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CompilerTest.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<title>CompilerTest: Testing Vminus: QuickChick in the Large</title>
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<a href='index.html'><li class='section_name'>VMinus Development</li></a>
<a href='toc.html'><li>Table of Contents</li></a>
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<div id="main">
<h1 class="libtitle">CompilerTest<span class="subtitle">Testing Vminus: QuickChick in the Large</span></h1>
<div class="doc">
<a name="lab156"></a><h1 class="section">QuickChick and Vellvm</h1>
One might expect a compiler (and correctness proof) for a language
as simple as IMP to be relatively straightforward.
<div class="paragraph"> </div>
However, LLVM — even in its simplified Vminus form — is a
somewhat complex IR, and a faithful formalization is necessarily
somewhat involved. Moreover, when a compiler is under
development, even <i>stating</i> its correctness can be difficult, much
less proving it.
<div class="paragraph"> </div>
Fortunately, <i>testing</i> the compiler and its correctness properties
is a much simpler affair, as we will see.
<div class="paragraph"> </div>
Our target language is the simplified SSA language Vminus defined
in the other files in this directory, and we use a variant of Imp
whose names are just memory addresses which can be interpreted in
the memory of Vminus states. Imp states and Vminus memory are
hence essentially the same, and this makes it easy to state
correct compilation: after running the source program and its
compiled version, every Imp variable/address is mapped to the same
<span class="inlinecode"><span class="id" type="var">nat</span></span> by both the final Imp state and the final Vminus memory.
<div class="paragraph"> </div>
A Vminus state consists of a memory (mapping addresses to <span class="inlinecode"><span class="id" type="var">nat</span></span>),
a program counter, an environment (mapping "locals" to <span class="inlinecode"><span class="id" type="var">nat</span></span>), a
"previous" program counter, and a "previous" environment; the
latter two are used for executing phi nodes. A configuration
consists of a Vminus state and a CFG, which holds Vminus
instructions organized in basic blocks.
<div class="paragraph"> </div>
<a name="lab157"></a><h1 class="section">Vminus Review</h1>
<div class="paragraph"> </div>
To make this case study chapter self contained, we begin with a
very brief review of the relevant parts of the Vminus intermediate
language and compiler. If you're already familiar with
Vminus (e.g., from reading the other files and working exercises),
you can safely skip to the next section.
<div class="paragraph"> </div>
A Vminus program in concrete syntax vs abstract syntax looks like
this (warning: program is meant to only show syntax, and is not
meant to make sense):
<div class="paragraph"> </div>
<pre>
block #1:
%1 = add 0 0
(* binary operations can work on numeric literals, i.e. [nat] *)
%2 = add %1 5 (* also, on variables *)
%3 = sub %2 %1
%4 = store @1 %3
(* store the value held by %3 into memory at address @1 *)
</pre>
<div class="paragraph"> </div>
Looking ahead a bit:
<div class="paragraph"> </div>
<ul class="doclist">
<li> The Imp store and Vminus memory are the same;
</li>
<li> Vminus addresses and Imp variables are the same (modulo the
<span class="inlinecode"><span class="id" type="var">AId</span></span> constructor).
</li>
</ul>
Hence Imp programs have direct access to Vminus memory, and
assignment statements correspond directly to such a store
instruction.
<div class="paragraph"> </div>
Also note that %4 is redundant here; the reason is to make the
abstract syntax uniform
<div class="paragraph"> </div>
<pre>
%5 = store @1 1 (* We can store literals too *)
%6 = load @1
%7 = cbr %6 #2 #3
(* conditional branch based on value held in %6, again with %7 being
redundant; we can also branch on [nat] *)
block #2:
%8 = load @1
%9 = jmp #4 (* unconditional jump *)
block #3:
%10 = load @1
%11 = jmp #4
block #4:
%12 = phi [(#2, %8) ; (#3, %10)]
(* %12 takes the value held by %8 if control came from block #2,
or the value held by %10 if control came from #3 *)
%13 = jmp #5
block #5:
%14 = ret (* again, variable %14 is unnecessary *)
</pre>
<div class="paragraph"> </div>
In the abstract syntax, every instruction is defined as:
<div class="paragraph"> </div>
<span class="inlinecode"><span class="id" type="keyword">Definition</span></span> <span class="inlinecode"><span class="id" type="var">insn</span></span> <span class="inlinecode">:=</span> <span class="inlinecode">(<span class="id" type="var">uid</span></span> <span class="inlinecode">*</span> <span class="inlinecode"><span class="id" type="var">cmd</span>)%<span class="id" type="var">type</span>.</span>
<div class="paragraph"> </div>
In the above, the <span class="inlinecode"><span class="id" type="var">uid</span></span> is the number on the left of the equal sign,
and <span class="inlinecode"><span class="id" type="var">cmd</span></span> is defined as follows:
<div class="paragraph"> </div>
<pre>
Inductive cmd : Set :=
| cmd_bop : bop -> val -> val -> cmd cmd_phi : list phiarg -> cmd
| cmd_tmn : tmn -> cmd cmd_load : addr -> cmd cmd_store : addr ->
| val -> cmd.
</pre>
<div class="paragraph"> </div>
Each individual command is either a binary operation like add, a
phi-node that joins a list of blocks, a terminator, a load, or a
store; all of these are illustrated in the program above.
<div class="paragraph"> </div>
Moreover, a program is represented as a control flow graph (see
<a href="ListCFG.html"><span class="inlineref">ListCFG</span></a> for implementation and \CHAP{CFG] for the interface).
A CFG is just a list of blocks, with one of
them distinguished (via <span class="inlinecode"><span class="id" type="var">lbl</span></span>) as the entry block.
<div class="paragraph"> </div>
<pre>
Definition t := (lbl * list block)%type.
Local Notation cfg := t.
</pre>
<div class="paragraph"> </div>
A program counter is a block label <span class="inlinecode"><span class="id" type="var">lbl</span></span> with an offset n within the
block.
<div class="paragraph"> </div>
<span class="inlinecode"></span> <span class="inlinecode"><span class="id" type="keyword">Definition</span></span> <span class="inlinecode"><span class="id" type="var">pc</span></span> <span class="inlinecode">:=</span> <span class="inlinecode">(<span class="id" type="var">lbl</span></span> <span class="inlinecode">*</span> <span class="inlinecode"><span class="id" type="var">nat</span>)%<span class="id" type="var">type</span>.</span> <span class="inlinecode"></span>
<div class="paragraph"> </div>
A block has its label <span class="inlinecode"><span class="id" type="var">lbl</span></span> and contains a list of instructions.
<div class="paragraph"> </div>
<span class="inlinecode"></span> <span class="inlinecode"><span class="id" type="keyword">Definition</span></span> <span class="inlinecode"><span class="id" type="var">block</span></span> <span class="inlinecode">:=</span> <span class="inlinecode">(<span class="id" type="var">lbl</span></span> <span class="inlinecode">*</span> <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">insn</span>)%<span class="id" type="var">type</span>.</span> <span class="inlinecode"></span>
<div class="paragraph"> </div>
As the concrete syntax shows, there are 3 forms of identifiers, and
they are all implemented as Atoms with the fresh interface.
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">uid</span></span>: The % prefix, e.g. %7.
</li>
<li> <span class="inlinecode"><span class="id" type="var">addr</span></span>: The @ prefix, e.g @1.
</li>
<li> <span class="inlinecode"><span class="id" type="var">lbl</span></span>: The prefix, e.g. 3. A potential point of confusion would
</li>
</ul>
be that <span class="inlinecode"><span class="id" type="keyword">Check</span></span>, <span class="inlinecode"><span class="id" type="keyword">Print</span></span>, etc. shows <span class="inlinecode"><span class="id" type="var">addr</span></span> instead of <span class="inlinecode"><span class="id" type="var">lbl</span></span> or <span class="inlinecode"><span class="id" type="var">uid</span></span>, because
<span class="inlinecode"><span class="id" type="var">addr</span></span> is last-defined.
<div class="paragraph"> </div>
<a name="lab158"></a><h1 class="section">Overview of the Compiler</h1>
<div class="paragraph"> </div>
<div class="paragraph"> </div>
One key part of compilation to Vminus, as an SSA form, is the need to generate
fresh labels as we go along. Hence the compiler is implemented as a Writer
monad, specifically as a CMD (and EXP) following the definitions below:
<div class="paragraph"> </div>
In <span class="inlinecode"><span class="id" type="var">Classes.v</span></span>:
<div class="paragraph"> </div>
<span class="inlinecode"></span> <span class="inlinecode"><span class="id" type="keyword">Definition</span></span> <span class="inlinecode"><span class="id" type="var">ST</span></span> <span class="inlinecode">(<span class="id" type="var">M</span>:<span class="id" type="keyword">Type</span>)</span> <span class="inlinecode">(<span class="id" type="var">A</span>:<span class="id" type="keyword">Type</span>)</span> <span class="inlinecode">:=</span> <span class="inlinecode"><span class="id" type="var">M</span></span> <span class="inlinecode">→</span> <span class="inlinecode">(<span class="id" type="var">M</span></span> <span class="inlinecode">*</span> <span class="inlinecode"><span class="id" type="var">A</span>).</span>
<div class="paragraph"> </div>
In <span class="inlinecode"><span class="id" type="var">Compiler.v</span></span>:
<pre>
Notation EXP := (ST (list uid)).
Definition cstate := (list lbl * list uid * list block)%type.
Notation CMD := (ST cstate).
</pre>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="var">CMD</span></span> monad is the compiler monad at the top level, handling in particular
control flow (that is why the block labels are threaded through). The <span class="inlinecode"><span class="id" type="var">EXP</span></span>
monad is for compilation within basic blocks. A subtlety: the compiler compiles
every Imp assignment statement to its own basic block, so basic blocks in the
compilation result are not maximal in the usual sense.
<div class="paragraph"> </div>
For both, <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">uid</span></span> is the set of uids already generated thus far; similar
interpretation for <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">lbl</span></span>. The <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">block</span></span> part holds the compilation
result thus far (recall that each block contains its own block label).
<div class="paragraph"> </div>
Compilation is hence activated by running the function <span class="inlinecode"><span class="id" type="var">CMD</span></span> on appropriate
initial (empty) lists:
<div class="paragraph"> </div>
<pre>
Definition compile (c:com) : ListCFG.t * lbl * lbl :=
let '(_, _, bs, (le, lr)) := comp_prog c ([], [], []) in
((le, bs), le, lr).
Definition comp_prog (c:com) : CMD (lbl * lbl) :=
do r <- fresh_lbl;
do x <- liftF fresh;
do _ <- add_insns r [(x, cmd_tmn tmn_ret)];
do e <- comp_com c r;
ret (e, r).
</pre>
<div class="paragraph"> </div>
Looking at just <span class="inlinecode"><span class="id" type="var">compile</span></span> and the signature of <span class="inlinecode"><span class="id" type="var">comp_prog</span></span>, the compiler
instantiates the value part of the state monad <span class="inlinecode"><span class="id" type="var">A</span></span> in <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">*</span> <span class="inlinecode"><span class="id" type="var">A</span></span> as <span class="inlinecode"><span class="id" type="var">lbl</span></span> <span class="inlinecode">*</span> <span class="inlinecode"><span class="id" type="var">lbl</span></span>.
The first component is the entry (block) label that can be used to start
executing the compilation result; the second component is the label for the
block that control goes to -after- executing the compilation result.
<div class="paragraph"> </div>
Note: <span class="inlinecode"><span class="id" type="var">le</span></span> and <span class="inlinecode"><span class="id" type="var">lr</span></span> are used consistently to mean as such.
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<div class="paragraph"> </div>
This is why the CFG (i.e. compiled program) returned has its entry label as
<span class="inlinecode"><span class="id" type="var">le</span></span>, and the compilation result is just <span class="inlinecode"><span class="id" type="var">bs</span></span>.
<div class="paragraph"> </div>
For <span class="inlinecode"><span class="id" type="var">comp_prog</span></span> itself:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">fresh_label</span></span> creates a new block with no instructions in the compilation
result, gives it a fresh label, and returns this as its value <span class="inlinecode"><span class="id" type="var">r</span></span>
</li>
<li> it generates a fresh <span class="inlinecode"><span class="id" type="var">uid</span></span> (%<number>) via <span class="inlinecode"><span class="id" type="tactic">fresh</span></span>, and returns this <span class="inlinecode"><span class="id" type="var">x</span></span>
</li>
<li> it sets, in the compilation result, the block labeled <span class="inlinecode"><span class="id" type="var">r</span></span> to hold a
return terminator instruction. To see this concretely, look at
<span class="inlinecode"><span class="id" type="var">block</span></span> <span class="inlinecode">#5:</span> <span class="inlinecode">%14</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">ret</span></span> above.
</li>
<li> it compiles the program <span class="inlinecode"><span class="id" type="var">c</span></span> via <span class="inlinecode"><span class="id" type="var">comp_com</span></span>. Note that the desired
return-of-control <span class="inlinecode"><span class="id" type="var">lbl</span></span> <span class="inlinecode"><span class="id" type="var">r</span></span> is passed to it. The <span class="inlinecode"><span class="id" type="var">lbl</span></span> <span class="inlinecode"><span class="id" type="var">e</span></span> returned by <span class="inlinecode"><span class="id" type="var">comp_com</span></span>
is in turn the entry-of-control label <span class="inlinecode"><span class="id" type="var">e</span></span>.
</li>
</ul>
As the documentation for <span class="inlinecode"><span class="id" type="var">comp_com</span></span> states:
The input <span class="inlinecode"><span class="id" type="var">lr</span></span> is the "continuation" label of the block to which control
should return after the command is done. The output is the label of the
entry block for the command.
<div class="paragraph"> </div>
Note how compiling a sequence <span class="inlinecode">[<span class="id" type="var">c<sub>1</sub></span>;;<span class="id" type="var">c<sub>2</sub></span>]</span> threads the control through the
commands in reverse order.
<div class="paragraph"> </div>
<a name="lab159"></a><h1 class="section">Overview of Relevant Files</h1>
<div class="paragraph"> </div>
<ul class="doclist">
<li> <a href="Vminus.html"><span class="inlineref">Vminus</span></a> defines the syntax of Vminus.
</li>
<li> <a href="VminusOpSem.html"><span class="inlineref">VminusOpSem</span></a> defines its semantics.
<div class="paragraph"> </div>
<span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> <span class="inlinecode"><span class="id" type="var">step</span></span> <span class="inlinecode">(<span class="id" type="var">g</span>:<span class="id" type="var">Cfg.t</span>)</span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">state</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" type="var">state</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" type="keyword">Prop</span></span> summarizes everything.
A Vminus state is described below, but concretely for now:
</li>
</ul>
<div class="paragraph"> </div>
<pre>
Record state := mkst { st_mem : mem
; st_pc : pc
; st_loc : loc
; st_ppc : pc (* predecessor pc *)
; st_ploc : loc (* predecessor "locals" *)
}.
</pre>
<div class="paragraph"> </div>
<ul class="doclist">
<li> <a href="Compiler.html"><span class="inlineref">Compiler</span></a> shows the compiler.
</li>
<li> <a href="CFG.html"><span class="inlineref">CFG</span></a> is the interface for CFGs; <a href="ListCFG.html"><span class="inlineref">ListCFG</span></a> is the implementation.
</li>
<li> <a href="CompilerProp.html"><span class="inlineref">CompilerProp</span></a> holds the relevant content for what follows, and is
described in-line.
</li>
</ul>
<div class="paragraph"> </div>
<a name="lab160"></a><h1 class="section">Testing Whole-Program Compiler Correctness</h1>
<div class="paragraph"> </div>
Here's a property we'd like to test (from <a href="CompilerProp.html"><span class="inlineref">CompilerProp</span></a>):
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">compile_program_correct_terminating</span>:<br/>
∀ <span class="id" type="var">c</span> <span class="id" type="var">m</span> <span class="id" type="var">m'</span> <span class="id" type="var">g</span> <span class="id" type="var">le</span> <span class="id" type="var">lr</span>,<br/>
(<span class="id" type="var">g</span>, <span class="id" type="var">le</span>, <span class="id" type="var">lr</span>) = <span class="id" type="var">compile</span> <span class="id" type="var">c</span> →<br/>
<span class="id" type="var">imp_terminates</span> <span class="id" type="var">c</span> <span class="id" type="var">m</span> <span class="id" type="var">m'</span> →<br/>
<span class="id" type="var">vminus_terminates</span> <span class="id" type="var">g</span> <span class="id" type="var">m</span> <span class="id" type="var">m'</span>.
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
This is one of the top-level correctness theorems for the
compiler: for any initial memory <span class="inlinecode"><span class="id" type="var">m</span></span>, if the source program <span class="inlinecode"><span class="id" type="var">c</span></span>
terminates with memory (Imp state) <span class="inlinecode"><span class="id" type="var">m'</span></span>, running the compilation
result <span class="inlinecode"><span class="id" type="var">g</span></span> (a control flow graph holding the instructions) on the
same initial memory <span class="inlinecode"><span class="id" type="var">m</span></span> also results in termination, and with its
final (Vminus) memory also being <span class="inlinecode"><span class="id" type="var">m'</span></span>.
<div class="paragraph"> </div>
(This is where the coincidence of Imp states and Vminus
memory comes into play.)
<div class="paragraph"> </div>
Recall: An Imp program takes memory <span class="inlinecode"><span class="id" type="var">m</span></span> to memory <span class="inlinecode"><span class="id" type="var">m'</span></span>, written
<span class="inlinecode"><span class="id" type="var">imp_terminates</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode"><span class="id" type="var">m'</span></span> if, when started in <span class="inlinecode"><span class="id" type="var">m</span></span>, it multi-steps to
just <span class="inlinecode"><span class="id" type="var">SKIP</span></span>, with new memory <span class="inlinecode"><span class="id" type="var">m'</span></span>.
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">imp_terminates</span> (<span class="id" type="var">c</span>: <span class="id" type="var">com</span>) (<span class="id" type="var">m</span> <span class="id" type="var">m'</span>: <span class="id" type="var">mem</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span class="id" type="var">star</span> <span class="id" type="var">Imp.step</span> (<span class="id" type="var">c</span>, <span class="id" type="var">m</span>) (<span class="id" type="var">SKIP</span>, <span class="id" type="var">m'</span>).
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
For Vminus programs <span class="inlinecode"><span class="id" type="var">g</span></span>, on the other hand, running on an initial
memory <span class="inlinecode"><span class="id" type="var">m</span></span> leads to with memory <span class="inlinecode"><span class="id" type="var">m'</span></span>, written <span class="inlinecode"><span class="id" type="var">vminus_terminates</span></span> <span class="inlinecode"><span class="id" type="var">g</span></span>
<span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode"><span class="id" type="var">m'</span></span> if execution starting from <span class="inlinecode"><span class="id" type="var">m</span></span> reaches a return terminator
<span class="inlinecode"><span class="id" type="var">tmn_ret</span></span>.
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">vminus_terminates</span> <br/>
(<span class="id" type="var">g</span>: <span class="id" type="var">ListCFG.t</span>) (<span class="id" type="var">m</span> <span class="id" type="var">m'</span>: <span class="id" type="var">mem</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
∃ <span class="id" type="var">x</span> <span class="id" type="var">st'</span>,<br/>
<span class="id" type="var">insns_at_pc</span> <span class="id" type="var">g</span> <span class="id" type="var">st'</span>.(<span class="id" type="var">st_pc</span>) [(<span class="id" type="var">x</span>, <span class="id" type="var">cmd_tmn</span> <span class="id" type="var">tmn_ret</span>)] ∧<br/>
<span class="id" type="var">st'</span>.(<span class="id" type="var">st_mem</span>) = <span class="id" type="var">m'</span> ∧<br/>
<span class="id" type="var">star</span> (<span class="id" type="var">step</span> <span class="id" type="var">g</span>) (<span class="id" type="var">init_state</span> <span class="id" type="var">g</span> <span class="id" type="var">m</span>) <span class="id" type="var">st'</span>.
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
The variables <span class="inlinecode"><span class="id" type="var">x</span></span> and <span class="inlinecode"><span class="id" type="var">st'</span></span> here are determined by the evaluation
relation, as indicated by <span class="inlinecode"><span class="id" type="var">star</span></span> <span class="inlinecode">(<span class="id" type="var">step</span></span> <span class="inlinecode"><span class="id" type="var">g</span>)</span> <span class="inlinecode">(<span class="id" type="var">init_state</span></span> <span class="inlinecode"><span class="id" type="var">g</span></span> <span class="inlinecode"><span class="id" type="var">m</span>)</span> <span class="inlinecode"><span class="id" type="var">st'</span></span>. So
checking for their existence can be thought of as verifying that
an evaluation function yields some <span class="inlinecode"><span class="id" type="var">x</span></span> and <span class="inlinecode"><span class="id" type="var">st'</span></span> satisfying the
constraints.
<div class="paragraph"> </div>
Let us try to write a <span class="inlinecode"><span class="id" type="var">Checker</span></span> for the property
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">compile_program_correct_terminating</span>:<br/>
∀ <span class="id" type="var">c</span> <span class="id" type="var">m</span> <span class="id" type="var">m'</span> <span class="id" type="var">g</span> <span class="id" type="var">le</span> <span class="id" type="var">lr</span>,<br/>
(<span class="id" type="var">g</span>, <span class="id" type="var">le</span>, <span class="id" type="var">lr</span>) = <span class="id" type="var">compile</span> <span class="id" type="var">c</span> →<br/>
<span class="id" type="var">imp_terminates</span> <span class="id" type="var">c</span> <span class="id" type="var">m</span> <span class="id" type="var">m'</span> →<br/>
<span class="id" type="var">vminus_terminates</span> <span class="id" type="var">g</span> <span class="id" type="var">m</span> <span class="id" type="var">m'</span>.
<div class="paragraph"> </div>
</div>
Looking naively at these "for all" quantifiers, it would appear that
we need generators for Imp commands (<span class="inlinecode"><span class="id" type="var">c</span></span>), Imp states / Vminus
memories (<span class="inlinecode"><span class="id" type="var">m</span></span>, <span class="inlinecode"><span class="id" type="var">m'</span></span>), control flow graphs (<span class="inlinecode"><span class="id" type="var">g</span></span>) and labels (<span class="inlinecode"><span class="id" type="var">le</span></span>,
<span class="inlinecode"><span class="id" type="var">lr</span></span>).
<div class="paragraph"> </div>
However, <span class="inlinecode"><span class="id" type="var">g</span></span>, <span class="inlinecode"><span class="id" type="var">le</span></span>, and <span class="inlinecode"><span class="id" type="var">lr</span></span> are computed by <span class="inlinecode"><span class="id" type="var">compile</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span>, so we
don't actually need generators for them. Moreover, we do not want
just any <span class="inlinecode"><span class="id" type="var">m'</span></span>; rather, <span class="inlinecode"><span class="id" type="var">m'</span></span> should be obtained by running an Imp
evaluator (which we have) on <span class="inlinecode"><span class="id" type="var">m</span></span>. Hence for generation, we only
need
(1) a generator for <span class="inlinecode"><span class="id" type="var">mem</span></span> and
(2) a generator for Imp programs.
<div class="paragraph"> </div>
And for checking that the property holds, we need
(1) an evaluator for Vminus that repeatedly steps until it reaches a
return terminator and
(2) a way to check that the final state <span class="inlinecode"><span class="id" type="var">st'</span></span> from the Vminus
evaluator has the desired memory contents.
<div class="paragraph"> </div>
One small technical issue with all of this is that memories are
total maps on an infinite domain, so we cannot actually check that
two memories are equal.
<div class="paragraph"> </div>
Fortunately, for compiler testing, we only care about variables
that appear in the source program. So as a simplification, we fix
a domain of variables from which we draw when generating Imp
programs, and we use the same set when checking memories.
<div class="paragraph"> </div>
The domain is fixed up front in <a href="ImpGen.html"><span class="inlineref">ImpGen</span></a> like this:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Print</span> <span class="id" type="var">id_store</span>.<br/>
<span class="comment">(* ===> <br/>
id_store = get_fresh_atoms 5 <span class="inlinecode"></span><br/>
*)</span><br/>
</div>
<div class="doc">
The generator <span class="inlinecode"><span class="id" type="var">gen_id</span></span> generates fresh identifiers (atoms) by sampling from
this fixed domain.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Print</span> <span class="id" type="var">gen_id</span>.<br/>
<span class="comment">(* ===> <br/>
gen_id = gen_fresh id_store<br/>
*)</span><br/>
</div>
<div class="doc">
Given the generator for <span class="inlinecode"><span class="id" type="var">id</span></span>s, we can generate <span class="inlinecode"><span class="id" type="var">aexp</span></span>, <span class="inlinecode"><span class="id" type="var">bexp</span></span>, and
<span class="inlinecode"><span class="id" type="var">com</span></span> as follows.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Instance</span> <span class="id" type="var">gen_aexp</span> : <span class="id" type="var">GenSized</span> <span class="id" type="var">aexp</span> :=<br/>
{<br/>
<span class="id" type="var">arbitrarySized</span> :=<br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">base_gen</span> := <span class="id" type="var">oneOf</span> [(<span class="id" type="var">n</span> <- <span class="id" type="var">arbitrary</span> ;; <span class="id" type="var">ret</span> (<span class="id" type="var">ANum</span> <span class="id" type="var">n</span>)) ;<br/>
(<span class="id" type="var">id</span> <- <span class="id" type="var">gen_id</span> ;; <span class="id" type="var">ret</span> (<span class="id" type="var">AId</span> <span class="id" type="var">id</span>))] <span class="id" type="keyword">in</span><br/>
<span class="id" type="var">fix</span> <span class="id" type="var">gen_aexp_func</span> <span class="id" type="var">n</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">n</span> <span class="id" type="keyword">with</span><br/>
| 0 ⇒ <span class="id" type="var">base_gen</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">n'</span> ⇒<br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">binop_gen</span> <span class="id" type="var">op</span> := <span class="id" type="var">liftGen2</span> <span class="id" type="var">op</span><br/>
(<span class="id" type="var">gen_aexp_func</span> <span class="id" type="var">n'</span>)<br/>
(<span class="id" type="var">gen_aexp_func</span> <span class="id" type="var">n'</span>) <span class="id" type="keyword">in</span><br/>
<span class="id" type="var">oneOf</span> [<span class="id" type="var">binop_gen</span> <span class="id" type="var">APlus</span> ;<br/>
<span class="id" type="var">binop_gen</span> <span class="id" type="var">AMinus</span> ;<br/>
<span class="id" type="var">binop_gen</span> <span class="id" type="var">AMult</span>;<br/>
<span class="id" type="var">base_gen</span>]<br/>
<span class="id" type="keyword">end</span><br/>
}.<br/>
<br/>
<span class="id" type="var">Derive</span> <span class="id" type="var">Arbitrary</span> <span class="id" type="keyword">for</span> <span class="id" type="var">bexp</span>.<br/>
<span class="id" type="var">Derive</span> <span class="id" type="var">Arbitrary</span> <span class="id" type="keyword">for</span> <span class="id" type="var">com</span>.<br/>
</div>
<div class="doc">
For convenience, <span class="inlinecode"><span class="id" type="keyword">Show</span></span> instances for <span class="inlinecode"><span class="id" type="var">aexp</span></span>, <span class="inlinecode"><span class="id" type="var">bexp</span></span>, and <span class="inlinecode"><span class="id" type="var">com</span></span>
have already been defined (in <a href="ImpGen.html"><span class="inlineref">ImpGen</span></a>). These give more
compact output than the ones that would automatically be
derived.
<div class="paragraph"> </div>
We also need a generator for <span class="inlinecode"><span class="id" type="var">mem</span></span>. Given the (global) list
<span class="inlinecode"><span class="id" type="var">id_store</span></span> of "interesting variables", we generate a list of
numeric values of the same length and use it to build a function
from atoms to values; other variables are set to 0.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">gen_mem</span> : <span class="id" type="var">G</span> <span class="id" type="var">mem</span> :=<br/>
<span class="id" type="var">nat_list</span> <- <span class="id" type="var">vectorOf</span> (<span class="id" type="var">List.length</span> <span class="id" type="var">id_store</span>) <span class="id" type="var">arbitrary</span> ;; <br/>
<span class="id" type="var">ret</span> (<span class="id" type="keyword">fun</span> (<span class="id" type="var">a</span> : <span class="id" type="var">Atom.t</span>) ⇒<br/>
<span class="id" type="keyword">match</span> (<span class="id" type="var">index_of_atom_in_list</span> <span class="id" type="var">a</span> <span class="id" type="var">id_store</span>) <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">None</span> ⇒ 0<br/>
| <span class="id" type="var">Some</span> <span class="id" type="var">i</span> ⇒<br/>
<span class="id" type="var">List.nth</span> <span class="id" type="var">i</span> <span class="id" type="var">nat_list</span> 0<br/>
<span class="id" type="keyword">end</span>).<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">show_memory</span> (<span class="id" type="var">mem</span>: <span class="id" type="var">mem</span>) : <span class="id" type="var">string</span> :=<br/>
<span class="id" type="var">show_memory_on_domain</span> <span class="id" type="var">mem</span> <span class="id" type="var">id_store</span>.<br/>
</div>
<div class="doc">
Next let's define an evaluator for Vminus that stops when a return
terminator is reached (or when it gets stuck or runs out of
fuel — we'll return different results in these cases so that we can
tell what happened later).
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">vminus_eval_result</span> :=<br/>
| <span class="id" type="var">Terminates</span> (<span class="id" type="var">st</span>: <span class="id" type="var">state</span>): <span class="id" type="var">vminus_eval_result</span><br/>
| <span class="id" type="var">GoesWrong</span> (<span class="id" type="var">s</span>: <span class="id" type="var">string</span>): <span class="id" type="var">vminus_eval_result</span><br/>
| <span class="id" type="var">Timeout</span> : <span class="id" type="var">vminus_eval_result</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">vminus_eval</span> (<span class="id" type="var">g</span>: <span class="id" type="var">ListCFG.t</span>) (<span class="id" type="var">st</span> : <span class="id" type="var">state</span>) (<span class="id" type="var">fuel</span>: <span class="id" type="var">nat</span>) :<br/>
<span class="id" type="var">vminus_eval_result</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">fuel</span> <span class="id" type="keyword">with</span><br/>
| 0 ⇒ <span class="id" type="var">Timeout</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">n'</span> ⇒<br/>
<span class="id" type="keyword">match</span> (<span class="id" type="var">ListCFG.fetch</span> <span class="id" type="var">g</span> (<span class="id" type="var">st_pc</span> <span class="id" type="var">st</span>)) <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">Some</span> <span class="id" type="var">instr</span> ⇒<br/>
<span class="id" type="keyword">if</span> <span class="id" type="var">eq_dec_cmd</span> (<span class="id" type="var">snd</span> <span class="id" type="var">instr</span>) (<span class="id" type="var">cmd_tmn</span> <span class="id" type="var">tmn_ret</span>) <span class="id" type="keyword">then</span> <span class="id" type="var">Terminates</span> <span class="id" type="var">st</span><br/>
<span class="id" type="keyword">else</span><br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">eval_step</span> <span class="id" type="var">g</span> <span class="id" type="var">st</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">inr</span> <span class="id" type="var">st'</span> ⇒ <span class="id" type="var">vminus_eval</span> <span class="id" type="var">g</span> <span class="id" type="var">st'</span> <span class="id" type="var">n'</span><br/>
| <span class="id" type="var">inl</span> <span class="id" type="var">err</span> ⇒ <span class="id" type="var">GoesWrong</span> <span class="id" type="var">err</span><br/>
<span class="id" type="keyword">end</span><br/>
| <span class="id" type="var">None</span> ⇒ <span class="id" type="var">GoesWrong</span> "no instr to fetch"<br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">end</span>.<br/>
</div>
<div class="doc">
With generation out of the way, we can now return to checking.
First, a checker for equality of memories on "interesting
variables":
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">memory_equal_checker_on_domain</span><br/>
(<span class="id" type="var">dom</span>: <span class="id" type="var">list</span> <span class="id" type="var">addr</span>) (<span class="id" type="var">mem1</span> <span class="id" type="var">mem2</span> : <span class="id" type="var">mem</span>) : <span class="id" type="var">Checker</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">dom</span> <span class="id" type="keyword">with</span><br/>
| [] ⇒ <span class="id" type="var">checker</span> <span class="id" type="var">true</span><br/>
| (<span class="id" type="var">a</span> :: <span class="id" type="var">l</span>) ⇒<br/>
<span class="id" type="keyword">if</span> <span class="id" type="var">Nat.eqb</span> (<span class="id" type="var">mem1</span> <span class="id" type="var">a</span>) (<span class="id" type="var">mem2</span> <span class="id" type="var">a</span>) <span class="id" type="keyword">then</span><br/>
<span class="id" type="var">memory_equal_checker_on_domain</span> <span class="id" type="var">l</span> <span class="id" type="var">mem1</span> <span class="id" type="var">mem2</span><br/>
<span class="id" type="keyword">else</span><br/>
<span class="id" type="var">whenFail</span><br/>
("memory_equal: memory at " ++ (<span class="id" type="var">show</span> <span class="id" type="var">a</span>)<br/>
++ " not equal:" ++ " mem1 has " ++ (<span class="id" type="var">show_nat</span> (<span class="id" type="var">mem1</span> <span class="id" type="var">a</span>))<br/>
++ "; mem2 has " ++ (<span class="id" type="var">show_nat</span> (<span class="id" type="var">mem2</span> <span class="id" type="var">a</span>))<br/>
)%<span class="id" type="var">string</span><br/>
<span class="id" type="var">false</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">memory_equal_checker</span> := <span class="id" type="var">memory_equal_checker_on_domain</span> <span class="id" type="var">id_store</span>.<br/>
</div>
<div class="doc">
Note that we define this operation to return a <span class="inlinecode"><span class="id" type="var">Checker</span></span> rather
than a <span class="inlinecode"><span class="id" type="var">bool</span></span>, so that, if we find that the two memories are <i>not</i>
equal, we can stash away a useful error message.
<div class="paragraph"> </div>
Now we can assemble the equal-final-state checker for Vminus.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Axiom</span> <span class="id" type="var">vminus_fuel</span> : <span class="id" type="var">nat</span>.<br/>
<span class="id" type="var">Extract</span> <span class="id" type="var">Constant</span> <span class="id" type="var">vminus_fuel</span> ⇒ "1000".<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">vminus_final_state_checker</span><br/>
(<span class="id" type="var">g</span>: <span class="id" type="var">ListCFG.t</span>) (<span class="id" type="var">m</span> <span class="id" type="var">m'</span>: <span class="id" type="var">mem</span>) : <span class="id" type="var">Checker</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">vminus_eval</span> <span class="id" type="var">g</span> (<span class="id" type="var">init_state</span> <span class="id" type="var">g</span> <span class="id" type="var">m</span>) <span class="id" type="var">vminus_fuel</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">Terminates</span> <span class="id" type="var">final_state</span> ⇒<br/>
<span class="id" type="var">whenFail</span> "vminus_final_state_checker: memories not equal"<br/>
(<span class="id" type="var">memory_equal_checker</span> (<span class="id" type="var">st_mem</span> <span class="id" type="var">final_state</span>) <span class="id" type="var">m'</span>)<br/>
| <span class="id" type="var">GoesWrong</span> <span class="id" type="var">err</span> ⇒ <br/>
<span class="id" type="var">whenFail</span> ("vminus_final_state_checker: " ++ <span class="id" type="var">err</span>) <span class="id" type="var">false</span><br/>
| <span class="id" type="var">Timeout</span> ⇒<br/>
<span class="id" type="var">checker</span> <span class="id" type="var">tt</span> <span class="comment">(* discard *)</span><br/>
<span class="id" type="keyword">end</span>.<br/>
</div>
<div class="doc">
And now, the <span class="inlinecode"><span class="id" type="var">Checker</span></span> for the whole correctness property.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">compile_program_correct_terminating_checker'</span>: <span class="id" type="var">Checker</span> :=<br/>
<span class="id" type="var">forAllShrink</span> <span class="id" type="var">arbitrary</span> <span class="id" type="var">shrink</span> (<span class="id" type="keyword">fun</span> (<span class="id" type="var">c</span> : <span class="id" type="var">Imp.com</span>) ⇒<br/>
<span class="id" type="var">forAllShrinkShow</span> <span class="id" type="var">gen_mem</span><br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">x</span> ⇒ []) <br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">m</span> ⇒ <span class="id" type="var">show_memory</span> <span class="id" type="var">m</span>) <br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">m</span> ⇒ <br/>
<span class="id" type="keyword">let</span> '(<span class="id" type="var">g</span>, <span class="id" type="var">le</span>, <span class="id" type="var">lr</span>) := <span class="id" type="var">compile</span> <span class="id" type="var">c</span> <span class="id" type="keyword">in</span><br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">imp_eval</span> <span class="id" type="var">c</span> <span class="id" type="var">m</span> 100 <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">Some</span> <span class="id" type="var">s'</span> ⇒<br/>
<span class="id" type="var">whenFail</span><br/>
("cfg is: " ++ <span class="id" type="var">show</span> <span class="id" type="var">g</span>)<br/>
(<span class="id" type="var">vminus_final_state_checker</span> <span class="id" type="var">g</span> <span class="id" type="var">m</span> <span class="id" type="var">s'</span>)<br/>
| <span class="id" type="var">None</span> ⇒ <span class="id" type="var">checker</span> <span class="id" type="var">tt</span> <span class="comment">(* discard the test *)</span><br/>
<span class="id" type="keyword">end</span>)).<br/>
</div>
<div class="doc">
There are a few things to note here.
<div class="paragraph"> </div>
Firstly, note the use of <span class="inlinecode"><span class="id" type="var">forAllShrinkShow</span></span> here, which lets us
choose the specific <span class="inlinecode"><span class="id" type="keyword">Show</span></span> we want to use; it is needed here
because we don't have a <span class="inlinecode"><span class="id" type="keyword">Show</span></span> instance for total maps (which are
essentially functions) like <span class="inlinecode"><span class="id" type="var">mem</span></span>.
<div class="paragraph"> </div>
Secondly, note the use of <span class="inlinecode"><span class="id" type="var">checker</span></span> <span class="inlinecode"><span class="id" type="var">tt</span></span>: This counts the test case
as a discard, for which we can see the count later. We could have
used <span class="inlinecode"><span class="id" type="var">checker</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span> to treat the test as a "pass," but then we
might be falsely reassured if most tests succeed because they just
hold vacuously.
<div class="paragraph"> </div>
And now, let's check it!
</div>
<div class="code code-tight">
<br/>
<span class="comment">(*! Extract Constant Test.defNumTests => "20".<br/>
QuickChick compile_program_correct_terminating_checker'. *)</span><br/>
<br/>
<span class="comment">(* ===> <br/>
+++ Passed 20 tests (7 discards)<br/>
*)</span><br/>
</div>
<div class="doc">
One thing to note is that we are generating non-terminating Imp
programs about 1/3 of the time, but the ones that are not
discarded all pass.
<div class="paragraph"> </div>
If we start getting too many discards, one way to find out why is
to change <span class="inlinecode"><span class="id" type="var">checker</span></span> <span class="inlinecode"><span class="id" type="var">tt</span></span> above to fail, so that we can examine the
counterexamples.
<div class="paragraph"> </div>
But for now we can live with the discard rate.
<div class="paragraph"> </div>
However, we also notice something else that's more problematic: Running
20 tests takes 15 seconds!
<div class="paragraph"> </div>
On reflection, this is not too surprising, as the grammar contains
a number of binary nodes and the <span class="inlinecode"><span class="id" type="var">size</span></span> parameter is treated as a
bound on <i>depth</i>, not total number of nodes. So our generated
tests are <i>huge</i>.
<div class="paragraph"> </div>
Let's see how huge...
<div class="paragraph"> </div>
We quickly whack together some functions for calculating
the sizes of Imp code.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">size_aexp</span> (<span class="id" type="var">a</span> : <span class="id" type="var">aexp</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">a</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">ANum</span> <span class="id" type="var">_</span> ⇒ 1<br/>
| <span class="id" type="var">AId</span> <span class="id" type="var">_</span> ⇒ 1<br/>
| <span class="id" type="var">APlus</span> <span class="id" type="var">a<sub>1</sub></span> <span class="id" type="var">a<sub>2</sub></span> <br/>
| <span class="id" type="var">AMinus</span> <span class="id" type="var">a<sub>1</sub></span> <span class="id" type="var">a<sub>2</sub></span> <br/>
| <span class="id" type="var">AMult</span> <span class="id" type="var">a<sub>1</sub></span> <span class="id" type="var">a<sub>2</sub></span> ⇒ <span class="id" type="var">size_aexp</span> <span class="id" type="var">a<sub>1</sub></span> + <span class="id" type="var">size_aexp</span> <span class="id" type="var">a<sub>2</sub></span> + 1<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">size_bexp</span> (<span class="id" type="var">b</span> : <span class="id" type="var">bexp</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">b</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">BTrue</span> | <span class="id" type="var">BFalse</span> ⇒ 1<br/>
| <span class="id" type="var">BEq</span> <span class="id" type="var">a<sub>1</sub></span> <span class="id" type="var">a<sub>2</sub></span><br/>
| <span class="id" type="var">BLe</span> <span class="id" type="var">a<sub>1</sub></span> <span class="id" type="var">a<sub>2</sub></span> ⇒ <span class="id" type="var">size_aexp</span> <span class="id" type="var">a<sub>1</sub></span> + <span class="id" type="var">size_aexp</span> <span class="id" type="var">a<sub>2</sub></span> + 1<br/>
| <span class="id" type="var">BAnd</span> <span class="id" type="var">b<sub>1</sub></span> <span class="id" type="var">b<sub>2</sub></span> ⇒ <span class="id" type="var">size_bexp</span> <span class="id" type="var">b<sub>1</sub></span> + <span class="id" type="var">size_bexp</span> <span class="id" type="var">b<sub>2</sub></span> + 1<br/>
| <span class="id" type="var">BNot</span> <span class="id" type="var">b</span> ⇒ <span class="id" type="var">size_bexp</span> <span class="id" type="var">b</span> + 1<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">size_com</span> (<span class="id" type="var">c</span> : <span class="id" type="var">com</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">c</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">SKIP</span> ⇒<br/>
1<br/>
| (<span class="id" type="var">x</span> ::= <span class="id" type="var">a</span>) ⇒<br/>
1 + <span class="id" type="var">size_aexp</span> <span class="id" type="var">a</span><br/>
| (<span class="id" type="var">c<sub>1</sub></span> ;; <span class="id" type="var">c<sub>2</sub></span>) ⇒<br/>
<span class="id" type="var">size_com</span> <span class="id" type="var">c<sub>1</sub></span> + <span class="id" type="var">size_com</span> <span class="id" type="var">c<sub>2</sub></span><br/>
| (<span class="id" type="var">IFB</span> <span class="id" type="var">b</span> <span class="id" type="var">THEN</span> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">ELSE</span> <span class="id" type="var">c<sub>2</sub></span> <span class="id" type="var">FI</span>) ⇒<br/>
<span class="id" type="var">size_bexp</span> <span class="id" type="var">b</span> + <span class="id" type="var">size_com</span> <span class="id" type="var">c<sub>1</sub></span> + <span class="id" type="var">size_com</span> <span class="id" type="var">c<sub>2</sub></span><br/>
| (<span class="id" type="var">WHILE</span> <span class="id" type="var">b</span> <span class="id" type="var">DO</span> <span class="id" type="var">c</span> <span class="id" type="var">END</span>) ⇒<br/>
<span class="id" type="var">size_bexp</span> <span class="id" type="var">b</span> + <span class="id" type="var">size_com</span> <span class="id" type="var">c</span><br/>
<span class="id" type="keyword">end</span>.<br/>
</div>
<div class="doc">
Now add a <span class="inlinecode"><span class="id" type="var">collect</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">compile_program_correct_terminating_checker_collect</span>: <span class="id" type="var">Checker</span> :=<br/>
<span class="id" type="var">forAllShrink</span> <span class="id" type="var">arbitrary</span> <span class="id" type="var">shrink</span> (<span class="id" type="keyword">fun</span> (<span class="id" type="var">c</span> : <span class="id" type="var">Imp.com</span>) ⇒<br/>
<span class="id" type="var">collect</span> (<span class="id" type="var">size_com</span> <span class="id" type="var">c</span>) (<br/>
<span class="id" type="var">forAllShrinkShow</span> <span class="id" type="var">gen_mem</span><br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">x</span> ⇒ []) <br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">m</span> ⇒ <span class="id" type="var">show_memory</span> <span class="id" type="var">m</span>)<br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">m</span> ⇒ <br/>
<span class="id" type="keyword">let</span> '(<span class="id" type="var">g</span>, <span class="id" type="var">le</span>, <span class="id" type="var">lr</span>) := <span class="id" type="var">compile</span> <span class="id" type="var">c</span> <span class="id" type="keyword">in</span><br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">imp_eval</span> <span class="id" type="var">c</span> <span class="id" type="var">m</span> 100 <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">Some</span> <span class="id" type="var">s'</span> ⇒<br/>
<span class="id" type="var">whenFail</span><br/>
("cfg is: " ++ <span class="id" type="var">show</span> <span class="id" type="var">g</span>)<br/>
(<span class="id" type="var">vminus_final_state_checker</span> <span class="id" type="var">g</span> <span class="id" type="var">m</span> <span class="id" type="var">s'</span>)<br/>
| <span class="id" type="var">None</span> ⇒ <span class="id" type="var">checker</span> <span class="id" type="var">tt</span> <br/>
<span class="id" type="keyword">end</span>))).<br/>
</div>
<div class="doc">
Now test:
</div>
<div class="code code-tight">
<br/>
<span class="comment">(*! Extract Constant Test.defNumTests => "20".<br/>
QuickChick compile_program_correct_terminating_checker_collect. *)</span><br/>
<span class="comment">(* ===><br/>
6 : 1<br/>
1 : 90<br/>
1 : 8<br/>
1 : 578<br/>
1 : 34<br/>
1 : 322<br/>
1 : 307<br/>
1 : 3<br/>
1 : 2264<br/>
1 : 208<br/>
1 : 2<br/>
1 : 171<br/>
1 : 115<br/>
1 : 11<br/>
1 : 10<br/>
+++ Passed 20 tests (6 discards)<br/>
*)</span><br/>
</div>
<div class="doc">
Yep, some of these look too big
<div class="paragraph"> </div>
To fix this, we can use the <span class="inlinecode"><span class="id" type="var">resize</span></span> combinator to change the
maximum size that will be passed to sized generators to something
other than the default (which is 10).
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">resize</span>.<br/>
<span class="comment">(* ===> <br/>
resize<br/>
: nat -> G ?A -> G ?A<br/>
where<br/>
?A : <span class="inlinecode"></span> <span class="inlinecode">|-</span> <span class="inlinecode"><span class="id" type="keyword">Type</span></span> <br/>
*)</span><br/>
</div>
<div class="doc">
Let's set it to 3.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">compile_program_correct_terminating_checker</span>: <span class="id" type="var">Checker</span> :=<br/>
<span class="id" type="var">forAllShrink</span> (<span class="id" type="var">resize</span> 3 <span class="id" type="var">arbitrary</span>) <span class="id" type="var">shrink</span> (<span class="id" type="keyword">fun</span> (<span class="id" type="var">c</span> : <span class="id" type="var">Imp.com</span>) ⇒<br/>
<span class="id" type="var">forAllShrinkShow</span> <span class="id" type="var">gen_mem</span><br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">x</span> ⇒ []) <br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">m</span> ⇒ <span class="id" type="var">show_memory</span> <span class="id" type="var">m</span>) <br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">m</span> ⇒ <br/>
<span class="id" type="keyword">let</span> '(<span class="id" type="var">g</span>, <span class="id" type="var">le</span>, <span class="id" type="var">lr</span>) := <span class="id" type="var">compile</span> <span class="id" type="var">c</span> <span class="id" type="keyword">in</span><br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">imp_eval</span> <span class="id" type="var">c</span> <span class="id" type="var">m</span> 100 <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">Some</span> <span class="id" type="var">s'</span> ⇒<br/>
<span class="id" type="var">whenFail</span><br/>
("cfg is: " ++ <span class="id" type="var">show</span> <span class="id" type="var">g</span>)<br/>
(<span class="id" type="var">vminus_final_state_checker</span> <span class="id" type="var">g</span> <span class="id" type="var">m</span> <span class="id" type="var">s'</span>)<br/>
| <span class="id" type="var">None</span> ⇒ <span class="id" type="var">checker</span> <span class="id" type="var">tt</span> <span class="comment">(* discard the test *)</span><br/>
<span class="id" type="keyword">end</span>)).<br/>
</div>
<div class="doc">
Now we can run 10000 tests in a few seconds:
</div>
<div class="code code-tight">
<br/>
<span class="comment">(*! QuickChick compile_program_correct_terminating_checkers. *)</span><br/>
<span class="comment">(* ===><br/>
+++ Passed 10000 tests (2841 discards)<br/>
*)</span><br/>
</div>
<div class="doc">
EX2 (GenComWithSmallExpr) Add a <span class="inlinecode"><span class="id" type="var">collect</span></span> to the checker that collects the exact Imp command
for each test. Set <span class="inlinecode"><span class="id" type="var">defNumTests</span></span> to 20 to avoid looking at too
many examples, and use QuickChick to take a look. Notice that the
generated programs tend to be quite short but that they tend to
involve quite large arithmetic and/or boolean expressions. This
distribution may not be giving us the most effective testing.
<div class="paragraph"> </div>
Write a generator for <span class="inlinecode"><span class="id" type="var">com</span></span> that takes a size bound <span class="inlinecode"><span class="id" type="var">n</span></span> and
generates commands in which <span class="inlinecode"><span class="id" type="var">bexp</span></span> and <span class="inlinecode"><span class="id" type="var">aexp</span></span> expressions have
size at most <span class="inlinecode"><span class="id" type="var">n</span></span>.
</div>
<div class="code code-tight">
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
EX<sub>1</sub> (IdentifyDiscards) By design of the Checker, we expect discards to be due to non-terminating
programs. To investigate this, change the Checker so that "out of fuel"
is treated as a failure. Is the counterexample what you expect? Can you
be absolutely sure that such counterexamples are due to "out of fuel"
rather than because of a genuine bug?
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
<a name="lab161"></a><h1 class="section">Mutation Testing</h1>
<div class="paragraph"> </div>
Beyond discarded tests, we are also interested in the tests that
<i>pass</i>. In particular, we want to make sure that they are passing
because the property really holds, rather than some oversight in
generation or formulation of the Checker. To do that, we can
<i>mutate</i> the compiler to falsify the property and check that the
Checker fails in this case.
<div class="paragraph"> </div>
<i>Mutation testing</i> is a powerful technique for increasing
confidence in the quality of our tests. QuickChick comes with a
command-line tool for mutation testing, but it isn't quite able
yet to handle the Vminus development. So for the moment we'll
just play with mutants manually.
<div class="paragraph"> </div>
We've left a suggested mutant in a comment in the <span class="inlinecode"><span class="id" type="var">APlus</span></span> case of the
<span class="inlinecode"><span class="id" type="var">comp_aexp</span></span> function in <a href="Compiler.html"><span class="inlineref">Compiler</span></a>.
<div class="paragraph"> </div>
Inserting this mutant gives us the following.
</div>
<div class="code code-tight">
<br/>
<span class="comment">(*! QuickChick compile_program_correct_terminating_checker. *)</span><br/>
</div>