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From Natural Deduction to Sequent Calculus (And Back) Reading Natural Deduction: Motivation Sequent Calculus: Motivation

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From Natural Deduction to Sequent Calculus (And Back) Reading Natural Deduction: Motivation Sequent Calculus: Motivation Reading F. Pfenning: Automated Theorem Proving, Course at Carnegie Mellon University. Draft. 1999. Introduction to Deduction Systems: A.S. Troelstra and H. Schwichtenberg: Basic Proof From Natural Deduction to Theory. Cambridge. 2nd Edition 2000. John Byrnes: Proof Search and Normal Forms in Natural Sequent Calculus Deduction. PhD Thesis. Carnegie Mellon University. 1999. (and back) . many more books on Proof Theory . Christoph Benzmüller Calculemus Autumn School, Pisa, Sep 2002 Calculemus Autumn School, Pisa, Sep 2002 Natural Deduction: Motivation Sequent Calculus: Motivation Frege, Russel, Hilbert Predicate calculus and type Gentzen had a pure technical motivation for sequent calculus theory as formal basis for mathematics Same theorems as natural deduction Gentzen ND as intuitive formulation of predicate calculus; Prove of the Hauptsatz (all sequent proofs can be found introduction and elimination rules for each logical connective with a simple strategy) Corollary: Consistency of formal system(s) The formalization of logical deduction, especially as it has been developed by The Hauptsatz says that every purely logical proof can be reduced to a defi- Frege, Russel, and Hilbert, is rather far removed from the forms of deduction nite, though not unique, normal form. Perhaps we may express the essential used in practice in mathematical proofs. In contrast I intended first to set properties of such a normal proof by saying: it is not roundabout. up a formal system which comes as close as possible to actual reasoning. In order to be able to prove the Hauptsatz in a convenient form, I had to The result was a calculus of natural deduction (NJ for intuitionist, NK for provide a logical calculus especially for the purpose. For this the natural classical predicate logic). [Gentzen: Investigations into logical deduction] calculus proved unsuitable. [Gentzen: Investigations into logical deduction] Calculemus Autumn School, Pisa, Sep 2002 Calculemus Autumn School, Pisa, Sep 2002 Sequent Calculus: Introduction Natural Deduction Sequent calculus exposes many details of fine Natural deduction rules operate on proof trees. structure of proofs in a very clear manner. Therefore it is well Example: ¢¡ ¢¡ suited to serve as a basic representation formalism for ¢¡¤£ ¢¥§¦ £ £ ¦ ¦ ¨ ¨ © ¨ ¨ ¨ £ £ ¦ ¦ many automation oriented search procedures ¨ Conjunction: Backward: tableaux, connection methods, matrix methods, some forms of resolution The presentation on the next slides treats the proof tree Forward: classical resolution, inverse method aspects implicit. Example: £ £ £ ¦ ¦ Don’t be afraid of the many variants of sequent calculi. ¦ ¨ ¨ © ¨ ¨ ¨ £ £ ¦ ¦ Choose the one that is most suited for you. Conjunction: ¨ Calculemus Autumn School, Pisa, Sep 2002 Calculemus Autumn School, Pisa, Sep 2002 Natural Deduction Rules Ia Natural Deduction Rules IIa £ £ £ ¦ ¦ ¦ £ ¨ ¨ ¡ © ¨ ¨ ¨ £ £ ¦ ¦ ¨ . Conjunction: . £ £ . ¡ © £ Disjunction: £ ¦ Negation: ¥ ¡ £ £ . ! . ! ! . © £ £ ! £ £ ¦ ¦ . " ¡ ¥ © © Universal Quantif.: £ £ ¦ ¦ £ £ ¡ . . £ . " £ . # . . # © # £ £ ¦ . ¦ £ # ¡ © £ ¦ ¦ Existential Quantif.: Implication: © *: parameter a must be new in context Truth and Falsehood: Calculemus Autumn School, Pisa, Sep 2002 Calculemus Autumn School, Pisa, Sep 2002 2002 2002 Sep Sep ant v Pisa, Pisa, used once the ND of School, School, be a irrele xts than utumn utumn in which A A is not us us on more Calculem Calculem order need © ¡ © ¥ Conte © used ) ¨ occurrence representation ) £ ) be xt. £ £ ula ¨ £ + ¨ ypothesis assumptions m ypotheses ¨ can £ £ h xplicit with h £ e or ( ¥ conte £ f + £ notation: ( * + xt £ ¥ £ each £ £ ( called + + conte or is f ypotheses ¡-, £ * h ¡ in £ ¡ £ (uncanceled) ypotheses; £ . h + ties the oof of oper Deduction Deduction pr pr depends ening action £ assumptions Localizing ultiset le eak m Exchange W Contr ula a m ailab is * v or Structural Idea: f a Example proof: Natural Natural & 2002 2002 ' Sep Sep $% £ £ Pisa, Pisa, . £ £ £ School, School, wing utumn utumn A A £ us us ollo f ¡ © Calculem Calculem IIIa the ¡ © © © ¨ ) ¨ of ) ¥ © £ ) ¡ ) £ © ¦ oofs £ ¨ £ ¨ ) ¨ one ) £ £ ( £ Rules Pr ¥ £ ¨ ¨ £ £ ( £ ¦ ¨ ( ( ( £ ¡ ¨ ¨ £ hoose £ ( c ¦ ) £ ¡ ¦ ¦ ¨ ¦ ¨ £ adiction £ £ logic ( Middle Contr Negation Deduction Deduction y b le lassical c Excluded Doub Proof For Natural Natural 2002 2002 ! # Sep Sep xt + + + + + + Pisa, Pisa, £ £ ¦ £ £ £ £ * £ + + + + * School, School, £ conte ¦ £ , , , , once) * * £ * * ! xts , in * * " * utumn utumn , , + A A , + + w us us £ © * than * * ne Calculem Calculem + + IIb © be * * , ! * more + ant) Conte v used) * ust £ £ © £ m be used a irrele # ! £ not is be + + + with Rules * + + £ * ensure can * * ameter need * order © , to # par £ *: £ .: ties .: # ypotheses ypothesis " + ypotheses (h + (h alsehood: oper Quantif * (h F Quantif * pr Deduction Deduction and ening action ersal eak uth r Exchange W Contr T Negation: Univ Existential Structural Natural Natural 2002 2002 £ © Sep Sep ¨ + ula of Pisa, Pisa, . ¦ ¦ £ m ¦ ¨ + ¦ + School, School, £ or , f + £ xts * + £ ¦ * utumn utumn * A A , + * + instead ¦ * us us , * ¦ * + which * Calculem Calculem © Ib © on + £ © ¨ Conte + £ + © ¦ ) ) * ¨ ¦ * * £ £ , ) ¨ £ assumptions £ £ ¨ + £ © ¦ + £ ¨ + of * + £ £ £ * with Rules + ( £ * £ ¨ + assumptions * ( £ ¦ sets * £ ¦ + £ + + £ © * ( £ ¨ £ £ + £ + + ¦ £ * £ £ , * ¦ + Consider + ¨ * (uncanceled) £ oof: of £ + Deduction Deduction + * pr . 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