Tarski's influence on computer science

The influence of Alfred Tarski on computer science was indirect but significant in a number of directions and was in certain respects fundamental. Here surveyed is the work of Tarski on the decision procedure for algebra and geometry, the method of elimination of quantifiers, the semantics of formal languages, modeltheoretic preservation theorems, and algebraic logic; various connections of each with computer science are taken up

Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?

Classical interpretations of Goedel's formal reasoning imply that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is essentially unverifiable. However, a language of general, scientific, discourse cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic is verifiably complete. We show how some paradoxical concepts of Quantum mechanics can be expressed, and interpreted, naturally under a constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht

Information enforcement in learning with graphics : improving syllogistic reasoning skills

This thesis is an investigation into the factors that contribute to good choices among graphical systems used in teaching, and the feasibility of implementing teaching software that uses this knowledge.The thesis describes a mathematical metric derived from a cognitive theory of human diagram processing. The theory characterises differences among representations by their ability to express information. The theory provides the factors and relationships needed to build the metric. It says that good representations are easily processed because they are more vivid, more tractable and less expressive, than poor representations.The metric is applied to abstract systems for teaching and learning syllogistic reasoning, TARSKI'S WORLD, EULER CIRCLES, VENN DIAGRAMS and CARROLL'S GAME OF LOGIC. A rank ordering reflects the value of each system predicted by the theory and the metric. The theory, the metric and the systems are then tested in empirical studies. Five studies involving sixty-eight learners, examined the benefit of software based on these abstract systems.Studies showed the theory correctly predicted learners' success with the circle systems and poorer performance with TARSKI'S WORLD. The metric showed small but clear differences in expressivity between the circle systems. Differences between results of the learners using the circle systems contradicted the predictions of the metric.Learners with mathematical training were better equipped and more successful at learning syllogistic reasoning with the systems. Performance of learners without mathematical training declined after using the software systems. Diagrams drawn by learners together with video footage collected during problem solving, led to a catalogue of errors, misconceptions and some helpful strategies for learning from graphical systems.A cognitive style test investigated the poor performance of non-mathematically trained learners. Learners with mathematics training showed serialist and versatile learning styles while learners without this training showed a holist learning style. This is consistent with the hypothesis that non-mathematically trained learners emphasise the use of semantic cues during learning and problem solving.A card-sorting task investigated learners' preferences for parts of the graphical lexicon used in the diagram systems. Preferences for the EULER lexicon increased difficulty in explaining the system's poor results in earlier studies. Video footage of learners using the systems in the final study illustrated useful learning strategies and improved performance with EULER while individual instruction was available.Further work describes a preliminary design for an adaptive syllogism tutor and other related work

Tarski and Lesniewski on Languages with Meaning versus Languages without Use: A 60th Birthday Provocation for Jan Wolenski

Constructivism, Judgement and Meanin

Visualisation and manipulation tools for Modal logic

In this thesis, an investigation into how visualisation and manipulation tools can provide better support for learners of Modal logic is described. Problems associated with learning Modal logic are also researched.Seven areas topics in Modal logic are investigated, as is the influence of domain independent factors (e. g. motivation) on learning. Studies show that students find concepts such as Modal proofs and systems difficult to learn, whilst possible worlds and Modes are fairly straightforward. Areas such as reference, belief and accessibility relations fall between these extremes.Two roles for representations in reasoning are identified: providing a concrete domain for students to reason about, and supporting the process of reasoning. Systems which make use of these complementary representations were found to be more effective for learners than either the syntactic or the diagrammatic representations traditionally used to teach Modal logic.A review of software used to support students learning logic highlights two important features: the use of examples, and automation of routine tasks. A learning environment for Modal logic was designed which incorporated these. The environment was developed using an adapted version of Smalltalk's Model-View-Controller mechanism, and incorporates complementary representations, enhance by direct manipulation.A further study investigates the added benefits of using this tool, as opposed to using the same representation but working with pen and paper. This confirms the importance of using 'concrete' content representations and minimising learners' cognitive load. Performance measures show that software users learnt more, had a deeper style of learning, and found the topics less abstract than their counterparts working with pen & paper.This research shows that complementary representations are an effective way of supporting students studying Modal logic, and that visualisation and manipulation tools which incorporate these systems will provide additional benefits for learners

How to interpret and establish consistency results for semantics of concurrent programming languages

It is meaningful that a language is provided with several semantic descriptions: e.g. one which serves the needs of the implementor, another one that is suitable for specification and yet another one that will be used to explain the language to the user. In this case one has to guarantee that the various semantics are 'consistent'. The attempt of this paper is to clarify the notion 'consistency' and to present a general framework and theorems for consistency results

Philosophy of mathematics education

PHILOSOPHY OF MATHEMATICS EDUCATION\ud This thesis supports the view that mathematics teachers should be aware of differing views of the nature of mathematics and of a range of teaching perspectives. The first part of the thesis discusses differing ways in which the subject 'mathematics' can be identified, by relying on existing philosophy of mathematics. The thesis describes three traditionally recognised philosophies of mathematics: logicism, formalism and intuitionism. A fourth philosophy is constructed, the hypothetical, bringing together the ideas of Peirce and of Lakatos, in particular. The second part of the thesis introduces differing ways of teaching mathematics, and identifies the logical and sometimes contingent connections that exist between the philosophies of mathematics discussed in part 1, and the philosophies of mathematics teaching that arise in part 2. Four teaching perspectives are outlined: the teaching of mathematics as aestheticallyorientated, the teaching of mathematics as a game, the teaching of mathematics as a member of the natural sciences, and the teaching of mathematics as technology-orientated. It is argued that a possible fifth perspective, the teaching of mathematics as a language, is not a distinctive approach. A further approach, the Inter-disciplinary perspective, is recognised as a valid alternative within previously identified philosophical constraints. Thus parts 1 and 2 clarify the range of interpretations found in both the philosophy of mathematics and of mathematics teaching and show that they present realistic choices for the mathematics teacher. The foundations are thereby laid for the arguments generated in part 3, that any mathematics teacher ought to appreciate the full range of teaching 4 perspectives which may be chosen and how these link to views of the nature of mathematics. This would hopefully reverse 'the trend at the moment... towards excessively narrow interpretation of the subject' as reported by Her Majesty's Inspectorate (Aspects of Secondary Education in England, 7.6.20, H. M. S. O., 1979). While the thesis does not contain infallible prescriptions it is concluded that the technology-orientated perspective supported by the hypothetical philosophy of mathematics facilitates the aims of those educators who show concern for the recognition of mathematics in the curriculum, both for its intrinsic and extrinsic value. But the main thrust of the thesis is that the training of future mathematics educators must include opportunities for gaining awareness of the diversity of teaching perspectives and the influence on them of philosophies of mathematics

A Metamodel for the Unified Modeling Language

Nowadays models, rather than code, become the key artifacts of software development. Consequently, this raises the level of requirements for modeling languages on which modeling practitioners should rely in their work. A minor inconsistency of a modeling language metamodel may cause major problems in the language applications; thus with the model driven systems development the solidness of modeling languages metamodels becomes particularly important. In its current state the UML metamodel leaves a significant area for improvement. We present an alternative metamodel that was inspired by the RM-ODP standard and that solves the problems of UML. RM-ODP was mentioned in UML specifications as a framework that has already influenced UML. Our metamodel was formalized, thus its resulting models can be simulated and checked for consistency. So, our proposed solution with constructive potential towards improvement of the UML metamodel, may have a significant practical impact on the UML specifications

An empirical study into COBOL type inferencing

AbstractIn a typical COBOL program, the data division consists of 50% of the lines of code. Automatic type inference can help to understand the large collections of variable declarations contained therein, showing how variables are related based on their actual usage. The most problematic aspect of type inference is pollution, the phenomenon that types become too large, and contain variables that intuitively should not belong to the same type. The aim of the paper is to provide empirical evidence for the hypothesis that the use of subtyping is an effective way for dealing with pollution. The main results include a tool set to carry out type inference experiments, a suite of metrics characterizing type inference outcomes, and the experimental observation that only one instance of pollution occurs in the case study conducted