32 lines
1.5 KiB
Markdown
32 lines
1.5 KiB
Markdown
---
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title: "Implementing and Verifying \"Static Program Analysis\" in Agda, Part 2: Combining Lattices"
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series: "Static Program Analysis in Agda"
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date: 2024-04-13T14:23:03-07:01
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draft: true
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tags: ["Agda", "Programming Languages"]
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---
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In the previous post, I wrote about how lattices arise when tracking, comparing
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and combining static information about programs. I then showed two simple lattices:
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the natural numbers, and the (parameterized) "above-below" lattice, which
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modified an arbitrary set with "bottom" and "top" elements (\(\bot\) and \(\top\)
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respectively). One instance of the "above-below" lattice was the sign lattice,
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which could be used to reason about the signs (positive, negative, or zero)
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of variables in a program.
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At the end of that post, I introduced a source of complexity: the "full"
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lattices that we want to use for the program analysis aren't signs or numbers,
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but maps of states and variables to lattices-based states. The full lattice
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for sign analysis might something in the form:
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{{< latex >}}
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\text{Info} \triangleq \text{ProgramStates} \to (\text{Variables} \to \text{Sign})
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{{< /latex >}}
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Thus, we have to compare and find least upper bounds (e.g.) of not just
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signs, but maps! Proving the various lattice laws for signs was not too
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challenging, but for for a two-level map like \(\text{info}\) above, we'd
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need to do a lot more work. We need tools to build up such complicated lattices!
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The way to do this, it turns out, is by using simpler lattices as building blocks.
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