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* **Sep 10** - Consistency through Monotonicity ([reading](https://dl-acm-org.gate.lib.buffalo.edu/citation.cfm?id=2391230) | [slides](slide/2019-09-10-BloomL.html))
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* **Sep 12** - Recursive View Maintenance ([reading](https://infoscience.epfl.ch/record/183767))
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* **Sep 17** - Program Slicing ([reading 1](https://cse.buffalo.edu/LRG/CSE705/Papers/Weiser-Static-Slicing.pdf) | [reading 2](http://sites.computer.org/debull/A07dec/cheney.pdf))
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* **Sep 19** - Learned Index Structures
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* **Sep 24** - Program Slicing ([slides](slide/2019-09-24-ProgramSlicing2.html))
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* **Oct 1** - Interpretable Deep Learning ([reading](https://ieeexplore-ieee-org.gate.lib.buffalo.edu/abstract/document/8022871))
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---
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431
src/teaching/cse-662/2019fa/slide/2019-09-24-ProgramSlicing2.erb
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431
src/teaching/cse-662/2019fa/slide/2019-09-24-ProgramSlicing2.erb
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---
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template: templates/cse662_2019_slides.erb
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title: Program Slicing (continued)
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date: September 24
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---
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<section>
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<section>
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<h3>Program Slicing Recap</h3>
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<ul>
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<li>Dead Code Elimination</li>
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<li>Highlight Dependent Instructions</li>
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<li>Delta Programs</li>
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<li>Optimization</li>
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</ul>
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</section>
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<section>
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<pre><code class="pascal">
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01: read(N)
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02: Z := 0
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03: I := 1
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04: while (I < N) do
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05: read(X)
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06: if (X < 0) then
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07: Y := f1(X)
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else
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08: Y := f2(X)
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end if
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09: Z = f3(Z, Y)
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10: I = I + 1
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end while
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11: write(Z)
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</code></pre>
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<attribution>Agrawal et. a. Dynamic Slicing in the Presence of Unconstrained Pointers</attribution>
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</section>
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<section>
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<h3>Control Flow Graph</h3>
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<img src="graphics/2019-09-24-FlowGraph.svg">
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</section>
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<section>
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<h3>Data Flow Dependency</h3>
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<img src="graphics/2019-09-24-FlowGraphUnbrokenPath.svg">
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<p>Instruction 4 has a data dependency on 1<p>
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<pre style="font-size: 30%"><code class="pascal">
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01: read(N)
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02: Z := 0
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03: I := 1
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04: while (I < N) do
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</code></pre>
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</section>
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<section>
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<h3>Data Flow Dependency</h3>
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<img src="graphics/2019-09-24-FlowGraphUnbrokenPath2.svg">
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<p>Instruction 11 has a data dependency on both 2 and 9<br/>(any unbroken path from write to read counts)<p>
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<pre style="font-size: 30%"><code class="pascal">
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02: Z := 0
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...
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04: while (I < N) do
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...
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09: Z = f3(Z, Y)
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...
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end while
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11: write(Z)
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</code></pre>
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</section>
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<section>
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<h3>Control Flow Dependency</h3>
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<img src="graphics/2019-09-24-FlowGraphControlDeps.svg">
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<p>Instruction 8 has a control flow dependency from 6<br/>Instruction 9 has a control flow dependency from 4<p>
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<pre style="font-size: 30%"><code class="pascal">
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04: while (I < N) do
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06: if (X < 0) then
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else
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08: Y := f2(X)
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09: Z = f3(Z, Y)
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</code></pre>
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</section>
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<section>
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<h3>Slicing</h3>
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<p><b>Target: </b> Start with a line and a set of variables.</p>
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<p><b>Goal: </b> Find the subset of the program that generates the same values for the specified variables.</p>
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</section>
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<section>
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<h3>Slicing Algorithm</h3>
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<ol>
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<li>Assume that the target line reads from the target variables</li>
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<li>Compute the transitive closure of all data dependencies</li>
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<li>Add in control dependencies to any available statement</li>
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<li>Repeat from step 2 until nothing new is added</li>
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</ol>
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</section>
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</section>
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<section>
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<section>
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<h3>Function Slices</h3>
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<ul>
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<li class="fragment"><b>Simple:</b> Treat $f(a, b, c)$ as a read/write to/from a, b, c.</li>
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<li class="fragment"><b>Moderate:</b> Expand out/inline $f(a, b, c)$ then go as normal.</li>
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<li class="fragment"><b>Up/Down Flow</b></li>
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</ul>
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</section>
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<section>
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<h3>Up/Down Flow</h3>
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For call $f(a, b, c)$ with $f$ defined as $\texttt{def } f(x, y, z)$.
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<pre><code class="pascal">
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define f(x, y, z):
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1: x = y + z
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2: z = y * z
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end
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</code></pre>
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<ol>
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<li>Add virtual write instructions for each: x, y, z at the start.</li>
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<li>For each variable, compute the slice targetting the variable on the last line of the function. Note down which virtual writes appear in the slice</li>
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<li>Treat each call site as one write for each variable, using the virtual writes to determine which variables the writes depend on.</li>
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</ol>
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</section>
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<section>
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<p>Worst case assumptions:
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<ul>
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<li>Every variable depends on every other variable</li>
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<li>Every call site depends on every other call sites (external state)</li>
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</ul>
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</p>
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</section>
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</section>
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<section>
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<section>
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<h3>Pointers</h3>
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<pre><code class="pascal">
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1: x = 2
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2: y = 3
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3: *z = 4
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4: write(x)
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</code></pre>
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<p>Does $x$ on the last line depend on line 3?</p>
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<p class="fragment">It depends on what $z$ is</p>
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</section>
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<section>
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<h3>Not just pointers</h3>
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<pre><code class="pascal">
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1: x[0] = 2
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2: x[1] = 3
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3: x[z] = 4
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4: write(x[0])
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</code></pre>
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<p>Does $x[0]$ on the last line depend on line 3?</p>
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<p class="fragment">It also depends on what $z$ is</p>
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</section>
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<section>
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<p>Which value gets written to can change at runtime.</p>
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</section>
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<!--
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<section>
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<p><b>lvalue: </b> An expression that defines a specific location in memory</p>
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<pre><code>
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x
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&x
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x.y
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x[2]
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</code></pre>
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</section>
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-->
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<section>
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<p><b>Idea 1: </b> Worst-case assumptions</p>
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<ul class="fragment">
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<li>Every variable access depends on all preceeding pointer assignments</li>
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<li>Every pointer reference depends on all preceding assignments</li>
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</ul>
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<p class="fragment">Slightly tighter restrictions possible if referring to an array/struct instead of a pointer</p>
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</section>
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<section>
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<p><b>Idea 2: </b> Determine possible program states</p>
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</section>
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<section>
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<h3>Example</h3>
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<pre><code class="pascal">
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struct { int curr; LL *next } LL;
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1: read(N)
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2: I := 0
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3: X := new LL { -1, null }
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4: Y := &X->curr
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5: while(I < N)
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6: X := new LL { I, X }
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7: Y := X->next->curr
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8: I := I + 1
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end while
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9: print_ll(X)
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</code></pre>
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<p class="fragment">What does X depend on as of line 6?</p>
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</section>
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<section>
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<p>Consider <b>possible</b> state trajectories</p>
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<ol>
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<li class="fragment">{ N = [read value] (line 1) }</li>
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<li class="fragment">+ { I = 0 (line 2) }</li>
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<li class="fragment">+ { X = ptr to $\ell_1$ (line 3); $\ell_1$ = { -1, null } (line 3) }</li>
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<li class="fragment">+ { Y = ptr to $\ell_1$.curr; (line 4) }</li>
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<li class="fragment">nothing added</li>
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<li class="fragment">+ { X = ptr to $\ell_2$ (line 6); $\ell_2$ = { 0, ptr to $\ell_1$ } (line 6) }<br/>
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<b>OR</b> nothing added</li>
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</ol>
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<p class="fragment">View each entry in the state trajectory as a set of possible states</p>
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</section>
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<section>
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<div style="vertical-align: middle">
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<table style="display: inline-block; font-size: 70%; vertical-align: middle; ">
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<tr><td>{</td><td align="left">N = [read value] (line 1);</td></tr>
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<tr><td> </td><td align="left">I = 0 (line 2); </td></tr>
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<tr><td> </td><td align="left">$\ell_1$ = { -1, null } (line 3); </td></tr>
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<tr><td> </td><td align="left">Y = ptr to $\ell_1$.curr; (line 4); </td></tr>
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<tr><td> </td><td align="left">X = ptr to $\ell_2$ (line 6); </td></tr>
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<tr><td> </td><td align="left">$\ell_2$ = { 0, ptr to $\ell_1$ } (line 6) }</td></tr>
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</table>
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<b>OR</b>
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<table style="display: inline-block; font-size: 70%; vertical-align: middle; ">
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<tr><td>{</td><td align="left">N = [read value] (line 1);</td></tr>
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<tr><td> </td><td align="left">I = 0 (line 2); </td></tr>
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<tr><td> </td><td align="left">X = ptr to $\ell_1$ (line 3); </td></tr>
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<tr><td> </td><td align="left">$\ell_1$ = { -1, null } (line 3); </td></tr>
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<tr><td> </td><td align="left">Y = ptr to $\ell_1$.curr; (line 4); }</td></tr>
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</table>
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</div>
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</section>
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<section>
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<table style="display: inline-block; font-size: 70%; vertical-align: middle; ">
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<tr><td>{</td><td align="left">N = [read value] (line 1);</td></tr>
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<tr><td> </td><td align="left">I = 0 (line 2); </td></tr>
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<tr><td> </td><td align="left">$\ell_1$ = { -1, null } (line 3); </td></tr>
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<tr><td> </td><td align="left">Y = ptr to $\ell_1$.curr; (line 4); </td></tr>
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<tr><td> </td><td align="left">X = ptr to $\ell_2$ <b>OR</b> $\ell_1$ (lines 3 or 6)</td></tr>
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<tr><td> </td><td align="left">$\ell_2$ = { 0, ptr to $\ell_1$ } (line 6) }</td></tr>
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</table>
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</section>
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<section>
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<p><b>Problem</b>: Infinite possible states!</p>
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<p class="fragment">Each time through the loop we get another $\ell$ assigned.</p>
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</section>
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<section>
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<p><b>Observation: </b> Only need to repeat states enough times to get dependencies</p>
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<p><b>Idea: </b> Collapse states into the minimum needed</p>
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</section>
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<section>
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<h3>Collapsing States</h3>
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<ul>
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<li>Ignore constants (0, -1, etc...)</li>
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<li>Merge values at unidentified locations (i.e. anything labeled by $\ell$), as long as the dependency data is the same</li>
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</ul>
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</section>
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<section>
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<h3>Simplified Example</h3>
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<pre><code class="pascal">
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struct { int curr; LL *next } LL;
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1: read(N)
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2: I := 0
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3: X := new LL { -1, null }
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4: while(I < N)
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5: X := new LL { I, X }
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6: I := I + 1
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end while
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7: print_ll(X)
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</code></pre>
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</section>
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<section>
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<h3>As of Line 4 (0 loops)</h3>
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<table style="display: inline-block; font-size: 70%; vertical-align: middle; ">
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<tr><td>{</td><td align="left">N = $\mathbb C$ (line 1);</td></tr>
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<tr><td> </td><td align="left">I = $\mathbb C$ (line 2); </td></tr>
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<tr><td> </td><td align="left">$\ell_1$ = { $\mathbb C$, null } (line 3); </td></tr>
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<tr><td> </td><td align="left">X = ptr to $\ell_1$ (line 3)</td></tr>
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</table>
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</section>
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<section>
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<h3>As of Line 4 (1 loop)</h3>
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<table style="display: inline-block; font-size: 70%; vertical-align: middle; ">
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<tr><td>{</td><td align="left">N = $\mathbb C$ (line 1);</td></tr>
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<tr><td> </td><td align="left">I = $\mathbb C$ (line 2); </td></tr>
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<tr><td> </td><td align="left">$\ell_1$ = { $\mathbb C$, null } (line 3); </td></tr>
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<tr><td> </td><td align="left">$\ell_2$ = { $\mathbb C$, ptr to $\ell_1$ } (line 5); </td></tr>
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<tr><td> </td><td align="left">X = ptr to $\ell_2$ (line 5)</td></tr>
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</table>
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</section>
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<section>
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<h3>As of Line 4 (2 loops)</h3>
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<table style="display: inline-block; font-size: 70%; vertical-align: middle; ">
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<tr><td>{</td><td align="left">N = $\mathbb C$ (line 1);</td></tr>
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<tr><td> </td><td align="left">I = $\mathbb C$ (line 2); </td></tr>
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<tr><td> </td><td align="left">$\ell_1$ = { $\mathbb C$, null } (line 3); </td></tr>
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<tr><td> </td><td align="left">$\ell_2$ = { $\mathbb C$, ptr to $\ell_1$ } (line 5); </td></tr>
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<tr><td> </td><td align="left">$\ell_3$ = { $\mathbb C$, ptr to $\ell_2$ } (line 5); </td></tr>
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<tr><td> </td><td align="left">X = ptr to $\ell_2$ (line 5)</td></tr>
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</table>
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<p class="fragment">$\ell_2$ and $\ell_3$ are "similar". Replace them with a new virtual $v_4$</p>
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</section>
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<section>
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<h3>As of Line 4 (2+ loops)</h3>
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<table style="display: inline-block; font-size: 70%; vertical-align: middle; ">
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<tr><td>{</td><td align="left">N = $\mathbb C$ (line 1);</td></tr>
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<tr><td> </td><td align="left">I = $\mathbb C$ (line 2); </td></tr>
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<tr><td> </td><td align="left">$\ell_1$ = { $\mathbb C$, null } (line 3); </td></tr>
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<tr><td> </td><td align="left">$v_4$ = { $\mathbb C$, ptr to $\ell_1$ <b>OR</b> $v_4$ } (line 5); </td></tr>
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<tr><td> </td><td align="left">X = ptr to $v_4$ (line 5)</td></tr>
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</table>
|
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</section>
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<section>
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<h3>Subsumption</h3>
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State $S_2$ subsumes state $S_1$ if you can reach $S_2$ from $S_1$ by any of...
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<ul>
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<li>Add alternatives for pointers</li>
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<li>Add new line dependencies</li>
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<li>Rename an un-identified (i.e., $\ell_i$) location</li>
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<li>Merge values at un-identified (i.e. anything labeled by $\ell$) into a virtual location.</li>
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</ul>
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</section>
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<section>
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<h3>Subsumption</h3>
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<ol>
|
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<li>0 loops</li>
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<li>1 loop</li>
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<li>2+ loops</li>
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<li>3+ loops</li>
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</ol>
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<p class="fragment">2+ loops subsumes both 1 loop and 3+ loops</p>
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<p class="fragment">Final state representation: { 0 loops; 2+ loops }</p>
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</section>
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<section>
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<h3>Flow Dependencies</h3>
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||||
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<ol>
|
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<li class="fragment">Replay state trajectory.</li>
|
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<li class="fragment">Apply virtual node replacements.</li>
|
||||
<li class="fragment">Use computed possible states to determine possible data dependencies.</li>
|
||||
<li class="fragment">Include every possible data dependency.</li>
|
||||
</ol>
|
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</section>
|
||||
</section>
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||||
<section>
|
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<section>
|
||||
<p>What if we want to slice a program <i>in the context of specific inputs</i>?</p>
|
||||
</section>
|
||||
<section>
|
||||
|
||||
<pre><code class="pascal">
|
||||
01: read(N)
|
||||
02: Z := 0
|
||||
03: I := 1
|
||||
04: while (I < N) do
|
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05: read(X)
|
||||
06: if (X < 0) then
|
||||
07: Y := f1(X)
|
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else
|
||||
08: Y := f2(X)
|
||||
end if
|
||||
09: Z = f3(Z, Y)
|
||||
10: I = I + 1
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end while
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11: write(Z)
|
||||
</code></pre>
|
||||
<p>... with N = 2, X = -4, 3 the trace of this program is<br/>
|
||||
$1^1$ $2^1$ $3^1$ $4^1$ $5^1$ $6^1$ $7^1$ $9^1$ $10^1$ $4^2$ $5^2$ $6^2$ $8^2$ $9^2$ $10^2$ $4^3$ $11^1$
|
||||
</p>
|
||||
<p>(superscripts distiguish repeated occurrences)</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h3>Dynamic Slicing</h3>
|
||||
|
||||
<p>Instead of considering all paths, focus on the trace under the provided inputs</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<img src="graphics/2019-09-24-DynamicSlicing.png">
|
||||
<p>(Dependencies for Y as of $8^2$ on N = 2, X = -4, 3)</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<ul>
|
||||
<li>Same as before, but allowed to have multiple copies of nodes.</li>
|
||||
<li>Each instruction instance may have different dependencies... only consider the immediately preceding assignemnt to each variable.</li>
|
||||
<li>First slice the trace</li>
|
||||
<li>Every instruction that appears at least once in the sliced trace is in the final slice</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>Added benefit: Can handle pointer tracking gracefully: Know exactly what the pointer's value will be.</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h3>Bibliography</h3>
|
||||
<ul>
|
||||
<li><a href="https://dl-acm-org.gate.lib.buffalo.edu/citation.cfm?id=802557">Program Slicing</a> (Weiser)</li>
|
||||
<li><a href="https://www.franktip.org/pubs/jpl1995.pdf">A Survey of Program Slicing Techniques</a> (Tip)</li>
|
||||
<li><a href="https://dl-acm-org.gate.lib.buffalo.edu/citation.cfm?id=74821">Dependence Analysis for Pointer Variables</a> (Horwitz, Pfeiffer, Reps)</li>
|
||||
<li><a href="http://www.academia.edu/download/30662068/tav91.pdf">Dynamic Slicing in the Presence of Unconstrained Pointers</a> (Agrawal, DeMillo, Spafford)</li>
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Reference in a new issue