TermsforClassical Sequents Proof Invariants & Strong Normalisation Greg Restall gothenburg logic seminar · 10 may 2016
My Aim To introduce a new invariant for classical propositional proofs and to show how they can be used. Greg Restall Terms for Classical Sequents 2 of 67
Today's Plan Background Preterms Derivations Terms Eliminating Cuts Strong Normalisation Further Work Greg Restall Terms for Classical Sequents 3 of 67
background
5 of 67 R Terms for Classical Sequents Greg Restall R L I E L When is π 1 the same proof as π 2 ? p � p p � p p ∧ q ∨ R ∧ L ∧ E p � p ∨ q p ∧ q � p p ∧ L ∨ R ∨ I p ∧ q � p ∨ q p ∧ q � p ∨ q p ∨ q
5 of 67 Greg Restall Terms for Classical Sequents When is π 1 the same proof as π 2 ? p � p p � p p ∧ q ∨ R ∧ L ∧ E p � p ∨ q p ∧ q � p p ∧ L ∨ R ∨ I p ∧ q � p ∨ q p ∧ q � p ∨ q p ∨ q q � q q � q p ∧ q ∨ R ∧ L ∧ E q � p ∨ q p ∧ q � q q ∧ L ∨ R ∨ I p ∧ q � p ∨ q p ∧ q � p ∨ q p ∨ q
6 of 67 Are these different proofs , or different ways of presenting the same proof ? Terms for Classical Sequents Greg Restall When is π 1 the same proof as π 2 ? [ p ] 1 [ q ] 1 [ p ] 1 [ q ] 1 ∨ I ∨ I ∨ I ∨ I p ∨ q q ∨ p q ∨ p q ∨ p q ∨ p ∨ E 1 ∨ I ∨ I ( q ∨ p ) ∨ r ( q ∨ p ) ∨ r q ∨ p p ∨ q ∨ E 1 ∨ I ( q ∨ p ) ∨ r ( q ∨ p ) ∨ r
Girard, Lafont and Taylor: ProofsandTypes , Chapter 2 Greg Restall Terms for Classical Sequents 7 of 67 Natural deduction is a slightly paradoxical system: it is limited to the intuitionistic case (in the classical case it has no particularly good properties) but it is only satisfactory for the ( ∧ , ⇒ , ∀ ) fragment of the language: we shall defer consideration of ∨ and ∃ until chapter 10. Yet disjunction and existence are the two most typically intuitionistic connectors! The basic idea of natural deduction is an asymmetry: a proof is a vaguely tree-like structure (this view is more a graphical illusion than a mathematical reality, but it is a pleasant illusion) with one or more hypotheses (possibly none) but a single conclusion. The deep symmetry of the calculus is shown by the introduction and elimination rules which match each other exactly. Observe, incidentally, that with a tree-like structure, one can always decide uniquely what was the last rule used, which is something we could not say if there were several conclusions.
LambdaTerms and Proofs Greg Restall Terms for Classical Sequents 8 of 67 [ x : p ⊃ ( q ⊃ r )] [ z : p ] [ y : p ⊃ q ] [ z : p ] ⊃ E ⊃ E xz : q ⊃ r yz : q ⊃ E ( xz )( yz ) : r ⊃ I λz ( xz )( yz ) : p ⊃ r ⊃ I λyλz ( xz )( yz ) : ( p ⊃ q ) ⊃ ( p ⊃ r ) ⊃ I λxλyλz ( xz )( yz ) : ( p ⊃ ( q ⊃ r )) ⊃ (( p ⊃ q ) ⊃ ( p ⊃ r ))
E Contraction and weakening are managed by variables E I Greg Restall Terms for Classical Sequents 9 of 67 [ x : p ] ⊃ I λy x : q ⊃ p ⊃ I λxλy x : p ⊃ ( q ⊃ p )
Contraction and weakening are managed by variables Greg Restall Terms for Classical Sequents 9 of 67 x : p ⊃ ( p ⊃ q ) [ y : p ] [ x : p ] ⊃ E xy : p ⊃ q [ y : p ] ⊃ I λy x : q ⊃ p ⊃ E ( xy ) y : q ⊃ I λxλy x : p ⊃ ( q ⊃ p ) ⊃ I λy ( xy ) y : p ⊃ q
Classical Sequent Derivations L Terms for Classical Sequents Greg Restall R L R 10 of 67 p � p p � p ¬ R ¬ L � p, ¬ p p, ¬ p � ∨ R ∧ L � p ∨ ¬ p p ∧ ¬ p �
Classical Sequent Derivations Greg Restall Terms for Classical Sequents 10 of 67 p � p p � p ¬ R ¬ L � p, ¬ p p, ¬ p � ∨ R ∧ L � p ∨ ¬ p p ∧ ¬ p � q � q r � r ∨ L p � p q ∨ r � q, r ∧ R p, q ∨ r � p ∧ q, r ∧ L p ∧ ( q ∨ r ) � p ∧ q, r ∨ R p ∧ ( q ∨ r ) � ( p ∧ q ) ∨ r
Sequents and Terms Where do you put the variables , and where do you put the terms ? Greg Restall Terms for Classical Sequents 11 of 67 X � Y X � A, Y X, A � Y
Our Choice Each premise and conclusion is decorated with variables. The sequent gets the term, showing how inputs & outputs are connected, with as much parallelism as possible. Greg Restall Terms for Classical Sequents 12 of 67 x 1 : A 1 , . . . , x n : A n � y 1 : B 1 , . . . , y m : B m
Our Choice Each premise and conclusion is decorated with variables. The sequent gets the term, showing how inputs & outputs are connected, with as much parallelism as possible. Greg Restall Terms for Classical Sequents 12 of 67 x 1 : A 1 , . . . , x n : A n � y 1 : B 1 , . . . , y m : B m
Our Choice Each premise and conclusion is decorated with variables. The sequent gets the term, showing how inputs & outputs are connected, with as much parallelism as possible. Greg Restall Terms for Classical Sequents 12 of 67 π ( x 1 , . . . , x n )[ y 1 , . . . , y m ] x 1 : A 1 , . . . , x n : A n � y 1 : B 1 , . . . , y m : B m
Example 1 Greg Restall Terms for Classical Sequents 13 of 67 y ⌢ y z ⌢ z y : q � y : q z : r � z : r ∨ L x ⌢ x L w ⌢ y R w ⌢ z x : p � x : p w : q ∨ r � y : q, z : r ∧ R x ⌢ F v L w ⌢ S v R w ⌢ z x : p, w : q ∨ r � v : p ∧ q, z : r ∧ L F u ⌢ F v LS u ⌢ S v RS u ⌢ z u : p ∧ ( q ∨ r ) � v : p ∧ q, z : r ∨ R F u ⌢ FL t LS u ⌢ SL t RS u ⌢ R t u : p ∧ ( q ∨ r ) � t : ( p ∧ q ) ∨ r
Example 2 Cut Terms for Classical Sequents Greg Restall 14 of 67 x ⌢ x x ⌢ x z ⌢ z x : p � x : p x : p � x : p z : p � z : p ∧ R ∧ L x ⌢ F y x ⌢ S y F w ⌢ z x : p � y : p ∧ p w : p ∧ p � z : p x ⌢ F • x ⌢ S • F • ⌢ z x : p � z : p
preterms
Variables and Cut Points . Terms for Classical Sequents Greg Restall and cut points, ommitting type superscripts where possible. as schematic letters for variables ; – We use are cut points of type , , , For each formula 16 of 67 ▶ For each formula A , x A 1 , x A 2 , . . . are variables of type A .
Variables and Cut Points – We use ; as schematic letters for variables and cut points, ommitting type superscripts where possible. Greg Restall Terms for Classical Sequents 16 of 67 ▶ For each formula A , x A 1 , x A 2 , . . . are variables of type A . ▶ For each formula A , • A 1 , • A 2 , . . . are cut points of type A .
and cut points, ommitting type superscripts where possible. Variables and Cut Points Greg Restall Terms for Classical Sequents 16 of 67 ▶ For each formula A , x A 1 , x A 2 , . . . are variables of type A . ▶ For each formula A , • A 1 , • A 2 , . . . are cut points of type A . – We use x, y, z, u, v, w, . . . ; • , ⋆ , ∗ , ♯ , ♭ as schematic letters for variables
Nodes and Subnodes node. is a node. If is a node, then N is an For each complex node L , R , F , S , A , C is an and N , is its immediate subnode, and the subnodes of are also subnodes of the original node. Greg Restall Terms for Classical Sequents node and C node, then A 17 of 67 node. If is an node, then L is an node and R is a If is an is an node, then F is an node and S is a node. If ▶ A variable x of type A and a cut point • of type A are both A nodes .
For each complex node L , R , F , S , A , C Nodes and Subnodes and N , is its immediate subnode, and the subnodes of are also subnodes of the original node. Greg Restall Terms for Classical Sequents 17 of 67 ▶ A variable x of type A and a cut point • of type A are both A nodes . ▶ If n is an A ∧ B node, then L n is an A node and R n is a B node. ▶ If n is an A ∨ B node, then F n is an A node and S n is a B node. ▶ If n is an A ⊃ B node, then A n is an A node and C n is a B node. ▶ If n is a ¬ A node, then N n is an A node.
Greg Restall Nodes and Subnodes Terms for Classical Sequents 17 of 67 ▶ A variable x of type A and a cut point • of type A are both A nodes . ▶ If n is an A ∧ B node, then L n is an A node and R n is a B node. ▶ If n is an A ∨ B node, then F n is an A node and S n is a B node. ▶ If n is an A ⊃ B node, then A n is an A node and C n is a B node. ▶ If n is a ¬ A node, then N n is an A node. ▶ For each complex node L n , R n , F n , S n , A n , C n and N n , n is its immediate subnode, and the subnodes of n are also subnodes of the original node.
Linkings, Inputs and Outputs If A Terms for Classical Sequents Greg Restall position of that linking. The inputs ( outputs ) of a linking are the variables in input ( output ) – A and N reverse position . is in input position . is in output position , or N If A is in output position . is in input position , or N – L , R , F , S and C each preserve position . is also in output position . are in output position , or C If L , R , F , S is also in input position . are in input position , or C If L , R , F , S Positions generalise to subnodes as follows: is in output position . is in input position , and , In 18 of 67 ▶ A linking is a pair n ⌢ m of nodes of the same type.
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