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24
AlgorithmSupported Mathematical Theory Exploration: A Personal View and Strategy
, 2004
"... We present a personal view and strategy for algorithmsupported mathematical theory exploration and draw some conclusions for the desirable functionality of future mathematical software systems. The main points of emphasis are: The use of schemes for bottomup mathematical invention, the algorit ..."
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Cited by 18 (5 self)
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We present a personal view and strategy for algorithmsupported mathematical theory exploration and draw some conclusions for the desirable functionality of future mathematical software systems. The main points of emphasis are: The use of schemes for bottomup mathematical invention, the algorithmic generation of conjectures from failing proofs for topdown mathematical invention, and the possibility to program new reasoners within the logic on which the reasoners work ("metaprogramming").
Predicate logic with sequence variables and sequence function symbols
 Proc. of the 3rd Int. Conference on Mathematical Knowledge Management. Vol. 3119 of LNCS
, 2004
"... Abstract. We describe an extension of firstorder logic with sequence variables and sequence functions. We define syntax, semantics and inference system for the extension so that Completeness, Compactness, LöwenheimSkolem, and Model Existence theorems remain valid. The obtained logic can be encoded ..."
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Cited by 11 (7 self)
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Abstract. We describe an extension of firstorder logic with sequence variables and sequence functions. We define syntax, semantics and inference system for the extension so that Completeness, Compactness, LöwenheimSkolem, and Model Existence theorems remain valid. The obtained logic can be encoded as a special ordersorted firstorder theory. We also define an inductive theory with sequence variables and formulate induction rules. The calculus forms a basis for the topdown systematic theory exploration paradigm. 1
Large Experimental Program Verification in the Theorema System
 In Proceedings ISOLA 2004, Cyprus
, 2004
"... Abstract We describe practical experiments of program verification in the frame of the Theorema system. This includes both imperative programs (using Hoare logic), as well as functional programs (using fixpoint theory). For a certain class of imperative programs we are able to generate automatically ..."
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Cited by 10 (8 self)
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Abstract We describe practical experiments of program verification in the frame of the Theorema system. This includes both imperative programs (using Hoare logic), as well as functional programs (using fixpoint theory). For a certain class of imperative programs we are able to generate automatically the loop invariants and then verification conditions, by using combinatorial and algebraic techniques. Verification conditions for functional recursive programs are derived and soundness theorem is proven. The verification conditions in both cases are generated as naturalstyle predicate logic formulae, which can be then proven by Theorema, by issuing naturalstyle proofs which are human–readable.
Context Sequence Matching for XML
, 2005
"... Context and sequence variables allow matching to explore termtrees both in depth and in breadth. It makes context sequence matching a suitable computational mechanism for a rulebased language to query and transform XML, or to specify and verify web sites. Such a language would have advantages of b ..."
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Cited by 10 (5 self)
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Context and sequence variables allow matching to explore termtrees both in depth and in breadth. It makes context sequence matching a suitable computational mechanism for a rulebased language to query and transform XML, or to specify and verify web sites. Such a language would have advantages of both pathbased and patternbased languages. We develop a context sequence matching algorithm and its extension for regular expression matching, and prove their soundness, termination and completeness properties.
An Environment for Building Mathematical Knowledge Libraries
 Proc. of the 3rd Int. Conference on Mathematical Knowledge Management, MKM’04
, 2004
"... Proving is an activity that makes use of mathematical knowledge. ..."
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Cited by 9 (4 self)
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Proving is an activity that makes use of mathematical knowledge.
Towards the Automated Synthesis of a Gröbner Bases Algorithm
, 2004
"... We discuss the question of whether the central result of algorithmic Gr obner bases theory, namely the notion of Spolynomials together with the algorithm for constructing Gr obner bases using Spolynomials, can be obtained by "artificial intelligence", i.e. a systematic (algorithmic) algorithm ..."
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Cited by 8 (5 self)
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We discuss the question of whether the central result of algorithmic Gr obner bases theory, namely the notion of Spolynomials together with the algorithm for constructing Gr obner bases using Spolynomials, can be obtained by "artificial intelligence", i.e. a systematic (algorithmic) algorithm synthesis method. We present the "lazy thinking" method for theorem and algorithm invention and apply it to the "critical pair / completion" algorithm scheme. We present a road map that demonstrates that, with this approach, the automated synthesis of the author's Gr obner bases algorithm is possible. Still, significant technical work will be necessary to improve the current theorem provers, in particular the ones in the Theorema system, so that the road map can be transformed into a completely computerized process.
Matching with Regular Constraints
 SUTCLIFFE G., VORONKOV A., Eds., Proceedings of LPAR’05
, 2005
"... We describe a sound, terminating, and complete matching algorithm for terms built over flexible arity function symbols and context, function, sequence, and individual variables. Context and sequence variables allow matching to move in term trees to arbitrary depth and breadth, respectively. The ..."
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Cited by 7 (7 self)
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We describe a sound, terminating, and complete matching algorithm for terms built over flexible arity function symbols and context, function, sequence, and individual variables. Context and sequence variables allow matching to move in term trees to arbitrary depth and breadth, respectively. The values of variables can be constrained by regular expressions which are not necessarily linear. We describe heuristics for optimization, and discuss applications.
Algorithm Synthesis by Lazy Thinking: Using Problem Schemes
 In [66
, 2004
"... Recently, as part of a general formal (i.e. logic based) methodology for mathematical knowledge management we also introduced a method for the automated synthesis of correct algorithms, which we called the lazy thinking method. For a given concrete problem specification (in predicate logic), the ..."
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Cited by 4 (0 self)
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Recently, as part of a general formal (i.e. logic based) methodology for mathematical knowledge management we also introduced a method for the automated synthesis of correct algorithms, which we called the lazy thinking method. For a given concrete problem specification (in predicate logic), the method tries out various algorithm schemes and derives specifications for the subalgorithms in the algorithm scheme.
Proof Based Synthesis of Sorting Algorithms Authors:
"... July, 2010We present some case studies in constructive synthesis of sorting algorithms. In order to synthesize some algorithms on tuples (like e. g. insertionsort, mergesort) we use an approach based on proving. Namely, we start from the specification of the problem (input and output condition) an ..."
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Cited by 3 (3 self)
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July, 2010We present some case studies in constructive synthesis of sorting algorithms. In order to synthesize some algorithms on tuples (like e. g. insertionsort, mergesort) we use an approach based on proving. Namely, we start from the specification of the problem (input and output condition) and we construct an inductive proof of the fact that for each input there exists a solution which satisfies the output condition. The problem will be reduced into smaller and smaller problems, the method will be applied like in a ”cascade” and finally the problem is so simple that the corresponding algorithm (function) already exists in the knowledge. The algorithm can be then extracted immediately from the proof. These experiments are paralleled with the exploration of the appropriate theory of tuples. The purpose of these experiments is multifold: to construct the appropriate knowledge base necessary for this type of proofs, to find the natural deduction inference rules and the necessary strategies for their application, and finally to implement the corresponding provers in the frame of the Theorema system. The novel specific feature of our approach is applying this method like in a ”cascade”
User interface features in Theorema: A summary
 In Mathematical UserInterfaces Workshop
, 2004
"... Abstract. This paper presents the main features of Theorema’s user interface. We briefly describe how mathematical knowledge can be expressed in the Theorema Formal Text Language and how the knowledge can be used for proving, solving, computing. We illustrate how the system presents the proofs it ge ..."
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Cited by 2 (0 self)
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Abstract. This paper presents the main features of Theorema’s user interface. We briefly describe how mathematical knowledge can be expressed in the Theorema Formal Text Language and how the knowledge can be used for proving, solving, computing. We illustrate how the system presents the proofs it generated and how the user can influence the proof search process interactively. 1