User:Bynne

= Frege Systems =

In propositional calculus a Frege system is an implicationally complete deduction system with a set of logical connectives and a set of sound inference rules.

Implicational completeness
We say that a deduction system $$I$$ is implicationally complete iff, for every formula $$S$$ that is a tautology, there exists a derivation from the axioms using the rules of the system.

Formal Definition
Let $$\mathcal{K}$$ be a set of boolean connectives on which we will define our system, let $$\mathcal{R}$$ be a set of sound inference rules. We denote the deduction system by $$\mathcal{F} = (\mathcal{K},\mathcal{R})$$, if $$\mathcal{F}$$ is implicationally complete we call it a Frege system.

F-Proof
Let $$A$$ and $$B$$ be formulas. Let $$F$$ be a sequence of formulas$$A_1, \ldots ,A_n, B$$ such that we can derive $$A_i$$ from $$A_1, \ldots ,A_{i-1},A$$ using a rule from $$\mathcal{R}$$, and such that we can derive $$B$$ from $$A_1, \ldots ,A_n,A$$ using a rule from $$\mathcal{R}$$. We call $$F$$, an $$\mathcal{F}$$-proof/derivation of the formula $$B$$ from $$A$$.

Refutational completeness
We define a Frege system to be refutationally complete if for every formula $$S$$ that is a contradiction, there exists an $$\mathcal{F}$$-derivation of False from $$S$$.

Length of formulas
We define the length of a $$\mathcal{F}$$-proof to be the number of applications of rules in the proof.

Examples
Rules are interpreted as follows, the two statements above the line entails the one below the line.
 * Common axioms are:
 * $$(\ ;A \lor \overline A)$$
 * $$(\ ;A \to A)$$
 * Common rules are:
 * $$\frac{A \quad A \to B}{B}$$ (Modus Ponens rule).
 * $$\frac{A \lor C \quad B \lor \overline C}{A \lor B}$$ (Resolution rule).
 * Resolution is not a Frege system because it is not implicationally complete, i.e. we cannot conclude $$A \lor B$$ from $$A$$- However adding the weakening rule: $$\frac{A}{A \lor B}$$ makes it implicationally complete and hence a Frege System.
 * Resolution is refutationally complete.
 * Natural deduction is a Frege system
 * Gentzen with cut is a Frege system
 * Frege's propositional calculus

Properties

 * A Frege proof can be converted to a sequent proof with at most a polynomial increase in length.
 * The minimal number of rounds in the prover-adversary game needed to prove a tautology $$\phi$$ is proportional to the logarithm of the minimal number of steps in a Frege proof of $$\phi$$.
 * Any Frege system is equivalent to a Gentzen with cut system, in the sense that there exists a translation of any frege proof/refutation to a Gentzen proof/refutation, with atmost a polynomial increase in length.
 * Same holds true for Natural deduction