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Formal methods use mathematically defined models, specifications and reasoning techniques to describe and analyze systems. They include writing a formal specification, proving properties through formal verification, exploring a finite model by model checking and deriving information through static program analysis. A method's guarantee is relative to its model and assumptions.

For example, a loop summing an array can maintain the invariant that the current accumulator equals the sum of the entries already visited. Initialization, preservation by one iteration and the exit condition establish the returned sum; a separate termination argument shows that an answer is reached. Arithmetic and array bounds must be specified, since a proof using unbounded integers need not describe overflowing machine arithmetic.^[ref]

Establishing specified properties of a system using formal proofs about a precise model. Formal verification connects an implementation to a formal specification; correctness of the model and its assumptions still matters when applying the result to a physical system.

For a small example, the summation loop verification below starts with executable steps and asks two separate questions: does any returned value satisfy the postcondition, and does execution reach that return? A loop invariant answers the first; a loop variant answers the second. This turns an unbounded family of permitted inputs into a few reusable proof obligations.

The model must specify programming language semantics, including the meaning of variable assignments, comparisons and integer arithmetic. A proof about exact mathematical integers need not describe a machine that wraps on integer overflow. Similarly, a correct formal proof of a weak or mistaken formal specification does not establish the intended behavior. Choosing the right formal specification and relating the model to the implemented system remain substantive tasks.

Software testing examines chosen executions and can expose a counterexample. A formal proof instead justifies a statement throughout its stated model; it does not require executing every permitted input. This makes formal verification valuable when rare boundary cases matter, while software testing can still reveal an incorrect model or misunderstood requirement.

In 1969, Tony Hoare presented rules for variable assignment, composition and iteration in a small programming language. His paper explicitly says those rules do not prove successful termination, and discusses the dependence of arithmetic reasoning on the chosen overflow behavior. These reservations already make the model's scope part of the argument.^[ref]

Hoare logic proves properties of programs through Hoare triples $\{P\}C\{Q\}$. A precondition $P$ constrains initial states and a postcondition $Q$ constrains returned states. The basic rules establish partial correctness; termination requires a further argument.^[ref]

For a program expression $E$ whose evaluation is defined and has no side effects, the assignment rule isThe substitution replaces the free occurrences of $x$ in the desired postcondition. For example, obtaining $x\ge 1$ after x := x+1 requires $x+1\ge 1$ beforehand, so $x\ge 0$ suffices. Integer overflow must be excluded or handled by the stated arithmetic model.

Sequential composition joins a first postcondition to the next precondition. For a while loop, proving $\{I\wedge b\}C\{I\}$ establishesHere $I$ is the loop invariant. For while x > 0 do x := x-1, starting with an exact nonnegative integer, $x\ge 0$ is preserved and the false exit guard yields $x=0$. The decreasing nonnegative integer $x$ is also a loop variant, establishing total correctness for this example.^[ref]

Partial correctness means that every terminating execution begun in a program state satisfying the precondition $P$ finishes in a program state satisfying the postcondition $Q$. For a sequential program $C$, the usual Hoare triple notation isHere the notation expresses partial correctness; it makes no promise that $C$ finishes. The precondition and postcondition describe permitted starting circumstances and required final circumstances, rather than tests performed automatically by the program.^[ref]

For a sharp example, take the precondition $n=4$ and the following sequential program over mathematical integers:The condition remains true, so execution never reaches the instruction after the loop. Consequently its partial correctness claim holds for the postcondition $2s=n(n+1)$, and even for the impossible postcondition $0=1$: there is no terminating execution with a wrong final result. This is vacuous truth, not evidence that the loop computes a sum. The same program fails total correctness for this permitted input.

The distinction is useful in a proof: first show that an answer, whenever produced, is right; then establish algorithm termination. The summation loop verification supplies both arguments for a loop that actually progresses. Its finite trace illustrates one execution, while its loop invariant covers every permitted input.

Tony Hoare is a computer scientist whose contributions include quicksort, Hoare logic and communicating sequential processes. His industrial work included an early ALGOL 60 compiler; his institutional biography also describes an unsuccessful operating system project, giving concrete context for the difficulty of reliable systems engineering.^[ref]

In his 1969 paper An Axiomatic Basis for Computer Programming by Tony Hoare, he set out assignment, composition and iteration rules for reasoning directly about program text. The assignment rule reasons backward from the required result, while a loop invariant compresses arbitrarily many iterations into a preservation obligation. The paper explicitly distinguishes these partial correctness results from successful termination and discusses how arithmetic and side effects affect the model.^[ref]