Scheme-Based Systematic Exploration of Natural Numbers

Hodorog, Mădălina; Crǎciun, Adrian

doi:10.1109/synasc.2006.67

Cited by 4 publications

(7 citation statements)

References 9 publications

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“…The knowledge schemes capture prior mathematical knowledge, and are then instantiated with symbols in the current theory in an attempt to produce new concepts or function definitions. A preliminary case-study of the natural number has been undertaken, but the process is not yet automated [12]. Systems using a deductive approach attempt to produce new theorems as logical consequences of known facts.…”

Section: An Overview Of Theory Formation Systemsmentioning

confidence: 99%

Conjecture Synthesis for Inductive Theories

2010

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We have developed a program for inductive theory formation, called IsaCoSy, which synthesises conjectures 'bottom-up' from the available constants and free variables. The synthesis process is made tractable by only generating irreducible terms, which are then filtered through counter-example checking and passed to the automatic inductive prover IsaPlanner. The main technical contribution is the presentation of a constraint mechanism for synthesis. As theorems are discovered, this generates additional constraints on the synthesis process.We evaluate IsaCoSy as a tool for automatically generating the background theories one would expect in a mature proof assistant, such as the Isabelle system. The results show that IsaCoSy produces most, and sometimes all, of the theorems in the Isabelle libraries. The number of additional un-interesting theorems are small enough to be easily pruned by hand.

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Section: An Overview Of Theory Formation Systemsmentioning

confidence: 99%

Conjecture Synthesis for Inductive Theories

2010

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“…However, we are not aware of any experimental evaluation of this algorithm. Finally, the MATHsAiD [20] and Theorema [14,4] systems have also been applied to inductive theory formation, although only to natural numbers. The similarity of their results in this domain suggests that if extended to other domains, they would provide similar results to IsaCoSy.…”

Section: Related Workmentioning

confidence: 99%

Dynamic Rippling, Middle-Out Reasoning and Lemma Discovery

Johansson

Dixon

Bundy

2010

Verification, Induction, Termination Analysis

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Abstract. We present a succinct account of dynamic rippling, a technique used to guide the automation of inductive proofs. This simplifies termination proofs for rippling and hence facilitates extending the technique in ways that preserve termination. We illustrate this by extending rippling with a terminating version of middle-out reasoning for lemma speculation. This supports automatic speculation of schematic lemmas which are incrementally instantiated by unification as the rippling proof progresses. Middle-out reasoning and lemma speculation have been implemented in higher-order logic and evaluated on typical libraries of formalised mathematics. This reveals that, when applied, the technique often finds the needed lemmas to complete the proof, but it is not as frequently applicable as initially expected. In comparison, we show that theory formation methods, combined with simpler proof methods, offer an effective alternative.

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“…• IR N consists of first order predicate logic calculus, equality reasoning, and the induction inference rule. For more details, see [7].…”

Section: Example 1 (The Theory Of Natural Numbers)mentioning

confidence: 99%

“…Developing Natural Numbers Theory in The Scheme Based Exploration Model. We already presented details of the initial development of the theory of natural numbers in [7]. We now give a "trace" ("script") in the form of scheme instantiations, for this development.…”

Section: Example 1 (The Theory Of Natural Numbers)mentioning

confidence: 99%

“…invent such that is-group[is-nat, +, 0, ]. We apply the lazy thinking synthesis method which gives that no such function exists, see [7]. In the same way, using the knowledge schemes we generate new notions and their properties: the multi-plication * , the exponentiation ∧ and the predecessor − function symbols.…”

Section: Example 1 (The Theory Of Natural Numbers)mentioning

confidence: 99%

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Decompositions of Natural Numbers: From a Case Study in Mathematical Theory Exploration

Crǎciun

Hodorog

2007

Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2007)

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In the context of a scheme based exploration model proposed by Bruno Buchberger, we investigate the idea of decomposition, applied in the exploration of natural numbers. The free decomposition problem (i.e. whether an element can always be decomposed with respect to an operation) can be arbitrarily difficult, and we illustrate this in the theory of natural numbers. We consider a restriction, the decomposition in domains with a well-founded partial ordering: we introduce the notions of irreducible elements, reducible elements w.r.t. a composition operation, decomposition of domain elements into irreducible ones, and also the problem of irreducible decomposition which we then solve. Natural numbers can be classified as a decomposition domain, in which we know how to solve the decomposition problem. This leads to the prime decomposition theorem.

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Scheme-Based Systematic Exploration of Natural Numbers

Cited by 4 publications

References 9 publications

Conjecture Synthesis for Inductive Theories

Conjecture Synthesis for Inductive Theories

Dynamic Rippling, Middle-Out Reasoning and Lemma Discovery

Decompositions of Natural Numbers: From a Case Study in Mathematical Theory Exploration

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