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Isar -- a Generic Interpretative Approach to Readable Formal Proof Documents
, 1999
"... We present a generic approach to readable formal proof documents, called Intelligible semi-automated reasoning (Isar). It addresses the major problem of existing interactive theorem proving systems that there is no appropriate notion of proof available that is suitable for human communication, or ..."
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Cited by 100 (16 self)
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We present a generic approach to readable formal proof documents, called Intelligible semi-automated reasoning (Isar). It addresses the major problem of existing interactive theorem proving systems that there is no appropriate notion of proof available that is suitable for human communication, or even just maintenance. Isar's main aspect is its formal language for natural deduction proofs, which sets out to bridge the semantic gap between internal notions of proof given by state-of-the-art interactive theorem proving systems and an appropriate level of abstraction for user-level work. The Isar language is both human readable and machine-checkable, by virtue of the Isar/VM interpreter. Compared to existing declarative theorem proving systems, Isar avoids several shortcomings: it is based on a few basic principles only, it is quite independent of the underlying logic, and supports a broad range of automated proof methods. Interactive proof development is supported as well...
Proof Planning with Multiple Strategies
- In Proc. of the First International Conference on Computational Logic
, 2000
"... . Humans have different problem solving strategies at their disposal and they can flexibly employ several strategies when solving a complex problem, whereas previous theorem proving and planning systems typically employ a single strategy or a hard coded combination of a few strategies. We introd ..."
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Cited by 60 (37 self)
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. Humans have different problem solving strategies at their disposal and they can flexibly employ several strategies when solving a complex problem, whereas previous theorem proving and planning systems typically employ a single strategy or a hard coded combination of a few strategies. We introduce multi-strategy proof planning that allows for combining a number of strategies and for switching flexibly between strategies in a proof planning process. Thereby proof planning becomes more robust since it does not necessarily fail if one problem solving mechanism fails. Rather it can reason about preference of strategies and about failures. Moreover, our strategies provide a means for structuring the vast amount of knowledge such that the planner can cope with the otherwise overwhelming knowledge in mathematics. 1 Introduction The choice of an appropriate problem solving strategy is a crucial human skill and is typically guided by some meta-level reasoning. Trained mathematicia...
The PROSPER Toolkit
, 2000
"... The Prosper (Proof and Specification Assisted Design Environments) project advocates the use of toolkits which allow existing verification tools to be adapted to a more flexible format so that they may be treated as components. A system incorporating such tools becomes another component that can be ..."
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Cited by 45 (2 self)
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The Prosper (Proof and Specification Assisted Design Environments) project advocates the use of toolkits which allow existing verification tools to be adapted to a more flexible format so that they may be treated as components. A system incorporating such tools becomes another component that can be embedded in an application. This paper describes the Prosper Toolkit which enables this. The nature of communication between components is specified in a language-independent way. It is implemented in several common programming languages to allow a wide variety of tools to have access to the toolkit.
Integrating computer algebra into proof planning
- Journal of Automated Reasoning
, 1998
"... Abstract. Mechanised reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two di erent tasks, proving and calculating. Even more importantly, proof and computation are often interwoven and not e ..."
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Cited by 41 (24 self)
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Abstract. Mechanised reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two di erent tasks, proving and calculating. Even more importantly, proof and computation are often interwoven and not easily separable. In this contribution we advocate an integration of computer algebra into mechanised reasoning systems at the proof plan level. This approach allows to view the computer algebra algorithms as methods, that is, declarative representations of the problem solving knowledge speci c to a certain mathematical domain. Automation can be achieved in many cases bysearching for a hierarchic proof plan at the methodlevel using suitable domain-speci c control knowledge about the mathematical algorithms. In other words, the uniform framework of proof planning allows to solve a large class of problems that are not automatically solvable by separate systems. Our approach also gives an answer to the correctness problems inherent insuch an integration. We advocate an approach where the computer algebra system produces high-level protocol information that can be processed by aninterface to derive proof plans. Such a proof plan in turn can be expanded to proofs at di erent levels of abstraction, so the approach iswell-suited for producing a high-level verbalised explication as well as for a low-level machine checkable calculus-level proof. We present an implementation of our ideas and exemplify them using an automatically solved example. Changes in the criterion of `rigour of the proof ' engender major revolutions in mathematics.
A Refinement of de Bruijn’s Formal Language of Mathematics
- Journal of Logic, Language and Information
, 2004
"... Abstract. We provide a syntax and a derivation system for a formal language of mathematics called Weak Type Theory (WTT). We give the metatheory of WTT and a number of illustrative examples. WTT is a refinement of de Bruijn’s Mathematical Vernacular (MV) and hence: WTT is faithful to the mathematici ..."
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Cited by 29 (15 self)
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Abstract. We provide a syntax and a derivation system for a formal language of mathematics called Weak Type Theory (WTT). We give the metatheory of WTT and a number of illustrative examples. WTT is a refinement of de Bruijn’s Mathematical Vernacular (MV) and hence: WTT is faithful to the mathematician’s language yet is formal and avoids ambiguities. WTT is close to the usual way in which mathematicians express themselves in writing. ¡ WTT has a syntax based on linguistic categories instead of set/type theoretic constructs. More so than MV however, WTT has a precise abstract syntax whose derivation rules resemble those of modern type theory enabling us to establish important desirable properties of WTT such as strong normalisation, decidability of type checking and subject reduction. The derivation system allows one to establish that a book written in WTT is well-formed following the syntax of WTT, and has great resemblance with ordinary mathematics books. WTT (like MV) is weak as regards correctness: the rules of WTT only concern linguistic correctness, its types are purely linguistic so that the formal translation into WTT is satisfactory as a readable, well-organized text. In WTT, logico-mathematical aspects of truth are disregarded. This separates concerns and means that WTT can be easily understood by either a mathematician, a logician or a computer scientist. acts as an intermediary between the language of mathematicians and that of logicians.
Proof Planning
- PROCEEDINGS OF THE 3RD INTERNATIONAL CONFERENCE ON AI PLANNING SYSTEMS, (AIPS
, 1996
"... We describe proof planning, a technique for the global control of search in automatic theorem proving. A proof plan captures the common patterns of reasoning in a family of similar proofs and is used to guide the search for new proofs in this family. Proof plans are very similar to the plans cons ..."
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Cited by 29 (2 self)
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We describe proof planning, a technique for the global control of search in automatic theorem proving. A proof plan captures the common patterns of reasoning in a family of similar proofs and is used to guide the search for new proofs in this family. Proof plans are very similar to the plans constructed by plan formation techniques. Some differences are the non-persistence of objects in the mathematical domain, the absence of goal interaction in mathematics, the high degree of generality of proof plans, the use of a meta-logic to describe preconditions in proof planning and the use of annotations in formulae to guide search.
Comparing approaches to the exploration of the domain of residue classes
- ARTICLE SUBMITTED TO JOURNAL OF SYMBOLIC COMPUTATION
, 2002
"... We report on a case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using different proof techniques, which are implemented as strategies in a multi-strategy ..."
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Cited by 25 (13 self)
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We report on a case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using different proof techniques, which are implemented as strategies in a multi-strategy proof planner. The search space of the proof planner can be drastically reduced by employing computations of two computer algebra systems during the planning process. To test the eectiveness of our approach we carried out a large number of experiments and also compared it with some alternative approaches. In particular, we experimented with substituting computer algebra by model generation and by proving theorems with a first order equational theorem prover instead of a proof planner.
A blackboard architecture for guiding interactive proofs
- Artificial Intelligence: Methodology, Systems and Applications
, 1998
"... Abstract. The acceptance and usability of current interactive theorem proving environments is, among other things, strongly influenced by the availability of an intelligent default suggestion mechanism for commands. Such mechanisms support the user by decreasing the necessary interactions during the ..."
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Cited by 22 (17 self)
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Abstract. The acceptance and usability of current interactive theorem proving environments is, among other things, strongly influenced by the availability of an intelligent default suggestion mechanism for commands. Such mechanisms support the user by decreasing the necessary interactions during the proof construction. Although many systems offer such facilities, they are often limited in their functionality. In this paper we present a new agent-based mechanism that independently observes the proof state, steadily computes suggestions on how to further construct the proof, and communicates these suggestions to the user via a graphical user interface. We furthermore introduce a focus technique in order to restrict the search space when deriving default suggestions. Although the agents we discuss in this paper are rather simple from a computational viewpoint, we indicate how the presented approach can be extended in order to increase its deductive power. 1
