1,033 research outputs found
Computer supported mathematics with Ωmega
AbstractClassical automated theorem proving of today is based on ingenious search techniques to find a proof for a given theorem in very large search spacesâoften in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited.The shift from search based methods to more abstract planning techniques however opened up a paradigm for mathematical reasoning on a computer and several systems of that kind now employ a mix of interactive, search based as well as proof planning techniques.The Ωmega system is at the core of several related and well-integrated research projects of the Ωmega research group, whose aim is to develop system support for a working mathematician as well as a software engineer when employing formal methods for quality assurance. In particular, Ωmega supports proof development at a human-oriented abstract level of proof granularity. It is a modular system with a central proof data structure and several supplementary subsystems including automated deduction and computer algebra systems. Ωmega has many characteristics in common with systems like NuPrL, CoQ, Hol, Pvs, and Isabelle. However, it differs from these systems with respect to its focus on proof planning and in that respect it is more similar to the proof planning systems Clam and λClam at Edinburgh
Using dialogue to learn math in the LeActiveMath project
We describe a tutorial dialogue system under development that assists students in learning how to differentiate equations. The system uses deep natural language understanding and generation to both interpret students â utterances and automatically generate a response that is both mathematically correct and adapted pedagogically and linguistically to the local dialogue context. A domain reasoner provides the necessary knowledge about how students should approach math problems as well as their (in)correctness, while a dialogue manager directs pedagogical strategies and keeps track of what needs to be done to keep the dialogue moving along.
A Subsumption Architecture for Theorem Proving?
Brooks has criticized traditional approaches
to artificial
intelligence as too ineffi-
cient. In particular, he has singled out techniques involving
search as
inadequate
to achieve the fast reaction times required by robots and other AI products
that
need to work in the real world. Instead he proposes
the subsumption architecture
as an overall organizing principle. This consists of
layers of behavioural modules,
each of which is capable of carrying out a complete (usually simple)
task. He has
employed
this architecture to build a series of simple mobile robots, but he claims
that it is appropriate
for all AI products. Brooks's proposal
is usually seen as an
example of nouvelle AI,
in contrast to good old-fashioned AI (GOFAI).
Automatic theorem proving
is the archetypal example of GOFAI. The resolution
theorem proving technique once served as the engine of AI. Of all areas of AI it
seems the most difficult to
implement using Brooks's ideas. It would thus serve
as a keen test of Brooks's proposal
to explore
to what extent the task of theorem
proving can be achieved by a subsumption architecture.
Tactics are programs
for guiding a theorem prover. They were introduced as
an efficient alternative to search-based techniques. In this paper
I compare recent
work on tactic-based theorem proving with Brooks's proposals and show that,
surprisingly,
there is a similarity between them. It thus seems that the distinction
between nouvelle AI and GOFAI is not so great as is sometimes claimed. However,
this exercise also identifies some criticisms of Brooks's proposal
Using features for automated problem solving
We motivate and present an architecture for problem solving where an abstraction
layer of "features" plays the key role in determining methods to apply. The system
is presented in the context of theorem proving with Isabelle, and we demonstrate
how this approach to encoding control knowledge is expressively different to
other common techniques. We look closely at two areas where the feature
layer may offer benefits to theorem proving â semi-automation and learning
â
and find strong evidence that in these particular domains, the approach shows
compelling promise. The system includes a graphical theorem-proving user
interface for Eclipse ProofGeneral and is available from the project web page,
http://feasch.heneveld.org
An Extensible User Interface for Lean 4
Contemporary proof assistants rely on complex automation and process libraries with millions of lines of code. At these scales, understanding the emergent interactions between components can be a serious challenge. One way of managing complexity, long established in informal practice, is through varying external representations. For instance, algebraic notation facilitates term-based reasoning whereas geometric diagrams invoke spatial intuition. Objects viewed one way become much simpler than when viewed differently. In contrast, modern general-purpose ITP systems usually only support limited, textual representations. Treating this as a problem of human-computer interaction, we aim to demonstrate that presentations - UI elements that store references to the objects they are displaying - are a fruitful way of thinking about ITP interface design. They allow us to make headway on two fronts - introspection of prover internals and support for diagrammatic reasoning. To this end we have built an extensible user interface for the Lean 4 prover with an associated ProofWidgets 4 library of presentation-based UI components. We demonstrate the system with several examples including type information popups, structured traces, contextual suggestions, a display for algebraic reasoning, and visualizations of red-black trees. Our interface is already part of the core Lean distribution
Knowledge Based Systems: A Critical Survey of Major Concepts, Issues, and Techniques
This Working Paper Series entry presents a detailed survey of knowledge based systems. After being in a relatively dormant state for many years, only recently is Artificial Intelligence (AI) - that branch of computer science that attempts to have machines emulate intelligent behavior - accomplishing practical results. Most of these results can be attributed to the design and use of Knowledge-Based Systems, KBSs (or ecpert systems) - problem solving computer programs that can reach a level of performance comparable to that of a human expert in some specialized problem domain. These systems can act as a consultant for various requirements like medical diagnosis, military threat analysis, project risk assessment, etc. These systems possess knowledge to enable them to make intelligent desisions. They are, however, not meant to replace the human specialists in any particular domain. A critical survey of recent work in interactive KBSs is reported. A case study (MYCIN) of a KBS, a list of existing KBSs, and an introduction to the Japanese Fifth Generation Computer Project are provided as appendices. Finally, an extensive set of KBS-related references is provided at the end of the report
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