## Reasoning Theories - Towards an Architecture for Open Mechanized Reasoning Systems (1994)

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Citations: | 47 - 11 self |

### BibTeX

@TECHREPORT{Giunchiglia94reasoningtheories,

author = {Fausto Giunchiglia and Paolo Pecchiari and Carolyn Talcott},

title = {Reasoning Theories - Towards an Architecture for Open Mechanized Reasoning Systems},

institution = {},

year = {1994}

}

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### Abstract

: Our ultimate goal is to provide a framework and a methodology which will allow users, and not only system developers, to construct complex reasoning systems by composing existing modules, or to add new modules to existing systems, in a "plug and play" manner. These modules and systems might be based on different logics; have different domain models; use different vocabularies and data structures; use different reasoning strategies; and have different interaction capabilities. This paper makes two main contributions towards our goal. First, it proposes a general architecture for a class of reasoning systems called Open Mechanized Reasoning Systems (OMRSs). An OMRS has three components: a reasoning theory component which is the counterpart of the logical notion of formal system, a control component which consists of a set of inference strategies, and an interaction component which provides an OMRS with the capability of interacting with other systems, including OMRSs and hum...

### Citations

696 | A framework for defining logics
- Harper, Honsell, et al.
- 1989
(Show Context)
Citation Context ...notational systems for presenting logics (more precisely, deductive or entailment systems using the classification of [53], as they treat only syntax and deduction, not models). Some examples are: LF =-=[37, 38]-=- -- a meta-logical system for describing and prototyping logics; lambdaprolog [55] -- a meta programming language for manipulating syntactic entities such as programs, formulas, and proof structures; ... |

691 |
Introduction to metamathematics
- Kleene
- 1952
(Show Context)
Citation Context ...y to the information that the user gets from the prover. This is shown in Figure 11. We use two presentations: linear and graphical. The first uses notation similar to Kleene's presentation of proofs =-=[39]-=-. The columns in the table of the figure correspond respectively to a sequent node, its label, the link node having the sequent node as conclusion, the subgoals of the link node and its justification.... |

537 | PVS: A Prototype Verification System
- OWRE, RUSHBY, et al.
- 1992
(Show Context)
Citation Context ..., rewriting, associative-commutative theories, linear arithmetic, and Groebner basis over boolean ring. The KADS system [73] uses resolution augmented with special purpose deciders. EHDM [78] and PVS =-=[61]-=- use a variety of ground decision procedures combined with rules for interactive proving. The Stanford Temporal Prover, STeP [24], combines model checking with deductive methods including contextual r... |

531 |
A Computational' Logic
- Boyer, Moore
- 1979
(Show Context)
Citation Context ...r composition and the second, prover enhancement. A few attempts in both directions have already been made. Some examples concerning prover composition are the following. The Boyer-Moore prover NQTHM =-=[10, 12]-=- combines techniques for propositional reasoning, equality reasoning, typeset inference, term rewriting, and linear arithmetic. (See [47] for a description of the new Boyer--Moore theorem prover, ACL2... |

475 |
Conditional rewriting logic as a unified model of concurrency
- Meseguer
- 1992
(Show Context)
Citation Context ...on from different languages (sequent systems), and bridge rules may discharge hypotheses across languages. There is a strong connection between the reasoning theory level of OMRSs and Rewriting Logic =-=[54]-=-. Sequent systems correspond to the equational part of rewriting logic theories and the rules component of reasoning theories correspond to rules component of rewriting logic theories. The initial mod... |

474 |
Institutions: abstract model theory for specification and programming
- Goguen, Burstall
- 1992
(Show Context)
Citation Context ...provides the starting point for this work. A very powerful notion of module composition is that based on theory mappings [53]. This is a 42 central idea in the work on the Clear specification language=-=[33]-=- and is the basis for the module level of the OBJ language[34]. Theory mappings (also called Views) insure not only syntactic, but also semantic composability of modules. Acknowledgements. The authors... |

395 |
A Computational Logic Handbook
- Boyer, Moore
- 1988
(Show Context)
Citation Context ...ut to the top Figure 6. The NQTHM waterfall formula in clausal form to a set of clauses (considered conjunctively) and can be considered as a derived rule of inference that runs backwards (cf. p. 129 =-=[12]-=-). The top level control can be described in analogy to a waterfall. (See Figure 6.) Conjectures to be proved are converted to clausal form and poured in at the top. Repeatedly, a clause is removed fr... |

395 | Simplification by cooperating decision procedures
- Nelson, Oppen
- 1979
(Show Context)
Citation Context ...CL2.) EKL [48, 50] has a highly developed rewriting component and also incorporates a decision procedure for direct logic [49]. The co-operating decision procedures (CDP) paradigm of Nelson and Oppen =-=[59]-=- is a method of combining solver/simplifier modules for disjoint theories to obtain solvers/simplifiers for combined theories. Shostak [72] gives an alternative algorithm for combining solvers, for th... |

287 | A logic programming language with lambda-abstraction, function variables, and simple unification
- Miller
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Citation Context ...ul in provers that use forms of higher-order unification. Schematic variables can also be used to give a first-order account of higher-order patterns in such as those of Combinatory Reduction Systems =-=[51, 55, 60, 76]-=-. Another interesting example of deduction using schematic variables is in the NQTHM linear arithmetic module, where a single derivation can serve multiple purposes by using place holders to represent... |

276 |
Symbolic Logic and Mechanical Theorem Proving
- Chang
- 1973
(Show Context)
Citation Context ...g theory machinery, and preliminary ideas on modeling control [21]. We plan to carry out additional case studies including: the Nelson-Oppen cooperating decision procedures and simplifier; resolution =-=[16]-=-, and Alan Bundy's rippling [14]. Another important area of application is the integration of computer algebra systems and theorem provers, an area which has lately received a lot of interest (see e.g... |

223 |
Natural Deduction - A proof theoretical study. Almquist and Wiksell
- Prawitz
- 1965
(Show Context)
Citation Context ...uent system for the logical calculus ND, a sequent system for the NQTHM simplification process. Example (ND sequent system): ND is a natural deduction system for classical first-order predicate logic =-=[63]-=-. Ssys ND is the sequent system underlying our representation of natural deduction as a reasoning theory. Its sequents are pairs \Gamma ` A consisting of a set of formulas, \Gamma, called assumptions,... |

216 |
Mathematical Logic
- Shoenfield
- 1967
(Show Context)
Citation Context ...end ND by adding a simple form of tautology rule, TAUT, TAUT if TautConseq(\Gamma ; A) \Gamma ` A where TautConseq(\Gamma ; A) holds just if the formula A is a tautological consequence of \Gamma (cf. =-=[71], p. -=-26). Example (An NQTHM equality rule): The following inference rule is used by the rewriter for equality reasoning NE if Ts(ti ; t 1 ) " Ts(ti ; t 2 )s; h `R ti ; pi ; (EQUAL t 1 t 2 ) ! I hF; ;;... |

200 |
HOL: A proof generating system for higher-order logic
- Gordon
- 1988
(Show Context)
Citation Context ... procedures for Presburger arithmetic, propositional and propositional temporal formulas, and a semi-decision procedure for first order logic. Concerning prover enhancement, recent proceedings of HOL =-=[35]-=- User meetings [17, 42] report a variety of experiments to incorporate additional inference capability, or to link HOL to other systems. FAUST [68] is a prover for full firstorder logic developed for ... |

185 |
Isabelle: the next 700 theorem provers
- Paulson
- 1990
(Show Context)
Citation Context ...ntegration of a model checker with PVS. Harrison and Th'ery [39], and Ballarin, Homann and Calmet [6] report experiments of integration of the computer algebra system Maple [15] with HOL and Isabelle =-=[62]-=-, respectively. Analytica [18] is a theorem prover, written in the Mathematica [66] environment, which uses the powerful algorithms embedded in Mathematica to prove sophisticated theorems in elementar... |

177 | Multilanguage Hierarchical Logics (or: How We Can Do Without Modal Logics - Giunchiglia, Serafini - 1994 |

167 |
General Logics
- Meseguer
- 1989
(Show Context)
Citation Context ... over S , c range over C , and ' range over I . (Recall our convention that, for example ec will range over P! (C ).) Satisfaction must obey the basic laws for a (classical) consequence relation (cf. =-=[5, 53]-=-): (mon) if ec ` ec 0 and ec j= c, then ec 0 j= c; (ax) if c 2 ec , then ec j= c; (cut) if ec j= c and fcg [ ec 0 j= c 0 , then ec [ ec 0 j= c 0 . We extend satisfaction to a relation between sets of ... |

162 | A.: Rippling: A Heuristic for Guiding Inductive Proofs
- Bundy, Stevens, et al.
- 1993
(Show Context)
Citation Context ...ary ideas on modeling control [21]. We plan to carry out additional case studies including: the Nelson-Oppen cooperating decision procedures and simplifier; resolution [16], and Alan Bundy's rippling =-=[14]-=-. Another important area of application is the integration of computer algebra systems and theorem provers, an area which has lately received a lot of interest (see e.g. [18, 6, 39, 70]). A first atte... |

158 |
Deciding combinations of theories
- Shostak
- 1984
(Show Context)
Citation Context ...ating decision procedures (CDP) paradigm of Nelson and Oppen [59] is a method of combining solver/simplifier modules for disjoint theories to obtain solvers/simplifiers for combined theories. Shostak =-=[72] give-=-s an alternative algorithm for combining solvers, for theories meeting certain constraints, to obtain decision procedures for combined first order theories. 1 In this paper we use the word "prove... |

147 | Rewriting logic as a logical and semantic framework
- Martí-Oliet, Meseguer
- 2000
(Show Context)
Citation Context ...s. To account uniformly for notions of binding we will build on the work on binding structures [76]. As a starting point we expect to use a notation for first-order theories such as OBJ [34] or Maude =-=[52]-=-. The module composition capability of these systems will also serve as a starting point for developing a richer calculus of reasoning theories. So far we have considered only the operational and proo... |

124 |
Higher-order critical pairs
- Nipkow
- 1991
(Show Context)
Citation Context ...ul in provers that use forms of higher-order unification. Schematic variables can also be used to give a first-order account of higher-order patterns in such as those of Combinatory Reduction Systems =-=[51, 55, 60, 76]-=-. Another interesting example of deduction using schematic variables is in the NQTHM linear arithmetic module, where a single derivation can serve multiple purposes by using place holders to represent... |

107 | Integrating decision procedures into heuristic theorem provers: A case study of linear arithmetic
- Boyer, Moore
- 1988
(Show Context)
Citation Context ...mu-calculus model checker inside PVS are two examples of black box interaction. Another example is the first integration of linear arithmetic inside the NQTHM simplification process (see Section 5 of =-=[11]-=-). The first two integrations are reported as successful, i.e. the resulting system is claimed to do better than the original system, while the last integration is described as largely unsuccessful (a... |

100 | Simple consequence relations
- Avron
- 1991
(Show Context)
Citation Context ... over S , c range over C , and ' range over I . (Recall our convention that, for example ec will range over P! (C ).) Satisfaction must obey the basic laws for a (classical) consequence relation (cf. =-=[5, 53]-=-): (mon) if ec ` ec 0 and ec j= c, then ec 0 j= c; (ax) if c 2 ec , then ec j= c; (cut) if ec j= c and fcg [ ec 0 j= c 0 , then ec [ ec 0 j= c 0 . We extend satisfaction to a relation between sets of ... |

100 |
Edinburgh LCF: a mechanized logic of computation
- Gordon, Milner, et al.
- 1979
(Show Context)
Citation Context ...d a set of control strategies used to search the space of possible inference rule applications. Control must be able to specify the usual backward or forward rule composition (`a la tactics, see e.g. =-=[36]-=-), but also all the usual tricks and annotations used in automated theorem provers, e.g. the fact that a formula has already been used, or that a formula should not be used. A reasoning theory is obta... |

88 | W.E.: The Semantics of Reflected Proof
- Allen, Constable, et al.
- 1990
(Show Context)
Citation Context ...rovide a hierarchical way of describing the search for or construction of a proof, but the resulting proofs are flat. In NuPrl proofs are terms corresponding to programs. The reflection rule of NuPrl =-=[2]-=- allows for application of meta-level proofs without explicit flattening. Some work on the presentation of proofs that addresses these issues is described in [4, 44]). Finally, some provers support pr... |

88 | An integration of model-checking with automated proof checking
- Rajan, Shankar, et al.
- 1995
(Show Context)
Citation Context ...translations of certain HOL goals to input for a resolution prover. Joyce and Seger [43] describe a link to the Voss model checking tool to be used in hardware verification. Rajan, Shankar and Srivas =-=[64]-=- describe the integration of a model checker with PVS. Harrison and Th'ery [39], and Ballarin, Homann and Calmet [6] report experiments of integration of the computer algebra system Maple [15] with HO... |

79 | IMPS: An interactive mathematical proof system
- Farmer
- 1993
(Show Context)
Citation Context ...cture. Each VL has an identifier, a formula, a set of dependencies, and a justification. A VL may be used as a premiss in several rule applications, providing a sharing of sub-deductions. In the IMPS =-=[25]-=- system, derivations are represented as deduction graphs -- labelled graphs with two sorts of nodes, sequent and rule nodes. A sequent may be linked as premiss to any number of rule nodes (possibly no... |

73 | Contextual reasoning
- Giunchiglia
- 1992
(Show Context)
Citation Context ...duction together. These rules have premisses and conclusions corresponding to different kinds of assertions. Examples of such rules in existing systems include: bridge rules in multi-language systems =-=[28, 30]-=-; the FOL reflection rule; and the inference rules for reading information off of a diagram, and for using linguistic assertions to extend diagrams, in Hyperproof [7, 3]. The Nelson-Oppen and Shostak ... |

50 | Theorems and algorithms: An interface between Isabelle and Maple
- Ballarin, Homann, et al.
- 1995
(Show Context)
Citation Context ...model checking tool to be used in hardware verification. Rajan, Shankar and Srivas [64] describe the integration of a model checker with PVS. Harrison and Th'ery [39], and Ballarin, Homann and Calmet =-=[6]-=- report experiments of integration of the computer algebra system Maple [15] with HOL and Isabelle [62], respectively. Analytica [18] is a theorem prover, written in the Mathematica [66] environment, ... |

44 |
Jos'e Meseguer, Kokichi Futatsugi, and Jean-Pierre Jouannaud. Introducing OBJ
- Goguen, Winkler
- 1988
(Show Context)
Citation Context ...sequent systems. To account uniformly for notions of binding we will build on the work on binding structures [76]. As a starting point we expect to use a notation for first-order theories such as OBJ =-=[34]-=- or Maude [52]. The module composition capability of these systems will also serve as a starting point for developing a richer calculus of reasoning theories. So far we have considered only the operat... |

40 |
Logic and Computation
- Paulson
- 1982
(Show Context)
Citation Context ...tency and completeness of equational specifications, and solving equations modulo an equational theory. The KADS system [61] uses resolution augmented with special purpose deciders. EHDM [66] and PVS =-=[51]-=- use a variety of ground decision procedures combined with rules for interactive proving. PVS has a rich type system and provides the ability to postpone type checking, by making presumptions, analogo... |

38 |
RRL: A Rewrite Rule Laboratory
- KAPUR, ZHANG
(Show Context)
Citation Context ...actic and semantic simplification, tautology checking based on a compiling formulas into a special representation, and a decision procedure for monadic predicate logic. The deductive machinery of RRL =-=[45, 46]-=- includes: a completion procedure, rewriting, associative-commutative theories, linear arithmetic, and Groebner basis over boolean ring. The KADS system [73] uses resolution augmented with special pur... |

38 | a Portable and Extensible Interface for Computer Algebra Systems
- Kajler
- 1992
(Show Context)
Citation Context ...domain is reported in [40]. Some preliminary work of ours in this direction is reported in [31]. In this area the work on the definition of the interaction level will exploit the results described in =-=[57]-=- and also the results of the OpenMath project [1, 79]. A formal notation is needed for presenting the syntax and semantics of sequent systems and for defining relations between sequent systems. To acc... |

36 | Analytica: A theorem prover for Mathematica
- Clarke, Zhao
- 1993
(Show Context)
Citation Context ...with PVS. Harrison and Th'ery [39], and Ballarin, Homann and Calmet [6] report experiments of integration of the computer algebra system Maple [15] with HOL and Isabelle [62], respectively. Analytica =-=[18]-=- is a theorem prover, written in the Mathematica [66] environment, which uses the powerful algorithms embedded in Mathematica to prove sophisticated theorems in elementary analysis. The SDVS system [2... |

36 | J.S.: Design goals of ACL2
- Kaufmann, Moore
- 1994
(Show Context)
Citation Context ...omposition are the following. The Boyer-Moore prover NQTHM [10, 12] combines techniques for propositional reasoning, equality reasoning, typeset inference, term rewriting, and linear arithmetic. (See =-=[47]-=- for a description of the new Boyer--Moore theorem prover, ACL2.) EKL [48, 50] has a highly developed rewriting component and also incorporates a decision procedure for direct logic [49]. The co-opera... |

35 | Axiomatizing Reflective Logics and Languages
- Clavel, Meseguer
- 1996
(Show Context)
Citation Context ...rize commonly used features of derivation histories in a general system independent 41 manner. This work has much in common with ongoing work developing general strategy languages for rewriting logic =-=[19, 20]-=-. The third aspect of an OMRS is interaction. First we need to characterize the kinds of interactions and the information needed to specify what the interactions mean. This includes making explicit al... |

33 | Extending the HOL theorem prover with a computer algebra system to reason about the reals
- Harrison, Théry
- 1993
(Show Context)
Citation Context ...eger [43] describe a link to the Voss model checking tool to be used in hardware verification. Rajan, Shankar and Srivas [64] describe the integration of a model checker with PVS. Harrison and Th'ery =-=[39]-=-, and Ballarin, Homann and Calmet [6] report experiments of integration of the computer algebra system Maple [15] with HOL and Isabelle [62], respectively. Analytica [18] is a theorem prover, written ... |

29 | A decidable fragment of predicate calculus
- Ketonen
- 1984
(Show Context)
Citation Context ...thmetic. (See [47] for a description of the new Boyer--Moore theorem prover, ACL2.) EKL [48, 50] has a highly developed rewriting component and also incorporates a decision procedure for direct logic =-=[49]-=-. The co-operating decision procedures (CDP) paradigm of Nelson and Oppen [59] is a method of combining solver/simplifier modules for disjoint theories to obtain solvers/simplifiers for combined theor... |

27 |
A theory of binding structures and applications to rewriting
- Talcott
- 1993
(Show Context)
Citation Context ...ul in provers that use forms of higher-order unification. Schematic variables can also be used to give a first-order account of higher-order patterns in such as those of Combinatory Reduction Systems =-=[51, 55, 60, 76]-=-. Another interesting example of deduction using schematic variables is in the NQTHM linear arithmetic module, where a single derivation can serve multiple purposes by using place holders to represent... |

25 |
The GETFOL Manual - GETFOL version 1
- Giunchiglia
- 1992
(Show Context)
Citation Context ...mer as systems, and to the latter as modules. A database of provers can be found in [75] along with links to related surveys and other information. 1 FOL [80, 81, 82] and its re-implementation GETFOL =-=[29]-=- provide for construction of and reasoning in an intended model, rewriting, mixed syntactic and semantic simplification, tautology checking based on a compiling formulas into a special representation,... |

24 | Finitary inductively presented logics
- FEFERMAN
- 1989
(Show Context)
Citation Context ...syntactic entities such as programs, formulas, and proof structures; and Feferman's formalism based on inductive definitions for specifying syntax, proofs, and in some cases models of logical systems =-=[26, 27]-=-. The closest in spirit to our work is the work on LF, as both approaches share the main goal of characterizing formal systems and provability at an abstract level. However there are important differe... |

21 |
A Verification Condition Generator for FORTRAN
- Boyer, Moore
- 1981
(Show Context)
Citation Context ...d as a component of various formal and informal systems where the system (possibly a person or ad hoc program) can reduce the problem to suitable form. Verification Condition checking is described in =-=[8]-=- (for Fortran) and [58] (for Pascal). In [13] a general theory, MLP, for defining semantics of synchronous circuits, was developed (informally) using function parameters. A Lisp front-end was used to ... |

19 |
The hol-voss system: Model-checking inside a generalpurpose theorem prover
- Joyce, Seger
- 1993
(Show Context)
Citation Context ...] is a prover for full firstorder logic developed for use with HOL. Archer [4] describes work in progress to define translations of certain HOL goals to input for a resolution prover. Joyce and Seger =-=[43]-=- describe a link to the Voss model checking tool to be used in hardware verification. Rajan, Shankar and Srivas [64] describe the integration of a model checker with PVS. Harrison and Th'ery [39], and... |

15 | Structures for symbolic mathematical reasoning and computation
- Calmet, Homann
- 1996
(Show Context)
Citation Context ...mputer algebra systems and theorem provers, an area which has lately received a lot of interest (see e.g. [18, 6, 39, 70]). A first attempt to apply our approach to this problem domain is reported in =-=[40]-=-. Some preliminary work of ours in this direction is reported in [31]. In this area the work on the definition of the interaction level will exploit the results described in [57] and also the results ... |

14 |
Rules and derived rules
- Scott
- 1974
(Show Context)
Citation Context ...uents) consisting of a list of assumption formulas and a conclusion formula, and proofs are tree-like structures. The notion of n-ary rule generator generalizes the notion of rule introduced by Scott =-=[69]-=-. Here (propositional) systems are determined by a set of sentences and a relation ` between finite sets of sentences. A rule is expressed using meta variables for sentences combined using finite set ... |

13 | Multilanguage systems
- Giunchiglia
- 1991
(Show Context)
Citation Context ...duction together. These rules have premisses and conclusions corresponding to different kinds of assertions. Examples of such rules in existing systems include: bridge rules in multi-language systems =-=[28, 30]-=-; the FOL reflection rule; and the inference rules for reading information off of a diagram, and for using linguistic assertions to extend diagrams, in Hyperproof [7, 3]. The Nelson-Oppen and Shostak ... |

13 | An Overview of the Tecton Proof System
- Kapur, Musser, et al.
- 1994
(Show Context)
Citation Context ...programs. The reflection rule of NuPrl [2] allows for application of meta-level proofs without explicit flattening. Some work on the presentation of proofs that addresses these issues is described in =-=[4, 44]-=-). Finally, some provers support provisional reasoning. In these situations, checking for the applicability of an inference rule is sometimes postponed until after the rule is applied. Some examples a... |

13 |
to a Theory of Formal Reasoning
- Weyhrauch
(Show Context)
Citation Context ...dded inside other software we refer to the former as systems, and to the latter as modules. A database of provers can be found in [75] along with links to related surveys and other information. 1 FOL =-=[80, 81, 82]-=- and its re-implementation GETFOL [29] provide for construction of and reasoning in an intended model, rewriting, mixed syntactic and semantic simplification, tautology checking based on a compiling f... |

13 |
Mechanizing Proof Theory: Resource-Aware Logics and Proof-Transformations to Extract Implicit Information
- Bellin
- 1990
(Show Context)
Citation Context ...d stores it as annotations to subexpressions in the form of rewrite rules. There is a language for expressing rewriting control strategies. EKL also incorporates a decision procedure for direct logic =-=[37, 5]-=- and some facility for representing terms as data and carrying out meta-level reasoning. RRL [34, 35] represents all axioms as equations and views theories as congruence relations. The deductive machi... |

11 | The use of proof plans for normalization
- Bundy
- 1991
(Show Context)
Citation Context ...mber algebra decision procedure, and AC unification. There is also a large amount of work in the area of hybrid systems. An overview of this work can be found in [30]. To mention some examples: Bundy =-=[12]-=- proposes using proof plans for combination of reasoning strategies; the blackboard model [31] is a mechanism for coordination of problem solving activities of planners and reasoning experts with a co... |

9 |
Structure and representation in LF
- Harper, Sannella, et al.
- 1989
(Show Context)
Citation Context ...notational systems for presenting logics (more precisely, deductive or entailment systems using the classification of [53], as they treat only syntax and deduction, not models). Some examples are: LF =-=[37, 38]-=- -- a meta-logical system for describing and prototyping logics; lambdaprolog [55] -- a meta programming language for manipulating syntactic entities such as programs, formulas, and proof structures; ... |