Christoph Benzmueller - Academia.edu (original) (raw)

Papers by Christoph Benzmueller

arXiv (Cornell University), Jul 28, 2012

A converter from first-order modal logics to classical higherorder logic is presented. This tool ... more A converter from first-order modal logics to classical higherorder logic is presented. This tool enables the application of off-the-shelf higher-order theorem provers and model finders for reasoning within firstorder modal logics. The tool supports logics K, K4, D, D4, T, S4, and S5 with respect to constant, varying and cumulative domain semantics.

arXiv (Cornell University), May 14, 2009

We present a straightforward embedding of quantified multimodal logic in simple type theory and p... more 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 higher-order theorem provers, to demonstrate that the embedding allows automated proofs of statements in these logics, as well as meta properties of them.

arXiv (Cornell University), Jul 28, 2012

A converter from first-order modal logics to classical higherorder logic is presented. This tool ... more A converter from first-order modal logics to classical higherorder logic is presented. This tool enables the application of off-the-shelf higher-order theorem provers and model finders for reasoning within firstorder modal logics. The tool supports logics K, K4, D, D4, T, S4, and S5 with respect to constant, varying and cumulative domain semantics.

Leo-III is an automated theorem prover for (polymorphic) higher-order logic which supports all co... more Leo-III is an automated theorem prover for (polymorphic) higher-order logic which supports all common TPTP dialects, including THF, TFF and FOF as well as their rank-1 polymorphic derivatives. It is based on a paramodulation calculus with ordering constraints and, in tradition of its predecessor LEO-II, heavily relies on cooperation with external (mostly first-order) theorem provers for increased performance. Nevertheless, Leo-III can also be used as a stand-alone prover without employing any external cooperation. Leo-III version 1.5 fixes some bugs of earlier versions. Used in CASC of 2020.

J. Formaliz. Reason., 2010

The Thousands of Problems for Theorem Provers (TPTP) problem library is the basis of a well known... more 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 first-order form (FOF) logic to typed higher-order form (THF) logic has provided a basis for new development and application of ATP systems for higher-order logic. Key developments have been the specification of the THF language, the addition of higher-order problems to the TPTP, the development of the TPTP THF infrastructure, several ATP systems for higher-order logic, and the use of higher-order ATP in a range of domains. This paper describes these developments.

Logic Journal of IGPL, 2010

We study straightforward embeddings of propositional normal multimodal logic and propositional in... more 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 non-classical logics. We report some experiments using the higher-order automated theorem prover LEO-II.

Deliverable 4.2: Case Studies in Formal MKM

arXiv preprint arXiv:0901.3608, Jan 23, 2009

Ь з Ы УС Ъ дгжи л з ви жв аан ж к л н а йзЙШ и ж Я жи Йб а д зКйв Йз К ЯЯЯ иид ЛЛлллК зКйв Йз К Л... more Ь з Ы УС Ъ дгжи л з ви жв аан ж к л н а йзЙШ и ж Я жи Йб а д зКйв Йз К ЯЯЯ иид ЛЛлллК зКйв Йз К Л дЛл а гб К иба ... Я з гл и и джгб в ви гйви ж м бда гж и гбда и в зз г жзи гж ж ЪЭ Йж згайи гв г з вги ддан иг и ж гж ж ЪЭ Йж згайи гв дджг ЪЭ К ... Р ж X з к ж а в f,g ж йв жн йв и гв знб газК Си з аайзиж и в Р О℄ и и и з г к гйзан в гвз зи ви а йз з и ввги ж йи в и жзи гж ж ЪЭ Йж згайи гв дджг г ж га К ... Ь ми вз гв а ж гж ж ЪЭ Йж згайи гв к ж ви ЪЭ з в джгдгз в в И в ℄ в гбда и в зз з в ано в в ℄К в ви ж зи в ей зи гв з л и ж и гк м бда з азг гйви ж м ...

Computing Research Repository, 2009

When mathematicians present proofs they usually adapt their explanations to their didac- tic goal... more When mathematicians present proofs they usually adapt their explanations to their didac- tic goals and to the (assumed) knowledge of their addressees. Modern automated theorem provers, in contrast, present proofs usually at a fixed level of detail (also called granularity). Often these presentations are neither intended nor suitabl e for human use. A challenge there- fore is to develop user-

FLAP, 2018

Computers may help us to better understand (not just verify) arguments. In this article we defend... more Computers may help us to better understand (not just verify) arguments. In this article we defend this claim by showcasing the application of a new, computer-assisted interpretive method to an exemplary natural-language argument with strong ties to metaphysics and religion: E. J. Lowe’s modern variant of St. Anselm’s ontological argument for the existence of God. Our new method, which we call computational hermeneutics, has been particularly conceived for use in interactive-automated proof assistants. It aims at shedding light on the meanings of words and sentences by framing their inferential role in a given argument. By employing automated theorem reasoning technology within interactive proof assistants, we are able to drastically reduce (by several orders of magnitude) the time needed to test the logical validity of an argument’s formalization. As a result, a new approach to logical analysis, inspired by Donald Davidson’s account of radical interpretation, has been enabled. In co...

Leo-III is an automated theorem prover for (polymorphic) higher-order logic which supports all co... more Leo-III is an automated theorem prover for (polymorphic) higher-order logic which supports all common TPTP dialects, including THF, TFF and FOF as well as their rank-1 polymorphic derivatives. It is based on a paramodulation calculus with ordering constraints and, in tradition of its predecessor LEO-II, heavily relies on cooperation with external (mostly first-order) theorem provers for increased performance. Nevertheless, Leo-III can also be used as a stand-alone prover without employing any external cooperation. Version 1.5.2 of Leo-III. New features: Improved TH1 proof output Faster and more robust parser for TPTP inputs

Journal of Applied Logic, 2006

In his autobiography 1 Bertrand Russell characterizes mathematics as follows: "It seems to me now... more In his autobiography 1 Bertrand Russell characterizes mathematics as follows: "It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect and yet true". Actually the perception of mathematical research as an artistic discipline has a long history and a significant number of today's mathematicians share this view. In contrast, however, Russell himself dedicated large parts of his life to defending logicism, that is, the view that mathematics is reducible to logic, and together with Alfred Whitehead he proposed in his influential Principia Mathematicae an axiomatic system to build all mathematics upon. The two viewpoints-mathematics as an art versus logicism-may appear contradictory at first. They are not though, if we separate the different aspects of mathematical practice. The invention and shaping of new mathematical structures based on mathematical knowledge as well as on aesthetic and social criteria or the discovery of the essential arguments in complex mathematical proof, for instance, are activities that typically require human ingenuity. On the other hand the verification and grounding of already pre-structured and established chunks of mathematics in foundational systems or the search for simple (sub-)proofs are examples of tasks that often require far less ingenuity. Some overoptimistic and improperly reflected predictions in the field of artificial intelligence and automated reasoning on the mechanization and automation of mathematics have unfortunately generally questioned the role of human ingenuity in mathematics without making the above distinction clear. Unlike in chess, however, where human intelligence is no longer dominating over machine intelligence, it seems to me that human ingenuity will remain dominant in many essential aspects in mathematics research and education for a long time to come. Taking our distinction above into account this does not mean, however, that there is no need for assistance systems for mathematics and Russell would presumably 1

Logical Methods in Computer Science, 2009

We investigate cut-elimination and cut-simulation in impredicative (higherorder) logics. We illus... more We investigate cut-elimination and cut-simulation in impredicative (higherorder) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic-in our case a sequent calculus for classical type theory-is like adding cut. The phenomenon equally applies to prominent axioms like Boolean-and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are built-in instead of being treated axiomatically.

This thesis focuses on equality and extensionality in automated higher-order theorem proving base... more This thesis focuses on equality and extensionality in automated higher-order theorem proving based on Church's simply typed lambda - calculus (classical type theory). First, a landscape of various semantical notions is preented that is motivated by the different roles equality adopts in them. Each of the semantical notions in this landscape - including Henkin semantics - is then linked with an abstract consistency principle that can be employed for analysing the connection between syntax and semantics of higer-order calculi. Apart from this proof theoretic tools, the main contributions of this are the three new calculi ER (extensional higher-order resolution), EP (extensoinal higher-order paramodulation) and ERUE (extensonal higher-order RUE-resolution) which improve the mechanisation of defined and primitvie equality in classical type theory. In contrast to the refutation approaches for classical type theory developed so far, these calculi reach Henkin completeness without requ...

In a case study we investigate whether off the shelf higher-order theorem provers and model gener... more In a case study we investigate whether off the shelf higher-order theorem provers and model generators can be employed to automate reasoning in and about quantified multimodal logics. In our experiments we exploit the new TPTP infrastructure for classical higher-order logic.

IFIP Advances in Information and Communication Technology, 2009

Garg and Abadi recently proved that prominent access control logics can be translated in a sound ... more 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 (which is also known as higher-order logic) and we have demonstrated that the higher-order theorem prover LEO-II 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 LEO-II can be applied to automate reasoning in prominent access control logics. embedding furthermore demonstrates that prominent access control logics as well as prominent multimodal logics can be considered and treated as natural fragments of STT. Using our embedding, reasoning in and about access control logic can be automated in the higher-order theorem prover LEO-II [9]. Since LEO-II generates proof objects the entire translation and reasoning process is in principle accessible for independent proof checking. This paper is structured as follows: Section 2 reviews background knowledge and Section 3 outlines the translation of access control logics into modal logic S4 as proposed by Garg and Abadi [15]. Section 4 restricts the general embedding of multimodal logics into STT [7] to an embedding of monomodal logics K and S4 into STT and proves its soundness and completeness. These results are combined in Section 5 in order to obtain a sound and complete embedding of access control logics into STT. Moreover, we present some first empirical evaluation of the approach with the higher-order automated theorem prover LEO-II. Section 6 concludes the paper. 2 Preliminaries We assume familiarity with the syntax and semantics and of multimodal logics and simple type theory and only briefly review the most important notions. The multimodal logic language ML is defined by s,t ::= p|¬ s|s ∨t|✷ r s where p denotes atomic primitives and r denotes accessibility relations (distinct from p). Other logical connectives can be defined from the chosen ones in the usual way. A Kripke frame for ML is a pair W, (R r) r∈I , where W is a non-empty set (called possible worlds), and the R r are binary relations on W (called accessibility relations). A Kripke model for ML is a triple W, (R r) r∈I , |= , where W, (R r) r∈I is a Kripke frame, and |= is a satisfaction relation between nodes of W and formulas of ML satisfying: w |= ¬ s if and only if w |= s, w |= s ∨t if and only if w |= s or w |= t, w |= ✷ r s if and only if for all u with R r (w, u) holds u |= s. The satisfaction relation |= is uniquely determined by its value on the atomic primitives p. A formula s is valid in a Kripke model W, (R r) r∈I , |= , if w |= s for all w ∈ W. s is valid in a Kripke frame W, (R r) r∈I if it is valid in W, (R r) r∈I , |= for all possible |=. If s is valid for all possible Kripke frames W, (R r) r∈I then s is called valid and we write |= K s. s is called S4-valid (we write |= S4 s) if it is valid in all reflexive, transitive Kripke frames W, (R r) r∈I , that is, Kripke frames with only reflexive and transitive relations R r. Classical higher-order logic or simple type theory STT [5, 12] is a formalism built on top of the simply typed λ-calculus. The set T of simple types is usually freely generated from a set of basic types {o, ι} (where o denotes the type of Booleans) using the function type constructor →. The simple type theory language STT is defined by (α, β , o ∈ T): s,t ::= p α |X α |(λ X α s β) α→β |(s α→β t α) β |(¬ o→o s o) o |(s o ∨ o→o→o t o) o |(Π (α→o)→o s α→o) o p α denotes typed constants and X α typed variables (distinct from p α). Complex typed terms are constructed via abstraction and application. Our logical connectives of choice are ¬ o→o ,

Cognitive Technologies, 2010

A notion of quantified conditional logics (QCLs) is provided that includes quantification over in... more A notion of quantified conditional logics (QCLs) is provided that includes quantification over individual and propositional variables. The former is supported with respect to constant and variable domain semantics. In addition, a sound and complete embedding of this framework in classical higher-order logic (HOL) is presented. Using prominent examples from the literature it is demonstrated how this embedding enables effective automation of reasoning within (object-level) and about (meta-level) quantified conditional logics with off-the-shelf higher-order theorem provers and model finders.

Logic Journal of IGPL, 2003

In this paper we present an approach to automated learning within mathematical reasoning systems.... more In this paper we present an approach to automated learning within mathematical reasoning systems. In particular, the approach enables proof planning systems to automatically learn new proof methods from well-chosen examples of proofs which use a similar reasoning pattern to prove related theorems. Our approach consists of an abstract representation for methods and a machine learning technique which can learn methods using this representation formalism. We present an implementation of the approach within the Ωmega proof planning system, which we call LearnΩmatic. We also present the results of the experiments that we ran on this implementation in order to evaluate if and how it improves the power of proof planning systems.

Annals of Mathematics and Artificial Intelligence, 2012

A sound and complete embedding of conditional logics into classical higher-order logic is present... more A sound and complete embedding of conditional logics into classical higher-order logic is presented. This embedding enables the application of off-the-shelf higher-order automated theorem provers and model finders for reasoning within and about conditional logics.

arXiv (Cornell University), Jul 28, 2012

A converter from first-order modal logics to classical higherorder logic is presented. This tool ... more A converter from first-order modal logics to classical higherorder logic is presented. This tool enables the application of off-the-shelf higher-order theorem provers and model finders for reasoning within firstorder modal logics. The tool supports logics K, K4, D, D4, T, S4, and S5 with respect to constant, varying and cumulative domain semantics.

arXiv (Cornell University), May 14, 2009

We present a straightforward embedding of quantified multimodal logic in simple type theory and p... more 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 higher-order theorem provers, to demonstrate that the embedding allows automated proofs of statements in these logics, as well as meta properties of them.

arXiv (Cornell University), Jul 28, 2012

A converter from first-order modal logics to classical higherorder logic is presented. This tool ... more A converter from first-order modal logics to classical higherorder logic is presented. This tool enables the application of off-the-shelf higher-order theorem provers and model finders for reasoning within firstorder modal logics. The tool supports logics K, K4, D, D4, T, S4, and S5 with respect to constant, varying and cumulative domain semantics.

Leo-III is an automated theorem prover for (polymorphic) higher-order logic which supports all co... more Leo-III is an automated theorem prover for (polymorphic) higher-order logic which supports all common TPTP dialects, including THF, TFF and FOF as well as their rank-1 polymorphic derivatives. It is based on a paramodulation calculus with ordering constraints and, in tradition of its predecessor LEO-II, heavily relies on cooperation with external (mostly first-order) theorem provers for increased performance. Nevertheless, Leo-III can also be used as a stand-alone prover without employing any external cooperation. Leo-III version 1.5 fixes some bugs of earlier versions. Used in CASC of 2020.

J. Formaliz. Reason., 2010

The Thousands of Problems for Theorem Provers (TPTP) problem library is the basis of a well known... more 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 first-order form (FOF) logic to typed higher-order form (THF) logic has provided a basis for new development and application of ATP systems for higher-order logic. Key developments have been the specification of the THF language, the addition of higher-order problems to the TPTP, the development of the TPTP THF infrastructure, several ATP systems for higher-order logic, and the use of higher-order ATP in a range of domains. This paper describes these developments.

Logic Journal of IGPL, 2010

We study straightforward embeddings of propositional normal multimodal logic and propositional in... more 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 non-classical logics. We report some experiments using the higher-order automated theorem prover LEO-II.

Deliverable 4.2: Case Studies in Formal MKM

arXiv preprint arXiv:0901.3608, Jan 23, 2009

Ь з Ы УС Ъ дгжи л з ви жв аан ж к л н а йзЙШ и ж Я жи Йб а д зКйв Йз К ЯЯЯ иид ЛЛлллК зКйв Йз К Л... more Ь з Ы УС Ъ дгжи л з ви жв аан ж к л н а йзЙШ и ж Я жи Йб а д зКйв Йз К ЯЯЯ иид ЛЛлллК зКйв Йз К Л дЛл а гб К иба ... Я з гл и и джгб в ви гйви ж м бда гж и гбда и в зз г жзи гж ж ЪЭ Йж згайи гв г з вги ддан иг и ж гж ж ЪЭ Йж згайи гв дджг ЪЭ К ... Р ж X з к ж а в f,g ж йв жн йв и гв знб газК Си з аайзиж и в Р О℄ и и и з г к гйзан в гвз зи ви а йз з и ввги ж йи в и жзи гж ж ЪЭ Йж згайи гв дджг г ж га К ... Ь ми вз гв а ж гж ж ЪЭ Йж згайи гв к ж ви ЪЭ з в джгдгз в в И в ℄ в гбда и в зз з в ано в в ℄К в ви ж зи в ей зи гв з л и ж и гк м бда з азг гйви ж м ...

Computing Research Repository, 2009

When mathematicians present proofs they usually adapt their explanations to their didac- tic goal... more When mathematicians present proofs they usually adapt their explanations to their didac- tic goals and to the (assumed) knowledge of their addressees. Modern automated theorem provers, in contrast, present proofs usually at a fixed level of detail (also called granularity). Often these presentations are neither intended nor suitabl e for human use. A challenge there- fore is to develop user-

FLAP, 2018

Computers may help us to better understand (not just verify) arguments. In this article we defend... more Computers may help us to better understand (not just verify) arguments. In this article we defend this claim by showcasing the application of a new, computer-assisted interpretive method to an exemplary natural-language argument with strong ties to metaphysics and religion: E. J. Lowe’s modern variant of St. Anselm’s ontological argument for the existence of God. Our new method, which we call computational hermeneutics, has been particularly conceived for use in interactive-automated proof assistants. It aims at shedding light on the meanings of words and sentences by framing their inferential role in a given argument. By employing automated theorem reasoning technology within interactive proof assistants, we are able to drastically reduce (by several orders of magnitude) the time needed to test the logical validity of an argument’s formalization. As a result, a new approach to logical analysis, inspired by Donald Davidson’s account of radical interpretation, has been enabled. In co...

Leo-III is an automated theorem prover for (polymorphic) higher-order logic which supports all co... more Leo-III is an automated theorem prover for (polymorphic) higher-order logic which supports all common TPTP dialects, including THF, TFF and FOF as well as their rank-1 polymorphic derivatives. It is based on a paramodulation calculus with ordering constraints and, in tradition of its predecessor LEO-II, heavily relies on cooperation with external (mostly first-order) theorem provers for increased performance. Nevertheless, Leo-III can also be used as a stand-alone prover without employing any external cooperation. Version 1.5.2 of Leo-III. New features: Improved TH1 proof output Faster and more robust parser for TPTP inputs

Journal of Applied Logic, 2006

In his autobiography 1 Bertrand Russell characterizes mathematics as follows: "It seems to me now... more In his autobiography 1 Bertrand Russell characterizes mathematics as follows: "It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect and yet true". Actually the perception of mathematical research as an artistic discipline has a long history and a significant number of today's mathematicians share this view. In contrast, however, Russell himself dedicated large parts of his life to defending logicism, that is, the view that mathematics is reducible to logic, and together with Alfred Whitehead he proposed in his influential Principia Mathematicae an axiomatic system to build all mathematics upon. The two viewpoints-mathematics as an art versus logicism-may appear contradictory at first. They are not though, if we separate the different aspects of mathematical practice. The invention and shaping of new mathematical structures based on mathematical knowledge as well as on aesthetic and social criteria or the discovery of the essential arguments in complex mathematical proof, for instance, are activities that typically require human ingenuity. On the other hand the verification and grounding of already pre-structured and established chunks of mathematics in foundational systems or the search for simple (sub-)proofs are examples of tasks that often require far less ingenuity. Some overoptimistic and improperly reflected predictions in the field of artificial intelligence and automated reasoning on the mechanization and automation of mathematics have unfortunately generally questioned the role of human ingenuity in mathematics without making the above distinction clear. Unlike in chess, however, where human intelligence is no longer dominating over machine intelligence, it seems to me that human ingenuity will remain dominant in many essential aspects in mathematics research and education for a long time to come. Taking our distinction above into account this does not mean, however, that there is no need for assistance systems for mathematics and Russell would presumably 1

Logical Methods in Computer Science, 2009

We investigate cut-elimination and cut-simulation in impredicative (higherorder) logics. We illus... more We investigate cut-elimination and cut-simulation in impredicative (higherorder) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic-in our case a sequent calculus for classical type theory-is like adding cut. The phenomenon equally applies to prominent axioms like Boolean-and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are built-in instead of being treated axiomatically.

This thesis focuses on equality and extensionality in automated higher-order theorem proving base... more This thesis focuses on equality and extensionality in automated higher-order theorem proving based on Church's simply typed lambda - calculus (classical type theory). First, a landscape of various semantical notions is preented that is motivated by the different roles equality adopts in them. Each of the semantical notions in this landscape - including Henkin semantics - is then linked with an abstract consistency principle that can be employed for analysing the connection between syntax and semantics of higer-order calculi. Apart from this proof theoretic tools, the main contributions of this are the three new calculi ER (extensional higher-order resolution), EP (extensoinal higher-order paramodulation) and ERUE (extensonal higher-order RUE-resolution) which improve the mechanisation of defined and primitvie equality in classical type theory. In contrast to the refutation approaches for classical type theory developed so far, these calculi reach Henkin completeness without requ...

In a case study we investigate whether off the shelf higher-order theorem provers and model gener... more In a case study we investigate whether off the shelf higher-order theorem provers and model generators can be employed to automate reasoning in and about quantified multimodal logics. In our experiments we exploit the new TPTP infrastructure for classical higher-order logic.

IFIP Advances in Information and Communication Technology, 2009

Garg and Abadi recently proved that prominent access control logics can be translated in a sound ... more 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 (which is also known as higher-order logic) and we have demonstrated that the higher-order theorem prover LEO-II 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 LEO-II can be applied to automate reasoning in prominent access control logics. embedding furthermore demonstrates that prominent access control logics as well as prominent multimodal logics can be considered and treated as natural fragments of STT. Using our embedding, reasoning in and about access control logic can be automated in the higher-order theorem prover LEO-II [9]. Since LEO-II generates proof objects the entire translation and reasoning process is in principle accessible for independent proof checking. This paper is structured as follows: Section 2 reviews background knowledge and Section 3 outlines the translation of access control logics into modal logic S4 as proposed by Garg and Abadi [15]. Section 4 restricts the general embedding of multimodal logics into STT [7] to an embedding of monomodal logics K and S4 into STT and proves its soundness and completeness. These results are combined in Section 5 in order to obtain a sound and complete embedding of access control logics into STT. Moreover, we present some first empirical evaluation of the approach with the higher-order automated theorem prover LEO-II. Section 6 concludes the paper. 2 Preliminaries We assume familiarity with the syntax and semantics and of multimodal logics and simple type theory and only briefly review the most important notions. The multimodal logic language ML is defined by s,t ::= p|¬ s|s ∨t|✷ r s where p denotes atomic primitives and r denotes accessibility relations (distinct from p). Other logical connectives can be defined from the chosen ones in the usual way. A Kripke frame for ML is a pair W, (R r) r∈I , where W is a non-empty set (called possible worlds), and the R r are binary relations on W (called accessibility relations). A Kripke model for ML is a triple W, (R r) r∈I , |= , where W, (R r) r∈I is a Kripke frame, and |= is a satisfaction relation between nodes of W and formulas of ML satisfying: w |= ¬ s if and only if w |= s, w |= s ∨t if and only if w |= s or w |= t, w |= ✷ r s if and only if for all u with R r (w, u) holds u |= s. The satisfaction relation |= is uniquely determined by its value on the atomic primitives p. A formula s is valid in a Kripke model W, (R r) r∈I , |= , if w |= s for all w ∈ W. s is valid in a Kripke frame W, (R r) r∈I if it is valid in W, (R r) r∈I , |= for all possible |=. If s is valid for all possible Kripke frames W, (R r) r∈I then s is called valid and we write |= K s. s is called S4-valid (we write |= S4 s) if it is valid in all reflexive, transitive Kripke frames W, (R r) r∈I , that is, Kripke frames with only reflexive and transitive relations R r. Classical higher-order logic or simple type theory STT [5, 12] is a formalism built on top of the simply typed λ-calculus. The set T of simple types is usually freely generated from a set of basic types {o, ι} (where o denotes the type of Booleans) using the function type constructor →. The simple type theory language STT is defined by (α, β , o ∈ T): s,t ::= p α |X α |(λ X α s β) α→β |(s α→β t α) β |(¬ o→o s o) o |(s o ∨ o→o→o t o) o |(Π (α→o)→o s α→o) o p α denotes typed constants and X α typed variables (distinct from p α). Complex typed terms are constructed via abstraction and application. Our logical connectives of choice are ¬ o→o ,

Cognitive Technologies, 2010

A notion of quantified conditional logics (QCLs) is provided that includes quantification over in... more A notion of quantified conditional logics (QCLs) is provided that includes quantification over individual and propositional variables. The former is supported with respect to constant and variable domain semantics. In addition, a sound and complete embedding of this framework in classical higher-order logic (HOL) is presented. Using prominent examples from the literature it is demonstrated how this embedding enables effective automation of reasoning within (object-level) and about (meta-level) quantified conditional logics with off-the-shelf higher-order theorem provers and model finders.

Logic Journal of IGPL, 2003

In this paper we present an approach to automated learning within mathematical reasoning systems.... more In this paper we present an approach to automated learning within mathematical reasoning systems. In particular, the approach enables proof planning systems to automatically learn new proof methods from well-chosen examples of proofs which use a similar reasoning pattern to prove related theorems. Our approach consists of an abstract representation for methods and a machine learning technique which can learn methods using this representation formalism. We present an implementation of the approach within the Ωmega proof planning system, which we call LearnΩmatic. We also present the results of the experiments that we ran on this implementation in order to evaluate if and how it improves the power of proof planning systems.

Annals of Mathematics and Artificial Intelligence, 2012

A sound and complete embedding of conditional logics into classical higher-order logic is present... more A sound and complete embedding of conditional logics into classical higher-order logic is presented. This embedding enables the application of off-the-shelf higher-order automated theorem provers and model finders for reasoning within and about conditional logics.