practice, this is an unattainable goal: ....

....procedure and a refutation procedure to fail to terminate; some contingent formulae are such. The fact that they indeed do exist follows from Church and Turing s theorem discussed in the previous chapter. Note that this terminology (uniformly used in publications in automated reasoning, such as [88, 78, 126, 77, 100]) may be confusing. A refutable formula is one that is not true in all possible worlds and interpretations. However, a refutation procedure does not detect refutable formulae but contradictions Higher order logic does not even have a proof procedure; this follows straightforwardly from ....

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Donald W. Loveland. Automated Theorem Proving: A Logical Basis, vol. 6 of Fundamental Studies in Computer Science. Amsterdam: North-Holland, 1978.

....Sim. trajectory break point at time t2 (due to agent 2 choice) Causes This exhaustive constraint based search over a range of possible trajectories makes it possible to establish the necessity of postulated emergent tendencies. Following a procedure similar to that used in theorem proving [3,10], a subset of the possible simulation parameterisations and agent choices is specified, the target emergent tendencies are prearranged in the form of negative constraints, and an automatic search over the possible trajectories is performed. Tendencies are shown to be necessary, with respect to the ....

Loveland, D. W., Automated Theorem-proving: A Logical Basis, North-Holland Pub., Amsterdam, 1978. 10

....of the instances. A better understanding of how this method di#ers from other methods would be useful. 3. 3 AI: Learning and restricted learning As the basic OR technique for solving satisfiability problems is branch andbound, the basic AI technique is the classic Davis Putnam Loveland method [11, 29]. 10 Davis Putnam Loveland The DPL procedure takes a valid partial assignment and attempts to extend it to a valid total assignment by incrementally assigning values to variables. This creates a binary search tree where each node corresponds to a set of variable assignments. If the algorithm ....

D. W. Loveland. Automated Theorem Proving: A Logical Basis. North Holland, 1978.

....Sim. trajectory break point at time t2 (due to agent 2 choice) Causes 5 This exhaustive constraint based search over a range of possible trajectories makes it possible to establish the necessity of postulated emergent tendencies. Following a procedure similar to that used in theorem proving [3,10], a subset of the possible simulation parameterisations and agent choices is specified, the target emergent tendencies are prearranged in the form of negative constraints, and an automatic search over the possible trajectories is performed. Tendencies are shown to be necessary, with respect to ....

Loveland, D. W., Automated Theorem-proving: A Logical Basis, North-Holland Pub., Amsterdam, 1978.

....not be changed. Clause Hardness Evaluation Many algorithms for the SAT problem have been proposed (see for instance [5,8,9,14] for extensive references) Most of complete methods are based on enumeration techniques and perform a tree search. They are also called Davis Putnam Loveland variants [7,12], and have the following general structure: DPL scheme (1) Choose a variable x according to a branching rule (see e.g. 10] Generally, priority is given to variables appearing in unit clauses (unit resolution) 2) Fix x to a truth value and remove all satisfied clauses and all falsified ....

D.W. Loveland. Automated Theorem Proving: a Logical Basis. North Holland, 1978.

....form, named SAT, is well known to be NP complete [7] and plays a protagonist role in mathematical logic and computing theory. A SAT formulation can be used to solve the problem of logical implication, that is to detect if a given proposition is logically implied by a set of propositions [8] 9] [10]. In the case of questionnaires, every edit can be encoded in a propo sitional logic clause. Moreover, since edits have a very precise syntax, encoding could be performed by means of the following automatic procedure. Edit propositional encoding procedure 1. Identification of the domains Dr for ....

....to 100. a) is clearly redundant. role head of the house) A (annual income 100) a) ual income 100 (b) A SAT formulation is used to solve the problem of logical implication. Given a set of statements and a single statement s, s if and only if J s is an unsatisfiable formula [9] [10]. Therefore, the following holds. Theorem 3.2. The clausal representation of an edit ej is implied by the clausal representation of a set of edits E if and only if E J ej i unsatist]able. It can be consequently checked if an edit with clausal representation ej is re dundant by testing if the ....

D.W.Loveland. Automated Theorem Proving: a Logical Basis. North Holland 1978.

....of the instances. A better understanding of how this method di ers from other methods would be useful. 3. 3 AI: Learning and restricted learning As the basic OR technique for solving satis ability problems is branch andbound, the basic AI technique is the classic Davis Putnam Loveland method [11, 29]. 10 Davis Putnam Loveland The DPL procedure takes a valid partial assignment and attempts to extend it to a valid total assignment by incrementally assigning values to variables. This creates a binary search tree where each node corresponds to a set of variable assignments. If the algorithm ....

D. W. Loveland. Automated Theorem Proving: A Logical Basis. North Holland, 1978.

.... Moreover consequence finding plays an important role in inductive logic programming [15, 22] Inoue [6] proposed SOL resolution for mechanically finding characteristic clauses within first order logic, which can be viewed as either an extension of Loveland s model elimination like calculus [13] with Skip operation or a generalization of Siegel s propositional production algorithm [16] Compared with other procedures, SOLresolution allows it to focus on generating only the characteristic clauses rather than all minimal logical consequences. SOL resolution is one of the most advanced and ....

....Newarc( Sigma; F; P) f2g (see also Proposition 2.5 in [6] for more details) Thus the consequence finding problem is a generalization of the ordinary refutational theorem proving problem. 3 SOL Tableau Calculus The original SOL resolution [6] was given in a model elimination like chain format [13]. In this paper, we reformulate SOL resolution within the framework of connection tableau calculus[10, 12] Compared with a chain format, a tableau format has several advantages, such as the retaining of solved parts in a tableau and etc. Definition 3 (Clausal tableau, Branch) 91 1. A clausal ....

D.W. Loveland, Automated Theorem Proving: a logical basis (North-Holland Publishing Company, Amsterdam, 1978).

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Loveland,D., 1978, Automated Theorem Proving: A Logical Basis, North Holland.

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D. Loveland. Automated Theorem Proving: A Logical Basis. North-Holland, Amsterdam, 1978.

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D. W. Loveland. Automated theorem proving: A logical basis. North Holland, Amsterdam, New York, 1978.

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Loveland, D. W. (1978). Automated theorem proving: a logical basis. Fundamental studies in computer science. North-Holland.

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D.W. Loveland. Automated Theorem Proving: a Logical Basis. (North Holland, 1978).

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D. Loveland. Automated Theorem Proving: A Logical Basis. North-Holland, Amsterdam, 1978.

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D. Loveland. Automated Theorem Proving - A Logical Basis. North Holland, 1978.

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D. W. Loveland. Automated Theorem Proving: A Logical Basis. North--Holland, 1978.

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D. Loveland. Automated Theorem Proving - A Logical Basis. North Holland, 1978.

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D.W. Loveland. Automated Theorem Proving: a Logical Basis. (North Holland, 1978).

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