MizarMode—an integrated proof assistance tool for the Mizar way of formalizing mathematics

The Emacs authoring environment for Mizar (MizarMode) is today the authoring tool of choice for many (probably the majority of) Mizar authors. This article describes the MizarMode and focuses on the proof assistance functions and tools available in it.

We start with the explanation of the design principles behind the Mizar system, and show how these design principles—mainly the concentration on simple and intuitive human-oriented proofs—have helped Mizar in developing and maintaining a very large body of formalized mathematics.

Mizar is a non-programmable and non-tactical verifier: the proofs are developed in the traditional “write—compile—correct” software programming loop. While this method is in the beginning more laborious than the methods employed in tactical and programmable proof assistants, it makes the “proof code” in the long-run more readable, maintainable and reusable. This seems to be a crucial factor for a long-term and large-scale formalization effort.

MizarMode has been designed with the aim to facilitate this kind of proof development by a number of “code-generating”, “code-browsing” and “code-searching” methods, and tools programmed or integrated within it. These methods and tools now include, e.g., the automated generation of proof skeletons, semantic browsing of the articles and abstracts, structured viewing, proof advice using trained machine learning tools like the Mizar Proof Advisor, deductive tools like MoMM, etc. We give an overview of these proof-assistance tools and their integration in the MizarMode, and also discuss some emerging and future extensions such as integration of external theorem proving assistance.     

The development of socialization skills in an electively mute child

In a booth designed especially for work with both autistic and electively mute children, a 4-yr-old girl named Dolly, who had no communicative speech or imitative skills, was given a preliminary session in which her verbal output was assessed. To elicit sounds from Dolly, an instrument called a “color organ” was used as a positive reinforcer. After this baseline assessment, in 40 half-hour sessions, Dolly was taught to make eye contact with E, and to obey instructions—although it was first necessary to extinguish her disruptive behavior, particularly her opérant crying. In addition, she learned non-verbal imitative behavior, such as hand clapping; and verbal imitative behavior, such as saying “Hi!” Social (play) sessions were begun after session 21, and continued for the remaining time. These were helpful in generalizing Dolly's learned skills to an environment other than the booth; and to other tasks, such as singing “Ee-eye-ee-eye-oh” in the refrain of the song, “Old McDonald”; and pointing to E's eyes, saying “ice”.     

Nonstutterers'' perceptions of stuttering and speech difficulty

Two versions of Woolf's (1967) Perception of Stuttering Inventory (PSI) were administered to nonstuttering college students in three experiments. In one (Woolf's published version) the word “stuttering” appeared and in another [a version developed by Daly, Oakes, Breen, and Mishler (1981)] “stuttering” was replaced by “speech difficulty.” Nonstutterers' scores on the PSI were found to be affected by knowledge of—or attention to—“stuttering” or “stutterers” as well as orientation toward speech or speech pathology. It was concluded that a total PSI score of 5 is presently a more defensible criterion of normality for stutterers than the more commonly used score of 10.     

Speech—Language clinicians'' and university students'' impressions of women and girls who stutter

To identify impressions speech—language clinicians and university students have of females who stutter, a 47-scale semantic differential form was administered to members of each group to obtain their responses to eight hypothetical constructs, i.e., “A Girl,” “A Girl Who Stutters,” “A Boy,” “A Boy Who Stutters,” “A Woman,” “A Woman Who Stutters,” “A Man,” and “A Man Who Stutters.” Both groups were found to possess negative stereotypes for all four categories of stutterers. The nature of the stereotypes appeared to be influenced by a stutterer's gender and relative age. Clinicians considered stuttering to exert a stronger negative impact on females and on children. Their strongest stereotype was of “A Girl Who Stutters.” University students considered stuttering to exert a stronger negative impact on males. Their stereotypes of stutterers seemed unaffected by the relative age of the stutterer. Their strongest stereotype was of “A Man Who Stutters.” Several theoretical and clinical implications of these findings are discussed.     

Are professors likely to report having “beliefs” about the intelligence and competence of students who stutter?

Some university students who stutter report that they avoid participating in class discussions and talking with professors because they believe that if their professors find out they stutter, the professors are likely to view the students as being less intelligent and/or competent than they otherwise would. The purpose of this study was to determine whether professors, as a group, are likely to report such beliefs. A random sample of 87 professors at Marquette University, Indiana University of Pennsylvania, and the State University of New York at Stonybrook rated “A Student Who Stutters” on each of 81 seven-point, bipolar semantic differential scales, including intelligent/unintelligent and competent/incompetent. Although there was considerable scatter in their ratings for the majority of the scales, 85 of the 87 rated the student at the center or “positive” end of the intelligent/unintelligent scale and 83 of the 87 did so on the competent/incompetent one. Assuming that the attitutes reported by the professors reflect their “real” ones, these data suggest that many—perhaps the majority—do not have negative attitudes about the intelligence and competence of their students who stutter.     

The influence of previous movement and posture on subsequent posture

An experiment with 21 subjects has confirmed the findings of other investigators that previous posture affects subsequent posture—the phenomenon of postural persistence. There seems also to be a tendency for the arm to be judged horizontal when in fact above the horizon, no matter which posture has been previously adopted—a “constant upwards effect.” It has also been found that the direction of previous movement affects subsequent posture. After movement, an overshooting effect adds to a “constant upwards effect.”

Various explanations of the phenomenon are discussed, and persistence is considered as an anomaly of postural recognition.     

A study of position habits induced by reward and “frustration”

Predictions from Maier's theory of “frustration”-instigated behaviour have been tested in an experimental situation differing significantly from that in which the theory was propounded yet containing the central element of “frustration”—the insoluble problem.

A water discrimination unit was employed in which the performance of rats would be observed during attacks on insoluble problems, position problems or symbol problems.

Two groups, each containing ten Wistar albino rats, served as subjects. The research design consisted of the following phases: preliminary training, development of position responses, exposure to a symbol-reward problem with 50 per cent, punishment and exposure to a symbol-reward problem with 100 per cent, punishment. The design differed for the two groups only at the phase in which the position responses were established. During this phase one group was exposed to a position-reward problem and the other to an insoluble problem.

Position responses were established as frequently under position-“frustration” (position stereotypes) as under position-reward (position habits) conditions. Position stereotypes were more rigid—more resistant to extinction—than position habits under conditions of 50 per cent, punishment. Position stereotypes were as readily extinguished under 100 per cent, punishment as were position habits under 30 per cent, punishment.

The first two observations conform to predictions made from Maier's theory. The third does not. That is to say, not all situations containing the basic elements of “frustration” give rise to stereotyped behaviour patterns which are as rigid or “fixated” as Maier's theory would predict. It is a reasonable hypothesis that the characteristics of stereotyped responses established in certain “frustration” situations may be described adequately in terms of conventional learning principles without the necessity of resorting to a distinction between “goal-motivated” and “frustration-instigated” behaviour.     

Modification of expectations in cases with obsessional rituals

Some theoretical issues in relation to the nature of obsessional rituals and the most commonly adopted method of behaviour therapy for this disorder are critically considered. On the basis of these considerations, a different method—“modification of expectations” or “reality testing” is put forward and its successful application to two patients described and discussed.     

Attitudes of school counsellors to some counselling issues

A questionnaire was constructed to ascertain the attitudes of school counsellors to guidance issues. 16 school counsellors in an urban area in the Midlands were asked to complete the questionnaire. The results show a wide area of agreement among the counsellors, particularly on issues concerned with practical and professional matters. Differences of opinion occur with respect to some of the other issues — including those concerned with the relationship of guidance and counselling to society — and possible implications of this lack of unanimity are discussed.     

The sad truth about depressive realism

In one form of a contingency judgement task individuals must judge the relationship between an action and an outcome. There are reports that depressed individuals are more accurate than are nondepressed individuals in this task. In particular, nondepressed individuals are influenced by manipulations that affect the salience of the outcome, especially outcome probability. They overestimate a contingency if the probability of an outcome is high—the “outcome-density effect”. In contrast, depressed individuals display little or no outcome-density effect. This apparent knack for depressives not to be misled by outcome density in their contingency judgements has been termed “depressive realism”, and the absence of an outcome-density effect has led to the characterization of depressives as “sadder but wiser”. We present a critical summary of the depressive realism literature and provide a novel interpretation of the phenomenon. We suggest that depressive realism may be understood from a psychophysical analysis of contingency judgements.*     

An experiment in psoblem solying

Trial-and-error problems are described in terms of “stimulus” difficulty, which is a measure of the number of possible modes of response left to the individual when all the information given is taken into account; and “phenomenal” difficulty, which is a measure derived from the individual's performance. An experiment is described in which three types of problem were presented to human subjects. In all three problems the stimulus difficulty was calculable, stage by stage, in the solution. The problems differed in this stimulus difficulty and also in the qualitative nature of the information provided—from unequivocal to conditional. It is shown that the qualitative difference of the nature of the information bears most relationship to phenomenal difficulty. Some observations are made on the modes of solution adopted, and further experimental work is suggested.     

Two modes of sensorimotor integration in intention-based and stimulus-based actions

Human actions may be driven endogenously (to produce desired environmental effects) or exogenously (to accommodate to environmental demands). There is a large body of evidence indicating that these two kinds of action are controlled by different neural substrates. However, only little is known about what happens—in functional terms—on these different “routes to action”. Ideomotor approaches claim that actions are selected with respect to their perceptual consequences. We report experiments that support the validity of the ideomotor principle and that, at the same time, show that it is subject to a far-reaching constraint: It holds for endogenously driven actions only! Our results suggest that the activity of the two “routes to action” is based on different types of learning: The activity of the system guiding stimulus-based actions is accompanied by stimulus-response (sensorimotor) learning, whereas the activity of the system controlling intention-based actions results in action-effect (ideomotor) learning.     

Acoustic factors in visual search

Previous work has shown that in searching for existing or absent “e.s” in printed prose, the presence or absence of silent “e.s” was less likely to be detected than that of pronounced “e.s.” It was suggested that the acoustic or kinaesthetic “image” was searched for evidence of an “e” in addition to the visual stimulus and that evidence from both sources was considered in making the appropriate response.

The present experiment employs mainly substitutive errors within words, which may or may not change their pronunciation. The results suggest that the form of the acoustic correlates has no bearing upon whether the words are detected as wrongly spelt, but that the presence or absence of an acoustic event corresponding in time to the spatial location of the error is important.     

Driye-leyel and flexibility in paired-associate nonsense-syllable learning

“High-anxiety” and “low-anxiety” subjects, selected for extreme scores on the Taylor Anxiety Scale, learned a list of paired-associate nonsense syllables in the belief that they were undergoing an intelligence test. Both groups were then given a second list of paired associates to learn, the stimulus-items being the same as those of the first list but the responses being changed. Before the presentation of the second list, half the subjects in each group were given anxiety-increasing instructions and the remaining half were given reassuring instructions.

The results verified two predictions made from Hull's behaviour theory, using the concept of fear or anxiety as a secondary drive:—“high anxiety” subjects took more trials to master the second learning task than “low-anxiety” subjects; and there was a significant interaction between initial anxiety-level and type of instructions, such that “high-anxiety” subjects who received drive-increasing instructions had a worse performance in the second part than all other sub-groups. There was no indication that “low-anxiety” subjects were significantly affected by the type of instructions received. The “high-anxiety” group had greater difficulty than the “low-anxiety” group in learning the first list, but the difference was non-significant.     

Remembering, Familiarity, and Source Monitoring

Two experiments investigated recollective experience in a source monitoring task. Subjects saw an array of objects and performed, watched, or imagined actions involving pairs of objects. In a subsequent recognition test, subjects indicated whether their recognition judgements were made on the basis of conscious recollective experience (“remember” responses), or on some other basis such as familiarity (“know” responses). The proportions of correct “remember” responses for both objects and actions decreased from performed, through watched, to imagined actions, whereas the proportions of correct “know” responses were uninfluenced by the source of the memories. In addition, the relationship between recollective experience and accuracy of source judgement varied across sources. Source accuracy for performed actions was obtained only in “remember” responses, whereas source accuracy for imagined actions was obtained only in “know” responses. Source accuracy for watched actions was obtained in both “remember” and “know” responses. The findings suggest that the types of memory attributes available at retrieval determine the quality of subsequent memory experience, and it is proposed that memories with strongly self-referential attributes (arising from performed actions) powerfully cue recollective experience during retrieval.     

Preferred tempo in the learning of a gross cyclical action

Seventy-five subjects, randomly assigned to one of five training conditions, were required to learn to make large-amplitude, high-tempo, fluent movements on a so-called ski-simulator over a period of four days. Subjects trained under different tempo conditions. In four of the conditions the tempo was prescribed (“preferred”, high, low, or increasing), augmented feedback being provided to enable subjects to stay on “target”. “Preferred” tempo was based on the weight of the subject and was derived from a regression equation based on previous empirical research. In a fifth condition, subjects trained on “discovery learning” principles, i.e. without the tempo being prescribed. The results obtained on the three parameters (amplitude, frequency, and fluency) during the daily test sessions (in which the tempo was not prescribed) formed the data for the analyses.

A learning effect was apparent on all three parameters over the four-day training period. Subjects who trained under the high or the low prescribed tempos, however, were shown to produce significantly smaller amplitude movements than subjects who trained under the other three conditions. Training under the low-tempo condition was also shown to have disadvantageous effects on the parameters tempo and fluency. It was concluded that, for these kinds of action, training at a high or a low tempo—and particularly the latter—has undesirable effects. Such disadvantageous effects, however, were shown to be avoidable if training is begun with the “preferred” tempo of the subject and increased successively by 7% over days.     

What can functional neuroimaging tell the experimental psychologist?

I argue here that functional neuroimaging data—which I restrict to the haemodynamic techniques of fMRI and PET—can inform psychological theorizing, provided one assumes a “systematic” function-structure mapping in the brain. In this case, imaging data simply comprise another dependent variable, along with behavioural data, that can be used to test competing theories. In particular, I distinguish two types of inference: function-to-structure deduction and structure-to-function induction. With the former inference, a qualitatively different pattern of activity over the brain under two experimental conditions implies at least one different function associated with changes in the independent variable. With the second type of inference, activity of the same brain region(s) under two conditions implies a common function, possibly not predicted a priori. I illustrate these inferences with imaging studies of recognition memory, short-term memory, and repetition priming. I then consider in greater detail what is meant by a “systematic” function-structure mapping and argue that, particularly for structure-to-function induction, this entails a one-to-one mapping between functional and structural units, although the structural unit may be a network of interacting regions and care must be taken over the appropriate level of functional/structural abstraction. Nonetheless, the assumption of a systematic function-structure mapping is a “working hypothesis” that, in common with other scientific fields, cannot be proved on independent grounds and is probably best evaluated by the success of the enterprise as a whole. I also consider statistical issues such as the definition of a qualitative difference and methodological issues such as the relationship between imaging and behavioural data. I finish by reviewing various objections to neuroimaging, including neophrenology, functionalism, and equipotentiality, and by observing some criticisms of current practice in the imaging literature.     

A memory-based, Simon-like, spatial congruence effect: Evidence for persisting spatial codes

The Simon effect refers to the finding that in a task where stimulus location is irrelevant, reaction time is faster when stimulus and response locations are congruent than when they are not. Dominant theories of the Simon effect have generally attributed this spatial congruence effect to a spatial code automatically generated upon stimulus presentation. A common assumption of these theories is that this spatial code decays in less than a few hundred milliseconds following stimulus onset. We report two working-memory experiments suggesting a reexamination of this assumption—a Simon-like spatial congruence effect persisted over a delay of as long as 2400 ms. We propose that, in addition to generating short-lived perceptual codes, spatial information may be coded in working memory as part of the context associated with stimulus events. When reactivated by cues from the original event, such information may influence response selection and produce spatial congruence effects (in this case, positive when participants made a “yes” response and negative when they made a “no” response).     

Auditory information is beneficial for adults with Down syndrome in a continuous bimanual task

Much recent research using discrete unimanual tasks has indicated that individuals with Down syndrome (DS) have more difficulty performing verbal-motor tasks as compared to visual-motor tasks (see Perceptual-Motor Behavior in Down Syndrome, Human Kinetics, Champaign, IL, 2000, p. 305 for a review). In continuous tasks, however, individuals with DS perform better when movement is guided by auditory information compared to visual information (Downs Syndr.: Res. Prac. 4 (1996) 25; J. Sport Exercise Psy. 22 (2000) S90). The aim of the present study was to investigate if there are any differences for adults with DS between visual, auditory and verbal guidance in a continuous bimanual task. Ten adults with DS, 10 adults without DS and 10 typically developing children drew lines bimanually towards the body (down) and away from the body (up) following three different guidance conditions: visual (flashing line), auditory (high tone, low tone), and verbal (“up”, “down”). All participants produced mostly in-phase movements and were close to the 1000 ms target time for all guidance conditions. The adults with DS, however, displayed greater variability in their movement time, movement amplitude and bimanual coordination than adults without DS. For all groups, the left hand was slower and more variable in producing the lateral movements than the right hand. The results regarding guidance information suggest that auditory information is beneficial for repetitive bimanual tasks for adults with DS. Possible mechanisms that cause these results will be discussed.     

Selective effects of an anaesthetic drug on cognitive behaviour

The prediction has been tested that under the influence of an anaesthetic drug, nitrous oxide, cognitive performances undergo differential impairment, the extent of which is positively correlated with the “complexity” of the task, Ten kinds of performance were investigated, ranging from speed of finger tapping to reasoning by analogy. The relative complexity of each task was determined, in accordance with conventional criteria, from its respective qualitative category or “level”—relational, associative, and motor—and within each category from qualitative analysis of the component processes involved in its execution. A simple group difference design was used, involving two groups of 50 subjects each, matched for age and sex.

Significant deterioration as a consequence of drug administration occurred in the performance of all tasks. On the whole, the more complex a task the more did it tend to be impaired. Motor performances were, however, impaired to a greater extent than had been predicted. The possible significance of these findings is discussed.