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| In [[proof complexity]], a '''Frege system''' is a [[propositional proof system]] whose proofs are sequences of [[propositional formula|formulas]] derived using a finite set of [[soundness|sound]] and implicationally complete [[inference rules]]. Frege systems (more often known as [[Hilbert system]]s in general [[proof theory]]) are named after [[Gottlob Frege]].
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| ==Formal definition==
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| Let ''K'' be a finite [[functional completeness|functionally complete]] set of Boolean connectives, and consider [[propositional formula]]s built from variables ''p''<sub>0</sub>, ''p''<sub>1</sub>, ''p''<sub>2</sub>, ... using ''K''-connectives. A Frege rule is an inference rule of the form
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| :<math>r=\frac{B_1,\dots,B_n}B,</math>
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| where ''B''<sub>1</sub>, ..., ''B<sub>n</sub>'', ''B'' are formulas. If ''R'' is a finite set of Frege rules, then ''F'' = (''K'',''R'') defines a derivation system in the following way. If ''X'' is a set of formulas, and ''A'' is a formula, then an ''F''-derivation of ''A'' from axioms ''X'' is a sequence of formulas ''A''<sub>1</sub>, ..., ''A<sub>m</sub>'' such that ''A<sub>m</sub>'' = ''A'', and every ''A<sub>k</sub>'' is a member of ''X'', or it is derived from some of the formulas ''A<sub>i</sub>'', ''i'' < ''k'', by a substitution instance of a rule from ''R''. An ''F''-proof of a formula ''A'' is an ''F''-derivation of ''A'' from the empty set of axioms (<math>X=\varnothing</math>). ''F'' is called a Frege system if
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| *''F'' is sound: every ''F''-provable formula is a tautology.
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| *''F'' is implicationally complete: for every formula ''A'' and a set of formulas ''X'', if ''X'' entails ''A'', then there is an ''F''-derivation of ''A'' from ''X''.
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| The length (number of lines) in a proof ''A''<sub>1</sub>, ..., ''A<sub>m</sub>'' is ''m''. The size of the proof is the total number of symbols.
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| A derivation system ''F'' as above is refutationally complete, if for every inconsistent set of formulas ''X'', there is an ''F''-derivation of a fixed contradition from ''X''.
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| == Examples ==
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| * [[Frege's propositional calculus]] is a Frege system.
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| * There are many examples of sound Frege rules on the [[Propositional calculus]] page.
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| * [[Resolution (logic)|Resolution]] is not a Frege system because it only operates on clauses, not on formulas built in an arbitrary way by a functionally complete set of connectives. Moreover, it is not implicationally complete, i.e. we cannot conclude <math>A \lor B</math> from <math>A</math>. However, adding the ''weakening'' rule: <math>\frac{A}{A \lor B}</math> makes it implicationally complete. Resolution is also refutationally complete.
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| == Properties ==
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| *Reckhow's theorem (1979) states that all Frege systems are [[propositional proof system#Relation with computational complexity theory|p-equivalent]].
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| * [[Natural deduction]] and [[sequent calculus]] (Gentzen system with cut) are also p-equivalent to Frege systems.
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| * There are polynomial-size Frege proofs of the [[pigeonhole principle]] (Buss 1987).
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| * Frege systems are considered to be fairly strong systems. Unlike, say, resolution, there are no known superlinear lower bounds on the number of lines in Frege proofs, and the best known lower bounds on the size of the proofs are quadratic.
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| * The minimal number of rounds in the ''prover-adversary'' game needed to prove a tautology <math>\phi</math> is proportional to the logarithm of the minimal number of steps in a Frege proof of <math>\phi</math>.
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| <gallery widths="350px" heights="350px">
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| File:Proofstrength.png|Proof strengths of different systems.
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| </gallery>
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| == References ==
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| *Krajíček, Jan (1995). "Bounded Arithmetic, Propositional Logic, and Complexity Theory", Cambridge University Press.
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| *{{cite article|first1=Stephen|last1=Cook|authorlink1=Stephen Cook|first2=Robert A.|last2=Reckhow|title=The Relative Efficiency of Propositional Proof Systems|journal=Journal of Symbolic Logic|volume=44|number=1|year=1979|pages=36–50|jstor=2273702}}
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| *Buss, S. R. (1987). "Polynomial size proofs of the propositional pigeonhole principle", Journal of Symbolic Logic 52, pp. 916–927.
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| *Pudlák, P., Buss, S. R. (1995). "How to lie without being (easily) convicted and the lengths of proofs in propositional calculus", in: Computer Science Logic'94 (Pacholski and Tiuryn eds.), Springer LNCS 933, 1995, pp. 151–162.
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| === Further reading ===
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| *MacKay, D. J. (2008). "Information Theory, Inference, and Learning Algorithms"
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| [[Category:Propositional calculus]]
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| [[Category:Logic in computer science]]
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I like Seashell Collecting.
I try to learn Norwegian in my spare time.
my website; Cannondale mountain bike sizing.