Hubbry Logo
Correctness (computer science)Correctness (computer science)Main
Open search
Correctness (computer science)
Community hub
Correctness (computer science)
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Correctness (computer science)
Correctness (computer science)
Correctness (computer science)
View on Wikipedia
from Wikipedia

In theoretical computer science, an algorithm is correct with respect to a specification if it behaves as specified. Best explored is functional correctness, which refers to the input–output behavior of the algorithm: for each input it produces an output satisfying the specification.[1]

Within the latter notion, partial correctness, requiring that if an answer is returned it will be correct, is distinguished from total correctness, which additionally requires that an answer is eventually returned, i.e. the algorithm terminates. Correspondingly, to prove a program's total correctness, it is sufficient to prove its partial correctness, and its termination.[2] The latter kind of proof (termination proof) can never be fully automated, since the halting problem is undecidable.

Partially correct C program to find
the least odd perfect number,
its total correctness is unknown as of 2023 
// return the sum of proper divisors of n
static int divisorSum(int n) {
   int i, sum = 0;
   for (i=1; i<n; ++i)
      if (n % i == 0)
         sum += i;
   return sum;
}
// return the least odd perfect number
int leastPerfectNumber(void) {
   int n;
   for (n=1; ; n+=2)
      if (n == divisorSum(n))
         return n;
}

For example, successively searching through integers 1, 2, 3, … to see if we can find an example of some phenomenon—say an odd perfect number—it is quite easy to write a partially correct program (see box). But to say this program is totally correct would be to assert something currently not known in number theory.

A proof would have to be a mathematical proof, assuming both the algorithm and specification are given formally. In particular it is not expected to be a correctness assertion for a given program implementing the algorithm on a given machine. That would involve such considerations as limitations on computer memory.

A deep result in proof theory, the Curry–Howard correspondence, states that a proof of functional correctness in constructive logic corresponds to a certain program in the lambda calculus. Converting a proof in this way is called program extraction.

Hoare logic is a specific formal system for reasoning rigorously about the correctness of computer programs.[3] It uses axiomatic techniques to define programming language semantics and argue about the correctness of programs through assertions known as Hoare triples.

Software testing is any activity aimed at evaluating an attribute or capability of a program or system and determining that it meets its required results. Although crucial to software quality and widely deployed by programmers and testers, software testing still remains an art, due to limited understanding of the principles of software. The difficulty in software testing stems from the complexity of software: we can not completely test a program with moderate complexity. Testing is more than just debugging. The purpose of testing can be quality assurance, verification and validation, or reliability estimation. Testing can be used as a generic metric as well. Correctness testing and reliability testing are two major areas of testing. Software testing is a trade-off between budget, time and quality.[4]

See also

[edit]

Notes

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Correctness (computer science)

View on Grokipedia
from Grokipedia
In , correctness is the property of a or that ensures it behaves as specified, transforming valid inputs into expected outputs while satisfying predefined preconditions and postconditions. This concept encompasses both partial correctness, where the program produces the correct result if it terminates, and total correctness, which additionally guarantees termination. of correctness is essential for reliable software, particularly in safety-critical systems, as it provides mathematical assurance against errors beyond what testing can achieve. The foundations of program correctness were established in the late 1960s through pioneering work by and C. A. R. Hoare. Floyd's 1967 paper introduced a method for assigning meanings to programs using assertions on flowcharts, enabling proofs that a program's execution paths satisfy intended properties. Building on this, Hoare's 1969 axiomatic approach formalized , which uses triples of the form {P} C {Q}, where P is a , C is a program command, and Q is a postcondition, to deduce correctness via inference rules and axioms for programming constructs like assignments and conditionals. These techniques shifted program verification from ad hoc to rigorous mathematical reasoning, influencing subsequent developments in semantics and automated tools. Modern approaches to proving correctness extend these ideas to handle complex systems, including for finite-state programs and theorem proving for infinite-state ones. Partial correctness proofs often rely on inductive invariants for loops, while total correctness incorporates separate termination arguments, such as variant functions that decrease over iterations. Despite undecidability challenges from the , correctness verification remains a cornerstone of , supporting applications from design to secure protocol analysis.

Overview

Definition

In computer science, correctness denotes the property of a program or algorithm whereby it produces outputs that conform precisely to its intended specification for every valid input, ensuring that the computed results match the expected behavior as defined by the program's requirements. This concept, foundational to software engineering and theoretical computing, emphasizes that a correct program accomplishes the intentions of its designers and users without deviation, no more and no less. A key distinction exists between program correctness and numerical accuracy in computing contexts, such as floating-point arithmetic. While numerical accuracy focuses on the precision and error bounds of computational results—often measured in terms of relative or absolute deviations—correctness pertains solely to the functional relationship between inputs and outputs, irrespective of any inherent approximation in the representation or calculation of values. For example, a program computing the square root might be numerically accurate to a specified precision but incorrect if it fails to handle edge cases like negative inputs as per its specification. The role of specifications is central to assessing correctness, serving as the unambiguous benchmark against which program behavior is evaluated. Informal specifications, typically expressed in or , provide high-level descriptions of expected functionality but may introduce ambiguity due to their non-mathematical nature. In contrast, formal specifications use rigorous mathematical notations, such as predicate logic or temporal logics, to define precise preconditions, postconditions, and invariants, enabling verifiable proofs of correctness. A representative example is a , which is deemed correct if it transforms any input of comparable elements into a sorted in non-decreasing order, preserving the of original elements without alteration. For instance, applying the algorithm to the [3, 1, 4, 1, 5] should yield [1, 1, 3, 4, 5], satisfying the specification for all possible inputs. Correctness in this sense is often analyzed through partial correctness (assuming termination) and total correctness (including guaranteed termination), concepts detailed in later sections.

Importance in Computing

Ensuring correctness in software is paramount due to the severe risks posed by incorrect programs, which can lead to catastrophic failures in real-world applications. For instance, the 1996 rocket explosion occurred 37 seconds after launch when a software error in the inertial reference system caused an operand error, resulting in the vehicle's self-destruction and the loss of a $370 million payload. Similarly, the radiation therapy machine in the 1980s delivered massive overdoses to at least six patients between 1985 and 1987 due to software bugs involving race conditions and inadequate error handling, leading to three fatalities and multiple severe injuries. These incidents underscore how software flaws can cause loss of life, financial devastation, and erosion of public trust in technology. The benefits of prioritizing correctness are especially pronounced in safety-critical systems, where reliability directly safeguards human lives and infrastructure. In avionics, standards like DO-178C mandate rigorous processes to verify software correctness, ensuring flight control systems operate without failure in environments like commercial aircraft. For medical devices, such as pacemakers and infusion pumps, correctness prevents malfunctions that could harm patients, with regulatory frameworks like IEC 62304 emphasizing software lifecycle activities to achieve this. Moreover, achieving correctness early in development yields substantial cost savings; a 2022 report estimates that poor software quality costs the U.S. economy $2.41 trillion annually, with nearly half of a large software system's long-term budget spent on finding and fixing bugs post-deployment, where such external failure costs are an order of magnitude higher than internal pre-release fixes. However, pursuing correctness often involves trade-offs with other priorities, such as , , and development time. Enhancing correctness through extensive testing or may increase computational overhead, potentially degrading system speed in resource-constrained environments like embedded devices. Similarly, rigorous correctness checks can extend timelines, as developers balance thorough validation against market pressures for rapid releases, sometimes compromising on edge-case handling to meet deadlines. These tensions highlight the need for strategic decisions in to mitigate impacts on , where overly restrictive correctness measures might hinder intuitive interfaces. Correctness plays an integral role throughout the lifecycle, from to deployment and maintenance, to preempt defects and ensure alignment with specifications. During requirements and design phases, defining precise behaviors facilitates downstream verification, reducing ambiguity that leads to errors. In implementation and testing, techniques like and integration checks embed correctness as a core practice, while post-deployment monitoring sustains it against evolving conditions. Verification methods, such as static analysis and , serve as key approaches to realize this integration across the lifecycle.

Types of Correctness

Partial Correctness

Partial correctness is a fundamental notion in program verification that concerns the behavior of a program upon termination, without requiring that the program actually terminates. It asserts that if a program starts in a state satisfying a given and executes to completion, then the resulting state satisfies the specified postcondition. This concept forms the basis of , introduced by C. A. R. Hoare, which provides a formal framework for reasoning about such properties. In Hoare logic, partial correctness is expressed using Hoare triples of the form {P} S {Q}, where P is the (a logical assertion true before executing statement S), S is the program statement or sequence of statements, and Q is the postcondition (a logical assertion that must hold after S terminates if P held initially). Semantically, the triple {P} S {Q} is valid if, whenever the initial state satisfies P and the execution of S terminates, the final state satisfies Q. Non-terminating executions are considered vacuously correct under this semantics, as they do not produce a final state to falsify Q. This formulation allows verification of functional correctness independently of termination concerns. A key technique for establishing partial correctness in loops is the use of loop invariants, which are assertions that remain true before and after each iteration of the loop. Consider a binary search algorithm on a sorted a of length n to find a key k. The might be that a is sorted in non-decreasing order and 1 ≤ lower ≤ upper ≤ n. The loop maintains the invariant that k is present in the subarray a[lower..upper] it is present anywhere in a, along with bounds lower ≤ mid ≤ upper. The loop body computes mid as the , compares k with a[mid], and narrows the search interval accordingly. Upon loop exit (when lower > upper), the postcondition holds: if k was found, its index is returned; otherwise, it is reported as absent. This invariant ensures partial correctness, as the search preserves the presence property throughout. In theoretical settings with infinite inputs, such as searching an unbounded sorted sequence, the algorithm may not terminate if k is absent or beyond the initial segment, but partial correctness still guarantees that any terminating execution yields the correct result. The primary limitation of partial correctness is that it disregards non-termination, treating infinite loops as acceptable outcomes rather than errors. This makes it particularly useful for verifying programs in infinite-state systems, where proving termination is generally undecidable due to the , allowing focus on behavioral properties when execution completes. Total correctness extends partial correctness by additionally requiring termination, but partial correctness alone suffices for many practical verification tasks.

Total Correctness

Total correctness extends partial correctness by additionally requiring that the program terminates in a finite number of steps for all initial states satisfying the . In the notation of , a triple {P} S {Q} denotes total correctness if, whenever the program S begins execution in a state where P holds, S is guaranteed to terminate and the resulting state satisfies postcondition Q. This formulation ensures both the functional behavior (as in partial correctness) and liveness through termination. Proving termination often relies on ranking functions, which map program states to elements of a well-founded set, such as the natural numbers, such that the ranking decreases strictly with each transition until reaching a base case. Well-founded orders, like the standard ordering on naturals, prevent infinite descent, thereby establishing that execution paths are finite. These functions are synthesized or inferred to cover loops and recursive calls, providing a rigorous argument for bounded execution time. A classic example is the recursive computation of the factorial function, where the precondition is n≥0n \geq 0 (an integer) and the postcondition is that the result equals n!n!. The following pseudocode illustrates the structure:

function fact(n): if n ≤ 1: return 1 else: return n * fact(n - 1)

function fact(n): if n ≤ 1: return 1 else: return n * fact(n - 1)

Total correctness holds because the recursion decreases nn strictly toward the base case n≤1n \leq 1, which can be formalized using a ranking function r(state)=nr(state) = n, ensuring termination after at most nn steps, after which the multiplicative invariant yields the correct factorial value. However, establishing total correctness faces fundamental limitations due to the undecidability of the , which shows that no general can determine whether an arbitrary program terminates on all inputs. Thus, proofs for total correctness are feasible only for restricted program classes or require manual intervention beyond automated tools.

Verification Methods

Informal Methods

Informal methods for assessing software correctness rely on empirical techniques that evaluate program behavior through observation and human judgment, rather than mathematical proofs. These approaches are widely adopted in practice due to their and integration into development workflows, allowing developers to identify defects early without requiring specialized formal tools. Software testing forms a of informal verification, involving the execution of programs with selected inputs to observe outputs and detect discrepancies. verifies the correctness of individual components, such as functions or modules, in isolation using tools like debuggers to ensure they perform as intended. examines interactions between these units, employing strategies like top-down or bottom-up assembly to confirm seamless collaboration and uncover interface errors. assesses the complete integrated system against specified requirements, focusing on overall functionality, performance, and non-functional attributes like reliability. Testing techniques are categorized as black-box or white-box based on the tester's knowledge of internal structure. treats the software as opaque, validating inputs and outputs against specifications without examining code, using methods like to target representative cases. In contrast, leverages code internals, such as control flows, to design tests that exercise specific paths, aiming for comprehensive coverage of logic branches. Code reviews and walkthroughs provide human-centric inspection to catch logical flaws and improve quality. A code walkthrough is an informal, developer-led session where the author presents the code to peers, explaining its design and soliciting feedback on potential issues like inefficiencies or errors. reviews, often asynchronous in modern tools, involve peers scrutinizing changes for adherence to standards, defect detection, and knowledge sharing, with studies showing they reduce post-release defects by identifying issues early. Simulation and prototyping enable behavioral observation by creating executable models of the software. runs abstract representations to mimic system responses under various conditions, helping verify expected behaviors without full implementation. Prototyping builds preliminary versions to test core functionalities, allowing iterative refinement based on observed outcomes and stakeholder input, which accelerates validation in complex systems. Despite their practicality, informal methods have inherent limitations, as they cannot exhaustively prove the absence of bugs—only demonstrate their presence in tested scenarios. Coverage metrics, such as coverage, quantify the proportion of paths exercised but often fall short of 100% due to infeasible paths or constraints, leaving untested areas vulnerable. These techniques complement by providing scalable, real-world checks but require integration with rigorous approaches for high-assurance systems.

Formal Methods

Formal methods in provide rigorous mathematical frameworks for verifying the correctness of programs and systems, offering guarantees beyond empirical testing by employing and automated analysis. These techniques model programs as mathematical objects and prove properties using logic, ensuring that implementations satisfy specified behaviors under all possible executions. Unlike informal methods that rely on observation, formal methods aim for exhaustive assurance through proofs or exploration, though they often require abstractions to manage complexity. Hoare logic, introduced in 1969, forms a foundational axiomatic system for reasoning about imperative programs using triples of the form {P} S {Q}, where P is a precondition, S a statement, and Q a postcondition; this asserts partial correctness, meaning that if P holds before executing S and S terminates, then Q holds afterward. The logic includes axioms for basic statements, such as assignment (x := e; {Q} becomes {Q[e/x]} x := e {Q}), and inference rules for composition and conditionals, enabling modular proofs via weakest preconditions (wp), where wp(S, Q) denotes the weakest condition guaranteeing that S establishes Q. Extensions to total correctness incorporate termination arguments, often by strengthening postconditions with variant functions that decrease over loops. Model checking automates verification by exhaustively exploring the state space of a finite-state model against a specification, typically in temporal logic, to detect violations of properties like safety or liveness. Pioneered in the early 1980s, it uses branching-time logics such as CTL for efficient algorithms that scale to moderate state spaces via symbolic representations like BDDs. Linear Temporal Logic (LTL) extends this for linear-time properties, with model checking reduced to automata emptiness problems; for instance, the SPIN tool implements on-the-fly LTL verification for concurrent systems, promoting and checking progress cycles to handle non-determinism. While powerful for protocol analysis, model checking suffers from state explosion, mitigated by partial-order reductions and abstraction. Testing serves as a precursor, identifying counterexamples to refine models before formal proof. Theorem proving supports interactive and automated deduction in , allowing verification of complex, infinite-state systems through user-guided proofs or machine search. Interactive provers like Coq, based on the Calculus of Inductive Constructions, and Isabelle/HOL, using , enable tactic-based proofs with machine-checked derivations. Automated variants, such as for applicative , apply decision procedures and induction for industrial-scale verification. The Curry-Howard underpins program extraction in constructive logics, mapping proofs of existential statements to executable code; in Coq, this yields certified programs from verified specifications by erasing proof irrelevancies. A representative application is proving in using invariants. For two processes sharing a , the {true} and postcondition {at most one in critical section} are established, ensuring no simultaneous entry.

Safety and Liveness

In concurrent and reactive systems, correctness extends beyond sequential programs by incorporating and properties, which together ensure both the avoidance of errors and the guarantee of . Safety properties assert that "nothing bad ever happens" during execution, such that any violation is detectable within a finite prefix of any trace. For instance, the absence of deadlock qualifies as a , as it prohibits the occurrence of a finite-time bad state where processes are permanently blocked waiting for each other. Similarly, preventing buffer overflows represents a , ensuring that the system never accesses beyond allocated buffer bounds, thereby avoiding or security breaches. In contrast, liveness properties require that "something good eventually happens," necessitating analysis over potentially infinite executions to confirm . An example is starvation-free scheduling in operating systems, which guarantees that every ready process will eventually receive under fair assumptions, preventing indefinite postponement. The relationship between and liveness is foundational: in reactive systems, total correctness—extending the sequential notion of partial correctness plus termination—typically decomposes into a property conjoined with a liveness property. This decomposition allows modular verification, where can be checked via finite traces and liveness via infinite behaviors, often under fairness conditions.

Robustness and Fault Tolerance

In , robustness refers to a system's capacity to maintain stable and predictable when confronted with erroneous or unexpected , such as malformed data or environmental stresses, thereby preventing catastrophic failures. This quality emphasizes graceful degradation, where the system reduces functionality to a sustainable level rather than halting entirely, ensuring continued partial operation under adverse conditions. A key mechanism for achieving robustness is input validation, which involves checking data against predefined criteria to detect anomalies early and respond appropriately, such as by rejecting invalid inputs or defaulting to safe states. For instance, robust software designs incorporate boundary checks and error-handling routines to mitigate risks from faulty inputs without compromising core operations. Fault tolerance builds on correctness by enabling systems to detect, isolate, and recover from failures, often through to preserve functionality despite component breakdowns. Replication techniques, such as duplicating computations or data across multiple nodes, allow the system to to healthy replicas when one fails, thereby masking the error and maintaining service availability. Similarly, error-correcting codes provide by adding structured to data, enabling the detection and automatic correction of transmission or storage errors; Reed-Solomon codes, for example, are widely used in storage systems like to recover from multiple disk failures by reconstructing lost information from parity symbols. In distributed systems, Byzantine faults represent a severe challenge to correctness, where faulty components may exhibit arbitrary or malicious behavior, such as sending conflicting messages, yet the system must still achieve agreement among honest nodes. tolerance protocols ensure this resilience by requiring a sufficient number of replicas—typically at least three times the number of faulty nodes—to tolerate such failures through mechanisms like authenticated voting and . A prominent hardware example of is (TMR), employed in safety-critical applications like systems, where three identical processing modules execute the same task in parallel, and a voter selects the output that appears twice, effectively masking single-point failures from transient errors or permanent faults. This approach has been implemented in field-programmable gate arrays (FPGAs) to enhance reliability in radiation-prone environments.

Historical Development

Early Foundations

The foundations of correctness in trace back to early theoretical work in logic and , which highlighted fundamental limits on verifying program behavior. In 1936, introduced the concept of computable numbers and abstract machines in his seminal paper, demonstrating through the that there exists no general to determine whether a given program will terminate on a particular input. This undecidability result established inherent boundaries for automated correctness verification, influencing subsequent efforts to formalize program semantics and proofs. Turing's work underscored that while specific programs could be analyzed, universal decision procedures for correctness—encompassing termination and output validity—were impossible, setting a theoretical constraint on the field. Influences from further shaped these ideas, particularly through the Curry-Howard correspondence, which established a deep between logical proofs and computational terms. Haskell Curry's 1934 exploration of functionality in laid initial groundwork by linking logical implications to functional abstractions, suggesting that proofs could be viewed as constructive programs. This connection was refined in William Howard's 1969 manuscript (published 1980), which explicitly formulated the formulae-as-types principle: propositions correspond to types, and proofs to typed programs, enabling the interpretation of program correctness as type inhabitability. These developments provided a logical framework for equating program execution with proof validation, bridging and in ways that anticipated techniques. By the mid-1960s, initial formalisms emerged to operationalize correctness for practical programming. Robert Floyd's 1967 paper introduced a method for assigning meanings to programs using symbolic assertions on flowcharts, allowing the verification of postconditions from preconditions through inductive invariants at program points. This approach enabled systematic reasoning about program behavior without execution, focusing on partial correctness by assuming termination. Concurrently, Peter Naur advocated for structured programming styles in his 1963 article, critiquing unrestricted goto statements in ALGOL 60 and promoting hierarchical control structures to enhance readability and verifiability. Naur's emphasis on disciplined program organization complemented Floyd's semantics, fostering environments where correctness could be more reliably assessed through human inspection. A pivotal advancement came in 1969 with C. A. R. Hoare's paper, which proposed an axiomatic basis for reasoning about programming languages using preconditions, postconditions, and inference rules. Hoare's formalism treated programs as mathematical predicates, with axioms for basic statements and rules for composition, enabling rigorous proofs of both partial and total correctness. This work formalized the idea that program verification could proceed deductively, much like mathematical theorem proving, and became a cornerstone for subsequent developments in program logics.

Modern Advances

Since the late , the field of automated verification has seen significant advancements through the development of (SMT) solvers, which extend Boolean (SAT) solvers to handle constraints over theories such as arithmetic and data structures, enabling more expressive checks for software correctness. These tools originated from early decision procedures in during the and but gained practical momentum in the with improvements in SAT solving algorithms, leading to their widespread adoption for tasks like bounded , where properties are verified up to a fixed number of execution steps by encoding the system as a SAT or SMT problem. A prominent example is Microsoft's Z3 solver, released in 2008, which integrates specialized decision procedures for efficient solving and has been applied in , static analysis, and across industries. In parallel, program extraction techniques have advanced in dependently typed languages, allowing proofs of correctness to yield executable code by erasing proof terms while preserving computational content. Agda, a based on Martin-Löf , exemplifies this by enabling the definition of types that encode program properties, followed by extraction to efficient code in languages like , thus bridging formal proofs with practical implementation. This approach ensures that verified programs are not merely abstract but deployable, with extraction preserving termination guarantees where specified. Real-world applications of these advances include the of operating system kernels and smart contracts. The seL4 , verified in 2009, represents a milestone as the first general-purpose OS kernel with a machine-checked proof of functional correctness from its abstract specification to C implementation, using Isabelle/HOL to confirm absence of bugs and enforcement of security invariants like isolation. In , the 2016 DAO hack, which exploited a reentrancy in an smart contract leading to $50 million in losses, spurred adoption of ; subsequent tools like those based on SMT solvers and model checkers have verified contract properties such as asset safety and termination, with surveys highlighting their role in preventing similar exploits. Despite these progresses, remains a core challenge in , as state-space explosion limits application to large-scale software, often requiring abstractions or modular proofs that introduce complexity. Recent trends as of 2025 integrate for assisted proving, where large language models guide theorem provers like Lean by generating proof sketches or tactics, improving automation on complex conjectures while human oversight ensures rigor; for instance, AI-enhanced systems have boosted success rates on mathematical benchmarks from 60% to 90%.

References

  1. https://pages.cs.wisc.edu/~cs240-1/readings/07_Program_Correctness.pdf
  2. https://www.cs.cornell.edu/gries/TechReports/84-641.pdf
  3. https://people.eecs.berkeley.edu/~necula/Papers/FloydMeaning.pdf
  4. http://sunnyday.mit.edu/16.355/Hoare-CACM-69.pdf
  5. https://dl.acm.org/doi/10.1145/3551665
  6. https://tildesites.bowdoin.edu/~allen/courses/cs260/readings/ch12.pdf
  7. https://www.cs.princeton.edu/~appel/papers/VerifiedODE.pdf
  8. https://www.cs.nmsu.edu/historical-projects/Projects/1920110527project.correctness.9.1.11.pdf
  9. http://www.cs.columbia.edu/~wing/publications/Wing90a.pdf
  10. https://courses.cs.duke.edu/summer06/cps001/lectures/lecture21.html
  11. https://www.esa.int/Newsroom/Press_Releases/Ariane_501_-_Presentation_of_Inquiry_Board_report
  12. http://sunnyday.mit.edu/papers/therac.pdf
  13. https://ldra.com/do-330/
  14. https://www.ncbi.nlm.nih.gov/books/NBK209656/
  15. https://www.it-cisq.org/wp-content/uploads/sites/6/2022/11/CPSQ-Report-Nov-22-2.pdf
  16. https://stackoverflow.blog/2022/01/17/plan-for-tradeoffs-you-cant-optimize-all-software-quality-attributes/
  17. https://betterprogramming.pub/the-false-trade-off-between-quality-and-speed-7f0f9e93fdd
  18. https://www.geeksforgeeks.org/software-engineering/software-engineering-verification-and-validation/
  19. https://dl.acm.org/doi/10.1145/363235.363259
  20. https://www.cs.cmu.edu/~aldrich/courses/654-sp06/notes/3-hoare-notes.pdf
  21. https://se.inf.ethz.ch/~meyer/publications/methodology/invariants.pdf
  22. https://www.csse.canterbury.ac.nz/walter.guttmann/publications/0025.pdf
  23. https://users.cecs.anu.edu.au/~jinbo/logic/Hoare.pdf
  24. https://www.cs.cornell.edu/courses/cs6860/2019sp/Handouts/Apt10years.pdf
  25. http://www.cs.cmu.edu/~hzarnani/fm-rg/cmuonly/podelski04complete.pdf
  26. https://www.cin.ufpe.br/~if721/intranet/TheFormalSemanticsofProgrammingLanguages.pdf
  27. https://cacm.acm.org/research/proving-program-termination/
  28. https://sceweb.uhcl.edu/helm/SWEBOK_IEEE/data/swebok_chapter_05.pdf
  29. https://www.khoury.northeastern.edu/home/lieber/courses/cs4500/f07/lectures/code-review-types.pdf
  30. https://www2.eecs.berkeley.edu/Pubs/TechRpts/2017/EECS-2017-217.pdf
  31. https://www.sae.org/papers/prototyping-simulation-a-means-software-verification-2006-21-0047
  32. https://ieeexplore.ieee.org/document/10344209/
  33. https://ieeexplore.ieee.org/document/588521
  34. https://www.cs.cmu.edu/~emc/15817-f08/lectures/lec-01_4up.pdf
  35. https://www.irif.fr/~letouzey/download/extraction2002.pdf
  36. http://www.cs.cmu.edu/~brookes/papers/AxTrPCDeadlock.pdf
  37. https://www.cs.cornell.edu/fbs/publications/RecSafeLive.pdf
  38. https://lamport.azurewebsites.net/pubs/liveness.pdf
  39. https://hongjin-liang.github.io/papers/concur13-prog.pdf
  40. https://www.sei.cmu.edu/blog/tactics-and-patterns-for-software-robustness/
  41. https://www.geeksforgeeks.org/system-design/graceful-degradation-in-distributed-systems/
  42. https://stackoverflow.blog/2025/01/22/why-all-developers-should-adopt-a-safety-critical-mindset/
  43. https://publications.lib.chalmers.se/records/fulltext/187161/187161.pdf
  44. https://www.netlib.org/lapack/lawnspdf/lawn289.pdf
  45. http://pmg.csail.mit.edu/papers/osdi99.pdf
  46. https://www.alexdelis.eu/M125/Papers/RS-Tutorial.pdf
  47. https://lamport.azurewebsites.net/pubs/byz.pdf
  48. https://dl.acm.org/doi/10.1145/357172.357176
  49. https://people.cs.pitt.edu/~melhem/courses/3410p/ft.pdf
  50. https://ieeexplore.ieee.org/document/8836210/
  51. https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf
  52. https://www.pnas.org/doi/pdf/10.1073/pnas.20.11.584
  53. https://www.cs.cmu.edu/~crary/819-f09/Howard80.pdf
  54. https://link.springer.com/chapter/10.1007/978-3-642-19583-9_2
  55. https://cacm.acm.org/research/the-silent-revolution-of-sat/
  56. https://cse.usf.edu/~haozheng/teach/cda5416/doc/Biere2003-bmc.pdf
  57. https://www.microsoft.com/en-us/research/publication/z3-an-efficient-smt-solver/
  58. https://www.cse.chalmers.se/~ulfn/papers/afp08/tutorial.pdf
  59. https://www.sigops.org/s/conferences/sosp/2009/papers/klein-sosp09.pdf
  60. https://ieeexplore.ieee.org/document/8763832
  61. https://www.csl.sri.com/users/shankar/VGC05/shankar-hcss.pdf
  62. https://ai.princeton.edu/news/2025/princeton-researchers-unveil-improved-mathematical-theorem-prover-powered-ai
Add your contribution
Related Hubs
User Avatar
No comments yet.