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Concept

Satisfiability

The property of a formula for which some assignment of values makes it true. Deciding it for a formula in many variables is the model hard search problem, since the assignments to try double with each variable.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

(p ∨ q) ∧ ¬r, drawn on the cube of 8 assignments. The assignments as corners of a cube, joined when they differ in one variable, with the satisfying corners filled.

A formula is a corner of a cube

A formula about three letters is a set of eight rows. Written as a table that is a list; drawn on a cube it is a shape — and the shape is what almost every later question in this field turns out to be about.

logic · Truth functions A tableau for ((p → q) ∧ (q → r)) → (p → r). A branching tree of formulas, each branch ending in a contradiction or in a description of a counterexample.

The tree that closes

To prove a formula, assume it false and take it apart. Every branch ends in a contradiction, or one of them describes exactly how it could have been false — and either way the tree is the answer, drawn.

logic · Proof systems A tree branching at most 3 ways, to depth 4, and the path through it. A tree drawn level by level, with the nodes that die out faint and a highlighted path that always steps to a node with descendants at the bottom.

An infinite tree has an infinite path

A tree that goes on forever, in which every node has only finitely many children, must contain a single branch that goes on forever. The proof is a rule for walking, and the rule is the whole of why finite information can decide an infinite question.

logic · Models A resolution refutation of four clauses on three variables. A derivation tree: the given clauses at the top, each later clause obtained by cancelling one variable between two clauses above it, ending in the empty clause.

A proof with one rule

Two clauses that disagree about exactly one variable can be combined into a third that forgets it; repeat, and if the clauses cannot all be true the empty clause eventually appears — a complete proof system with a single move.

logic · Resolution Closed after 3 uses of the universal. The Herbrand expansion of a first-order question at 5 stages, with the number of remaining models at each. It reaches nought after 3 instantiations.

The instance that has to be guessed

Every rule of a propositional tableau replaces a formula by shorter ones, which is why it stops. The rule for a universal claim does not replace it — it keeps it and adds an instance — and one word changing turns a decision procedure into a search that may run forever.

logic · Proof systems

Named alongside it

The objects these essays reach for when they reach for this one.

CompletenessProof systemDecision procedureLiteralNormal formRefutationSoundnessTableauAssignmentBranchingCompactnessConnective

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