Results 1  10
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13
On Equivalents of Wellfoundedness  An experiment in Mizar
, 1998
"... Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be w ..."
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Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies wellfoundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. The theory of cardinals in Mizar was developed earlier by G. Bancerek. With the current state of the Mizar system, the proofs turned out to be an exercise with only minor additions at the fundamental level. We would like to stress the importance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Project.
Natural deduction calculus for lineartime temporal logic
 In To be publsihed in the Proceedings of Jelia2006, LNAI 4160
, 2006
"... Abstract. We present a natural deduction calculus for the propositional lineartime temporal logic and prove its correctness. The system extends the natural deduction construction of the classical propositional logic. This will open the prospect to apply our technique as an automatic reasoning tool ..."
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Abstract. We present a natural deduction calculus for the propositional lineartime temporal logic and prove its correctness. The system extends the natural deduction construction of the classical propositional logic. This will open the prospect to apply our technique as an automatic reasoning tool in a deliberative decision making framework across various AI applications. 1
Mathematical Knowledge Management in MIZAR
 Proc. of MKM 2001
, 2001
"... We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proofchecking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowled ..."
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We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proofchecking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowledge. The language and the checking software evolve and the evolution is driven by the growing MML.
Automated Natural Deduction for Propositional Lineartime Temporal Logic ∗
"... We present a proof searching technique for the natural deduction calculus for the propositional lineartime temporal logic and prove its correctness. This opens the prospect to apply our technique as an automated reasoning tool in a number of emerging computer science applications and in a deliberat ..."
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We present a proof searching technique for the natural deduction calculus for the propositional lineartime temporal logic and prove its correctness. This opens the prospect to apply our technique as an automated reasoning tool in a number of emerging computer science applications and in a deliberative decision making framework across various AI applications. 1
Proof Assistants: history, ideas and future
"... In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assista ..."
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In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assistants are used and how we envision their extended use in the future. While being an introduction into the world of proof assistants and the main issues behind them, this paper is also a position paper that pushes the further use of proof assistants. We believe that these systems will become the future of mathematics, where definitions, statements, computations and proofs are all available in a computerized form. An important application is and will be in computer supported modelling and verification of systems. But their is still along road ahead and we will indicate what we believe is needed for the further proliferation of proof assistants.
Redirecting proofs by contradiction
"... This paper presents an algorithm that redirects proofs by contradiction. The input is a refutation graph, as produced by an automatic theorem prover (e.g., E, SPASS, Vampire, Z3); the output is a direct proof expressed in natural deduction extended with case analyses and nested subproofs. The algori ..."
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This paper presents an algorithm that redirects proofs by contradiction. The input is a refutation graph, as produced by an automatic theorem prover (e.g., E, SPASS, Vampire, Z3); the output is a direct proof expressed in natural deduction extended with case analyses and nested subproofs. The algorithm is implemented in Isabelle’s Sledgehammer, where it enhances the legibility of machinegenerated proofs.
A Note on Intersection Types
, 1995
"... : Following J.L. Krivine, we call D the type inference system introduced by M. Coppo and M. Dezani where types are propositional formulae written with conjunction and implication from propositional letters  there is no special constant !. We show here that the wellknown result on D, stating th ..."
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: Following J.L. Krivine, we call D the type inference system introduced by M. Coppo and M. Dezani where types are propositional formulae written with conjunction and implication from propositional letters  there is no special constant !. We show here that the wellknown result on D, stating that any term which possesses a type in D strongly normalises does not need a new reducibility argument, but is a mere consequence of strong normalization for natural deduction restricted to the conjunction and implication. The proof of strong normalization for natural deduction, and therefore our result, as opposed to reducibility arguments, can be carried out within primitive recursive arithmetic. On the other hand, this enlightens the relation between & and & that G. Pottinger has already wondered about, and can be applied to other situations, like the lambda calculus with multiplicities of G. Boudol. Keywords: Lambda calculus , intersection types , strong normalization. Logic, proof th...
A History of Natural Deduction and Elementary Logic Textbooks
"... this article is to give a history of the development of this method of doing logic and to characterize what sort of thing is meant nowadays by the name. My view is that the current connotation of the term functions rather like a prototype: there is some exemplar that the term most clearly applies to ..."
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this article is to give a history of the development of this method of doing logic and to characterize what sort of thing is meant nowadays by the name. My view is that the current connotation of the term functions rather like a prototype: there is some exemplar that the term most clearly applies to and which manifests a number of characteristics. But there are other proof systems that differ from this prototypical natural deduction system and are nevertheless correctly characterized as being natural deduction. It is not clear to me just how many of the properties that the prototype exemplifies can be omitted and still have a system that is correctly characterized as a natural deduction system, and I will not try to give an answer. Instead I will focus on a number of features that are manifested to different degrees by the various natural deduction systems. My picture is that if a system ranks `low' on one of these features, it can `make up for it' by ranking high on different features. And it is somehow an overall rating of the total amount of conformity to the entire range of these different features that determines whether any specific logical system will be called a natural deduction system. Some of these features stem from the initial introduction of natural deduction in 1934; but even more strongly, in my opinion, is the effect that elementary textbooks from the 1950s had. There were of course some more technical works that brought the notion of natural deduction into the consciousness of the logical world of the 1950s and 1960s, but I will not consider them in this shortened article. In any case the `ordinary philosopher' of the time would have been little influenced by these works because the huge sway that natural deduction holds over current philosophy is most...
Distributed Reasoning in a PeertoPeer Setting: Application to the Semantic Web
"... In a peertopeer inference system, each peer can reason locally but can also solicit some of its acquaintances, which are peers sharing part of its vocabulary. In this paper, we consider peertopeer inference systems in which the local theory of each peer is a set of propositional clauses defined ..."
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In a peertopeer inference system, each peer can reason locally but can also solicit some of its acquaintances, which are peers sharing part of its vocabulary. In this paper, we consider peertopeer inference systems in which the local theory of each peer is a set of propositional clauses defined upon a local vocabulary. An important characteristic of peertopeer inference systems is that the global theory (the union of all peer theories) is not known (as opposed to partitionbased reasoning systems). The main contribution of this paper is to provide the first consequence finding algorithm in a peertopeer setting: DeCA. It is anytime and computes consequences gradually from the solicited peer to peers that are more and more distant. We exhibit a sufficient condition on the acquaintance graph of the peertopeer inference system for guaranteeing the completeness of this algorithm. Another important contribution is to apply this general distributed reasoning setting to the setting of the Semantic Web through the Somewhere semantic peertopeer data management system. The last contribution of this paper is to provide an experimental analysis of the scalability of the peertopeer infrastructure that we propose, on large networks of 1000 peers. 1.