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Unification under a mixed prefix
 Journal of Symbolic Computation
, 1992
"... Unification problems are identified with conjunctions of equations between simply typed λterms where free variables in the equations can be universally or existentially quantified. Two schemes for simplifying quantifier alternation, called Skolemization and raising (a dual of Skolemization), are pr ..."
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Cited by 136 (14 self)
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Unification problems are identified with conjunctions of equations between simply typed λterms where free variables in the equations can be universally or existentially quantified. Two schemes for simplifying quantifier alternation, called Skolemization and raising (a dual of Skolemization), are presented. In this setting where variables of functional type can be quantified and not all types contain closed terms, the naive generalization of firstorder Skolemization has several technical problems that are addressed. The method of searching for preunifiers described by Huet is easily extended to the mixed prefix setting, although solving flexibleflexible unification problems is undecidable since types may be empty. Unification problems may have numerous incomparable unifiers. Occasionally, unifiers share common factors and several of these are presented. Various optimizations on the general unification search problem are as discussed. 1.
PROOFS IN HIGHERORDER LOGIC
, 1983
"... Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higherorder logic based on Church’s simple theory of types. Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either ..."
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Cited by 84 (18 self)
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Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higherorder logic based on Church’s simple theory of types. Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either critical variables or skolem terms used to instantiate quantifiers in the original formula and those resulting from instantiations. An expansion tree is called an expansion tree proof (ETproof) if it encodes a tautology, and, in the form not using skolem functions, an “imbedding ” relation among the critical variables be acyclic. The relative completeness result for expansion tree proofs not using skolem functions, i.e. if A is provable in higherorder logic then A has such an expansion tree proof, is based on Andrews ’ formulation of Takahashi’s proof of the cutelimination theorem for higherorder logic. If the occurrences of skolem functions in instantiation terms are restricted appropriately, the use of skolem functions in place of critical variables is equivalent to the requirement that the imbedding relation is acyclic. This fact not only resolves the open question of what
Quantified multimodal logics in simple type theory
, 2009
"... We present a straightforward embedding of quantified multimodal logic in simple type theory and prove its soundness and completeness. Modal operators are replaced by quantification over a type of possible worlds. We present simple experiments, using existing higherorder theorem provers, to demonstr ..."
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Cited by 27 (16 self)
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We present a straightforward embedding of quantified multimodal logic in simple type theory and prove its soundness and completeness. Modal operators are replaced by quantification over a type of possible worlds. We present simple experiments, using existing higherorder theorem provers, to demonstrate that the embedding allows automated proofs of statements in these logics, as well as meta properties of them.
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 24 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
COMPARING APPROACHES TO RESOLUTION BASED HIGHERORDER THEOREM PROVING
, 2002
"... We investigate several approaches to resolution based automated theorem proving in classical higherorder logic (based on Church’s simply typed λcalculus) and discuss their requirements with respect to Henkin completeness and full extensionality. In particular we focus on Andrews’ higherorder res ..."
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Cited by 18 (12 self)
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We investigate several approaches to resolution based automated theorem proving in classical higherorder logic (based on Church’s simply typed λcalculus) and discuss their requirements with respect to Henkin completeness and full extensionality. In particular we focus on Andrews’ higherorder resolution (Andrews 1971), Huet’s constrained resolution (Huet 1972), higherorder Eresolution, and extensional higherorder resolution (Benzmüller and Kohlhase 1997). With the help of examples we illustrate the parallels and differences of the extensionality treatment of these approaches and demonstrate that extensional higherorder resolution is the sole approach that can completely avoid additional extensionality axioms.
Exploring properties of normal multimodal logics in simple type theory with LEOII
 Festschrift in Honor of Peter B. Andrews on His 70th Birthday, Studies in Logic and the Foundations of Mathematics, IFCoLog
"... There are two well investigated approaches to automate reasoning in modal logics: the direct approach and the translational approach. The direct approach [6, 7, 14, 27] develops specific calculi and tools for the task; the translational approach [29, 30] transforms modal logic formulas into firstord ..."
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Cited by 18 (7 self)
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There are two well investigated approaches to automate reasoning in modal logics: the direct approach and the translational approach. The direct approach [6, 7, 14, 27] develops specific calculi and tools for the task; the translational approach [29, 30] transforms modal logic formulas into firstorder
A Structured Set of HigherOrder Problems
 Theorem Proving in Higher Order Logics: TPHOLs 2005, LNCS 3603
, 2005
"... Abstract. We present a set of problems that may support the development of calculi and theorem provers for classical higherorder logic. We propose to employ these test problems as quick and easy criteria preceding the formal soundness and completeness analysis of proof systems under development. Ou ..."
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Cited by 14 (9 self)
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Abstract. We present a set of problems that may support the development of calculi and theorem provers for classical higherorder logic. We propose to employ these test problems as quick and easy criteria preceding the formal soundness and completeness analysis of proof systems under development. Our set of problems is structured according to different technical issues and along different notions of semantics (including Henkin semantics) for higherorder logic. Many examples are either theorems or nontheorems depending on the choice of semantics. The examples can thus indicate the deductive strength of a proof system. 1 Motivation: Test Problems for HigherOrder Reasoning Systems Test problems are important for the practical implementation of theorem provers as well as for the preceding theoretical development of calculi, strategies and heuristics. If the test theorems can be proven (resp. the nontheorems cannot) then they ideally provide a strong indication for completeness (resp. soundness). Examples for early publications providing firstorder test problems are [21,29,23]. For more than decade now the TPTP library [28] has been developed as a systematically structured electronic repository of
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 ..."
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Cited by 14 (12 self)
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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.
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 ..."
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Cited by 13 (11 self)
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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