Additional notes on assignment and textual substitution.
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"fields_dict": {
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@ -8,4 +8,6 @@ title: "2024-07-20"
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- [ ] Sheet Music (10 min.)
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- [ ] Korean (Read 1 Story)
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* Begin notes on [[hashing/index#Random Hashing|random hashing]] and, more specifically, [[hashing/index#Universal Hashing|universal hashing]].
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* Begin notes on [[hashing/index#Random Hashing|random hashing]] and, more specifically, [[hashing/index#Universal Hashing|universal hashing]].
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* Notes on generalized [[equiv-trans#General|textual substitution]] and [[pred-trans#General|assignment]].
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* Finished chapter 9 "Assignment Command" of "The Science of Programming".
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@ -591,6 +591,30 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
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<!--ID: 1714395640953-->
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END%%
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%%ANKI
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Basic
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Consider selector update syntax. Is precedence left-to-right or right-to-left?
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Back: Right-to-left.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721497014090-->
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END%%
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%%ANKI
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Basic
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What does it mean for selector update syntax to have right-to-left precedence?
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Back: Rightmost selectors overwrite duplicate selectors.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721497014094-->
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END%%
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%%ANKI
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Basic
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How is $(b; s_1{:}e_1; s_2{:}e_2; s_1{:}e_3)$ simplified?
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Back: As $(b; s_2{:}e_2; s_1{:}e_3)$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721497014098-->
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END%%
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## Substitution
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**Textual substitution** refers to the replacement of a [[pred-logic#Identifiers|free]] identifier with an expression, introducing parentheses as necessary. This concept amounts to the [[#Equivalence Rules|Substitution Rule]] with different notation.
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@ -751,16 +775,150 @@ Substitution is defined recursively as follows:
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1. If each $x_i$ is a distinct identifier with a null selector, then $E_{\bar{e}}^{\bar{x}}$ is the simultaneous substitution of $\bar{x}$ with $\bar{e}$.
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2. Adjacent reference-expression pairs may be permuted as long as they begin with different identifiers. That is, for all distinct $b$ and $c$, $$\Large{E_{\bar{e}, \,f, \,h, \,\bar{g}}^{\bar{x}, \,b, \,c, \,\bar{y}} = E_{\bar{x}, \,h, \,f, \,\bar{g}}^{\bar{x}, \,c, \,b, \,\bar{y}}}$$
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3. Multiple assignments to subparts of an object $b$ can be viewed as a single assignment to $b$. That is, provided $b$ does not begin any of the $x_i$, $$\Large{E_{e_1, \,\ldots, \,e_m, \,\bar{g}}^{b \circ s_1, \,\ldots, \,b \circ s_m, \,\bar{x}} = E_{(b; \,s_1{:}e_1; \,\cdots; \,s_m{:}e_m), \,\bar{g}}^{b, \,\bar{x}}}$$
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3. Multiple assignments to subparts of an object $b$ can be viewed as a single assignment to $b$. That is, provided $b$ does not begin any of the $x_i$, $$\Large{E_{e_1, \,\ldots, \,e_m, \,\bar{g}}^{b \,\circ\, s_1, \,\ldots, \,b \,\circ\, s_m, \,\bar{x}} = E_{(b; \,s_1{:}e_1; \,\cdots; \,s_m{:}e_m), \,\bar{g}}^{b, \,\bar{x}}}$$
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Note that simultaneous substitution is different from sequential substitution.
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TODO
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%%ANKI
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Basic
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Consider $E_{\bar{e}}^{\bar{x}}$. What is each $x_i$ in $\bar{x}$?
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Back: An identifier concatenated with a selector.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879842-->
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END%%
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%%ANKI
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Basic
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Consider $E_{\bar{e}}^{\bar{x}}$. What is each $e_i$ in $\bar{e}$?
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Back: An expression.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879845-->
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END%%
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%%ANKI
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Basic
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What is the base case in the evaluation of $E_{\bar{e}}^{\bar{x}}$?
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Back: If $\bar{x}$ are distinct identifiers with null selectors, direct simultaneous substitution.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879846-->
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END%%
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%%ANKI
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Basic
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Which of $E_{\bar{e}}^{\bar{x}}$'s reference-expression pairs may be moved?
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Back: Adjacent pairs with distinct identifiers.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879847-->
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END%%
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%%ANKI
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Basic
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When is $b_1 \circ s_1$ and $b_2 \circ s_2$ said to have distinct identifiers?
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Back: When $b_1 \neq b_2$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879848-->
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END%%
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%%ANKI
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Basic
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When is $b_1 \circ s_1$ and $b_2 \circ s_2$ said to have distinct selectors?
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Back: When $s_1 \neq s_2$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879850-->
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END%%
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%%ANKI
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Basic
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Suppose $x$ and $y$ are distinct. Is the following a tautology? $$\large{E_{e_1, e_2}^{x, y} = E_{e_2, e_1}^{y, x}}$$
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Back: Yes.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879851-->
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END%%
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%%ANKI
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Basic
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When is the following a tautology? $$\large{E_{e_1, e_2}^{x, y} = E_{e_2, e_1}^{y, x}}$$
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Back: When $x$ and $y$ refer to distinct identifiers.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879852-->
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END%%
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%%ANKI
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Basic
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Suppose $x = y$. When is the following a tautology? $$\large{E_{e_1, e_2}^{x, y} = E_{e_2, e_1}^{x, y}}$$
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Back: When $e_1 = e_2$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879853-->
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END%%
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%%ANKI
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Basic
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Suppose $x$, $y$, and $z$ are distinct. What is the result of a single evaluation step? $$\large{E_{e_1, e_2, e_3}^{x, y, z}}$$
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Back: N/A.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879854-->
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END%%
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%%ANKI
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Basic
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Suppose $x \neq y$. What is the result of a single evaluation step? $$\large{E_{e_1, e_2, e_3}^{x, y, x}}$$
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Back: $$\large{E_{e_1, e_3, e_2}^{x, x, y}}$$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879855-->
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END%%
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%%ANKI
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Basic
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Suppose $x \neq y$. What is the result of a single evaluation step? $$\large{E_{e_1, e_3, e_2}^{x, x, y}}$$
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Back: $$\large{E_{(x; \,\epsilon{:}e_1; \,\epsilon{:}e_3), e_2}^{x, y}}$$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879856-->
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END%%
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%%ANKI
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Basic
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Suppose $x \neq y$. What is the result of a single evaluation step? $$\large{E_{(x; \,\epsilon{:}e_1; \,\epsilon{:}e_3), e_2}^{x, y}}$$
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Back: $$\large{E_{e_3, e_2}^{x, y}}$$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879857-->
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END%%
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%%ANKI
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Basic
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Suppose $x \neq y$. *Why* isn't the following a tautology? $$\large{E_{e_1, e_2, e_3}^{x, y, x}} = E_{(x; \epsilon{:}e_1), e_2, e_3}^{x, y, x}$$
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Back: N/A. It is.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879858-->
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END%%
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%%ANKI
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Basic
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Suppose $x \neq y$. *Why* isn't the following a tautology? $$\large{E_{e_1, e_2, e_3, e_4}^{x, x, y, x}} = E_{(x; \epsilon{:}e_1; \epsilon{:}e_2), e_3, e_4}^{x, y, x}$$
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Back: Because not every $x$ was made adjacent before grouping.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879859-->
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END%%
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%%ANKI
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Basic
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Consider array $b$ and $i \in \mathop{domain}(b)$. What is the result of a single evaluation step? $$\large{E_{e}^{b[i]}}$$
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Back: $$\large{E_{(b; [i]{:}e)}^{b}}$$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879860-->
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END%%
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%%ANKI
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Basic
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Consider identifier $x$, array $b$ and $i \in \mathop{domain}(b)$. What is the result of a single evaluation step? $$\large{E_{b[i]}^{x}}$$
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Back: N/A.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721495879861-->
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END%%
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### Theorems
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* $(E_u^x)_v^x = E_{u_v^x}^x$
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* The only possible free occurrences of $x$ that may appear after the first of the sequential substitutions occur in $u$.
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* The only possible free occurrences of $x$ that may appear after the first of the substitutions occur in $u$.
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%%ANKI
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Basic
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@ -780,13 +938,13 @@ END%%
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%%ANKI
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Basic
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Why is $(E_u^x)_v^x = E_{u_v^x}^x$ an equivalence?
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*Why* is $(E_u^x)_v^x = E_{u_v^x}^x$ an equivalence?
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Back: After the first substitution, the only possible free occurrences of $x$ are in $u$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707762304146-->
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END%%
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* If $y$ is not free in $E$, then $(E_u^x)_v^y = E_{u_v^y}^x$.
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* If $y \not\in FV(E)$, then $(E_u^x)_v^y = E_{u_v^y}^x$.
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* $y$ may not be free in $E$ but substituting $x$ with $u$ can introduce a free occurrence. It doesn't matter if we perform the substitution first or second though.
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%%ANKI
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@ -807,7 +965,7 @@ END%%
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%%ANKI
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Basic
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Why should $y$ not be free in $E$ for $(E_u^x)_v^y = E_{u_v^y}^x$ to be an equivalence?
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Why does $y \not\in FV(E)$ ensure $(E_u^x)_v^y = E_{u_v^y}^x$ is an equivalence?
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Back: If it were, a $v$ would exist in the LHS that doesn't in the RHS.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707762304152-->
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@ -918,7 +1076,7 @@ END%%
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%%ANKI
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Basic
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When is $(E_{\bar{u}}^{\bar{x}})_{\bar{x}}^{\bar{u}} = E$ guaranteed to be an equivalence?
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Back: When $\bar{x}$ and $\bar{u}$ refer to distinct identifiers.
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Back: When $\bar{x}$ and $\bar{u}$ refer to distinct identifiers (concatenated with selectors).
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707939006297-->
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END%%
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@ -622,6 +622,14 @@ END%%
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### Assignment
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%%ANKI
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Basic
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What equivalence transformation rule is assignment related to?
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Back: Substitution.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721497014026-->
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END%%
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#### Simple
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The assignment command has form $x \coloneqq e$, provided the types of $x$ and $e$ are the same. This command is read as "$x$ becomes $e$" and is defined as $$wp(''x \coloneqq e'', R) = domain(e) \textbf{ cand } R_e^x$$
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@ -757,7 +765,112 @@ END%%
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#### General
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TODO
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The multiple assignment command has form $$x_1 \circ s_1, \cdots, x_n \circ s_n \coloneqq e_1, \cdots, e_n$$
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where each $x_i$ is an identifier, each $s_i$ is a [[equiv-trans#Selectors|selector]], and each expression $e_i$ has the same type as $x_i \circ s_i$. We denote this assignment more compactly as $\bar{x} \coloneqq \bar{e}$. We define multiple assignment as $$wp(''\bar{x} \coloneqq \bar{e}'', R) = domain(\bar{e}) \textbf{ cand } R_{\bar{e}}^\bar{x}$$
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%%ANKI
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Basic
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How is $x \coloneqq e$ expressed in more general form $x_1 \circ s_1, \ldots, x_n \circ s_n \coloneqq e_1, \ldots, e_n$?
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Back: As $x \circ \epsilon \coloneqq e$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721497014033-->
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END%%
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%%ANKI
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Basic
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How is $b[i] \coloneqq e$ expressed in more general form $x_1 \circ s_1, \ldots, x_n \circ s_n \coloneqq e_1, \ldots, e_n$?
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Back: As $b \circ [i] \coloneqq e$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721497014038-->
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END%%
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%%ANKI
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Basic
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Consider assignment command $\bar{x} \coloneqq \bar{e}$. In what order must $\bar{e}$ be evaluated?
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Back: In any order.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721497014043-->
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END%%
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%%ANKI
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Basic
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Consider assignment command $\bar{x} \coloneqq \bar{e}$. In what order must assignment be performed?
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Back: Left-to-right.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721497014047-->
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END%%
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%%ANKI
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Basic
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*Why* must assignment in $\bar{x} \coloneqq \bar{e}$ happen left-to-right?
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Back: Because update selector syntax has right-to-left precedence.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721497014051-->
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END%%
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%%ANKI
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Basic
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Consider assignment command $\bar{x} \coloneqq \bar{e}$. When can assignment be performed in any order?
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Back: When the identifiers in $\bar{x}$ are distinct.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721497014055-->
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END%%
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%%ANKI
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Basic
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The general assignment command has what form?
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Back: $\bar{x} \coloneqq \bar{e}$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1721497014059-->
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END%%
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%%ANKI
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Basic
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How is assignment "$\bar{x} \coloneqq \bar{e}$" defined in terms of $wp$?
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Back: $wp(''\bar{x} \coloneqq \bar{e}'', R) = domain(\bar{e}) \textbf{ cand } R_{\bar{e}}^{\bar{x}}$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1721497014064-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
In the $wp$ definition of "$\bar{x} \coloneqq \bar{e}$", what does $domain(\bar{e})$ refer to?
|
||||
Back: A predicate that holds if each member of $\bar{e}$ is well-defined.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1721497014068-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
In the $wp$ definition of "$\bar{x} \coloneqq \bar{e}$", $domain(\bar{e})$ must exclude which states?
|
||||
Back: Those in which evaluation of any member of $\bar{e}$ would be undefined.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1721497014072-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What assumption is made when defining assignment as "$wp(''\bar{x} \coloneqq \bar{e}'', R) = R_{\bar{e}}^{\bar{x}}$"?
|
||||
Back: $domain(\bar{e})$, i.e. evaluation of each member of $\bar{e}$ is well-defined.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1721497014076-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How is definition "$wp(''\bar{x} \coloneqq \bar{e}'', R) = R_{\bar{e}}^{\bar{x}}$" more completely stated?
|
||||
Back: $wp(''\bar{x} \coloneqq \bar{e}'', R) = domain(\bar{e}) \textbf{ cand } R_{\bar{e}}^{\bar{x}}$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1721497014081-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Given $\bar{e} = \langle e_1, \ldots, e_n \rangle$, how is $\mathop{domain}(\bar{e})$ defined in predicate logic?
|
||||
Back: $\forall i, 1 \leq i \leq n \Rightarrow \mathop{domain}(e_i)$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1721497014085-->
|
||||
END%%
|
||||
|
||||
## Bibliography
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue