[HN Gopher] The Hoare Cube
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The Hoare Cube
Author : matt_d
Score : 46 points
Date : 2024-12-04 18:09 UTC (4 hours ago)
(HTM) web link (johnwickerson.wordpress.com)
(TXT) w3m dump (johnwickerson.wordpress.com)
| MeetingsBrowser wrote:
| I'm just glad to know I'm not the only one who finds some of
| Cousot's writing to be overwhelming.
| mhb wrote:
| As will I be to know that about writings about his writing.
| PaulHoule wrote:
| I've learned the hard way that logic is one of the most
| depressing subjects.
|
| (Exhibit 1: No single completely satisfactory form of logical
| negation)
| ecmm wrote:
| What do you mean?
| thechao wrote:
| My advisor used to go drinking with the Cousots'. He said
| Cousot becomes more scrutable with more wine. I asked my
| advisor if he meant Cousots' _papers_ or Cousots '
| _conversation_ ; his answer was "more French".
| BtM909 wrote:
| Can someone ELI5 this, please?
| aaronsnoswell wrote:
| Bing Copilot's transcript from me asking it to ELI5. Take with
| a massive grain of salt, as it's an LLM etc. etc.
| Me: Can you please explain this webpage like I'm five!
| Copilot: Sure! Let's break it down simply: The
| Hoare Cube: Imagine a magic cube that helps us understand how
| to check if computer programs work correctly or not.
| Commands and States: Think of commands as actions and states as
| places. When you do an action in one place, you might end up in
| another place. Hoare Triples: These are like rules that
| say, "If you start in a good place and do an action, you'll end
| up in another good place." Flipping Switches: By changing
| some rules (like flipping switches), you can get different ways
| to check if programs are correct. Three Dimensions:
| Reach: Can you get to a place or must you get to a place?
| Direction: Do you go forward or backward? Approximation:
| Do you guess too much or too little about what happens?
| The Cube: All these different ways of checking can be put
| together into a cube, showing how they relate to each other.
| It's like a puzzle that helps computer scientists make sure
| programs do what they're supposed to do! Me:
| Explain more about the Hoare Triples for me? That bit doesn't
| make sense Copilot: Sure! Let's break down Hoare Triples
| in a simple way: What is a Hoare Triple? A
| Hoare Triple is a way to describe what a computer program does.
| It looks like this: {P} C {Q}. Parts of a Hoare Triple:
| P: This is the precondition. It describes what needs to be true
| before the program (or command) runs. C: This is the
| command. It's the part of the program that does something.
| Q: This is the postcondition. It describes what should be true
| after the command runs. Example: Imagine you have a
| toy box (P) and you want to put a toy car in it (c). After you
| put the toy car in the box, the box should have the toy car
| inside (Q). In Hoare Triple terms: {Box is empty} Put toy
| car in box {Box has toy car}. Why is it useful?
| Hoare Triples help us make sure that programs do what we expect
| them to do. They are like rules that tell us if we start
| with certain conditions and do a specific action, we will end
| up with the desired result. Does that help clarify
| things? Me: So where do the hoare triples fit
| geometrically on this cube representation? Copilot: Great
| question! Let's break it down: Hoare Triples:
| These are represented in the cube by different combinations of
| three dimensions: Reach: Whether the states can reach
| (may) or must reach (must) a certain condition.
| Direction: Whether the logic is forward (from start to end) or
| backward (from end to start). Approximation: Whether the
| behavior is over-approximated (includes all possible outcomes)
| or under-approximated (includes only certain outcomes).
| Cube Representation: Each corner (vertex) of the cube
| represents a different combination of these dimensions.
| For example, partial correctness is represented by the
| combination of may, forward, and over-approximate.
| Incorrectness logic is represented by may, forward, and under-
| approximate. Visualizing the Cube: Imagine a 3D
| cube where each axis represents one of the dimensions (Reach,
| Direction, Approximation). The corners of the cube show
| how different combinations of these dimensions form different
| types of Hoare Triples. Equivalences: Some corners
| are equivalent, meaning they represent the same logic but from
| different perspectives. For instance, may+forward+over is
| equivalent to must+backward+over. So, Hoare Triples fit
| into this cube by occupying different corners based on their
| specific combination of reach, direction, and approximation.
| This helps in understanding and comparing different program
| logics systematically! Me: So is "partial
| correctness" an example of a Hoare Triple? What is the 'P',
| 'C', and 'Q' in this case? Copilot: Yes, "partial
| correctness" is an example of a Hoare Triple. Let's break it
| down: Partial Correctness Hoare Triple: {P}
| C {Q} means that if the precondition P is true before running
| the command C, and if C terminates, then the postcondition Q
| will be true after C finishes. Components: P
| (Precondition): The condition that must be true before the
| command starts. For example, if you're sorting a list, P could
| be "the list is not empty." C (Command): The action or
| program being executed. In our example, C would be "sort the
| list." Q (Postcondition): The condition that must be true
| after the command finishes, assuming it terminates. For the
| sorting example, Q would be "the list is sorted."
| Example: {List is not empty} Sort the list {List is
| sorted}. In the context of the Hoare Cube, partial
| correctness is represented by the combination of may, forward,
| and over-approximate. This means: May: The command
| may reach the postcondition. Forward: The logic is
| applied from the start to the end of the command. Over-
| approximate: The behavior includes all possible outcomes.
| This helps ensure that if the program starts in a good state
| (precondition) and runs correctly, it will end in a good state
| (postcondition). Me: So partial correctness isn't
| one specific hoare triple then, it's referring to a class of
| hoare triples that share the 'may, forward, over' properties,
| right? Copilot: Exactly! Partial correctness refers to a
| class of Hoare Triples that share the properties of may,
| forward, and over-approximate. Here's a quick
| recap: May: The command may reach the
| postcondition. Forward: The logic is applied from the
| start to the end of the command. Over-approximate: The
| behavior includes all possible outcomes. So, any Hoare
| Triple that fits these criteria is considered part of the
| partial correctness class. It's not just one specific triple,
| but rather a group of them that all follow these rules.
| Me: And I can travel between vertices of the cube by altering
| one of the properties I want my hoare triples to have?
| Copilot: Yes, exactly! You can move between the vertices of the
| Hoare Cube by changing one of the properties (reach, direction,
| or approximation) of your Hoare Triples. Here's how it works:
| Reach: May: The command may reach the postcondition.
| Must: The command must reach the postcondition.
| Direction: Forward: The logic is applied from the start
| to the end of the command. Backward: The logic is applied
| from the end to the start of the command. Approximation:
| Over-approximate: Includes all possible outcomes. Under-
| approximate: Includes only certain outcomes. By changing
| one of these properties, you move to a different vertex on the
| cube. For example: If you start with partial
| correctness (may, forward, over-approximate) and change the
| reach property from may to must, you move to total correctness
| (must, forward, over-approximate). This way, you can
| explore different types of program logics and understand their
| relationships by navigating through the cube. Me:
| Why isn't there a point on the approximation dimension for 'no
| approximation'? I thought logic was an exact mathematical
| science? Copilot: That's a great question! Let's break it
| down: Approximation in Logic: In program
| verification, approximation helps us deal with the complexity
| of programs. Over-approximation includes all possible
| behaviors, even some that might not actually happen.
| Under-approximation includes only certain behaviors, possibly
| missing some that could happen. Why No 'Exact' Point?:
| Exact logic would mean capturing every possible behavior of a
| program perfectly, which is often impractical due to the
| complexity and undecidability of many program properties.
| Approximations allow us to reason about programs in a
| manageable way, even if it's not perfect. Mathematical
| Logic: While logic is exact in mathematics, applying it
| to real-world programs often requires dealing with
| uncertainties and complexities. Approximations help
| bridge the gap between theoretical logic and practical program
| verification. So, the approximation dimension in the
| Hoare Cube reflects the practical need to simplify and manage
| the complexity of verifying real-world programs.
| thechao wrote:
| That's a shockingly good Socratic dialog that maps fairly
| well! I've not read your dialog in detail; nor have I read
| this blog article enough times to know if the mapping is
| correct. However, a quick glance shows that it's in the same
| ballpark!
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(page generated 2024-12-04 23:00 UTC)