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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 14 (12 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.
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 ..."
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Cited by 14 (10 self)
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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.
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 9 (9 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.
Combining Logics in Simple Type Theory
, 2010
"... Simple type theory is suited as framework for combining classical and nonclassical logics. This claim is based on the observation that various prominent logics, including (quantified) multimodal logics and intuitionistic logics, can be elegantly embedded in simple type theory. Furthermore, simple ..."
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Cited by 3 (3 self)
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Simple type theory is suited as framework for combining classical and nonclassical logics. This claim is based on the observation that various prominent logics, including (quantified) multimodal logics and intuitionistic logics, can be elegantly embedded in simple type theory. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Offtheshelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about combinations of logics. Combinations of modal logics and other logics are particularly relevant for multiagent systems.
Simple type theory as framework for combining logics
 in Contest paper at the World Congress and School on Universal Logic III (UNILOG’2010
, 2010
"... Abstract. Simple type theory is suited as framework for combining classical and nonclassical logics. This claim is based on the observation that various prominent logics, including (quantified) multimodal logics and intuitionistic logics, can be elegantly embedded in simple type theory. Furthermore ..."
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Cited by 2 (2 self)
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Abstract. Simple type theory is suited as framework for combining classical and nonclassical logics. This claim is based on the observation that various prominent logics, including (quantified) multimodal logics and intuitionistic logics, can be elegantly embedded in simple type theory. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Offtheshelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about combinations of logics. 1
Sigma: An Integrated Development Environment for Formal Ontology
"... Abstract. Sigma is an open source environment for the development of logical theories. It has been under development and regular release for nearly a decade, and has been the principal environment under which the open source Suggested Upper Merged Ontology (SUMO) has been created. We discuss its fea ..."
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Cited by 1 (0 self)
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Abstract. Sigma is an open source environment for the development of logical theories. It has been under development and regular release for nearly a decade, and has been the principal environment under which the open source Suggested Upper Merged Ontology (SUMO) has been created. We discuss its features and evolution, and explain why it is an appropriate environment for the development of expressive ontologies in first and higher order logic. 1
Combining and Automating Classical and NonClassical Logics in Classical HigherOrder Logics
 ANNALS OF MATHEMATICS AND ARTIFICIAL INTELLIGENCE (PREFINAL VERSION)
"... Numerous classical and nonclassical logics can be elegantly embedded in Church’s simple type theory, also known as classical higherorder logic. Examples include propositional and quantified multimodal logics, intuitionistic logics, logics for security, and logics for spatial reasoning. Furthermor ..."
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Cited by 1 (1 self)
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Numerous classical and nonclassical logics can be elegantly embedded in Church’s simple type theory, also known as classical higherorder logic. Examples include propositional and quantified multimodal logics, intuitionistic logics, logics for security, and logics for spatial reasoning. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Offtheshelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about embedded logics and logics combinations. In this article we focus on combinations of (quantified) epistemic and doxastic logics and study their application for modeling and automating the reasoning of rational agents. We present illustrating example problems and report on experiments with offtheshelf higherorder automated theorem provers.
The THFTPTP Project – An Infrastructure for Typed Higherorder Form Automated Theorem Proving
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"... LEOII is a standalone, resolutionbased higherorder theorem prover that is designed for fruitful cooperation with specialist provers for firstorder and propositional logic. The idea is to combine the strengths of the different systems. On the other hand, LEOII itself, as an external reasoner, wa ..."
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LEOII is a standalone, resolutionbased higherorder theorem prover that is designed for fruitful cooperation with specialist provers for firstorder and propositional logic. The idea is to combine the strengths of the different systems. On the other hand, LEOII itself, as an external reasoner, wants to support interactive proof assistants such as Isabelle/HOL, HOL, and OMEGA by efficiently automating subproblems and thereby reducing user effort. LEOII predominantly addresses higherorder aspects in its reasoning process with the aim to quickly remove higherorder clauses from the search space and to turn them into essentially firstorder clauses which can then be refuted with a firstorder prover. For this LEOII cooperates with the firstorder theorem provers E, Spass or Vampire. LEOII’s Data Structures LEOII provides efficient term data structures based on a perfectly shared term graph, i.e., syntactically equal terms are represented by a single instance. Ideas from firstorder term sharing are adapted to higherorder logic by (i) keeping indexed terms in βη normal form (i.e., η short and β normal) and (ii) using de Bruijn indices to allow λabstracted terms to be shared. LEOII also provides an interactive mode in which user and system can interact to produce resolution proofs in simple type theory. LEOII is implemented in Objective Caml and it can be download from the LEOII website
A Topdown Approach to Combining Logics
"... The mechanization and automation of combination of logics, expressive ontologies and notions of context are prominent current challenge problems. I propose to approach these challenge topics from the perspective of classical higherorder logic. From this perspective these topics are closely related ..."
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The mechanization and automation of combination of logics, expressive ontologies and notions of context are prominent current challenge problems. I propose to approach these challenge topics from the perspective of classical higherorder logic. From this perspective these topics are closely related and a common, uniform solution appears in reach. 1