# Rice theorem in automata torrent

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Interesting point. That would be many. But that support has been dropped in Lean 3, and Lean 4 does not support it either. Lean 4 itself seems to be a radical rewrite, and libraries written for Lean 3 do not work in Lean 4. Mathematicians are using LEAN today to prove learning edge mathematics correct. See my reply to you deeper below. Total functional programming with types is literally the same as writing proofs in intuitionistic logic, in a technical sense the Curry Howard correspondence.

This isn't a new fad. It's a deep result that was known more than half a century ago. As the article mentions, formal verification techniques are primarily used today for two things: - Creating secure "core" code -- library functions and kernels and stuff, where the things they're supposed to do are very well-defined. I'm not sure formal techniques will be as useful when expanded to other areas.

Most of the bugs I encounter day-to-day happen because the programmer had the wrong goal in mind -- if you asked them to create a formal proof that their code worked, they would be able to do that, but it would be a proof that their function did a thing which was not actually the thing we wanted. Similarly to, e. Has anyone successfully applied proof techniques to reduce defects in UI development, "business logic", or similarly fuzzy disciplines?

This often surfaces these misunderstandings before a proof is even necessary. I've only done a bit of formal verification but I'd estimate that writing that spec was x harder than writing the actual program and was more complicated than the code. In the end I had lower confidence that the spec lacked bugs than the program. This was after expending a huge amount of effort on a pretty tiny program.

I dont think this was a tooling thing. I think it spoke to the fundamental limits of formal verification. I think it'll always remain kinda niche. I've used formal verification in anger, as well as working with a vendor to formaly verify some critical code for us. This rings very true. It is extraordinarily difficult to write correct rules. It ends up being the same problem as wishing against an evil genie. Most rules that you come up with at first end up having a class of obvious exceptions in the real world, which the verifier finds, and then even more unobvious exceptions, and soon your logic around the exceptions to the rules become at least as complicated as the code you are attempting to verify.

And in this any wrong assumptions that falsely allow bad behavior are not caught or flagged because they pass. Even giving perfect proving software, it's still a far harder challenge to write rules than to write code. And current software is still far from perfect - you are likely to spend a lot of your rules time fighting with your prover.

I think this depends on the spec language and the target system. Separately the spec can often have its own properties which can be verified as a means to interrogate its correctness. For example state machines as spec, temporal logic properties and model checking where the state machine is the abstraction for a concrete system.

Worth noting that proving state machines are an abstraction of a concrete system is a going research concern though. Sure, there are a lot of formal languages for specifying logic with checkers that ensure no bugs in the input specification exist, but AFAIK none of them are useful enough to emit programs. Needing a human to translate one formal language the formal spec into another formal language is pointless and useless, because then the human may as well just translate human language specification into formal language.

If you're writing a sorting algorithm or a hash table implementation or something, then the spec is meaningfully different from the code. The spec says "the output array is sorted", the program describes some particular strategy for sorting it, and then you use the proof tools to make sure that the strategy actually works to sort the array in all cases. But for things like UI code, I too am having trouble imagining a spec that is concrete enough to be useful for formal verification and does not have some trivial correspondence to the implementation.

If anyone knows of an example, I'd really be interested in seeing it! I dunno. Thinking more deeply about the specification for a sorting algorithm, it makes sense that the specification includes the O n runtime or memory usage, or both , or else it's an informal specification. If the spec is really nailed down formally then the specification language really would be the implementation language too. But that doesn't have much to do with formal methods. You can achieve the same effect grabbing a colleague and explaining your spec to them, it will trigger the same rigorous thought because you want them to understand you.

I would love to see a fuzzer applied to business logic. It should take design requests from PMs and execucritters and ask pointed questions about edge cases. Transfinity 29 days ago root parent next [—]. I love this idea. I bet you could get surprisingly useful results just using a language model like GPT I am strongly inclined toward verifying my software to the extent possible but there are many practical challenges.

I think academic formal verification methods look elegant, which appeals to me, but are extremely human intensive when what I really want to do is throw machines at the problem to the extent possible. There are also some important types of software correctness that are still difficult to capture with these methods, though the state-of-the-art has improved with time. I've toyed with many methods, tools, techniques, and approaches to get a sense of where the ROI maxima is for my own purposes.

In practice, I've found that sophisticated and comprehensive application of less elegant methods amenable to throwing hardware at them, like exhaustive functional testing, thorough fuzzing infrastructure, systematic fault injection coverage, various types of longevity testing, etc when done well often found all the same design flaws as a tractable level of more academic formal verification.

Also easier to maintain as code evolves. Furthermore, these less elegant approaches also found the occasional compiler and hardware bug that more elegant formal verification methods typically do not. I have wondered if developing and standardizing this less elegant tooling to a high level, so that it is easier to be lazy and throw hardware at the problem, would have at least as much impact on software quality as trying to get everyone to apply very academic formal verification methods, with their current limitations and theoretical constraints.

As much as I like the concept of very pure formal verification, I lean toward whatever makes maximizing software quality practical and economic. Outside of mission critical applications, if the cost involved to make software "provably correct" time, salaries is greater than the cost of the bugs, it will never be adopted. Believe me, I see the appeal, but it's kind of like demanding your house have all perfect right angles and completely level surfaces.

Living with manageable imperfection is far more realistic. KSteffensen 30 days ago parent next [—]. Why does it always have to be all or nothing? You don't have to make sure all the angles of your house are perfect right angles, but there's probably a couple that really have to be. Formally verify those and live with manageable imperfection for the rest. I'm arguing that the effort involved in creating formal proofs for any part of your software is almost never worth the benefits, outside of mission critical applications, when a dozen simple unittests could provide And certainly not worth it enough to become "mainstream", which is the title of the post.

When the defect happens, the impact to the customer and our business is not so bad. If you want to get paid to play with formal verification in the day job, you'd best find a business context where it is business critical to identify and remove design and implementation defects early, and it is worth spending a lot of money trying complementary techniques for doing so. For my above example: - the defect is not related to a core capability of the product.

Defects, and potential defects, tend to cause a background of mental overhead, and communication overhead, throughout the organization. People have to remember what kinds of assumptions are broken under what circumstances, and relate that to a business impact.

There's a mental unburdening when you can just say "situation X won't happen". We use static type systems all the time, as well as specialized checkers and linters, and none of those showed themselves to have "costs greater than the cost of the bugs". And none of them are even nearly similar to "demanding your house have all perfect right angles and completely level surfaces".

Do you have any reason to believe that all the rest of the verification theory is completely impractical when every piece that was packaged in a usable context became a hit? AnimalMuppet 29 days ago root parent next [—]. This is kind of like AI. When something succeeds in general practice, it is no longer "formal verification".

Now it's "robust type systems" and "static analysis tools". Those provide formal verification of some aspects of the code. And that's great! It's progress! Full formal verification is probably? My definition would be one where Knuth's observation that "it's amazing how many bugs there can be in a formally verified program" is no longer true.

Yes indeed. That is insightful. Is there a name for this process? Well, in AI, the process is called "AI can never succeed". So maybe the general thing is "X can never succeed", where X is some hugely ambitious and ambiguous thing, like "AI" or "formal software verification" or "curing cancer". We're probably never going to cure cancer - that is, have some treatment that conquers all cancers. Instead, we get "for this specific type of cancer, for these specific conditions, this treatment has a higher survival rate than the ones we had before".

Over time, that adds up to a lot of people living out their days rather than dying early. And maybe software verification is the same. Enough ways of verifying specific aspects of software, and bugs have fewer places to hide. It won't find all bugs, but we'll still get better software. But "X can never succeed" isn't a very catchy phrase.

Can anyone coin a better one? Or, is there already a better one that I don't know about? Well, the article has quite a list of "some aspects" of your code the author is working on. I have no reason why any of them could not be successful.

I don't expect all of them to be, but anyone that gets there is already a huge advance. That's not what Rice's theorem states. Rice's theorem states that interesting properties are undecidable, not that they can't be proven. Undecidability is not relevant when you are providing the proofs to the computer. Checking proofs is decidable.

Coming up with proofs is undecidable. This tool does the former, leaving the latter up to humans. Using a language with provides static typed! I think the cost of provably correct software is actually much lower, but you pay more of it up front. The perceived incentives of being first to market are higher than that of quality software. I suspect eventually there will be a big lawsuit where the blame can be laid on negligence in the software development and the incentives might change somewhat.

In most cases, that "later" never arrives anyway. AnimalMuppet 29 days ago root parent prev next [—]. The first such successful lawsuit is going to change the landscape. Not just "somewhat" - it's going to be a massive change.

I'd argue that, if there was going to be a needle-mover lawsuit, it would have happened by now. Until there is evidence that it will happen, we can continue assuming that it won't. This sounds like motivated reasoning to me. Have you heard of Therac? You might be inclined to suggest that that could have been prevented if only they'd used formal methods.

Perhaps that's true. It's something that could have been prevented in many different ways, though. Yet it still happened. If I recall correctly, Therac destroyed the company. It and incidents like it led the FDA to gradually be more stringent on scrutinizing software in medical devices. And yet, to the best of my knowledge, even the FDA has not mandated formal verification of the embedded software in medical devices.

And yet, there almost certainly were lawsuits. So maybe you found evidence that perfectly counters my theory. It changed software, but only a little bit. Very little, given the magnitude of what happened. There seems to have been something akin to an "accident chain", where a large number of things went wrong.

Had any one of these things not happened, there might have been much less harm caused, or even no harm at all. I will admit to being peevish about stuff like this. Some of the failures with Therac were systems failures that had nothing to do with software per se I'm not counting "software hubris" as a software problem.

They were failures of process, problems with hardware interlocks, and even UI bugs that made the software confusing to operators. I have nothing against formal methods, but they're no substitute for a deep and abiding paranoia. The company is still around and still makes hundreds of millions in revenue.

Maybe it would be a bigger deal now. It may already be. Where's the research? It may not be happening just because of quarterly cycles, other misaligned incentives, culture or all kinds of other reasons. The difference is that those things are all within tolerances and serve their purposes and function correctly.

Software often doesn't. And yet the world keeps turning, tech companies keep profiting, and customers are generally happy with the value provided, all without formally provable code bases. How does "provably correct" improve on this without extending timelines and costing more?

People are used to computer systems just breaking and being the root of various problems e. The fact that they accept this flaky and unreliable state as the status quo doesn't mean they're happy with it - they just don't understand that better is actually possible. I work in the security and assurance world. The biggest obstacle we face isn't technical - it's social.

Developers want the route of least effort and least time to get products to market, and end users are largely ignorant of the fact that the world doesn't have to be full of garbage software. At this point, I'm rooting for a massive change in the legal landscape to start treating software defects the way we do engineering defects in physical systems.

Developers and businesses aren't going to do the right thing by choice, so a giant hammer in the form of the legal system is likely to be the only thing to force change. I am fully aware of the consequences of that e. Ever heard of this thing called ransomware, for example? Identity theft? And you must know, this stuff is only the beginning Just wait until the day everyone's private Facebook chats are available on torrent. Software can still be provably correct and have security holes resulting from an insecure definition of "correct.

Please explain how a company could profit more if formal verification does not bring more revenue than it costs? You seem to be assuming that revenue will appear that is greater than the costs. Where is this revenue coming from, exactly? Nirvana fallacy. The point is that it can be much better and eliminate ALL non-design bugs. The revenue could come from savings on fixing bugs, paying for ransomed assets and all other costs that come from bugs.

You're just assuming that doesn't add up and that there's no other reason that we don't do formal verification. That's just stupid. Show me the studies. Your claim is just as strong as the claim you think I'm making, but you missed my point entirely. Serious question: do you really believe that?

There is a kind of HAL quality to many of these arguments. Formal verification is perfect by definition. The fact that it hasn't had very much impact in the real world is all the more evidence of the world being full of wicked people. I mean it's a fact, so yes. You can prove programs are correct.

The only possible flaw they can have is the specification is wrong. That's very much not what I said. There may be many reasons. Assuming some conclusion without actual research is braindead. It's an empirical question that requires actual research, not a priori jacking off. So specifications are kind of like programs? Have you heard of logical positivism? My point was that it always seems to be some external factor. That strikes me as being very convenient. I didn't think I assumed anything.

Like anybody else, I have many things that I need to assess in my day to day life, and often deal with considerable uncertainty. Well, seL4 has verifications of it's mixed-criticality hard-real-time guarantees sufficiently tight bounds on scheduling latency and such to be useful and data diode functionalities, and it's isolation properties have been verified not just at a fine-grained specification level but at a high-level human-readable level of invariant description.

It doesn't cover timing and maybe some kinds of similar, other, side channels, but it's still extremely useful. Formal verification shines in two situations: complicated optimized algorithms with a naive reference implementation you want to confirm equivalent, and high-level behavioral invariants of complex systems like seL4's capability system, or a cluster database's consistency during literally all possible failover scenarios. For example, you can just state that an implementation takes a list of X and that it outputs a sorted list of X.

Nothing more is necessary in cases like that in such a system, no code, just the single type. Well, yes, that falls under the second case: behavioral invariants of complex systems. And the reference for "sorting" could likely be a deterministic bogosort and certainly a primitive bubblesort. Even if you're just looking at sorting stability, you're past what your simple "sorted" type would cover.

Most things are far less trivial than "sorted list", including almost? It's a tradeoff though - you could spend the time looking for bugs between the design and implementation, or you could get the implementation out sooner and get feedback and iterations on the design.

Even regular static type-checking is seen as a burden by many programmers. Genbox 29 days ago prev next [—]. Microsoft developed the excellent Code Contracts[1] project. From the user's perpective, tt was a simple class called Contracts with a few methods such as Require , Ensure and Invariant Underneath the hood it used the Z3 Solver[2], which is both intuitive, flexible and fast.

It validated the contracts while coding and highlighted in the Visual Studio IDE when a contract was broken. You could write something like: Contracts. Unfortunately, Code Contracts has been dead for years now, and it was even removed entirely from. NET[3] due to being hard to maintain, and the verifier stopped working in newer versions of VS. Luckily, C developers now have a small taste of contracts due to nullability analysis[4], but even more exciting is that contracts is making its way into C as a first-level standard[5].

ChrisMarshallNY 29 days ago prev next [—]. While I laud the goals, I am skeptical of the ability to met them. I very much believe that there is an industry-wide crisis of terrible software, but I don't believe that it's practical to go directly from "garbage to gold. Best Practices are how engineering disciplines, throughout history, have achieved progress in Quality.

Currently, Best Practices aren't really a "thing," in software development, and it shows. People like Steve McConnell are not really respected, and a general culture of "move fast and break things" is still pervasive. Engineers flit around companies like mayflies, techniques and libraries come and go, and there's an enormous reliance on dependencies with very little vetting.

We spend so much time, trying to perfect our tools, without trying to perfect ourselves. Academics and theorists have been proposing languages, libraries, infrastructure, and management practices that are designed to change lead into gold for decades, yet it never seems to happen. I have always been a fan of self-Discipline, and the apprenticeship model.

That requires a lot of social infrastructure that does not currently exist. It's as old as human history, and absolutely proven to achieve results. Edwards Deming "The significant problems we face cannot be solved by the same level of thinking that created them. Let's start with the easiest part, and rewrite everything in Rust or a comparably safe language. We'll need something like this anyway to give our software a workable semantics that's free of UB at least wrt. Then we can work on the harder problem of using proof to establish that the unchecked parts do not invoke UB, and that intended specifications are not violated even in the "safe" parts.

ChrisMarshallNY 29 days ago root parent next [—]. The tools shouldn't matter. A good engineer can use whatever tools are at hand, to achieve their ends, as long as they take a Disciplined approach, informed by industry Best Practices. Tools can help us to work faster, and abstract some of the "day to day" trivia, but, at the end of the day, we are still left with ourselves. If you want a lesson in limited tools, try working on embedded software.

Embedded development systems often have languages that are incredibly dangerous, and extremely limited It's not uncommon to be working in a subset of ANSI C. Good embedded engineers are usually trained in hardware practices, as opposed to CS ones. They understand the core fundamentals of what they are doing, so their work is not just rote.

I know that it's an incredibly unpopular stance, but I don't see any alternate path to becoming a better engineer, other than through patience, practice, persistence, and Discipline. Are you saying C and Java are safe languages or which ones did you have in mind? Formal verification needs machine readible formal specifications, but any kind of written specification, informal or not was pretty hard to find in my career at internet giants. Maybe you can get a formal spec in aerospace or FDA regulated implanted devices, but cost to write the spec, let alone to follow the spec is way too high when the spec needs to change at the whim of a hat.

In the SPARK subset of Ada, the specifications and contracts live alongside your code in the same language, then you can prove that the specs are satisfied. You can also leave out the contracts and just prove absence of behaviors like divide by zero, out-of-bounds array access and integer overflow.

Proving that code meets the specification can be really difficult but proving absence of bad behavior is usually a straightforward endeavor. What are other good resources in formal verification? I did more digging and discovered Alloy[0], a "lightweight" formal verification system, which is much simpler, and has industry adoption as well.

It's working nicely for me so far. OOP is just poorly organized state machines. But it also seems quite dead. The latest link is from It is still alive, it has just moved to github! It is a big language and it can prove useful programs. Apparently, part of the Ethereum 2 specification was verified using it.

As a software developer this makes it much more approachable than Coq. The proof statements also feel more like the math I learned in college rather than the weird magic keywords of Coq. Pretty cool. If we apply careful mathematical reasoning, we could find solutions to anything. Computer science is the very discipline that proved that essential complexity can arise even in the smallest of systems. Yet sometimes it is computer scientists and software developers who attempt to challenge the very foundation of their own discipline.

Complexity is essential. It cannot be tamed, and there is no one big answer. I think there is a project quite similar to this one called Verifiable Software Toolchain VST in which you can write a C program, convert it into a massive Coq expression, and then write theorems about that expression in Coq.

The Software Foundations series has a volume about it [1], which I found to be an order of magnitude harder to understand than the other volumes. It feels like the magmide project aims to the same goal as VST. It's unclear how it will improve on what VST has done. It may just be that formal verification of real world languages is inherently complex. Cloudef 29 days ago prev next [—]. Do we have formal verification for formal verification yet? I want to make sure my verification does not have bugs.

Yes, it's an active area of research. If so, can I please get an ELI5 why there is a salient formal verification outcomes difference between using the AST and "directly reading the original source"? So this actually looks really neat if it works. Now to the author's credit!

Further down, under "Do you think this language will make all software perfectly secure? And I think the writer actually does appreciate the limits of what this can actually do, and I very much appreciate them explaining that in what I'd call clear terms. Honestly, to me this project is a means seeking an end, the same way JS devs love to play around with frontend frameworks, the author saw a bunch of shiny powerful highly complex tools and decided that combining them all was the solution to our problems.

I don't want to discourage them from learning Iris, or designing a dependently typed language, but I really think that's missing the difficulty in formal verification. I think the two areas that need focus are: ease of specification and automation. In short, we need to lower the cost of verifying a line of code, by at least an order of magnitude. These two objectives are also directly opposed to the direction Magmide sets as the goal. Ease of specification means we want to use the least amount of seperation logic possible, and hide it from the user if possible.

Automation means favoring simpler logics, specifically we want to stick as much as possible to FOL since that's where we have good automation. By doing everything in a rich dependently typed language from the start it also makes it harder to do incremental verification, I think there is a lot of value in having a 'pyramid of trust' with more and more powerful tools which take you up a level of trust and verification, potentially requiring more input from engineers as they go up.

Finally, I think there's a lot of potential to explore in the interfaces we use to write, read, and debug proofs. I don't think tactic languages as exist today are the last word, and I think we should be doing a lot more interesting things to interface with and explore the proofs. IngoBlechschmid 29 days ago prev next [—]. We then transform the input program i. But the benefit is that programs that do meet the constraints C are provably verified. This builds on the success of Rust, but Rust has not been a success when it comes to [number of engineers writing professional code in the language].

By that measure it's still incredibly niche compared to interpreted languages. The main reason why formal verification has not had even the success of Rust is that most developers myself included don't know enough about the area to take an interest, and certainly don't know enough about the area to pursuade skeptical managers. Unless a big company comes forward with a bunch of case studies about how they used formal verification successfully I can't see the developer mindset changing. Formal verification predates Rust by decades.

I meant to say "this hopes to build on the success of Rust" — Rust is explicitly called out in the readme. For functional stuff, sure, but I don't think this is achievable within the UI domain. The most practical solution for UI is visual regression testing across browsers. This provable correctness of programs, at the expense of performance is one of the explicit design goals or urbit. I know HN hates urbit and don't want to rehash that, but it seems like a good goal for some use cases and I'm not sure it's possible to achieve without building the OS around it.

Urbit has not put much effort into security, for some reason. To be fair, they don't claim their runtime is secure yet [1]. The process downloading and executing code from the network is not sandboxed with seccomp or anything similar, and its "jets" are unverified C libraries which any of this code is allowed to call into.

They could sandbox it pretty easily the worker process which runs third-party code only talks to a host process and not the rest of the world, so it could probably be run under seccomp, not even seccomp-bpf which makes it all the more surprising that they haven't. Urbit has also had and almost certainly still has bugs where jets give different results than the code they're supposed to accelerate a "jet mismatch" [2].

I agree that its "axiomatic" bytecode would lend itself well to verification theoretically, but Urbit as she is spoke is not anywhere close. They also at least historically seemed somewhat hostile towards academic CS research including formal methods probably for weird Moldbug reasons. AnimalMuppet 29 days ago parent prev next [—]. Let's say that Urbit claims to be formally, provably secure.

But approximately nobody actually understands Urbit, which means that nobody knows whether the proof is solid. So as an outsider, I have to either take it on faith that it's secure, or I have to spend a fair amount of time immersing myself in this hard-to-learn system to see if the claimed benefits are really there.

But it's not just Urbit. Rust has essentially the same problem. In fact, perhaps all of formal verification has this kind of problem. How do you prove the benefits to someone who doesn't know the tools? You don't have to understand the linux kernel to buffer overlow a linux application; If they do something like "Here's the IP of a running urbit with BTC in it, good luck! But more generally, if it's true that the only way to make a provably secure app is to design the OS and language around that purpose, then the problem you describe is general too - it will always be a challenge to find auditors.

Buttons 30 days ago prev next [—]. I applaud the research. Of course, those organizations creating and suffering from the most bugs will be the least able to utilize such a language. I suspect that verifying software is a lot like the termination problem of Turing machines: the more useful properties you want to verify, the closer it is to NP completeness. All of those proof assistants are using languages that are not Turing complete. So being Turing complete is not really necessary for writing complex software.

Both termination and verification go beyond NP completess, in that they are undecidable. Also, "verifying so[m]ething with Termination is undecidable in general but that is not a problem in practice. And if not, the developer has to prove termination by hand.

Which is not hard in most cases I have encountered. I want to understand what is verification process? How would programming will take new step forward if this is achieved? I have been programmer for a while but I don't understand context and discussion around verification. Please point me any useful resources which can give me deep understanding of what's being discussed here.

It makes some promises that are quite attractive. One of the benefits of having a DSL rather than something general purpose is that it can make these promises in a more comprehensive and focused manner. Many engineers and teams are aware that they write bad code, and they love it.

You can get very far as a clumsy code vendor. Even if formal verification was practical, it would be pretty difficult to make it mainstream. You can prove that an algorithm is correct most of the time yes, halting and decidability but for practical purposes you mostly can. How do you prove an event driven application is correct? Thiez 28 days ago parent next [—]. Modelling your application as a state machine will allow you to use model checking to prove various properties about your model.

Proving that your program is equivalent for some chosen definition of equivalence to the state machine is left as an exercise for the reader. Wow, the language here is even more optimisitc than the rosiest descriptions you see from young researchers, which prompted me to check if the author has had much experience deductively verifying interesting "deep" functional properties of non-trivial programs. The answer seems to be no.

Like a newcomer to the field, he focuses on "first-day" problems such as language convenience, but the answer to his question of why this hasn't been done before is because that's not the hard problem, something he'll know once he obtains more experience. One of the biggest issues — indeed, the problem separation logic tries to address, but does so successfully only for relatively simple properties — is that "correctness" does not feasibly affordably compose.

The difficulty of proving the correctness of a program made out of components, each of which has already been proven correct for any desired property is not easier than proving the correctness of the program from scratch, without concern for its decomposition. This has been shown to be the case not only in the theoretical worst case, but also in practice. Here's an example to make things concrete. In fact, we only happen to know that this particular question is extremely hard because it is one that has interested mathematicians for years and remains unanswered.

While most verification tasks don't neatly correspond to well-known mathematical problems, and most require far less than years to figure out, this kind of difficulty is encountered by anyone who tries to deductively verify non-trivial programs for anything but very specific properties such as "there's no memory corruption", which separation logic does help with. Various kinds of non-determinism, such as the result of concurrency or any kind of interaction, only makes the possible compositions more complex.

In short, the effort it takes to verify a program does not scale nicely with its size, even when it is neatly decomposed, and it is this practical affordability — which is not a result of the elegance of the tools used — that makes this subject so challenging and interesting , and requires some humility and lowering of expectations even when it is useful and it can be certainly useful when yielded properly and at the right scope.

Another problem is an incorrect model of how programs are constructed. One might think that if a programmer has written a program, then they must have some informal but deductive model of it in their mind, and all that's missing is "just" formally specifying it.

But that is not how programs are constructed over time when many people are involved. In practice, programmers often depend on inductive properties in their assumptions, such as "if the software survived for many years, then local changes are unlikely to have global effects that aren't caught by existing tests.

That is why much of the contemporary research focuses on less sound approaches, that aren't fully deductive, such as concolic testing e. Klee , that allow better scaling for both specification and verification at the cost of "perfection".

The reason why both research and industry don't all do what is proposed here is because they know that's not where to real problems are. There are bigger issues to tackle before making the languages more beginner-friendly. And that's just looking at how things work within the engineering teams. What is the perfect software for "we need an identity card system for physical and logical authorization of two million military personnel"?

Can we formally verify the software cannot be used for evil? Bjartr 30 days ago parent next [—]. Sure, given a formal definition of evil. Evil: the privation of a good that should be present. This implies that v and y are each completely contained within one block and that v and y cannot contain s from all three blocks. The third case is when v and y don t contain any s from the first two blocks. The fourth case is when v consists of s the first block and y consists of s the third block.

The fifth and final case is when v consists of s the second block and y consists of s the third block. In all cases, we have a contradiction. This proves that L is not context-free. The string w consists of three blocks of symbols. Since v x y p, v and y are completely contained within two consecutive blocks.

Suppose that v. Then uv 2 x y 2 z has additional symbols of one type but not the other. Therefore, this string is not in L. Now suppose that v and y touch two consecutive blocks, the first two, for example.

This string is clearly not in L. The same is true for the other blocks. Therefore, in all cases, we have that w cannot be pumped. This contradicts the Pumping Lemma and proves that L is not context-free Let L denote that language and suppose that L is context-free.

For the remaining cases, assume that neither v or y contains a. This implies that v and y are each completely contained within one block and that v and y cannot touch all three blocks. The second case is when v and y are contained within the first two blocks. The third case is when v and y are both within the third block. The fourth case is when v consists of s from the first block and y consists of s from the third block. This case cannot occur since vx y p. The language on the left is regular and, therefore, context-free.

We have just shown that the language on the right is context-free. Therefore, the. Focus on one of the positions where x and y differ. The idea behind a CFG that derives w is to generate u as followed by s 2 bv 2. We know that L is not context-free.

This would contradict the fact that L is not even context-free. Introduction The idea is to repeatedly cross off one a, one b and one c.. If the input is empty, accept. Scan the input to verify that it is of the form a b c. If not, reject. Return the head to the beginning of the memory. Cross off the first a. Move right to the first b and cross it off. If no b can be found, reject.

Move right to the first c and cross it off. If no c can be found, reject. Repeat Steps 2 to 5 until all the a s have been crossed off. When that happens, scan right to verify that all other symbols have been crossed off. If so, accept. In what follows, when we. Similarly for marking with an R..

Mark the first unmarked symbol with an L. Move right to the last unmarked symbol. If none can be found, reject because the input is of odd length. Otherwise, mark it with an R and move left. Repeat Steps 2 and 3 until all the symbols have been marked.

For b and c, we will use y and z, respectively. Missing transitions go to the rejecting state. We construct a basic Turing machine M that simulates M as follows. Let w be the input string. Shift w one position to the right. Place a before and after w so the tape contains w.

Move the head to the first symbol of w and run M. Whenever M moves to the rightmost , replace it with a blank and write a in the next position. Return to the blank and continue running M. Whenever M moves to the leftmost , shift the entire contents of the memory up to the rightmost one position to the right. Write a and a blank in the first two positions, put the head on that blank and continue running M.

Repeat Steps 2 to 4 until M halts. Accept if M accepts. Otherwise, reject Suppose that L and L 2 are decidable languages. Let M and M 2 be TM s that decide these languages. Here s a TM that decides L :. Run M on the input. If M accepts, reject. If M rejects, accept. Here s a TM that decides L L 2 :. Copy the input to a second tape. Run M on the first tape. If M accepts, accept. Otherwise, run M 2 on the second tape. If M 2 accepts, accept. Otherwise, reject,. If M rejects, reject.

If the input is empty, run M on the first tape and M 2 on a blank second tape. If both accept, accept. Otherwise, reject. Mark the first symbol of the input. With an underline, for example. Copy the beginning of the input, up to but not including the marked symbol, to a second tape. Copy the rest of the input to a third tape. Run M on the second tape and M 2 on the third tape. Otherwise, move the mark to the next symbol of the input.

While the mark has not reached a blank space, repeat Steps 3 to Delete the mark from the first tape. Run M on the first tape and M 2 on a blank second tape. Otherwise, reject.. Verify that the input is of the form x y z where x, y and z are strings of digits of the same length.

Write a in the first position of tapes 2, 3 and Copy x, y and z to tapes 2, 3 and 4, respectively. Set the carry to. Remember the carry with the states of the TM. Scan those numbers simultaneously from right to left, using the initial to know when to stop. For each position, compute the sum n of the carry and the digits of x and y using the transition function. If n mod is not equal to the digit of z, reject.

If the carry is, accept. Move the memory head to location i. Copy 32 bits starting at that memory location to an extra tape. Move the memory head to location j. Copy the 32 bits from the extra tape to the 32 bits that start at the current memory location. Here s a TM for the add instruction:. Copy 32 bits starting at that memory location to a second extra tape.

Add the 32 bits from the second extra tape to the 32 bits that start at the current memory location. This can be done by adapting the solution to an exercise from the previous section. Discard any leftover carry. Here s a TM for the jump-if instruction:. If they re all, transition to the first state of the group of states that implement the other instruction. Otherwise, continue to the next instruction Suppose that L is a decidable language. Let M be a TM that decides this language. Here s a TM that decides L.

Note that this is a high-level description. Regular Languages This can be done by simply switching the acceptance status of every state in M. Accept if that algorithm accepts. Otherwise, reject Verify that the input string is of the form R, R 2 where R and R 2 are regular expressions. This can be done by converting R and R 2 to DFA s and then combining these DFA s using the constructions for closure under complementation and intersection.

Let L ODD denote the language of strings of odd length. Since L ODD is regular, this leads to the following algorithm:. Reject if that algorithm accepts. Otherwise, accept.. Here s an algorithm:. This leads to the following algorithm:. Otherwise, accept. An Unrecognizable Language Suppose, by contradiction, that L is recognized by some Turing machine M.

In other words, for every string w, M accepts w if and only if w L. Therefore, M cannot exist and L is not recognizable.. Here s a Turing machine that recognizes D:. Generate the encoding of machine M i. Simulate M i on s i. We use this algorithm to design an algorithm S for the acceptance problem:.

Verify that the input string is of the form M, w where M is a Turing machine and w is a string over the input alphabet of M. Without loss of generality, suppose that is a symbol not in the tape alphabet of M. Otherwise, pick some other symbol. Construct the following Turing machine M : a Let x be the input string. Shift x one position to the right. Place a before x so the tape contains x. If R accepts, accept. Then when M runs on w, it attempts to move left from the first position of its tape where the is.

This implies that R accepts M, w and that S accepts M, w, which is what we want. Then when M runs on w, it never attempts to move left from the first position of its tape. This implies that R rejects M, w and that S rejects M, w. Therefore, S decides A TM. Since A TM is undecidable, this is a contradiction.

Run R on M, w, q accept, where q accept is the accepting state of M. It s easy to see that S decides A TM because M accepts w if and only if M enters its accepting state while running on w. Run R on M. On the other hand, suppose that M does not accept w. Construct the following Turing machine M : a Run M on w. Note that M ignores its input and always runs M on w.

First, Step 4 in the description of S should be changed to the following: If R accepts, reject. Second, the paragraph that follows the description of S as follows: Suppose that M accepts w. We use this algorithm to design an algorithm S for A TM :.

An undecidable problem. In , Hilbert stated a. Pushdown Automata In the last section we found that restricting the computational power of computing devices produced solvable decision problems for the class of sets accepted by finite automata. But along. Automata and Formal Languages Winter Yacov Hel-Or 1 What this course is all about This course is about mathematical models of computation We ll study different machine models finite automata,.

Finite Automata Reading: Chapter 2 1 Finite Automaton FA Informally, a state diagram that comprehensively captures all possible states and transitions that a machine can take while responding to a stream. They provide a precise,. Model 2. Cook and T. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my.

Verhoeff TUE. You may have used a Web search engine with a pattern like travel cancun. Chapter 7 Uncomputability 7. First undecidable problem obtained by diagonalisation. Other undecidable problems obtained by means of the reduction. Finite Automata Reading: Chapter 2 1 Finite Automata Informally, a state machine that comprehensively captures all possible states and transitions that a machine can take while responding to a stream or. Larsen Odense University Abstract For many years, regular expressions with back referencing have been used in a variety.

Students who have not may want to look. What is a class of languages? Regular Languages and Finite State Machines Plan for the Day: Mathematical preliminaries - some review One application formal definition of finite automata Examples 1 Sets A set is an unordered collection.

International Journal of Information and Computation Technology. Rodger rodger cs. On well-ordering and induction: a Prove the induction principle from the well-ordering principle. Prime numbers 1. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing. Binary numbers The reason humans represent numbers using decimal the ten digits from 0,1, There is no other reason than that. There is nothing special otherwise about.

Control Engineering Prof. Last time we were. Mathematical Induction Handout March 8, 01 The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,. August 3, What this course is about What we study. CS Advanced Algorithms Lecture Online algorithms We now shift focus to a different kind of algorithmic problem where we need to perform some optimization without knowing the input in advance.

Robert H. Computer Science Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two. Basics of Counting 22C, Chapter 6 Hantao Zhang 1 The product rule Also called the multiplication rule If there are n 1 ways to do task 1, and n 2 ways to do task 2 Then there are n 1 n 2 ways to do.

Simulating The Simpletron Computer 50 points 1 Description of The Simpletron In this assignment you will write a program to simulate a fictional computer that we will call the Simpletron. As its name implies. Lecture 11 Dynamic Programming Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing. Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature.

However, before getting to it, let us look at some very simple. Prove that the empty set is a subset of every set. We can reach. Math Workshop October Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,. Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. Lesson The Binary Number System.

Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base- system. When you. In this short paper we give the definition of Markov algorithm and also. Bonsangue and J. Kleijn Fall Let L be a language. It is clear. Pigeonhole Principle Solutions 1. Solution: Note that consecutive numbers such.

Mathematical Induction 3. First Principle of Mathematical Induction. States and Automata A finite-state machine or finite automaton the noun comes from the Greek; the singular is automaton, the Greek-derived. Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F. These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications 3rd edition These notes are intended primarily for in-class presentation.

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Clannad ost torrent | For each position, compute the sum n of the carry and the digits of x and y using the transition function. It changed software, but only a little bit. We then transform the input program i. Ref: Weiss, page 1. If M accepts, accept. |

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