Results 21  30
of
88
HigherOrder Semantics and Extensionality
 Journal of Symbolic Logic
, 2004
"... Abstract. In this paper we reexamine the semantics of classical higherorder logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higherorder models with respect to various combinations of Boolean extensionality and three forms of func ..."
Abstract

Cited by 15 (9 self)
 Add to MetaCart
Abstract. In this paper we reexamine the semantics of classical higherorder logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higherorder models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we develop a methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of (machineoriented) higherorder calculi with respect to these model classes. §1. Motivation. In classical firstorder predicate logic, it is rather simple to assess the deductive power of a calculus: firstorder logic has a wellestablished and intuitive settheoretic semantics, relative to which completeness can easily be verified using, for instance, the abstract consistency method (cf. the introductory textbooks [6, 22]). This well understood metatheory has supported the development of calculi adapted to special applications—such as automated theorem proving (cf. [16, 47] for an overview). In higherorder logics, the situation is rather different: the intuitive settheoretic standard semantics cannot give a sensible notion of completeness, since it does
Extensional higherorder paramodulation and RUEresolution
 AUTOMATED DEDUCTION — CADE16 INTERNATIONAL CONFERENCE, LNAI 1632
, 1999
"... This paper presents two approaches to primitive equality treatment in higherorder (HO) automated theorem proving: a calculus EP adapting traditional firstorder (FO) paramodulation [RW69] , and acalculusERUE adapting FO RUEResolution [Dig79] to classical type theory, i.e., HO logic based on Church ..."
Abstract

Cited by 14 (8 self)
 Add to MetaCart
This paper presents two approaches to primitive equality treatment in higherorder (HO) automated theorem proving: a calculus EP adapting traditional firstorder (FO) paramodulation [RW69] , and acalculusERUE adapting FO RUEResolution [Dig79] to classical type theory, i.e., HO logic based on Church’s simply typed λcalculus. EP and ERUE extend the extensional HO resolution approach ER [BK98a]. In order to reach Henkin completeness without the need for additional extensionality axioms both calculi employ new, positive extensionality rules analogously to the respective negative ones provided by ER that operate on unification constraints. As the extensionality rules have an intrinsic and unavoidable differencereducing character the HO paramodulation approach loses its pure termrewriting character. On the other hand examples demonstrate that the extensionality rules harmonise quite well with the differencereducing HO RUEresolution idea.
A comprehensive framework for combined decision procedures
 In FroCos
, 2005
"... Abstract. We define a general notion of a fragment within higher order type theory; a procedure for constraint satisfiability in combined fragments is outlined, following NelsonOppen schema. The procedure is in general only sound, but it becomes terminating and complete when the shared fragment enj ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Abstract. We define a general notion of a fragment within higher order type theory; a procedure for constraint satisfiability in combined fragments is outlined, following NelsonOppen schema. The procedure is in general only sound, but it becomes terminating and complete when the shared fragment enjoys suitable noetherianity conditions and allows an abstract version of a ‘KeislerShelah like ’ isomorphism theorem. We show that this general decidability transfer result covers as special cases, besides applications which seem to be new, the recent extension of NelsonOppen procedure to nondisjoint signatures [16] and the fusion transfer of decidability of consistency of ABoxes with respect to TBoxes axioms in local abstract description systems [9]; in addition, it reduces decidability of modal and temporal monodic fragments [32] to their extensional and onevariable components. 1
Automated reasoning in higherorder logic using the TPTP THF infrastructure
 J. of Formalized Reasoning
, 2010
"... Articulate Software The Thousands of Problems for Theorem Provers (TPTP) problem library is the basis of a well known and well established infrastructure that supports research, development, and deployment of Automated Theorem Proving (ATP) systems. The extension of the TPTP from firstorder form (F ..."
Abstract

Cited by 14 (10 self)
 Add to MetaCart
Articulate Software The Thousands of Problems for Theorem Provers (TPTP) problem library is the basis of a well known and well established infrastructure that supports research, development, and deployment of Automated Theorem Proving (ATP) systems. The extension of the TPTP from firstorder form (FOF) logic to typed higherorder form (THF) logic has provided a basis for new development and application of ATP systems for higherorder logic. Key developments have been the specification of the THF language, the addition of higherorder problems to the TPTP, the development of the TPTP THF infrastructure, several ATP systems for higherorder logic, and the use of higherorder ATP in a range of domains. This paper surveys these developments. 1.
A Notation for Lambda Terms II: Refinements and Applications
, 1994
"... Issues that are relevant to the representation of lambda terms in contexts where their intensions have to be manipulated are examined. The basis for such a representation is provided by the suspension notation for lambda terms that is described in a companion paper. This notation obviates ffconver ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
Issues that are relevant to the representation of lambda terms in contexts where their intensions have to be manipulated are examined. The basis for such a representation is provided by the suspension notation for lambda terms that is described in a companion paper. This notation obviates ffconversion in the comparison of terms by using the `nameless' scheme of de Bruijn and also permits a delaying of substitutions by including a class of terms that encode other terms together with substitutions to be performed on them. The suspension notation contains a mechanism for `merging' substitutions so that they can be effected in a common structure traversal. The mechanism is cumbersome to implement in its full generality and a simplification to it is considered. In particular, the old merging operations are eliminated in favor of new ones that capture some of their functionality and that permit a simplified syntax for terms. The resulting notation is refined by the addition of annotations ...
Combined reasoning by automated cooperation
 JOURNAL OF APPLIED LOGIC
, 2008
"... Different reasoning systems have different strengths and weaknesses, and often it is useful to combine these systems to gain as much as possible from their strengths and retain as little as possible from their weaknesses. Of particular interest is the integration of firstorder and higherorder tech ..."
Abstract

Cited by 11 (7 self)
 Add to MetaCart
Different reasoning systems have different strengths and weaknesses, and often it is useful to combine these systems to gain as much as possible from their strengths and retain as little as possible from their weaknesses. Of particular interest is the integration of firstorder and higherorder techniques. Firstorder reasoning systems, on the one hand, have reached considerable strength in
some niches, but in many areas of mathematics they still cannot reliably solve relatively simple problems, for example, when
reasoning about sets, relations, or functions. Higherorder reasoning systems, on the other hand, can solve problems of this kind
automatically. But the complexity inherent in their calculi prevents them from solving a whole range of problems. However, while
many problems cannot be solved by any one system alone, they can be solved by a combination of these systems.
We present a general agentbased methodology for integrating different reasoning systems. It provides a generic integration
framework which facilitates the cooperation between diverse reasoners, but can also be refined to enable more efficient, specialist
integrations. We empirically evaluate its usefulness, effectiveness and efficiency by case studies involving the integration of first
order and higherorder automated theorem provers, computer algebra systems, and model generators.
Extensional higherorder resolution
 In Kirchner and Kirchner [KK98
, 1998
"... Abstract. In this paper we present an extensional higherorder resolution calculus that is complete relative to Henkin model semantics. The treatment of the extensionality principles – necessary for the completeness result – by specialized (goaldirected) inference rules is of practical applicabilit ..."
Abstract

Cited by 11 (6 self)
 Add to MetaCart
Abstract. In this paper we present an extensional higherorder resolution calculus that is complete relative to Henkin model semantics. The treatment of the extensionality principles – necessary for the completeness result – by specialized (goaldirected) inference rules is of practical applicability, as an implentation of the calculus in the LeoSystem shows. Furthermore, we prove the longstanding conjecture, that it is sufficient to restrict the order of primitive substitutions to the order of input formulae. 1
Automating access control logics in simple type theory with LEOII
 FB Informatik, U. des Saarlandes
, 2008
"... Abstract Garg and Abadi recently proved that prominent access control logics can be translated in a sound and complete way into modal logic S4. We have previously outlined how normal multimodal logics, including monomodal logics K and S4, can be embedded in simple type theory and we have demonstrate ..."
Abstract

Cited by 11 (9 self)
 Add to MetaCart
Abstract Garg and Abadi recently proved that prominent access control logics can be translated in a sound and complete way into modal logic S4. We have previously outlined how normal multimodal logics, including monomodal logics K and S4, can be embedded in simple type theory and we have demonstrated that the higherorder theorem prover LEOII can automate reasoning in and about them. In this paper we combine these results and describe a sound (and complete) embedding of different access control logics in simple type theory. Employing this framework we show that the off the shelf theorem prover LEOII can be applied to automate reasoning in and about prominent access control logics. 1
Multimodal and Intuitionistic Logics in Simple Type Theory
"... We study straightforward embeddings of propositional normal multimodal logic and propositional intuitionistic logic in simple type theory. The correctness of these embeddings is easily shown. We give examples to demonstrate that these embeddings provide an effective framework for computational inve ..."
Abstract

Cited by 9 (9 self)
 Add to MetaCart
We study straightforward embeddings of propositional normal multimodal logic and propositional intuitionistic logic in simple type theory. The correctness of these embeddings is easily shown. We give examples to demonstrate that these embeddings provide an effective framework for computational investigations of various nonclassical logics. We report some experiments using the higherorder automated theorem prover LEOII.
Semantic cut elimination in the intuitionistic sequent calculus
 Typed Lambda Calculi and Applications, number 3461 in Lectures
, 2005
"... Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to extend the cut elimination result to other intuitionistic deduction systems, in particular to deduction modulo provided the rewrite system verifies some properties. We also give an example of rewrite system for which cut elimination holds but that doesn’t enjoys proof normalization.