On Two Notions of Computation in Transparent Intensional Logic

Abstract

In Transparent Intensional Logic we can recognize two distinct notions of computation that loosely correspond to term rewriting and term interpretation as known from lambda calculus. Our goal will be to further explore these two notions and examine some of their properties.

Appendix

1.1 Constructions

The traditional definition of TIL constructions goes as follows (original formulation by Tichý 1988, we follow Duží et al. 2010 with minor deviations):¹⁷

Definition 5

(Constructions)

• 1. The variablex is a construction that constructs an object O of the respective type dependently on a valuation. We say that it v-constructs O.

• 2. Where X is any object, $^0 X$ is the construction trivialization. It constructs X without any change.

• 3. The composition$[X_0 \, X_1 \ldots X_m]$ is the following construction. If Xv-constructs a function f of type $(\alpha \beta _1 \ldots \beta _m)$ and $X_1 \ldots X_m$v-construct objects $b_1 , \ldots , b_m$ of types $\beta _1 , \ldots , \beta _m$, respectively, then the composition$[X_0 \, X_1 \ldots X_m]$v-constructs the value (an object of type $\alpha$, if any) of f on the tuple-argument $\langle b_1 , \ldots ,b_m \rangle$. Otherwise, it is a v-improperconstruction, i.e., construction that does not construct anything.

• 4. The closure$[\lambda x_1 \ldots x_m \, Y]$ is the following construction. Let $x_1 , \ldots x_m$ be pairwise distinct variables v-constructing objects of types $\beta _1 , \ldots , \beta _m$ and Y a construction v-constructing an object of type $\alpha$. Then $[\lambda x_1 \ldots x_m \, Y]$ is the constructionclosure. It v-constructs the following function f of type $(\alpha \beta _1 \ldots \beta _m)$: let $\langle b_1, \ldots , b_m \rangle$ be a tuple of objects of types $\beta _1 \ldots \beta _m$, respectively, and $v'$ be a valuation that associates $x_i$ with $b_i$ and is identical to v otherwise. Then the value of function f on argument tuple $\langle b_1, \ldots , b_m \rangle$ is the object of type $\alpha$$v'$-constructed by Y. If Y is $v'$-improper, then f is undefined on $\langle b_1, \ldots , b_m \rangle$.

• 5. The single execution$^1 X$ is the construction that either v-constructs the object v-constructed by X or, if Xv-constructs nothing, is v-improper.

• 6. The double execution$^2 X$ is the following construction: let X be any object, the double execution$^2 X$ is v-improper if X is a non-construction or if X does not v-construct a construction or if Xv-constructs a v-improper construction. Otherwise, let Xv-construct a construction $X'$ and let $X'$v-construct and object $X''$, then $^2 K$v-constructs $X''$.

• 7. Nothing other is a construction, unless it follows from 1–6.

1.2 Ramified Type Theory

We follow the specification from Tichý (1988):

Definition 6

(Ramified type theory of TIL)

Let B be a base, i.e., a set of atomic types.

• 1.

  • ($\hbox{t}_1$i) Every member of B is a type of order 1 over B.

  • ($\hbox{t}_1$ii) If $0<m$ and $\alpha , \beta _1 \ldots \beta _m$ are types of order 1 over B, then the collection $(\alpha \beta _1 \ldots \beta _m)$ of all m-ary (total and partial) mappings from $\beta _1 \ldots \beta _m$ to $\alpha$ is also a type of order 1 over B.

  • ($\hbox{t}_1$iii) Nothing is a type of order 1 over B unless it follows from ($\hbox{t}_1$i) and ($\hbox{t}_1$ii).

• 2.

  • ($\hbox{c}_k$i) Let $\alpha$ be any type of orderkover B. Every variable ranging over $\alpha$ is a construction of order k over B. If X is of (i.e., belongs to) type $\alpha$, then $^0 X$, $^1 X$, and $^2 X$ are constructions of order k over B. Every variable ranging over $\alpha$ is a construction of orderkover B.

  • ($\hbox{c}_k$ii) If $0<m$ and $X_0, X_1, \ldots , X_m$ are constructions of orderk, then $[X_0 \; X_1 \ldots \, X_m]$ is a construction of order k over B. If $0<m$, $\alpha$ is a type of orderkover B, and Y as well as the distinct variables $x_1, \ldots , x_m$ are constructions of orderkover B, then $[\lambda _\alpha \, x_1 \ldots x_m \; Y]$ is a construction of orderkover B.

  • ($\hbox{c}_k$iii) Nothing is a construction of orderkover B unless it follows from ($\hbox{c}_k$i) and ($\hbox{c}_k$ii).

Let $*_k$ be the collection of constructions of order k over B. The collection of types of order$k+1$over B is defined as follows:

• ($\hbox{t}_{k+1}$i) $*_n$ and every type of order n is a type of order$k+1$.

• ($\hbox{t}_{k+1}$ii) If $0<m$ and $\alpha , \beta _1 , \ldots , \beta _m$ are types of order$k+1$over B, then the collection $(\alpha \beta _1 \ldots \beta _m)$ of all m-ary (total and partial) mappings from $\beta _1 , \ldots , \beta$ to $\alpha$ is also a type of order$k+1$over B.

• ($\hbox{t}_{k+1}$iii) Nothing is a type of order$k+1$over B unless it follows from ($\hbox{t}_{k+1}$i) and ($\hbox{t}_{k+1}$ii).

1.3 Collisionless Substitution

Originally defined by Duží et al. (2010):

Definition 7

(Collisionless substitution) Let x be a variable and C, D any kinds of construction. If x is not free in C, then the result of substitutingDforxinC is C. Assume now that x is free in C. Then:

• (a) If C is x, then the result of substitutingDforxinC is D. If C is $^1 X$ or $^2 X$, then the result of substitutingDforxinC is $^1 Y$ or $^2 Y$, where Y is the result of substituting D for x in X.

• (b) If C is $[X \, X_1 \ldots X_m]$, then the result of substitutingDforxinC is $[Y \, Y_1 \ldots Y_m]$, where $Y , Y_1 , \ldots , Y_m$ are the results of substituting D for x in $X , X_1 , \ldots , X_m$, respectively.

• (c) Let C be of the form $[\lambda x_1 \ldots x_m \, Y]$; for $1 \le i \le m$, let $y_i = x_i$ if $x_i$ is not free in D, and otherwise the first variable v-constructing entities of the same type as $x_i$, not occurring in C, not free in D, and distinct from $y_1 , \ldots , y_{i+1}$. Then the result of substitutingDforxinC is $[\lambda y_1 \ldots y_m \, Z]$, where Z is the result of substituting D for x in the result of substituting $y_i$ for $x_i(1 \le i \le m)$ in Y.

To simplify the notation, let $C, D_1 , \ldots , D_n$ be arbitrary constructions, $x_1 , \ldots , x_n$ variables. Then, for $1 \le i \le n$, $C(D_i/x_i)$ will represent the result of substituting $D_i$ for $x_i$ in C.

1.4 $\beta$-Reduction

We follow the specification of $\beta$-reduction from Duží (2017):¹⁸

Definition 8

($\beta$-reduction) Let C be a construction typed to v-construct object of type $\alpha$, $x_1$ and $D_1$ constructions typed to v-construct objects of type $\beta _1$, ..., $x_n$ and $D_n$ constructions typed to v-construct objects of type $\beta _n$ and $[\lambda x_1 \ldots x_n \, C]$ is typed to v-construct object of type $(\alpha \beta _1 \ldots \beta _n )$, then

$$\begin{aligned}{}[[\lambda x_1 \ldots x_n \, C] \; D_1 \ldots D_n] \;\; {\longrightarrow _{\beta }} \;\; C(D_1 / x_1 \ldots D_n / x_n) \end{aligned}$$

where $C(D_1 / x_1 \ldots D_n / x_n)$ is the construction that arises from C by collisionless substitution of $D_1$ for all the occurrences of $x_1$, ..., $D_n$ for all the occurrences of $x_n$.

1.5 n-Execution

In this section, we introduce the construction of multiple execution we will call n-execution (for $n \ge 0$), which is a generalization of TIL’s native constructions execution and double execution (see Sect. 1 above). Further, we will show that trivialization can be understood as its limiting case when $n = 0$. Our generalization is based on the following three simple observations that keep reappearing in TIL literature:

Observation 1

Intuitively, if we have single execution and double execution, then we should be able to consider even triple execution, quadruple execution and so on. And if that is the case, then we can generalize this observation and introduce the notion of n-execution where n is the number of consecutive executions.¹⁹

Observation 2

If we should understand $^1 X$ as ‘execute X’ (i.e., ‘$^1$’ = one execution) and $^2 X$ as ‘execute X and then execute its result $X'$’ (i.e., ‘$^2$’ = two executions), then, arguably the most natural reading of $^0 X$—if we have never heard of trivialization—is ‘do not execute $X'$ (i.e., ‘$^0$’ = zero executions).²⁰

Observation 3

Multiple execution for $n > 2$ is reducible to iterated double execution.²¹

Definition 9

(n-execution) For any object X we shall speak of the n-execution of X and symbolize it as $^n X$, where $n \ge 0$.

• 1. If $n = 0$, then $^0 X$ is X (where X is either a construction or a non-construction). More specifically, to carry out $^0 X$, we start with X and do nothing further with it, not even valuation. It is never v-improper.

• 2. If $n \ge 1$, there is a need to distinguish two main cases (the second has three further subcases):

  • (a) X is a non-construction: then $^n X$ is v-improper construction.

  • (b) X is a construction:

    • i. n = 1: then ⁿXv-constructs the same object as does X (if any).

    • ii. n = 2: then ⁿXv-constructs the object v-constructed by X′ (assuming ⁿXv-constructs X′). Otherwise, it is v-improper.

    • iii. n > 2: then ⁿXv-constructs the same object (if any) as does ^sX where s is a sequence of n − 1 iterated n-executions with n = 2. For example, ³X can be reduced to ²(²X).

From the type-theoretical perspective, if X is of (i.e., belongs to) some type $\alpha$ of order k over B, then $^n X$ (where $n \ge 0$) is a construction of orderkover B.²²

To get a better feeling of how n-execution works for cases with $n > 1$, consider the following cases:

• 1. $n = 2$: same as Tichý’s original definition above.

• 2. $n = 3$: analogously to the case above, but the construction $X'$v-constructed by X has to be v-constructing another (proper) construction $X''$, otherwise it is v-improper.

• 3. $n = 4$: analogously to the case above, but $X''$ has to be v-constructing yet another (proper) construction $X'''$, otherwise it is v-improper.

  ...  etc.

Specific instances of n-execution (for $n = 0, 1 , 2, \ldots$) will be called 0-execution, 1-execution, 2-execution, etc. For example, $^0 5$ is an example of 0-execution, $^1 x$ is an example of 1-execution, $^2 5$ is an example of 2-execution, etc.

To conclude, on this approach, trivialization, execution, and double execution are just specific instances of n-execution for some $n \ge 0$, with trivialization being an alternative name for the degenerate case when $n = 0$ (informally: ‘do not execute’, ‘do nothing’ or—as Tichý put it—‘leave it as it is’ (Tichý 1988, p. 63).

Remark 7

The case of $n = 0$ is a special one also from another point of view – the appearance of ‘$^0$’, i.e., 0-execution, binds free variables in contrast to 1-execution, 2-execution, etc. (see e.g., Duží et al. 2010, p. 47). As we mentioned, 0-execution $^0 X$ is essentially an instruction to ‘do nothing with X’ and by binding the free variables we are making sure that nothing is indeed done with X, not even valuation or substitution.