Stickel, M. E. (1985). Automated deduction by theory resolution. Journal of Automated Reasoning, 1:333-355. 40  Home/Search   Document Details and Download   Summary   Related Articles   Check

....generalities and referring to categories or kinds: as such, they are often desirable. In a system that provides generic answers, the need to introduce individuals just so that an answer can be found (e.g. a particular fox, a particular wolf, etc. in formulations of Schubert s Steamroller [Stickel, 1985]) is obviated. There are many circumstances under which a generic answer is preferable to a specific answer. Even when there is not a preference, a generic answer may be given in the absence of a specific answer, conveying information that would otherwise be lost. When backward chaining is ....

Mark E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1:333--355, 1985.

....system and the technical e ort to implement it are considerable. On the other hand, in black box approaches a theory module is linked in a modular fashion to a general deductive system and the interaction between both is limited to a relatively small interface. Examples include theory resolution [Sti85], constraint resolution [B ur90] hierarchic superposition [BGW94] and the combination procedures of Nelson and Oppen [NO79] and Shostak [Sho84] Shostak introduced a congruence closure procedure that can be combined with decision procedures for other theories, provided that these theories have ....

M. E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1(4):333-355, 1985.

....the cycle, and X = t 1 ; r = t n g is the set of exit equations after k iterations through the cycle. A cycle unification problem should not be confused with a theory unification problem hG =C F i , i.e. the problem whether there exists a substitution oe such that oeG =C oeF [1, 19]. The following proposition is an immediate consequence of the completeness and soundness of the connection method [1] or SLD resolution, e.g. 9] Due to lack of space we had to omit the proof of this proposition and all further theorems. They can be found in detail in [4] Proposition 1 oe is a ....

M. E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasonsing, 1:333--356, 1985.

....consider cycle unification problems of this form. Furthermore, as the case P 6= P is trivial, we assume P = P . A cycle unification problem should not be confused with a theory unification problem hG =C F i , i.e. the problem whether there exists a substitution oe such that oeG =C oeF [Bib87, Sti85]. To solve a cycle unification problem hG Gamma F i we have to find a substitution which either unifies G and F or unifies viz. simultaneously unifies each equation in = N [ Y [ X where N = fs 1 1 ; s n n g is the set of entry equations, 1 ; ....

M.E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1:333--355, 1985.

....from a specific calculus without incorporating already known facts, mathematicians try to draw as much as possible on facts that have already been derived for a particular mathematical domain. This insight gave rise to the investigation of theory reasoning (as for instance, of theory resolution [18] and constrained resolution [3] and to knowledge based proof planning [9] In knowledge based proof planning external reasoners can be integrated. In particular, a domain specific constraint solver can help to construct mathematical objects that are elements of a specific domain. As long as ....

M.K. Stickel, `Automated deduction by theory resolution', in Proc. of the 9th International Joint Conference on Artificial Intelligence, (1985).

....deduction paradigm in which a general purpose reasoner is complemented by a specialized procedure, the background reasoner, which (semi )decides formula satisfiability with respect to a certain background theory. After the pioneering work of Stickel who devised a theory version of resolution [8], nearly all existing calculi for automatic reasoning have been extended to theory reasoning (see [2] for a survey) Essentially all of them however consider the integration of just one background reasoner into the main one. The usual explanation for such a limitation is that, although ....

....and scalability purposes, it is hard to get them to cooperate in a sound and complete way. We argue in this paper that the cooperation of background reasoners in theory reasoning is instead conceptually simple and can be easily described in terms of partial theory reasoning in the sense of [8]. We will appeal to a variant of a well known interpolation result, the Craig Interpolation Lemma, to show that background reasoner cooperation can be achieved as a form of constraint propagation, much in the spirit of existing combination methods for decision procedures [5] The main idea in this ....

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M. E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1(4):333--355, 1985.

....on an algorithmic representation of the theory, its decision procedure, and use it as a background reasoner. Although the main idea of theory reasoning can be found in several early works (such as [Bib82, Plo72] to name just a few) the first systematic treatment of it was given by Stickel in [Sti85] which describes a theory version of the resolution calculus and the matings calculus. After that work, nearly all existing calculi for automatic reasoning have been extended to theory reasoning (see [BFP92] for a survey) Essentially all of them, however, consider the integration of just one ....

....cooperation in a sound and complete way is a non trivial task. We introduce one such way in this paper. We show that the cooperation of background reasoners in theory reasoning is actually conceptually simple, and can be easily described in terms of partial theory reasoning in the sense of [Sti85] We appeal to a variant of a well known interpolation result, the Craig Interpolation Lemma, to show that background reasoner cooperation can be achieved as a form of constraint propagation, much in the spirit of well known combination methods for decision procedures [NO79] The main idea is to ....

[Article contains additional citation context not shown here]

Mark E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1(4):333--355, 1985.

....deduction paradigm in which a general purpose reasoner is complemented by a specialized procedure, the background reasoner, which (semi )decides formula satis ability with respect to a certain background theory. After the pioneering work of Stickel who devised a theory version of resolution [8], nearly all existing calculi for automatic reasoning have been extended to theory reasoning (see [2] for a survey) Essentially all of them however consider the integration of just one background reasoner into the main one. The usual explanation for such a limitation is that, although ....

....and scalability purposes, it is hard to get them to cooperate in a sound and complete way. We argue in this paper that the cooperation of background reasoners in theory reasoning is instead conceptually simple and can be easily described in terms of partial theory reasoning in the sense of [8]. We will appeal to a variant of a well known interpolation result, the Craig Interpolation Lemma, to show that background reasoner cooperation can be achieved as a form of constraint propagation, much in the spirit of existing combination methods for decision procedures [5] The main idea in this ....

[Article contains additional citation context not shown here]

M. E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1(4):333-355, 1985.

....to be constructed. More often than not, uni cation o ers little support for this task and logic proofs, say of linear inequalities, can be very long and infeasible for purely logical theorem proving. This situation was a reason to develop theory reasoning approaches, e.g. in theory resolution [19], constrained resolution [6] and constraint logic programming [8] and to integrate linear arithmetic decision procedures into provers such as Nqthm [4] Boyer and Moore, e.g. report how dicult such as integration may be. In knowledge based proof planning [12] external reasoners can be ....

M.K. Stickel. Automated Deduction by Theory Resolution. In Proc. of the 9th International Joint Conference on Articial Intelligence, 1985.

....More often than not, unification offers little support for this task and logic proofs, say of linear inequalities, can be very long and infeasible for purely logical automated theorem proving. This situation was a reason to develop theory reasoning approaches, e.g. in theory resolution [16], constrained resolution [4] and constraint logic programming [5] and to integrate linear arithmetic decision procedures into provers such as Nqthm [3] The construction of mathematical objects with theory specific properties is necessary, e.g. in proving theorems about limits in the theory of ....

M.K. Stickel. Automated deduction by theory resolution. In Proc. of the 9th International Joint Conference on Artificial Intelligence, pages 1181--1186, 1985. 10

....subject to certain constraints over the background theory. The satis ability of these constraints is not veri ed by the main reasoner itself but is instead delegated to the more ecient background reasoner. After Stickel s pioneering work on a theory reasoning version of the resolution calculus [Sti85] nearly all existing calculi for automatic reasoning have been extended to theory reasoning (see [BFP92] for a survey) Essentially all of them, however, consider the integration of just one background reasoner into the main one. The reason for such a restriction, despite the clear desirability ....

....cooperation in a sound and complete way is a non trivial task. We introduce one such way in this paper. 1 We show that the cooperation of background reasoners in theory reasoning is actually conceptually simple, and can be easily described in terms of partial theory reasoning in the sense of [Sti85] We appeal to a variant of a well known interpolation result, the Craig Interpolation Lemma, to show that background reasoner cooperation can be achieved as a form of constraint propagation, much in the spirit of existing combination methods for decision procedures [NO79] The main idea is to ....

[Article contains additional citation context not shown here]

Mark E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1(4):333-355, 1985.

....procedures instead of algorithms computing complete sets of uni ers may be advantageous for theories where the complete sets are large or even in nite. E uni cation algorithms that compute complete sets of uni ers are, for example, applied in theorem proving with built in theories (see, e.g. [55, 68]) in generalizations of the Knuth Bendix completion procedure to rewriting modulo theories (see, e.g. 34, 13] and in logic programming with equality (see, e.g. 32] With the development of constraint approaches to theorem proving (see, e.g. 18, 51] term rewriting (see, e.g. 41] and ....

M.E. Stickel. Automated deduction by theory resolution. J. Automated Reasoning, 1(4):333-355, 1985.

....the concepts of T Wrappers and logic programs (Lloyd 1987) to be combined more easily and to guide the extraction process with interfering inference processes. Therefore we interpret T Wrappers as special theories and handle Logic Programs extended with T Wrappers by means of theory reasoning (Stickel 1985). These methods are discussed in (Thomas 1999b) In general a T Wrapper defines a relation, where its instances are obtained by extracting information from a document with an associated pattern (T Pattern) A T Wrapper is said to be successful if its set of instances is not empty. The ....

Stickel, M. 1985. Automated Deduction by Theory Resolution.

....of a theory, where the calculation of the theory (the facts) are performed by a background reasoner. In the following we refer to normal logic programs when we talk about logic programs. We assume the reader to be familiar with the fields of logic programming [11] and theory reasoning [13]. 3.1 Template Theories A token template theory 9 T is the set of all template ground atoms, that we obtain by applying all templates in T . For example, consider the template set t D v p . Assume p to be an arbitrary token pattern and v an extraction tupel. A template ....

M. Stickel. Automated Deduction by Theory Resolution. Journal of Automated Reasoning, 1:333--355, 1985.

....and chapter I.2.6 on 243 W. Bibel, P. H. Schmitt (eds. Automated Deduction. A basis for applications. Vol. III c 1998 Kluwer Academic Publishers. Printed in the Netherlands 244 FRIEDER STOLZENBURG AND BERND THOMAS theory reasoning. This may be traced back to the work of Bibel (1982) and Stickel (1985). It is interesting to notice that the success of the application to be presented here is due to the combination of both approaches. 2.1. The PTTP Technique Constraint logic programming (CLP) is a field of active research. There, logic programming with Horn clauses is enhanced with an interface ....

Stickel, M. E.: 1985, `Automated Deduction by Theory Resolution'. Journal of Automated Reasoning 1(3), 333--355.

....1986] he describes a resolution based proof system R for L Bq . In addition to Robinson s binary resolution rule [Robinson, 1965] this system contains the B resolution rule to deal with belief literals. B resolution is a special case of total narrow theory resolution developed by Stickel [Stickel, 1985]. Here is its definition: S i ] Phi 1 A 1 [S i ] Phi 2 A 2 . S i ] Phi n A n : S i ] A A 1 A 2 Delta Delta Delta A n A; when Phi ffl 1 ; Phi ffl n ae(i) ffl The rule is justified by the attachment lemma; it basically says that if we can deduce from ....

M. E. Stickel. Automated deduction by theory resolution. In Proceedings of the Ninth International Conference on Artificial Intelligence, pages 1181--1186, Palo Alto, CA, 1985. Morgan Kaufmann.

....5 Constraints in Logic Formalisms We survey and discuss the addition of constraints to several logic formalisms. Although logic programming was the first logic formalism to which constraints, in the sense of this paper, were added [19] there was a significant precursor in automated reasoning [53]. For this reason, and because work on automated reasoning extends naturally the work of Herbrand. we begin the discussion with automated reasoning. 5.1 Automated Reasoning Many of the comments of the previous section can be applied directly in the field of automated reasoning. A substantial ....

M. Stickel, Automated Deduction by Theory Resolution, Journal of Automated Reasoning 1, 333--355, 1984.

....in an inference C; A G D; A H C[A= D which is related to (a sequence of two) type B inferences of the inverse method but with slight di erences in the way multiple occurrences of the resolved atom A are handled. 7.6. Ordered Theory Resolution Theory resolution, a concept introduced by Stickel [1985], refers to resolution inferences that have been specially designed for a given consistent set of clauses T , called the theory. Clauses not in T are also called goal clauses. A minimal requirement for a theory resolution calculus is that explicit inferences within the theory be Resolution ....

Stickel M. E. [1985], Automated deduction by theory resolution, in A. Joshi, ed., `Proceedings of the 9th International Joint Conference on Articial Intelligence', Morgan Kaufmann, Los Angeles, CA, pp. 1181-1186.

....or initialize new sub searching processes. By merging token templates and logic programs we gain a mighty inference mechanism that allows us to search the web with deductive methods. We emphasize the well defined theoretical background for this integration, which is given by theory reasoning [2] [26] in logical calculi, whereas token templates are interpreted as theories. This article is organized as follows: in Section 2 we describe the language of tokentemplates for the fact retrieval from web pages. In Section 3 the integration of tokentemplates into logic programs and the underlying T ....

....with template theories, which will lead to the T T Calculus, is defined in Section 3.2. Section 4.1 and 4.2 will give some small examples for the use of token templates in logic programs. We assume the reader to be familiar with the fields of logic programming [16] and theory reasoning [2] [26]. 3.1. Template Theories In the context of first order predicate logic (PL1) we interpret a set of token templates to be an axiomatization of a theory. A token template theory T T is the set of all template ground atoms, that we obtain by applying all templates in T . For example, consider the ....

M.E. Stickel. Automated Deduction by Theory Resolution. Journal of Automated Reasoning, 1:333--355, 1985.

.... its proof progression (the system designer uses this information to change the theory or to detect problems in it) needs to allow for proof strategies (to be specified either by a human or by another layer) needs to allow for easy embedding of semantic attachments (and possibly theory resolution [Stickel, 1985]) and needs to allow for nonmonotonic reasoning (or at least some approximation of it) 3 The first implementation of the LSA can be found at http: www formal.stanford.edu eyal lsa lsaV1.0.tar.gz. Layer 0 Layer 1 Layer 2 Layer 3 Time Infer. Time Infer. Time Infer. Time Infer. Mean SD Mean ....

Mark E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1:333--355, 1985.

....need to be complemented with axioms of interest, e.g. axioms for arithmetic, or for associative commutative operations. It is then interesting to build these axioms inside the prover itself, for performance reasons. This has been done for a variety of theories, whether equational [Plo72] or not [Sti85] Recently, Baaz, Egly and Fermuller [BEF97] have proposed to build in various forms of induction in tableaux theorem provers, by coding minimality principles as Skolem functions. One form of theory that has not yet been investigated is that of datatype constructors. Consider for instance the ....

Mark E. Stickel. Automated deduction by theory resolution. In Proceedings of the 9th International Joint Conference on Artificial Intelligence, pages 1181--1186, Los Angeles, 1985.

....address the problem of enriching an interactive theorem prover with complex proof procedures. Most of the notable works concerned with the integration of decision procedures inside a theorem proving system have been developed in the setting of completely automated theorem proving (see for example [NO78, BM88, Sti85]) All these works are concerned with building and integrating procedures specialized to deal with particular subproblems w.r.t. a given general and uniform proof strategy. In interactive theorem proving a change of perspective is required. The heuristic strategy is not fixed and decision ....

M. Stickel. Automated Deduction by Theory Resolution. Journal of Automated Reasoning, 4:333--356, 1985.

....systems. The knowledge from a given domain (or theory) is made use of by applying efficient methods for reasoning in that domain. The general purpose foreground reasoner calls a special purpose background reasoner to handle problems from a certain theory. Following the pioneering work of Stickel [22], sound and complete theory reasoning methods have been described for various calculi; e.g. path resolution [17] the connection method [20] model elimination [2] In addition, background reasoners have been designed for various theories, in particular for equality reasoning [4] an overview can ....

M. E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1:333--355, 1985. 15 16

....a speci c logic calculus without incorporating already known facts, mathematicians try to draw as much as possible on facts that have already been derived for a particular mathematical domain. This insight gave rise to the investigation of theory reasoning (as for instance, of theory resolution [Sti85] and constrained resolution [B ur94] and to knowledge based proof planning [MS99] In particular, diculties with purely logical proofs can be caused by the need to construct mathematical objects (witnesses) either by computation or restricting the range of variables. Uni cation o ers little ....

M.K. Stickel. Automated deduction by theory resolution. In Proceedings of the 9th International Joint Conference on Articial Intelligence, pages 1181-1186, 1985.

....sense) The languages support a term syntax which is not first order, although every term can be interpreted through first order constraints. Unlike other CLP languages domains, 1 We note, however, some work combining constraints and resolution in first order automated theorem proving [242, 44]. 2 The language Absys [82] which was very similar to Prolog, used equations explicitly, making it more obviously a CLP language. 6 Prolog like trees are essentially part of this domain, instead of being built on top of the domain. CIL [192] computes over a domain similar to feature trees. ....

M. Stickel, Automated Deduction by Theory Resolution, Journal of Automated Reasoning 1, 333-355, 1984.

.... this would require an enumeration over the infinite domain of the simulation structure (ff and fi are formulas in first order logic, prime and even are unary predicates with their standard interpretation, and expressions are universally quantified) For a generalization on Weyrauch s approach, see Stickel (1985). 3.3 Spatial Reasoning An important early example of the use of an analogical representation can be found in Funt s spatial reasoning system, called whisper (Funt 1976, 1980, and 1983) whisper deals with simple stability problems in a blocks world. It is able to predict how an unstable ....

Stickel, M.E. (1985). "Automated Deduction by Theory Resolution." Proceedings of the Ninth International Joint Conference on Artificial Intelligence, Los Angeles, CA, 1985, 1181--6.

....1996c) can be used 17 as a basis for developing extensible modular constraint languages, i.e. languages combining constraint solvers over different data types in truly modular fashion a la Clear OBJ tradition. This needs further 17 In a remotely similar way to the theory resolution of (Stickel, 1985). Also, recent advances in making paramodulation based techniques more effcient (see (Bachmair et al. 1995) for example) have to be incorporated in any system implementing constraint paramodulation. Constraint Logic 31 development of the topic of Section 7.2 in conjunction with constraint ....

Stickel, M. (1985). Automated deduction by theory resolution. Journal of Automated Reasoning, 1(4):333-- 355.

....reasoner with a background reasoner. The foreground reasoner takes care of the general logical structure of a formula to be proved or refuted. The background reasoner is consulted whenever the meaning of special built ins has to be considered. Many instances of this general scheme are known (see [9, 7, 2] for overviews) A number of general results [2, 3, 4, 7] form a framework for building in theories into quit different theorem proving procedures. However, those approaches consider the built in theory as homogeneous. Nevertheless, in certain applications we have to take care of the internal ....

....easier. Further question is, what can we learn for proving in hybrid theories from combination techniques for unification algorithms (see [1] Related work. M. Stickel extended resolution to theory resolution and showed many improvements of resolution as special kinds of theory resolution [9]. 2] presents an alternative view exploring possibilities of computing sets of theory connections. An approach to theories given by classes of models has been presented by H. J. Burckert [4] ....

M.E. Stickel. Automated deduction by theory resolution. J. of Automated Reasoning, 4(1):333--356, 1985. 112

....axioms are now redundant: as the terms 0 y and y are congruent, the axiom 8y 0 y = y is congruent to the equality axiom 8y y = y. Hence, it can be dropped. Using the terminology introduced by Plotkin, these axioms have been builtin (Plotkin, 1972; Andrews, 1971; Peterson and Stickel, 1981; Stickel, 1985; Jouannaud and Kirchner, 1986; March e, 1994; Viry, 1995; Viry, 1998) In the example above, the congruence is just the congruent closure of the relation induced on terms by the term rewriting system. In many situations, it is also natural to consider congruences de ned directly at the ....

Stickel, M. (1985). Automated deduction by theory resolution. Journal of Automated Reasoning, 1(4):285-289.

....constraints and of the relation to rewriting techniques. It supports a concise evaluation and comparison of the proof search complexity of chaining calculi with related methods. For unordered resolution, the set of support strategy has been developed in [17] a scenario for theory resolution in [15]. Both methods are based on semantic considerations. Ordered chaining calculi have been proposed in [2, 5] and later without the opaque Transitivity Resolution rule in [6] based on additional coding. Although such coding is beyond our simple and natural approach, for which the Transitivity ....

M. Stickel. Automated deduction by theory resolution. In A. Joshi, editor, 9th International Joint Conference on Articial Intelligence, pages 1181-1186. Morgan Kaufmann, 1985.

....extends naturally to general 98 sentences of 2LS(L) not necessarily restricted. Notice that the satis ability problem for pure rst order 98 sentences with equality has been rst solved by Bernays and Sch on nkel (cf. 9] A more general approach to theory reasoning has been rst described in [12] (in the context of resolution) and, more recently, in [1] in the context of semantic tableaux) In particular, 1] proposes a general incremental approach to apply theory reasoning to the framework of free variable semantic tableaux. Though our approach is more restricted in scope, in some ....

M.E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1:333-355, 1985.

.... translationfunction, performingtherequiredadditive modiflcationstotheaxiomof Deflnition2.W earecurrentlyexploringthepossibility ofan e cientimplementationofthisframeworkby usingthe theoryresolution techniqueforflrst ordertheoremprovingdevisedby Stickel[Sti85],which wouldallow usto wire theLT theoryintoa resolution basedtheoremprover,thereby freeingtheKB managementsystemfromtheneedtohandletheaxiomsofLT directly. ....

Mark E.Stickel.Automateddeductionby theoryresolution.Journal of Automated Reasoning,1:333-355,1985.

....theorem provers is deemed in part responsible for their low eciency in dealing even with simple theorems; as a remedy, the speci c nature of the problems being tackled should enter explicitly into play. Within the T resolution framework (as well as in other similar proposals such as [Sti85]) the reasoning activity develops at two levels; combining a decider expert on a speci c domain with a general foreground inference engine (cf. Section 5) In this report we extend to T resolution the last of the above mentioned approaches to completeness proofs. The paper is organized as ....

....in general, very large sentences must be manipulated to prove even simple theorems; moreover, no speci c knowledge relative to the particular problem is used in the process. In order to overcome these disadvantages, the T resolution rule (as well as, for instance, the theory resolution rule, cf. [Sti85]) was proposed in [PS95] This rule permits one to eliminate the axioms of the theory from the theorem to be tested, and exploits a decider at each step of the inference process. Theoretical results were presented that make it possible to express in a natural manner, for a speci ed underlying ....

M. E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning,

....of a theory, where the calculation of the theory (the facts) are performed by a background reasoner. In the following we refer to normal logic programs when we talk about logic programs. We assume the reader to be familiar with the fields of logic programming [11] and theory reasoning [13]. 3.1 Template Theories A token template theory T T is the set of all template ground atoms, that we obtain by applying all templates in T . For example, consider the template set ft(D;v; p)g. Assume p to be an arbitrary token pattern and v an extraction tupel. A template theory for T is given ....

M. Stickel. Automated Deduction by Theory Resolution. Journal of Automated Reasoning, 1:333--355, 1985.

....deduction paradigm in which a general purpose reasoner is complemented by a specialized procedure, the background reasoner, which (semi )decides formula satisfiability with respect to a certain background theory. After the pioneering work of Stickel who devised a theory version of resolution [8], nearly all existing calculi for automatic reasoning have been extended to theory reasoning (see [2] for a survey) Essentially all of them however consider the integration of just one background reasoner into the main one. The usual explanation for such a limitation is that, although ....

....and scalability purposes, it is hard to get them to cooperate in a sound and complete way. We argue in this paper that the cooperation of background reasoners in theory reasoning is instead conceptually simple and can be easily described in terms of partial theory reasoning in the sense of [8]. We will appeal to a variant of a well known interpolation result, the Craig Interpolation Lemma, to show that background reasoner cooperation can be achieved as a form of constraint propagation, much in the spirit of existing combination methods for decision procedures [5] The main idea in this ....

[Article contains additional citation context not shown here]

M. E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1(4):333--355, 1985.

....in general, very large sentences must be manipulated to prove even simple theorems; moreover, no speci c knowledge relative to the particular problem is used in the process. In order to overcome these disadvantages, the T resolution rule (as well as, for instance, the theory resolution rule, cf. [Sti85]) was proposed in [PS95] This rule permits one to eliminate the axioms of the theory from the theorem to be tested, and exploits a decider at each step of the inference process. Theoretical results were presented that make it possible to express in a natural manner, for a speci ed underlying ....

....the major di erences between these two proposals for theory reasoning consists in the manner new literals are introduced (into the resolvents) during the derivation steps. T resolution uses loading operation while theory resolution reaches the same result by means of the so called residues (cf. [Sti85]) which are a set of literals (to be added to the resolvent) suggested by the T decider. Hence, we have a di erent behavior of the T decider: in theory resolution it must play an active role in developing the derivations. On the other hand, in T resolution the two levels of reasoning (namely, ....

[Article contains additional citation context not shown here]

M. E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning,

.... thechargeofimplementingtheT translationfunction,performingtherequiredadditive modiflcationstoaxioms(8) 9)and(10) W earecurrentlyexploringthepossibility ofan e cientimplementationofthisframeworkby usingthe theoryresolution techniquefor flrstordertheoremprovingdevisedby Stickel[17],which allows usto wire theLT theory intoa resolution basedtheoremprover,thereby freeingtheknowledgebasemanagement systemfromtheneedtohandletheaxiomsofLT directly. ....

Mark E.Stickel.Automateddeductionby theoryresolution.JournalofAutomated Reasoning,1:333-355,1985.

....the only constructor equations are the partiality axioms (see 10.5) From a proof theoretic viewpoint, however, initial semantics is less appropriate. Efficient resolution and rewriting oriented proof methods treat constructor equations CE on a lower level than other axioms (see, e.g. Plo72, Sti85,JK86] Normal forms are replaced by equivalence classes of normal forms modulo the equivalence relation jCE induced by CE . Resolution and rewriting modulo jCE work well if CE is restricted to particular axioms such as associativity, commutativity, idempotence, etc. Otherwise corresponding ....

....from standard functions and predicates on natural numbers, lists, bags, sets and maps. Here normalization is actually (partial) evaluation with respect to fixed parameter models (see Section 10.1) Inference rules with built in standard type evaluating solving mechanisms are presented in, e.g. Sti85] JK86] and [AB92] CLP ( constraint logic programming) integrates goal solvers tailored to standard types into logiclanguage interpreters or compilers. The efficiency of a theorem prover that uses partial evaluators is increased considerably if the formulas to be evaluated are compiled into, ....

M. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1(4):333--355, 1985.

....sake of flexible and transparent theorem proving we clearly separate deduction rules, which are fixed, but applied through user interaction, from the simplifier, which is applied automatically, but amenable to modifications due to particular base theories. This differs from theory resolution (cf. [41]) and rewriting modulo where theories or models are built into inference rules. The theory that is built into our rules is always the inductive theory, i.e. the theory of the initial model of the specification we actually deal with. This means that we may apply lemmas as in natural proofs , but ....

....simplifies equations by applying particular lemmas, which hold true whenever 17 This includes the case that g is not SP solvable (cf. 4.1) and Sigma(g) FALSE. 44 SP is canonical and free constructor based (cf. 5. 2) Further rule based simplifications are given by theory resolution (cf. [41]) rewriting modulo equational theories (cf. 20] and rewriting modulo algebras (cf. 2] In contrast to these approaches we distinguish between the rule of Sigma simplification (cf. Sect. 4) and more sensitive inferences such as induction steps, which produce descent conditions (cf. Sect. ....

M. Stickel, Automated Deduction by Theory Resolution, J. Automated Reasoning 1 (1985) 333-356

....systems dedicated to modal logics has been thoroughly developed in [ Fitting, 1983 ] whereas our treatment of the condition C (see e.g. Sections 4, 5, 6) can be viewed as a means to parametrize our calculi by the theory of the accessibility relations. Hence, the idea of theory resolution [ Stickel, 1985 ] in which a theory is separately dealt with from the rest of the calculus is present in our calculi. This idea is not new in the realm of the mechanization of modal logics (see e.g. Frisch and Scherl, 1990; Gent, 1993 ] but the originality of our work is related to the conditions satisfied ....

M. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1:333--355, 1985.

.... Besides being used as a stand alone theorem prover, PTTP can play a useful subordinate role in the proof of difficult theorems if the theorem can be decomposed into manageable chunks [44] by performing fast refutation checks on newly derived clauses [1] or by executing the theory resolution [38] or linked inference principle [47] procedures. We are currently investigating the latter approach by developing an extension of PTTP that, instead of proving a query outright, finds single literal assumptions that would suffice to complete a proof. This Unit Resulting PTTP can then perform by ....

Stickel, M.E. Automated deduction by theory resolution. Journal of Automated Reasoning 1, 4 (1985), 333--355.

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Stickel, M. E. (1985). Automated deduction by theory resolution. Journal of Automated Reasoning, 1:333-355. 40

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M.E. Stickel. Automated Deduction by Theory Resolution. Journal of Automated Reasoning, 1:333--355, 1985.

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Mark E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1(4):333-355, 1985.

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M. Stickel, Automated Deduction by Theory Resolution, J. Automated Reasoning 1 (1985) 333-356

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M. Stickel, Automated Deduction by Theory Resolution, J. Automated Reasoning 1 (1985) 333356

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M. Stickel: "Automated Deduction by Theory Resolution," J. Automated Reasoning, 1, 1985

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Mark E. Stickel. Automated deduction by theory resolution. Journal of AutomatedReasoning, 1#4#:333#355, 1985.

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M. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1(4):333--355, 1985.

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