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Extending Sledgehammer with SMT Solvers
"... Abstract. Sledgehammer is a component of Isabelle/HOL that employs firstorder automatic theorem provers (ATPs) to discharge goals arising in interactive proofs. It heuristically selects relevant facts and, if an ATP is successful, produces a snippet that replays the proof in Isabelle. We extended Sl ..."
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Abstract. Sledgehammer is a component of Isabelle/HOL that employs firstorder automatic theorem provers (ATPs) to discharge goals arising in interactive proofs. It heuristically selects relevant facts and, if an ATP is successful, produces a snippet that replays the proof in Isabelle. We extended Sledgehammer to invoke satisfiability modulo theories (SMT) solvers as well, exploiting its relevance filter and parallel architecture. Isabelle users are now pleasantly surprised by SMT proofs for problems beyond the ATPs ’ reach. Remarkably, the best SMT solver performs better than the best ATP on most of our benchmarks. 1
Automatic Proof and Disproof in Isabelle/HOL
, 2011
"... Isabelle/HOL is a popular interactive theorem prover based on higherorder logic. It owes its success to its ease of use and powerful automation. Much of the automation is performed by external tools: The metaprover Sledgehammer relies on resolution provers and SMT solvers for its proof search, the c ..."
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Isabelle/HOL is a popular interactive theorem prover based on higherorder logic. It owes its success to its ease of use and powerful automation. Much of the automation is performed by external tools: The metaprover Sledgehammer relies on resolution provers and SMT solvers for its proof search, the counterexample generator Quickcheck uses the ML compiler as a fast evaluator for ground formulas, and its rival Nitpick is based on the model finder Kodkod, which performs a reduction to SAT. Together with the Isar structured proof format and a new asynchronous user interface, these tools have radically transformed the Isabelle user experience. This paper provides an overview of the main automatic proof and disproof tools.
The Naproche Project Controlled Natural Language Proof Checking of Mathematical Texts
"... Abstract. This paper discusses the semiformal language of mathematics and presents the Naproche CNL, a controlled natural language for mathematical authoring. Proof Representation Structures, an adaptation of Discourse Representation Structures, are used to represent the semantics of texts written ..."
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Abstract. This paper discusses the semiformal language of mathematics and presents the Naproche CNL, a controlled natural language for mathematical authoring. Proof Representation Structures, an adaptation of Discourse Representation Structures, are used to represent the semantics of texts written in the Naproche CNL. We discuss how the Naproche CNL can be used in formal mathematics, and present our prototypical Naproche system, a computer program for parsing texts in the Naproche CNL and checking the proofs in them for logical correctness.
Escape to ATP for Mizar
 PxTP2011 (2011
"... An interactive ATP service is a new feature in the Mizar proof assistant. The functionality of the service is in many respects analogous to the Sledgehammer subsystem of Isabelle/HOL. The ATP service requires minimal user configuration and is accessible via a few keystrokes from within Mizar mode in ..."
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An interactive ATP service is a new feature in the Mizar proof assistant. The functionality of the service is in many respects analogous to the Sledgehammer subsystem of Isabelle/HOL. The ATP service requires minimal user configuration and is accessible via a few keystrokes from within Mizar mode in Emacs. In return, for a given goal formula, the ATP service, when it succeeds, finds premises sufficient to prove the goal. The “escape ” to ATP uses a sound translation from Mizar’s language to that of firstorder provers, the same translation that has been used in the more batch oriented Automated Reasoning for Mizar (MizAR) web services presented in [16]. We briefly present the interactive ATP service followed by an account of initial experiments with the tool. We claim with some confidence that the tool will substantially ease the process of preparing new Mizar articles. 1
The Naproche system
 In Intelligent Computer Mathematics
, 2009
"... was initiated by Bernhard Schröder and Peter Koepke at the University of Bonn to focus on an interdisciplinary study of the semiformal language of mathematics. A central goal of Naproche is to develop a controlled natural language (CNL) for mathematical texts and adapted proof checking software whi ..."
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was initiated by Bernhard Schröder and Peter Koepke at the University of Bonn to focus on an interdisciplinary study of the semiformal language of mathematics. A central goal of Naproche is to develop a controlled natural language (CNL) for mathematical texts and adapted proof checking software which checks texts written in the CNL for syntactical and mathematical correctness. The project is still at a prototypical stage, further information is available at www.naproche.net. This paper describes the Naproche system, an implementation of the ideas developed by the Naproche project. The Naproche system accepts L ATEXstyle texts, consisting of mathematical formulas imbedded in a controlled natural language. Texts written in the controlled natural language are parsed using techniques from computational linguistics and transformed into firstorder formulas. The formulas are given to an automatic theorem prover which checks whether each formula of an argument is a logical consequence of the preceding formulas or axioms.
Testing FirstOrder Logic Axioms in Program Verification
"... Abstract. Program verification systems based on automated theorem provers rely on userprovided axioms in order to verify domainspecific properties of code. However, formulating axioms correctly (that is, formalizing properties of an intended mathematical interpretation) is nontrivial in practice, ..."
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Abstract. Program verification systems based on automated theorem provers rely on userprovided axioms in order to verify domainspecific properties of code. However, formulating axioms correctly (that is, formalizing properties of an intended mathematical interpretation) is nontrivial in practice, and avoiding or even detecting unsoundness can sometimes be difficult to achieve. Moreover, speculating soundness of axioms based on the output of the provers themselves is not easy since they do not typically give counterexamples. We adopt the idea of modelbased testing to aid axiom authors in discovering errors in axiomatizations. To test the validity of axioms, users define a computational model of the axiomatized logic by giving interpretations to the function symbols and constants in a simple declarative programming language. We have developed an axiom testing framework that helps automate model definition and test generation using offtheshelf tools for metaprogramming, propertybased random testing, and constraint solving. We have experimented with our tool to test the axioms used in AutoCert, a program verification system that has been applied to verify aerospace flight code using a firstorder axiomatization of navigational concepts, and were able to find counterexamples for a number of axioms. Key words: modelbased testing, program verification, automated theorem proving, propertybased testing, constraint solving
Noname manuscript No. (will be inserted by the editor) Extending Sledgehammer with SMT Solvers
"... the date of receipt and acceptance should be inserted later Abstract Sledgehammer is a component of Isabelle/HOL that employs resolutionbased firstorder automatic theorem provers (ATPs) to discharge goals arising in interactive proofs. It heuristically selects relevant facts and, if an ATP is succe ..."
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the date of receipt and acceptance should be inserted later Abstract Sledgehammer is a component of Isabelle/HOL that employs resolutionbased firstorder automatic theorem provers (ATPs) to discharge goals arising in interactive proofs. It heuristically selects relevant facts and, if an ATP is successful, produces a snippet that replays the proof in Isabelle. We extended Sledgehammer to invoke satisfiability modulo theories (SMT) solvers as well, exploiting its relevance filter and parallel architecture. The ATPs and SMT solvers nicely complement each other, and Isabelle users are now pleasantly surprised by SMT proofs for problems beyond the ATPs ’ reach. 1
Automated Higherorder Reasoning about
"... Originally developed as an algebraic characterisation for quantum mechanics, the algebraic structure of quantales nowadays finds widespread applications ranging from (noncommutative) logics to hybrid systems. We present an approach to bring reasoning about quantales into the realm of (fully) autom ..."
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Originally developed as an algebraic characterisation for quantum mechanics, the algebraic structure of quantales nowadays finds widespread applications ranging from (noncommutative) logics to hybrid systems. We present an approach to bring reasoning about quantales into the realm of (fully) automated theorem proving. This will yield automation in various (new) fields of applications in the future. To achieve this goal and to receive a general approach (independent of any particular theorem prover), we use the TPTP Problem Library for higherorder logic. In particular, we give an encoding of quantales in the typed higherorder form (THF) and present some theorems about quantales which can be proved fully automatically. We further present prospective applications for our approach and discuss practical experiences using THF. 1
Abstract
, 2010
"... Originally developed as an algebraic characterisation for quantum mechanics, the algebraic structure of quantales nowadays finds widespread applications ranging from (noncommutative) logics to hybrid systems. We present an approach to bring reasoning in quantales into the realm of (fully) automated ..."
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Originally developed as an algebraic characterisation for quantum mechanics, the algebraic structure of quantales nowadays finds widespread applications ranging from (noncommutative) logics to hybrid systems. We present an approach to bring reasoning in quantales into the realm of (fully) automated theorem proving. Hence the paper paves the way for automatisation in various (new) fields of applications. To achieve this goal and to receive a general approach (independent of any particular theorem prover), we use the TPTP Problem Library for higherorder logic. In particular, we give an encoding of quantales in the typed higherorder form (THF) and present some theorems about quantales which can be proved fully automatically. We further present prospective applications for our approach and discuss practical experiences using THF. 1