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b/semantics-1/semantics.typ index e69de29..954bf83 100644 --- a/semantics-1/semantics.typ +++ b/semantics-1/semantics.typ @@ -0,0 +1,197 @@ +// ============================================================ +// Operational Semantics for BoolInt* Language — Typst version +// ============================================================ + +// ----- page setup (fullpage equivalent) --------------------- +#set page(margin: 1in) +#set text(size: 11pt) +#set par(justify: true) + +// ----- helper functions ------------------------------------- + +// Language keywords rendered in monospace +#let kw(body) = text(font: "DejaVu Sans Mono", size: 0.85em, body) + +#let tr = kw[true] +#let fl = kw[false] +#let ife(e1, e2, e3) = [#kw[if] #e1 #kw[then] #e2 #kw[else] #e3] +#let suc(e) = [#kw[succ] #e] +#let prd(e) = [#kw[pred] #e] + +// Big-step evaluation relation: e ⇓ v +#let bstep(e, v) = $#e arrow.b.double #v$ + +// Rule name label in small-caps +#let rel(name) = text(size: 0.9em, smallcaps[\[#name\]]) + +// Big-step rule: fraction with name on the left +// premises (content), conclusion (content) +#let bsrule(name, premises, conclusion) = { + grid( + columns: (auto, 1fr), + column-gutter: 1em, + align: (right + horizon, left + horizon), + rel(name), + math.equation(block: true, numbering: none, + math.frac(premises, conclusion) + ), + ) + v(0.4em) +} + +// Definition head: "e ::= ..." with label +#let mydefhead(lhs, label) = { + grid( + columns: (1fr, auto), + [#lhs], + emph(label), + ) +} + +// Definition case: indented alternative with description +#let mydefcase(alt, desc) = { + grid( + columns: (2em, 1fr, auto), + [], [#alt], [#desc], + ) +} + +// ----- title block ------------------------------------------ + +#align(center)[ + #text(size: 17pt, weight: "bold")[Operational Semantics for BoolInt\* Language] + + #v(0.8em) + + Yuri Tatishchev \ + San José State University \ + iurii.tatishchev\@sjsu.edu +] + +#v(1em) + +// ----- abstract --------------------------------------------- + +#align(center)[ + #block(width: 85%)[ + #set text(size: 10pt) + *Abstract.* In this paper, we will provide a review of the big-step operational semantics + for the BoolInt\* language we discussed in class. + ] +] + +#v(0.5em) + +// ----- body text -------------------------------------------- + +BoolInt\* is a very minimal language that allows us to experiment with operational semantics. + +First, we define the valid expressions in our language. +These expressions dictate the possibilities of what expressions +we may have in our source programs. +(Note that other language might also have _statements_. +Statements might not evaluate to a value; +expressions always will.) + +// ----- Figure 1: language definition ------------------------ + +#figure( + block(inset: (left: 2em))[ + #set align(left) + #mydefhead[$e ::=$ #h(6em)][Expressions] + #mydefcase(tr, [true value]) + #mydefcase(fl, [false value]) + #mydefcase($i$, [integers]) + #mydefcase(ife[$e$][$e$][$e$], [conditional expressions]) + #mydefcase(suc[$e$], [successor]) + #mydefcase(prd[$e$], [predecessor]) + + #v(0.5em) + + #mydefhead[$v ::=$ #h(6em)][Values] + #mydefcase(tr, [true value]) + #mydefcase(fl, [false value]) + #mydefcase($i$, [integers]) + ], + caption: [The BoolInt\* language], +) + +@fig:lang shows the list of expressions and values for the BoolInt\* language. +Expressions can be the value $#tr$, the value $#fl$, +or the conditional expression $#ife[$e$][$e$][$e$]$. +Note that conditional expressions have a recursive structure, +with 3 sub-expressions. + +After evaluating a program, we should be able to produce a value in this language. +(In other languages, we might hit a bad situation and need to crash instead; +that won't happen in this language.) +The valid values for BoolInt\* are $#tr$, $#fl$, and $i$. + +With our expressions and values defined, +we can now specify the semantics for our language. +To do so, we will use the following big-step evaluation relation: + +$ #bstep[$e$][$v$] $ + +The above line should be read as "the expression $e$ evaluates to the value $v$". + +// ----- Figure 2: big-step semantics ------------------------- + +#figure( + block(inset: (left: 1em))[ + #set align(left) + #text(weight: "bold")[Evaluation Rules:#h(1em)]#box(stroke: 0.5pt, inset: 4pt, $#bstep[$e$][$v$]$) + + #v(0.6em) + + #bsrule("B-Value", + $emptyset$, + $#bstep[$v$][$v$]$, + ) + + #bsrule("B-IfTrue", + $#bstep[$e_1$][$#tr$] #h(2em) #bstep[$e_2$][$v$]$, + $#bstep[#ife[$e_1$][$e_2$][$e_3$]][$v$]$, + ) + + #bsrule("B-IfFalse", + $#bstep[$e_1$][$#fl$] #h(2em) #bstep[$e_3$][$v$]$, + $#bstep[#ife[$e_1$][$e_2$][$e_3$]][$v$]$, + ) + + #bsrule("B-Succ", + $#bstep[$e$][$i$]$, + $#bstep[#suc[$e$]][$i + 1$]$, + ) + + #bsrule("B-Pred", + $#bstep[$e$][$i$]$, + $#bstep[#prd[$e$]][$i - 1$]$, + ) + ], + caption: [Big-step semantics for BoolInt\*], +) + +@fig:bigstep shows the big-step evaluation rules for the BoolInt\* language. +Of course, there are additional possible rules. + +The #rel("B-Value") rule applies when the expression (to the left of "$arrow.b.double$") +is also a value, as defined in @fig:lang. +There are no premises for this rule (above the line), +meaning that it is an _axiom_. +This rule states that a value evaluates to itself, +so that #tr evaluates to #tr and #fl evaluates to #fl. + +Two different rules are needed for handling conditional expressions. +Which rule applies depends on the premises. + +For the #rel("B-IfTrue") rule, +the premise states that $e_1$ evaluates to #tr +and $e_2$ evaluates to some value $v$. +If the premise holds, then the result of evaluating the expression is the value $v$. +The structure of #rel("B-IfFalse") is similar. + +We also have rules for handling the integer operations. +The #rel("B-Succ") rule states that if the operand $e$ evaluates to an integer $i$, +then the expression $#suc[$e$]$ evaluates to $i + 1$. +Similarly, #rel("B-Pred") evaluates $#prd[$e$]$ to $i - 1$ if $e$ evaluates to $i$.