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30
Complete sequent calculi for induction and infinite descent
 Proceedings of LICS22
, 2007
"... This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing induct ..."
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Cited by 18 (6 self)
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This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cutfree complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (nonwellfounded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple casesplit rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cutfree complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.
The Inverse Method
, 2001
"... this paper every formula is equivalent to a formula in negation normal form ..."
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Cited by 13 (1 self)
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this paper every formula is equivalent to a formula in negation normal form
Algorithms, datastructures, and other issues in efficient automated deduction
 Automated Reasoning. 1st. International Joint Conference, IJCAR 2001, number 2083 in LNAI
, 2001
"... Abstract. Algorithms and datastructures form the kernel of any efficient theorem prover. In this abstract we discuss research on algorithms and datastructures for efficient theorem proving based on our experience with the theorem prover Vampire. We also briefly overview other works related to algori ..."
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Cited by 11 (0 self)
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Abstract. Algorithms and datastructures form the kernel of any efficient theorem prover. In this abstract we discuss research on algorithms and datastructures for efficient theorem proving based on our experience with the theorem prover Vampire. We also briefly overview other works related to algorithms and datastructures, and to efficient theorem proving in general. 1
Knowledge Representation and Classical Logic
"... Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspe ..."
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Cited by 10 (4 self)
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Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields. The language of classical logic that is most widely used in the theory of knowledge representation is the language of firstorder (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of firstorder logic; recent
Customised induction rules for proving correctness of imperative programs
, 2004
"... This thesis is aimed at simplifying the userinteraction in semiinteractive theorem proving for imperative programs. More specifically, we describe the creation of customised induction rules that are tailormade for the specific program to verify and thus make the resulting proof simpler. The conce ..."
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Cited by 8 (1 self)
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This thesis is aimed at simplifying the userinteraction in semiinteractive theorem proving for imperative programs. More specifically, we describe the creation of customised induction rules that are tailormade for the specific program to verify and thus make the resulting proof simpler. The concern is in user interaction, rather than in proof strength. To achieve this, two different verification techniques are used. In the first approach, we develop an idea where a software testing technique, partition analysis, is used to compute a partition of the domain of the induction variable, based on the branch predicates in the program we wish to prove correct. Based on this partition we derive mechanically a partitioned induction rule, which then inherits the divideandconquer style of partition analysis, and (hopefully) is easier to use than the standard (Peano) induction rule. The second part of the thesis continues with a more thorough development of the method. Here the connection to software testing is completely removed
Ordinal Arithmetic: A Case Study for Rippling in a Higher Order Domain
 In TPHOLs’01, volume 2152 of LNCS
, 2001
"... This paper reports a case study in the use of proof planning in the context of higher order syntax. Rippling is a heuristic for guiding rewriting steps in induction that has been used successfully in proof planning inductive proofs using first order representations. Ordinal arithmetic provides a nat ..."
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Cited by 5 (0 self)
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This paper reports a case study in the use of proof planning in the context of higher order syntax. Rippling is a heuristic for guiding rewriting steps in induction that has been used successfully in proof planning inductive proofs using first order representations. Ordinal arithmetic provides a natural set of higher order examples on which transfinite induction may be attempted using rippling. Previously BoyerMoore style automation could not be applied to such domains. We demonstrate that a higherorder extension of the rippling heuristic is sufficient to plan such proofs automatically. Accordingly, ordinal arithmetic has been implemented in Clam, a higher order proof planning system for induction, and standard undergraduate text book problems have been successfully planned. We show the synthesis of a fixpoint for normal ordinal functions which demonstrates how our automation could be extended to produce more interesting results than the textbook examples tried so far.
Proving properties of constraint logic programs by eliminating existential variables
 In Proc. ICLP ’06, LNCS 4079
, 2006
"... Abstract. We propose a method for proving rst order properties of constraint logic programs which manipulate nite lists of real numbers. Constraints are linear equations and inequations over reals. Our method consists in converting any given rst order formula into a strati ed constraint logic progra ..."
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Cited by 5 (5 self)
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Abstract. We propose a method for proving rst order properties of constraint logic programs which manipulate nite lists of real numbers. Constraints are linear equations and inequations over reals. Our method consists in converting any given rst order formula into a strati ed constraint logic program and then applying a suitable unfold/fold transformation strategy that preserves the perfect model. Our strategy is based on the elimination of existential variables, that is, variables which occur in the body of a clause and not in its head. Since, in general, the rst order properties of the class of programs we consider are undecidable, our strategy is necessarily incomplete. However, experiments show that it is powerful enough to prove several nontrivial program properties. 1
What is a proof
 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
, 2005
"... To those brought up in a logicbased tradition there seems to be a simple and clear definition of proof. But this is largely a 20 th century invention; many earlier proofs had a different nature. We will look particularly at the faulty proof of Euler’s Theorem and Lakatos ’ rational reconstruction o ..."
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Cited by 4 (0 self)
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To those brought up in a logicbased tradition there seems to be a simple and clear definition of proof. But this is largely a 20 th century invention; many earlier proofs had a different nature. We will look particularly at the faulty proof of Euler’s Theorem and Lakatos ’ rational reconstruction of the history of this proof. We will ask: how is it possible for the errors in a faulty proof to remain undetected for several years – even when counterexamples to it are known? How is it possible to have a proof about concepts that are only partially defined? And can we give a logicbased account of such phenomena? We introduce the concept of schematic proofs and argue that they offer a possible cognitive model for the human construction of proofs in mathematics. In particular, we show how they can account for persistent errors in proofs.
Formalised inductive reasoning in the logic of bunched implications
 In SAS14, volume 4634 of LNCS
, 2007
"... Abstract. We present a framework for inductive definitions in the logic of bunched implications, BI, and formulate two sequent calculus proof systems for inductive reasoning in this framework. The first proof system adopts a traditional approach to inductive proof, extending the usual sequent calcul ..."
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Cited by 4 (4 self)
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Abstract. We present a framework for inductive definitions in the logic of bunched implications, BI, and formulate two sequent calculus proof systems for inductive reasoning in this framework. The first proof system adopts a traditional approach to inductive proof, extending the usual sequent calculus for predicate BI with explicit induction rules for the inductively defined predicates. The second system allows an alternative mode of reasoning with inductive definitions by cyclic proof. In this system, the induction rules are replaced by simple casesplit rules, and the proof structures are cyclic graphs formed by identifying some sequent occurrences in a derivation tree. Because such proof structures are not sound in general, we demand that cyclic proofs must additionally satisfy a global trace condition that ensures soundness. We illustrate our inductive definition framework and proof systems with simple examples which indicate that, in our setting, cyclic proof may enjoy certain advantages over the traditional induction approach. 1
CaseAnalysis for Rippling and Inductive Proof
"... Abstract. Rippling is a heuristic used to guide rewriting and is typically used for inductive theorem proving. We introduce a method to support caseanalysis within rippling. Like earlier work, this allows goals containing ifstatements to be proved automatically. The new contribution is that our me ..."
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Cited by 4 (1 self)
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Abstract. Rippling is a heuristic used to guide rewriting and is typically used for inductive theorem proving. We introduce a method to support caseanalysis within rippling. Like earlier work, this allows goals containing ifstatements to be proved automatically. The new contribution is that our method also supports caseanalysis on datatypes. By locating the caseanalysis as a step within rippling we also maintain the termination. The work has been implemented in IsaPlanner and used to extend the existing inductive proof method. We evaluate this extended prover on a large set of examples from Isabelle’s theory library and from the inductive theorem proving literature. We find that this leads to a significant improvement in the coverage of inductive theorem proving. The main limitations of the extended prover are identified, highlight the need for advances in the treatment of assumptions during rippling and when conjecturing lemmas. 1