Dynamic epistemic logic

Ever wondered how an ai knows when you've changed your mind, or how a self-driving car "realizes" a pedestrian just stepped off the curb? It's all about how information moves and changes, which is where dynamic epistemic logic—or DEL—comes into the picture.

Honestly, traditional logic is pretty boring because it usually looks at things that never change. You know, "if A is true, then B is true," end of story. But in the real world, and especially in digital transformation, things are messy.

Static logic is like a photograph of what an agent knows at one specific second. Dynamic epistemic logic, on the other hand, is the movie. It's the logic of "becoming informed." According to the Internet Encyclopedia of Philosophy, this field really took off when researchers realized they needed to model how knowledge shifts after an event happens.

• Static vs. Dynamic: Static logic tells you what is true right now. Dynamic logic (DEL) describes the action of updating that knowledge. Think of a retail inventory system; it's one thing to know you have ten shirts (static), but it's another for the system to process a sale and "know" it now has nine (dynamic).
• Agent Awareness: In multi-agent systems, like a high-frequency trading floor in finance, it’s not enough for one bot to know a price. It needs to know if the other bots know that price too. This is what we call higher-order information.
• Information Change: We’ve moved from just worrying about "the truth" to worrying about "information change." As noted by Wikipedia, DEL handles "epistemic events"—like a private api call—that change what an agent knows without changing the actual physical facts of the world.

"DEL is a young field of research... it really started in 1989 with Plaza's logic of public announcement."

It all kind of started with a guy named Hintikka back in 1962. He wrote this book called Knowledge and Belief that laid the groundwork. But the real "aha!" moment for computer scientists came much later.

In 1989, Jan Plaza introduced the "logic of public communications." This was huge because it gave us a mathematical way to say: "Everyone just heard this, so now everyone knows that everyone knows it." It sounds like a tongue twister, but it’s vital for things like blockchain or secure healthcare data sharing.

To understand the diagrams, you gotta look at the "accessibility relations"—those are the lines or arrows between the circles. They represent "indistinguishability." If there's a line between two situations, it means the agent can't tell them apart. When an event happens, we cut those lines so the agent finally "knows" which circle they are actually in.

Puzzles actually played a massive role in the math here. You might’ve heard of the "Muddy Children" puzzle or the "Sum and Product" riddle. These weren't just for fun; they were used to test if the math could handle complex situations where agents learn things just by watching what others don't know.

Imagine a hospital where an ai agent is monitoring patient vitals.

1. The AI knows the patient's heart rate is high (Static).
2. A nurse enters a "public announcement" into the system that the patient just finished exercising.
3. The AI updates its belief: the high heart rate isn't a crisis, it's a result of the exercise (Dynamic Update).

If the system didn't have a way to model this "change of state," it would just keep screaming about a heart attack.

Anyway, it's pretty cool stuff once you get past the jargon. Next, we’re gonna dive into how these agents actually "reason" through these updates.

Public Announcement Logic (pal) and why it matters for marketing teams

Ever wonder why a single "flash sale" email can cause absolute chaos in a marketing department if the inventory bot doesn't get the memo at the exact same time? It's because information isn't just data—it's a trigger that changes what every "agent" in your system believes to be possible.

In the world of pal—or Public Announcement Logic—we treat a message as a way to prune the "impossible." Imagine a Kripke model (basically a map of all possible situations). When a truthful announcement happens, we delete every world where that announcement is false.

• Eliminating Worlds: If a retail system announces "Item X is out of stock," the ai agent instantly deletes all "worlds" where it thought it could promise two-day shipping. It's not just updating a number; it's shifting its entire reality.
• Trustworthy Workflows: Automated workflows rely on the "Truth" axiom (Axiom T). As noted by Stanford Encyclopedia of Philosophy, if an agent knows something, it must be true. In a digital transformation project, this is the difference between a system that "thinks" a payment went through and one that knows it did because of a verified api callback.
• Moore Sentences: Here is where it gets trippy. Some announcements fail because they're "self-defeating." If I tell you, "It's raining but you don't know it," the moment I say it, you do know it. The second half of my sentence becomes false the instant it's uttered. In pal, the update is successful, but the proposition itself becomes false in the new model. Marketing teams hit this when they send "exclusive" offers that become non-exclusive the moment they're blast-emailed to a million people—the "exclusivity" dies the second it's announced.

You've probably heard of the Muddy Children Puzzle. It’s the classic pal test case. Imagine three kids playing in the dirt. Dad says, "At least one of you is dirty." He says it publicly, so now everyone knows that everyone knows there's mud.

In pal, we formalize this as Common Knowledge, which is basically a fixed-point operator. It means the information is "out there" and everyone has factored it into their reasoning. This is how group notifications work on platforms like Slack or Microsoft Teams. When a ceo posts a company-wide update, it creates that common knowledge bridge.

1. Round 1: I know the news.
2. Round 2: I know that you know the news.
3. Round 3: I know that you know that I know... (and so on, ad infinitum).

In marketing, this is huge for synchronized launches. If your social media bot, your email server, and your ad manager don't have "common knowledge" of the launch time, you get a fragmented customer experience. pal gives us the math to ensure these agents are "reasoning" from the same pruned model of the world.

As discussed by Wikipedia, pal is unique because the modalities are interpreted by transforming the very structures used to interpret knowledge.

If your marketing stack doesn't have a way to handle these "epistemic events," you're basically running a movie with a bunch of still photos that don't talk to each other. It’s messy, and honestly, it’s why so many automated campaigns go off the rails.

Next up, we’re going to look at what happens when the info isn't public—the sneaky world of private suspicions and how ai handles secrets.

Complex interactions and action models in agent orchestration

Ever tried to whisper a secret in a crowded room and realized halfway through that everyone is actually staring at you? In the world of ai agent orchestration, that’s the difference between a private handoff and a public broadcast—and getting it wrong is how data leaks happen.

While we've mostly talked about everyone hearing the same news, the real world of digital transformation is way more secretive. Most business processes rely on agents knowing things that their "colleagues" don't. This is especially true in Hybrid Cloud and Edge Computing architectures, where an edge device might have local data that the main cloud shouldn't see yet.

In a complex multi-agent system, information isn't always for everyone. Think about an automated b2b procurement flow. When a vendor sends a private api key or a sensitive price quote to your procurement bot, you don't want every other bot in the network—like the public-facing inventory tracker—to see that data.

• Modeling Secrets: We use something called action models to map out who knows what. If Agent A sends a secret to Agent B, we model this as an event where Agent B "sees" the real info, but every other agent thinks a "null" event happened.
• Secure Handoffs: When a healthcare ai passes patient data to a billing bot, it's a private announcement. The billing bot gets the diagnosis code (the truth), but the scheduling bot only knows that a handoff happened, not what was in it.
• Technokeens helps businesses build these complex b2b automation solutions with high precision, making sure the math behind these "private events" actually holds up so secrets stay secret.

But sometimes, it's not just about secrets—it's about suspicion. Imagine a high-frequency trading bot that sees a sudden price dip. It doesn't know if this is a "public announcement" (a real market crash) or a "private misdirection" (one big player dumping stock to trigger stop-losses).

Action models are basically the "Kripke models" of events. Instead of mapping out possible worlds, they map out possible actions. This is how we handle things like deception or suspicion in ai.

If an agent receives a message but isn't sure if it's been tampered with, it creates a model where two events are possible: one where the message is true, and one where it's a lie. This is huge for cybersecurity. A security bot might "suspect" an incoming instruction is a spoofed command.

As noted by the Stanford Encyclopedia of Philosophy, these action models allow us to analyze the epistemic consequences of these events without "hard-wiring" the results. It's about letting the agents reason through the uncertainty themselves.

So, how does an ai actually merge what it used to know with a new, complex event? It uses something called a product update. This is the math that combines the current world model with the action model.

According to Wikipedia, the product update defines how epistemic models are updated by executing actions described through event models.

It’s basically a multiplication of possibilities. If you have 3 possible worlds and 2 possible ways an event could have happened, the resulting model might have up to 6 new "candidate" worlds. The agent then prunes the ones that are logically inconsistent.

1. Initial State: The bot knows its current environment (World Model).
2. Event Occurs: The bot sees an action (Action Model).
3. Cross-Reference: The bot pairs every possible world with every possible event.
4. Consistency Check: If a world doesn't fit the "precondition" of an event (e.g., you can't "announce" a red card if the card is white), that pair is deleted.

This is how a finance bot handles a "limit order." It sees the order (event) and updates its world model only for the scenarios where that order makes sense. If the order contradicts its current data, it might flag it as an anomaly.

Honestly, it's a bit like playing a game of Cluedo. You're constantly crossing off "who, where, and what" based on new info, but you're also keeping track of who else saw the same clues you did.

Next, we're going to look at what happens when agents don't just "know" things, but actually have beliefs—and what happens when they realize they were wrong. We'll dive into the messy world of belief revision and how ai handles "changing its mind."

Belief revision and changing your mind in ai systems

Ever had that moment where you're 100% sure you left your keys on the counter, but then you find them in the door? Your brain does a quick "belief revision" – you don't just add new info, you throw the old, wrong idea in the trash.

In ai systems, this is actually a massive headache. Most basic logic is great at adding new facts (monotonic reasoning), but it's terrible at handling contradictions. If a system "knows" something is true and then gets told it's false, the whole thing usually just breaks or starts believing everything is true at once, which is obviously useless for digital transformation.

We've mostly looked at "knowledge" so far, which in logic-speak usually means something that is guaranteed to be true. But real-world ai agents—like a logistics bot or a healthcare diagnostic tool—often operate on beliefs. And beliefs can be flat-out wrong.

• Static vs Dynamic change: As mentioned earlier in the section on epistemic logic, static logic is like a snapshot. Static belief change is about looking at a past state and saying "okay, based on what I know now, my belief back then was wrong." Dynamic change is much more "live"—it's the agent actually updating its current worldview because the situation itself is shifting.
• The Trivialization Trap: If an ai agent thinks a server is up and an api call says it's down, a rigid system hits a wall. Without a revision strategy, the bot might enter an "inconsistent state." In the world of belief revision, we try to avoid this by letting the agent rank its beliefs so it knows which ones to ditch first when things get messy.
• Revision Strategies: For autonomous agents, we use "success postulates." Essentially, if the system gets a piece of completely trustworthy info that contradicts its current brain, it must accept the new info and reorganize its old beliefs to make room.

So how does a bot actually "change its mind" without crashing? It uses something called plausibility models. Instead of just saying things are "true" or "false," the agent ranks "possible worlds" by how likely it thinks they are.

It’s like a target: the center is what the ai believes most strongly. The outer rings are things it thinks are possible but unlikely. When new info comes in—say, a patient's vitals spike unexpectedly—the ai shifts its focus to a different "sphere" that fits the new data.

This is called an action-priority update. It’s a fancy way of saying the agent gives priority to the most recent, reliable "action" it observed. If a warehouse bot "believed" a shelf was full but its camera sees it's empty, the visual data (the action) overrides the old inventory log (the belief).

According to Wikipedia, these transformations usually don't change the physical facts of the world, just the "accessibility relations"—basically the paths the ai takes to decide what it thinks is going on.

In finance, a trading bot might believe a stock price will rise based on historical data. If a sudden regulatory announcement drops, the bot undergoes belief revision. It doesn't just "add" the news; it has to demote its old "price-will-rise" world to a less plausible sphere and promote the "market-crash" world to the center of its model.

Healthcare ai does this too. A diagnostic bot might believe a patient has a common cold. If a lab result comes back with a specific marker, it has to "revise" that belief immediately. It uses the doxastic logic (the logic of belief) to ensure the new diagnosis doesn't just sit alongside the old one, but actually replaces it.

Anyway, it's a messy process because human info is messy. But without these "spheres of belief," our ai would be stuck in the past, unable to admit when it's made a mistake.

Next, we’re going to look at Security and Governance to see how we keep these agents from doing something stupid while they're busy updating their brains.

Security, Governance, and IAM for ai agents using del

Ever tried to explain to a security auditor why an ai agent "thought" it had permission to move a file? It’s usually a nightmare of logs and vague api calls that don't really tell you the why behind the action.

That is where things get interesting with dynamic epistemic logic. Instead of just checking if a token is valid, we can actually look at the "logical state" of the agent to see if it even knows it should be doing that task. It’s moving security from a simple "yes/no" gate to a full understanding of an agent's situational awareness.

In most digital transformation projects, we give agents permissions like they're just fancy service accounts. But an ai agent isn't a static script; it’s constantly updating its worldview.

If we define permissions based on what an agent knows (its epistemic state), we create a much tighter security loop. For instance, a finance bot shouldn't just have "write access" to a ledger. It should only have that access when it knows—through a verified public announcement—that a specific invoice has been approved by a human mgr.

• Knowledge-Based Permissions: You can actually restrict an agent's api access so it only functions when certain logical preconditions are met. If the agent doesn't "know" the current security protocol has been updated, the system can automatically revoke its ability to execute high-risk actions.
• Zero Trust and Epistemic States: In a zero trust setup, we assume everything is a threat. By using del, we can require agents to prove they have the right "information state" before they get access to sensitive data. It’s like a logical handshake that happens before the actual data transfer.
• Logical Audit Trails: This is my favorite part. Instead of just seeing that an agent accessed a database, a del-based audit trail shows you exactly what the agent knew at that specific second. You can track "who knew what and when," which makes forensic analysis a whole lot easier when things go sideways.

Compliance is usually a box-checking exercise, but for ai, it’s about making sure the "movie" of its information change doesn't include any illegal scenes. We've talked before about how information moves, and in a b2b environment, leaking secrets is the fastest way to kill a partnership.

Using del action models, we can mathematically verify that an agent isn't accidentally leaking private info through public channels. If Agent A gets a private quote from a vendor, the "product update" math (as mentioned earlier) ensures that the observer agents—like a general reporting bot—literally cannot see that data in their version of the world model.

As noted by the Stanford Encyclopedia of Philosophy, the advantage of this dynamic perspective is that we can analyze the consequences of actions—like private announcements—without having to "hard wire" the results from the start.

This means your compliance team doesn't have to write a million "if-then" rules. Instead, they can use logic to verify that the agent's internal reasoning always follows safety policies. If a healthcare ai is about to share a patient's record, the system checks: "Does the agent know this is a public or private event?" If the agent's model thinks it's a public broadcast, the governance layer kills the process before the leak happens.

Imagine a retail chain where an ai manages dynamic pricing.

1. The Secret: The ceo sends a private message to the pricing bot about a secret upcoming merger that will drop costs.
2. The Risk: A marketing bot is watching the pricing bot to adjust ad spend.
3. The Logic: Using an action model, the pricing bot updates its belief to "lower prices soon," but the marketing bot's model remains unchanged.
4. The Result: The pricing bot doesn't accidentally "announce" the merger through its pricing behavior because its logical state is partitioned.

It sounds a bit like sci-fi, but honestly, it’s just better math for a messier world. By treating security as a part of the agent's "knowledge," we stop guessing what they might do and start proving what they can do.

Anyway, this whole setup only works if the agents can handle the "flow" of time—because permissions and beliefs aren't just snapshots; they’re a constant stream. Next, we’re going to look at the temporal side of things and how del handles the passage of time.

Epistemic Temporal Logic (ETL) and the flow of time

So, we've talked a lot about what agents know, but we haven't really talked about when they know it. In the real world, time doesn't stand still while an ai thinks. This is where Epistemic Temporal Logic (ETL) comes in. It’s basically DEL's cousin that cares about the clock.

While DEL looks at how a single event changes a model, ETL looks at the whole "forest" of possible futures. Imagine a tree where every branch is a different path the world could take. As an agent moves forward in time, some branches get chopped off because they didn't happen.

• Protocols: In ETL, we use "protocols" to define what actions are even allowed to happen over time. It’s like the rules of a game. A bot can't "know" it won the game before the game even starts.
• Perfect Recall: This is a big one for ai. Does the agent remember everything it knew in the past? In a forest model, we can mathematically track if an agent is "forgetting" things or if it's carrying its past knowledge into every new branch.
• Synchronicity: ETL helps us model what happens when two agents are on different clocks. If a server in New York sends a message to a bot in London, there's a delay. ETL handles the "temporal gap" where one agent knows something that the other hasn't received yet.

Without a temporal layer, an ai might get stuck in a loop because it doesn't realize that a "fact" it learned five minutes ago might be expired now. By combining DEL's updates with ETL's time-tracking, we get a system that can reason about the past, present, and future all at once.

Future-proofing your digital transformation with formal logic

So, you've made it to the end of this deep dive into the logic of "becoming informed." If there is one thing I hope you take away, it's that digital transformation isn't just about moving data from point A to point B—it's about managing how your entire system reasons through change.

When we move from simple chatbots to sophisticated agent clusters, things get messy fast. In a huge enterprise setup, you don't just have one ai; you have dozens of them talking to each other, accessing apis, and making decisions that affect your bottom line.

Dynamic epistemic logic (del) is basically the secret sauce for making sure these clusters don't hallucinate their way into a disaster. It allows us to move beyond "if-this-then-that" and into a world where agents understand the context of information.

• Hybrid Cloud and Edge Computing: As mentioned earlier in the section on action models, information isn't always for everyone. In a hybrid cloud setup, del helps manage what an edge device "knows" versus what the central server "knows," preventing data bloat and keeping secrets secure at the periphery.
• B2B Precision: For businesses doing heavy b2b automation, formal verification is a lifesaver. Instead of just hoping your procurement bot understands a price update, you use the math of del to prove it has updated its belief state correctly before it places a million-dollar order.
• Beyond Trial-and-Error: Honestly, most people just "test" their ai agents by throwing stuff at them and seeing what sticks. Formal logic lets you verify the logical state of the agent, so you know it won't fail when a weird, never-before-seen edge case pops up.

I've seen so many projects fail because the "brain" of the system couldn't handle a simple contradiction. A server goes down, an api returns a weird error, and suddenly the whole workflow is stuck in a loop because it doesn't have a strategy for belief revision.

By building your digital transformation on a foundation of formal logic, you're basically giving your system a set of rules for how to think. It's the difference between a toddler who just reacts to things and a grown-up who can say, "Wait, that doesn't make sense, I need to update my plan."

Look at healthcare. A diagnostic ai isn't just looking at a lab result; it's revising its belief about a patient's health based on a "private announcement" from a specialist. If it can't handle that update logically, it might stick to a wrong diagnosis.

In retail, a pricing bot needs to know if a competitor's price drop is a "public event" (a market shift) or just a glitch. According to Springer Nature, del provides the family of logics needed to specify these dynamic aspects of multi-agent systems, ensuring that your bots don't start a price war over a typo.

And in finance, where every millisecond counts, having agents that can perform "product updates" on their world models without crashing is the only way to stay competitive. It's not just about speed; it's about the quality of the reasoning.

At the end of the day, del is just a tool. But it's a powerful one. It moves us away from static, boring snapshots of data and into a world where information is alive and constantly shifting.

If you're leading a digital transformation, don't just settle for "smart" tools. Aim for logically sound ones. Because when the world changes—and it always does—you want a system that knows how to change its mind without losing its head.

It’s been a bit of a ride through the math and the puzzles, but hopefully, you see the value now. Anyway, thanks for sticking with me through the jargon and the "muddy children"—it’s time to go build something that actually thinks.