# Explicit substitutions

## Summary (2 min read)

### Introduction

- Knowledge management (KM) represents the process of effectively capturing, documenting, assimilating, sharing, and deploying organizational knowledge [13,16].
- Focused aggregation of such knowledge to maximize the organizational objectives is critical for the efficient and effective functioning of any enterprise [16].
- A main challenge for companies is the reluctance of their knowledge experts to share their intellectual capital [13,21].
- While KM systems provide the information technology to store, retrieve, and share knowledge, users often lack the motivation to engage with them [30].

### CLEVER: A gameful KM system

- CLEVER is an online KMS that incorporates game elements.
- The system is composed of two parts: (1) an online knowledge repository, where employees can provide important knowledge to the company, and (2) a trivia strategy game that motivates players to interact with content from the knowledge repository.
- Next, the authors describe this game, its implementation, and the exploratory study they conducted to test the prototype of the learning game component.

### Game Description

- Inspired by traditional board games such as Risk [22], Antike II [24], and Diplomacy [12], CLEVER is a strategic, turn-based trivia game in a digital play space.
- The game can be played by a minimum of two and a maximum of four players who compete against each other on a single digital map, constructed from tiles (see Figure 1).
- The collected energy is used to perform an action on a unit as part of the action phase which follows the trivia phase.
- Units represented as a token on the map are present as different types of units – archer, fighter, and tank.
- CLEVER’s game interface (see Figure 3) features panels for each player showing the player’s username, race, stars, energy, domination points, a number of units, and available actions.

### Motivational Elements

- Trivia questions trigger player interaction with knowledge from the repository, which fosters learning.
- CLEVER facilitates the players’ intrinsic and extrinsic motivation, as suggested by self-determination theory [26,28] in the following manner: Competence: Players receive immediate feedback after answering a question correctly, in the form of energy and stars, which helps them feel competent.
- Players can freely choose which units they will use as well as the category of questions they will answer on each round, also known as Autonomy.
- Players can play together with peers from their company, to establish a social connection which provides the feeling of relatedness, also known as Relatedness.
- Additionally, performing actions can be seen as a reward for answering questions during the trivia phase.

### Evaluation

- The authors conducted an exploratory focus group study to gather players’ thoughts, experiences, and motivations to use CLEVER.
- While interest-enjoyment, perceived competence, perceived choice, and pressure-tension are the main categories of the Task Evaluation Questionnaire from the Intrinsic Motivation Inventory (IMI) [27], due to the nature of playing the game in groups and the learning objective of their system, the authors also added questions for the following categories: relatedness, perceived learning, and extrinsic motivation.
- Nine participants (four females, five males), aged 22– 46 years (M=28 years), who were employees of neusta software development GmbH, played the game in a conference room arranged as shown in Figure 4.
- The authors then conducted a deductive analysis of the focus group sessions using a standardized form with the categories of the IMI.
- Finally, the authors compared the clustered items from the three researchers for reliability and collated the results into a single document.

### Results

- The authors analyzed the focus groups’ answers to identify insights related to participants’ motivation to interact with knowledge through the game.
- Experimental setup for the exploratory focus group study.
- Participants felt that the game would be better to learn smaller things or to recap content they already knew rather than to learn something new and complex (G1 P2; G2 P3; G3 P3; G3 P2), also known as Perceived Learning.

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### "Explicit substitutions" refers background in this paper

...(The problem arises in full generality with dependent types [14], and some readers may find it helpful to think about calculi of substitutions with dependent types....

[...]

919 citations

772 citations

### "Explicit substitutions" refers methods in this paper

...By a well-known lemma, local confluence suffices [11]; it can be checked by examining critical pairs, according to the Knuth-Bendix test....

[...]

662 citations

##### Related Papers (5)

##### Frequently Asked Questions (15)

###### Q2. What is the term of the form a[s]?

If s represents the infinite substitution {a1/1, a2/2, a3/3, . . .}, the authors write a[s] for a with the substitution s. A term of the form a[s] is called a closure.

###### Q3. What is the basic strategy for typechecking a term?

In order to typecheck a term a[s], the basic strategy is to analyze simpler and simpler components of a while accumulating more and more complex substitutions in s.

###### Q4. What is the effect of decrementing a?

all the remaining free indices in a must be decremented; the desired effect is obtained with an infinite substitution:(λx.a)b →β a{b/x} becomes (λa)b →β a{b/1, 1/2, 2/3, . . .

###### Q5. What can be removed from the rule Clos?

The inference rules = s′ t = t′s ◦ t = s′ ◦ t′can be removed, and the inference rule for the closure operator can be restricted tos = s′1[s] = 1[s′]

###### Q6. What is the role of the calculus in the study of substitutions?

The calculus is a vehicle in designing, understanding, verifying, and comparing implementations of the λ-calculus, from interpreters to machines.

###### Q7. What are other applications of the calculus?

Other applications are in the analysis of typechecking algorithms for higher-order languages and, potentially, in the mechanization of logical systems.

###### Q8. What is the advantage of the -calculus over the combinator calculi?

¿From their perspective, the advantage of the λσ-calculus over combinator calculi is that it remains closer to the original λ-calculus.

###### Q9. What is the alternative solution to the first-order calculus?

An alternative (but heavy) solution would be to have separate index sets for ordinary term variables and for type variables, and to manipulate separate term and type environments as well.

###### Q10. What does the theory of S2 mean?

The theory S2 is formulated with equivalence judgments, for example judgments of the form E ` a ∼ b : A. This judgment means that in the environment E the terms a and b both have type A and are equivalent.

###### Q11. What is the definition of a weak head normal form?

In their setting, weak head normal forms are defined as follows:Definition 3.7 A weak head normal form (whnf for short) is a λσ term of the form λa or na1 · · · am.

###### Q12. What is the invariant for the relation between environment lengths and substitutions sizes?

(The precise relation between environment lengths, and substitutions sizes, as defined in section 2, obeys the invariant: if E ` s substp and | s | = (m,n) then p = m + | E | − n ≥ 0.)

###### Q13. What is the syntax for the typed first-order -calculus?

For the typed first-order λσ-calculus, the syntax becomes:Types A ::= K |A → B Environments E ::= nil |A,E Terms a ::= 1 | ab |λA.a | a[s]

###### Q14. What is the syntax of the first-order typed -calculus?

The first-order λσ-calculus has the following syntax:Types A ::= K |A → B Environments E ::= nil |A,E Terms a ::= 1 | ab |λA.a | a[s]

###### Q15. What is the correct technique for the version of calculi with explicit substitutions?

The rule for application takes this into account; a substitution is applied to B to “unshift” its indices:E ` b : A → B E ` a : AE ` b(a) : B[a:A · id ]The B[a:A · id ] part is reminiscent of the rule found in calculi for dependent types, and this is the correct technique for the version of such calculi with explicit substitutions.