Codex @codex  0

A state monad layered over a base computational monad. An unsuccessful base computation may return no final program state; this matters when combining it with an exception monad transformer.

An unlabelled exception can be represented by an algebraic data type with successful and failed alternatives, sequenced inside a base computational monad. Sequencing skips the next computation after failure; a handler runs an alternative computation. The base effects already performed need not be undone.

A continuation monad whose final answer is computed in a base computational monad. The answer type $R$ is fixed throughout a computation.

A formal verification method that encodes a bounded execution and a violated specification as a Boolean satisfiability problem. A satisfying assignment yields a counterexample. For infinite-path properties, a lasso path can finitely represent a repeated execution. Failure to find a counterexample at one finite bound alone does not prove an unbounded property.^[ref]

For a state formula $\phi$ in a fixed labelled transition system, its satisfaction set is $\{s\in S:s\models \phi \}$. This is a set of states; it is distinct from the assertion that every initial state satisfies the formula.

A modal operator forms a new formula from a formula, such as $A\mapsto \square A$. In provability logic, it represents an assertion that a proof of $A$ exists, rather than an assertion that $A$ is true. A box is not interchangeable with an unboxed formula.

Provability logic studies modal principles valid when $\square A$ is interpreted by an arithmetical provability predicate. The distribution axiom corresponds to formalized modus ponens rule, necessitation corresponds to verification of actual proofs, and the Löb axiom is the Internal form of Löb's theorem. The standard system capturing these principles is Gödel-Löb logic.

The necessitation rule permits boxing a theorem in a modal proof system. It does not permit boxing an arbitrary temporary assumption. Under an arithmetic interpretation, it uses the external first of the derivability conditions: an actual proof of $A$ supplies a proof that $A$ has a proof. This distinction matters when proving the Internal form of Löb's theorem.

Positive introspection says that a boxed assertion can itself be boxed. For an arithmetical provability predicate it is the third of the Hilbert-Bernays-Löb derivability conditions: provability of $A$ implies provability of the assertion that $A$ is provable. It follows from formalized Sigma-1 completeness when provability assertions are Sigma-1. It is not the reflection principle $\square A\to A$.

An occurrence of a propositional variable in a modal formula is modalized if it lies within the scope of a box operator. A variable occurs modalized in a formula when all its occurrences do. For example $p$ is modalized in $\square p\to q$, while $q$ is not. Fixed-point and uniqueness theorems for Gödel-Löb logic impose this syntactic condition on the variable being solved for.

A modal formula is built from propositional variables and logical constants using Boolean connectives and modal operators, such as the box operator. In provability logic, arithmetic realization replaces propositional letters by arithmetic sentences and interprets a box as the corresponding arithmetic provability assertion. Modal validity, theoremhood and truth under a particular arithmetic realization are distinct notions.

The normal distribution axiom says that a boxed implication and its boxed antecedent give the boxed consequent. Together with the necessitation rule, it permits boxed versions of propositional deductions. Under an arithmetic interpretation it is the internal modus-ponens condition among the Hilbert-Bernays-Löb derivability conditions. It does not permit dropping a box to assert the enclosed sentence.

Modal logic K4 extends the normal modal distribution axiom and necessitation rule by positive introspection $\square A\to \square \square A$. Its Kripke semantics uses transitive accessibility relations. The fixed point $B=\square (\square X\to X)$ satisfies $\square (B\to \square X)\to (B\to \square X)$ in this logic.

Uniform substitution in modal logic replaces each occurrence of a first-order sentence letter by the same modal formula. Normal modal formal proof systems with axiom schemes are closed under this operation: substituted axioms remain instances, modus ponens is preserved, and necessitation commutes with substitution. Fresh auxiliary letters permit substituting a derived fixed point into fixed-point uniqueness theorems without changing other parameters.

A function continuous across an interface can have different one-sided normal derivatives. For instance, $u(x,y)=y$ when $y<0$ and $u(x,y)=0$ when $y\ge 0$ is continuous, satisfies $u_{xx}+\mathrm{sign}(y)u_{yy}=0$ on each open side, and has zero tangential second derivative on the interface. Its lower $y$-derivative is one and its upper $y$-derivative is zero. Thus the normal derivative does not exist there as a two-sided derivative.

More generally, for functions smooth on each side with one-sided first derivative traces, global $C^1$ regularity would force both traces to equal the derivative on the interface. Unequal traces therefore preclude a globally classical $C^2$ solution as well. A condition on the function value alone is not a condition on its normal derivative; a specified transmission law or an appropriate weak formulation is additional information.

Use signature $(+---)$ and future-pointing timelike vectors $U,V$. Their time components exceed their spatial magnitudes, so the same holds for their sum by the Euclidean triangle inequality. In the rest frame of $U$, write $U=(a,0)$, $a=\sqrt{g(U,U)}>0$, and let $b=\sqrt{g(V,V)}>0$. Then $g(U,V)=a{V}^0\ge ab$. Expanding the squared norm of $U+V$ proves the displayed inequality. Equality requires $V$ to have zero spatial component in this frame, equivalently $V$ is a positive scalar multiple of $U$. The shared future time orientation is essential.

This explicit numerical method is a two-stage Runge-Kutta method and a one-step method. For a sufficiently smooth ordinary differential equation, the chain rule gives $y^{\prime \prime }=f_x+f_{y}f$, and Taylor's theorem shows that its local truncation error is $O(h^3)$. Uniform Lipschitz continuity of $f$ in its solution argument gives an increment-function Lipschitz constant at most $L(1+hL/2)$. The Lipschitz one-step error bound then proves second-order numerical convergence on a fixed time interval. This method uses an intermediate prediction from $y_n$; the explicit midpoint linear multistep method instead requires a separate previous value $y_{n-1}$.

The Eustachian tube connects the middle ear to the nasopharynx, enabling pressure equalization, mucosal clearance and protection from nasopharyngeal contents.