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Conjecture Synthesis for Inductive Theories
 JOURNAL OF AUTOMATED REASONING
, 2010
"... We have developed a program for inductive theory formation, called IsaCoSy, which synthesises conjectures ‘bottomup’ from the available constants and free variables. The synthesis process is made tractable by only generating irreducible terms, which are then filtered through counterexample checkin ..."
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We have developed a program for inductive theory formation, called IsaCoSy, which synthesises conjectures ‘bottomup’ from the available constants and free variables. The synthesis process is made tractable by only generating irreducible terms, which are then filtered through counterexample 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 uninteresting theorems are small enough to be easily pruned by hand.
QuickSpec: Guessing formal specifications using testing
 In Tests and Proofs, Fourth International Conference, TAP
, 2010
"... Abstract. We present QuickSpec, a tool that automatically generates algebraic specifications for sets of pure functions. The tool is based on testing, rather than static analysis or theorem proving. The main challenge QuickSpec faces is to keep the number of generated equations to a minimum while ma ..."
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Abstract. We present QuickSpec, a tool that automatically generates algebraic specifications for sets of pure functions. The tool is based on testing, rather than static analysis or theorem proving. The main challenge QuickSpec faces is to keep the number of generated equations to a minimum while maintaining completeness. We demonstrate how QuickSpec can improve one’s understanding of a program module by exploring the laws that are generated using two case studies: a heap library for Haskell and a fixedpoint arithmetic library for Erlang. 1
Seventy four minutes of mathematics: An analysis of the third MiniPolymath project
"... Abstract. Alan Turing proposed to consider the question, “Can machines think? ” in his famous article [40]. We consider the question, “Can machines do mathematics, and how? ” Turing suggested that intelligence be tested by comparing computer behaviour to human behaviour in an online discussion. We h ..."
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Abstract. Alan Turing proposed to consider the question, “Can machines think? ” in his famous article [40]. We consider the question, “Can machines do mathematics, and how? ” Turing suggested that intelligence be tested by comparing computer behaviour to human behaviour in an online discussion. We hold that this approach could be useful for assessing computational logic systems which, despite having produced formal proofs of the Four Colour Theorem, the Robbins Conjecture and the Kepler Conjecture, have not achieved widespread take up by mathematicians. It has been suggested that this is because computer proofs are perceived as ungainly, bruteforce searches which lack elegance, beauty or mathematical insight. One response to this is to build such systems which perform in a more humanlike manner, which raises the question of what a “humanlike manner ” may be. Timothy Gowers recently initiated Polymath [4], a series of experiments in online collaborative mathematics, in which problems are posted online, and an open invitation issued for people to try to solve them collaboratively, documenting every step of the ensuing discussion. The resulting record provides an unusual example of fully documented mathematical activity leading to a proof, in contrast to typical research papers which record proofs, but not how they were obtained. We consider the third MiniPolymath project [3], started by Terence Tao and published online on July
forthcoming)). Applying lakatosstyle reasoning to ai problems
 Thinking Machines and the philosophy of computer science: Concepts and principles. IGI Global
, 2010
"... One current direction in AI research is to focus on combining different reasoning styles such as deduction, induction, abduction, analogical reasoning, nonmonotonic reasoning, vague and uncertain reasoning. The philosopher Imre Lakatos produced one such theory of how people with different reasoning ..."
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One current direction in AI research is to focus on combining different reasoning styles such as deduction, induction, abduction, analogical reasoning, nonmonotonic reasoning, vague and uncertain reasoning. The philosopher Imre Lakatos produced one such theory of how people with different reasoning styles collaborate to develop mathematical ideas. Lakatos argued that mathematics is a quasiempirical, flexible, fallible, human endeavour, involving negotiations, mistakes, vague concept definitions and disagreements, and he outlined a heuristic approach towards the subject. In this chapter we apply these heuristics to the AI domains of evolving requirement specifications, planning and constraint satisfaction problems. In drawing analogies between Lakatos’s theory and these three domains we identify areas of work which correspond to each heuristic, and suggest extensions and further ways in which Lakatos’s philosophy can inform AI problem solving. Thus, we show how we might begin to produce a philosophicallyinspired AI theory of combined reasoning. 1
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 10 (23) 2007 Automated Discovery of Inductive Theorems ⋆
"... Abstract. Inductive mathematical theorems have, as a rule, historically been quite difficult to prove – both for mathematics students and for automated theorem provers. That said, there has been considerable progress over the past several years, within the automated reasoning community, towards prov ..."
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Abstract. Inductive mathematical theorems have, as a rule, historically been quite difficult to prove – both for mathematics students and for automated theorem provers. That said, there has been considerable progress over the past several years, within the automated reasoning community, towards proving some of these theorems. However, little work has been done thus far towards automatically discovering them. In this paper we present our methods of discovering (as well as proving) inductive theorems, within an automated system. These methods have been tested over the natural numbers, with regards to addition and multiplication, as well as to exponents of group elements. 1
IsaCoSy: Synthesis of Inductive Theorems
"... Abstract. We have implemented a program for inductive theory formation, called IsaCoSy, which synthesises conjectures about recursively defined datatypes and functions. Only irreducible terms are generated, which keeps the search space tractably small. The synthesised terms are filtered through coun ..."
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Abstract. We have implemented a program for inductive theory formation, called IsaCoSy, which synthesises conjectures about recursively defined datatypes and functions. Only irreducible terms are generated, which keeps the search space tractably small. The synthesised terms are filtered through counterexample checking and then passed on to the automatic inductive prover IsaPlanner. Experiments have given promising results, with high recall of 83 % for natural numbers and 100 % for lists when compared to libraries for the Isabelle theorem prover. However, precision is somewhat lower, 3863%. 1
Applying Lakatosstyle reasoning to AI problems
"... Lakatos (1976) argued that mathematics develops in a much more organic way than its rigid textbook presentation of definitiontheoremproof would suggest. He outlined a heuristic approach which holds that mathematics progresses by a series of primitive conjectures, proofs, counterexamples, proofgen ..."
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Lakatos (1976) argued that mathematics develops in a much more organic way than its rigid textbook presentation of definitiontheoremproof would suggest. He outlined a heuristic approach which holds that mathematics progresses by a series of primitive conjectures, proofs, counterexamples, proofgenerated concepts, modified conjectures and modified proofs. The purpose of this chapter is to apply these heuristics to the AI domains of evolving requirement specifications, planning and constraint satisfaction problems. In drawing analogies between Lakatos’s theory and these three domains, we identify areas of work which correspond to each heuristic, and suggest extensions and further ways in which Lakatos’s philosophy can inform AI problem solving. As well as the implications for added flexibility in AI, these analogies extend both the power and our understanding of Lakatos’s theory, thus this work contributes to the philosophy of mathematics. In addition, we start to build a bridge between these largely disparate AI domains. 1
A Calculus for Conjecture Synthesis
"... Abstract. IsaCoSy is a theory formation system which synthesises and proves conjectures in order to produce a background theory for a new formalisation within a proof assistant. The key idea we employ to make synthesis tractable is to only consider synthesis of terms that are not more complex versio ..."
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Abstract. IsaCoSy is a theory formation system which synthesises and proves conjectures in order to produce a background theory for a new formalisation within a proof assistant. The key idea we employ to make synthesis tractable is to only consider synthesis of terms that are not more complex versions of already known terms. IsaCoSy identifies such undesirable terms as those that match the lefthand sides of rewrite rules. In this paper, we slightly generalise this idea to present a formal language for constraining synthesis such that it does not construct terms that can be matched by a given set of constraintterms. We give a mathematical account of the algorithms involved, and prove their correctness. In particular, we prove the correctness property for IsaCoSy’s approach to synthesis: when given a set of rewrite rules as input, it only produces irreducible terms. 1