Glossary: objects, guarantees, and common confusions

Algorithm: An abstract procedure with a specification and reasoning about its behavior; one implementation adds language, representation, and platform choices.
Amortized analysis: A bound on total cost across a sequence of operations, not a probability statement or a per-operation worst-case bound.
Approximation: A value or method close to a target under a stated error measure; an approximation sign should not silently become exact equality.
Atomic transaction: A unit of database changes that commits or rolls back together under the system’s guarantees; it does not automatically include external services.
Authentication / authorization: Establishing identity versus deciding which actions that identity may perform on a resource.
Basis: A linearly independent spanning set; coordinates describe a vector relative to a chosen basis, not an intrinsic list independent of representation.
Conditioning / stability: Sensitivity of a problem’s answer to input changes versus how an algorithm propagates numerical error.
Concurrency / parallelism: Multiple activities with overlapping progress versus simultaneous execution; a single-core event loop can be concurrent without parallel CPU execution.
Confidence interval: A procedure with a long-run coverage interpretation under assumptions, not a claim that a fixed frequentist parameter randomly moves inside the observed interval.
Continuity / differentiability: Matching local function values to limits versus admitting the specified derivative or linear approximation. Differentiability implies continuity in the relevant setting, not conversely.
Correctness / testing: Correctness is satisfaction of a specification; a finite test suite supplies evidence on selected cases and does not exhaust arbitrary input or schedules.
Data race: Conflicting concurrent accesses without the required synchronization under a memory model; it is not merely a slow program or an unfortunate output order.
Derivative: A limit of a rate of change when it exists. $f(a)$ is a function value; $f'(a)$ is a derivative value and may have different units.
Domain: The allowed inputs of an object or expression. Simplifying a formula does not automatically fill holes in its original domain.
Expected complexity: Average cost with respect to a specified probability model, often randomness internal to an algorithm; not the same as an unqualified typical runtime.
Gradient: A vector of partial derivatives under chosen coordinates and inner product; it determines directional change and locally steepest increase in the Euclidean setting.
Idempotency: Repeating an operation preserves the relevant effect under its contract. Request-key deduplication is one implementation technique with storage and retention assumptions.
Invariant: A property preserved across a sequence of valid transitions, useful for proving loops, data structures, transactions, and simulations correct.
Jacobian: A derivative matrix for a vector map; its determinant’s absolute value scales volume locally in suitable coordinate changes, with injectivity/multiplicity conditions.
Lifetime: The interval during which an object exists under language rules. A pointer’s numerical address remaining unchanged does not preserve a destroyed object’s lifetime.
Linearizability: A concurrent-object contract in which operations can be ordered as if each took effect at one instant between invocation and response while respecting real-time precedence.
Model error: Error due to how the mathematical or statistical model represents the real phenomenon, distinct from computing that model inaccurately.
NP: Decision problems with polynomial-size certificates verifiable in polynomial time; not an abbreviation for “not polynomial.”
Ownership: Responsibility for a resource’s lifetime and permitted access; C++ RAII ties cleanup to object lifetime but still requires valid sharing rules.
Partial failure: A situation in which some components or communications fail while others continue, leaving different participants with different knowledge.
Proof: A logically justified argument from stated assumptions; numerical agreement and exhaustive testing of a finite proxy are not automatically proofs of a general theorem.
Reduction: A transformation preserving answers so a solver for one problem would solve another; direction and transformation cost are essential.
Reference implementation: An implementation intended to make a specification and method inspectable, not a promise of production hardening, optimal performance, or broad platform validation.
Singular value: A nonnegative scale factor in an orthogonal decomposition of a linear map, useful for rank, conditioning, and approximation.
State: Information needed to determine later behavior under a transition rule; hidden mutable state can invalidate an otherwise simple function-like explanation.
Tangent line: A line through $(a,f(a))$ with slope $f'(a)$ when the derivative exists: $y-f(a)=f'(a)(x-a)$ . The point’s height is not generally the vertical-axis intercept.
Validation: Checking that a model or system is suitable for its intended purpose against appropriate evidence; distinct from checking that an implementation follows a specified equation.