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Introducing Substitution in Proof Theory

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Introducing Substitution in Proof Theory Introducing Substitution in Proof Theory Alessio Guglielmi University of Bath 20 July 2014 This talk is available at http://cs.bath.ac.uk/ag/t/ISPT.pdf It is an abridged version of this other talk: http://cs.bath.ac.uk/ag/t/PCMTGI.pdf Deep inference web site: http://alessio.guglielmi.name/res/cos/ Outline Problem: compressing proofs. Solution: proof composition mechanisms beyond Gentzen. Open deduction: composition by connectives and inference, smaller analytic proofs than in Gentzen. Atomic flows: geometry is enough to normalise. Composition by substitution: more geometry, more efficiency, more naturality. ON THE PROOF COMPLEXITY OF DEEP INFERENCE 21 SKSg can analogously be extended, but there is no need to create a special rule; we only need to broaden the criterion by which we recognize a proof. Definition 5.3. An extended SKSg proof of α is an SKSg derivation with conclusion α ¯ ¯ ¯ ¯ ¯ ¯ and premiss [A1 β1] [β1 A1] [Ah βh ] [βh Ah ],whereA1, A1,...,Ah , Ah ∨ ∧ ∨ ∧ ···∧ ∨ ∧ ∨ are mutually distinct and A1 / β1,α and ... and Ah / β1,...,βh ,α.WedenotebyxSKSg the proof system whose proofs∈ are extended SKSg proofs.∈ Theorem 5.4. For every xFrege proof of length l and size n there exists an xSKSg proof of 2 the same formula and whose length and size are, respectively, O(l ) and O(n ). Proof. Consider an xFrege proof as in Definition 5.1. By Remark 5.2 and Theorem 4.6, there exists the following xSKSg proof, whose length and size are yielded by 4.6: ¯ ¯ ¯ ¯ [A1 β1] [β1 A1] [Ah βh ] [βh Ah ] ∨ ∧ ∨ ∧ ···∧ ∨ ∧ ∨ $ SKSg . $ αk ! Although not strictly necessary to establish the equivalence of the four extended for- malisms (see diagram in the Introduction), the following theorem is very easy to prove. Theorem 5.5. For every xSKSg proof of size n there exists an xFrege proof of the same 4 5 formula and whose length and size are, respectively, O(n ) and O(n ). Proof. Consider an xSKSg proof as in Definition 5.3. The statement is an immediate consequence of Theorem 4.11, after observing that there is an O(h)-length and O(hn)- size xFrege proof A β ,...,A β ,..., A β A β . Problem: compressing1 ↔ 1 h ↔ proofsh ( 1 ↔ 1) ∧ ∧ ( h ↔ h ) ON THE PROOF COMPLEXITY··· OF DEEP INFERENCE 7 ! CorollaryHow can 5.6.we makeSystems proofsxFrege smaller?and xSKSgKnownareStructural mechanisms:p-equivalent.rules Logical rule We1. Re-use now move the to same thesubstitution sub-proof: cut rule. rule. Proof theory. A A¯ A A Definition2. Re-use 5.7. theA samesubstitution sub-proof:i Frege∧ dagness(proof,) orsystem cocontractionw is a Frege: systemc augmented. with ↑ f ↑ t ↑ A A A ∧ the3.substitution Substitution rule: sub cointeraction.WedenotebyIn Frege, equivalentsFrege coweakeningthe to (4). proof system cocontraction where a proof is a Aσ or cut derivation with no premisses, conclusion α ,andshape 4. Tseitin extensionSKSg : p A (where p kis a fresh atom). Optimal? t f A A A B C $αi αi ∨ ∧ [ ∨ ] 5. Higher orders (including 1i 2nd order propositional).w h c s ↓≡A A¯ ↓ A ≡ ↓ A A B C α ,...,α , α σ ,∨α ,...,α , α σ ,α ,...,α , ( ∧ ) ∨ KSg 1 i1 1 j1 1 i1+1 ih 1 jh h ih +1 k − interaction weakening− contraction switch where all the conclusions of substitution instances α ,...,α are singled out, α We will see that 1– 4 (and !or a"# bit$ also 5) have a lot! to"#i1 do$ withih j1 identity ∈ αproof,..., compositionα ,...,α: thisα is our,..., mainα tool.,andtherestoftheproofisasinFrege. 1 i1 1 jh 1 ih 1 { − } ∈{ − } We rely on the following result. FIGURE 2. Systems SKSg and KSg. Our main objective is providing for small spaces of canonical proofs Theorem(= eliminating 5.8. (Cook-Reckhow bureaucracy = getting and Krají goodˇcek-Pudlák, proof semantics).[CR79, KP89]) Systems xFrege and sFrege are p-equivalent. Atomic structural rules Logical rules We can extend SKSg witha a¯ the same substitutiona rule asa for Frege.Theruleisusedlike ai ∧ aw ac other proper rules of system↑ f SKSg,soitsinstancesareinterleavedwith↑ t ↑ a a =-rule instances. ∧ Definition 5.9. AncointeractionsSKSg proof is coweakening a proof of SKSg cocontractionwhere, in addition to the inference steps generated by rulesor ofcutSKSg,weadmitinferencestepsobtainedasinstancesofthe SKS A substitution rulesub . t f a a A [B C ] (A B) (C D) Aaiσ aw ac ∨ s ∧ ∨ m ∧ ∨ ∧ ↓ a a¯ ↓ a ↓ a (A B) C [A C ] [B D] ∨ ∧ ∨ ∨ ∧ ∨ KS interaction weakening contraction switch medial or identity FIGURE 3. Systems SKS and KS. We can now define some deep-inference proof systems. System SKS is the most impor- tant for the proof theory of classical logic, because of its atomic structural rules. System SKSg relates SKS to proof systems in other formalisms, like Frege. Definition 2.13. CoS proof systems KSg = i ,w ,c ,s , SKSg = KSg i ,w , { ↓ ↓ ↓ } ∪ { ↑ ↑ c , KS = ai ,aw ,ac ,s,m and SKS = KS ai ,aw ,ac are defined in Figures 2 and↑} 3, for a{ language↓ ↓ containing↓ } f, t, disjunction,∪ { and↑ conjunction.↑ ↑} Proof systems where none of the rules i , ai , w ,andaw appear are said to be analytic. ↑ ↑ ↑ ↑ Example 2.14. This is a valid derivation in all CoS proof systems defined previously (and it plays a role in the proof of Lemma 3.11): γ [(([α¯ α] c) (α d)) δ] = ∨ ∨ ∧ ∧ ∧ ∨ γ [(((α d) c) [α α¯]) δ] s ∨ ∧ ∧ ∧ ∨ ∨ . γ [[(((α d) c) α) α¯] δ] = ∨ ∧ ∧ ∧ ∨ ∨ α¯ γ α c α d δ [ ∨ ] ∨ [(( ∧ ) ∧ ( ∧ )) ∨ ] Note that SKSg, KSg, SKS,andKS are closed under renaming and substitution (see Remark 2.11). This is so because of the distinction between atoms and formula variables. Obtaining the closure of these and other systems under renaming and substitution is one of the main technical reasons for distinguishing between atoms and variables. The following theorem is proved in [Brü04],andfollowsimmediatelyfromSec- tion 3.1, where we prove that CoS systems p-simulate Gentzen systems. Solution: proof composition mechanisms beyond Gentzen Less is more. Let's make better use of what we have already. Given two proofs φ : A B and : C D: ) ) 1. For a logical connective ? we have: φ ? :(A ? C) (B ? D) : ) Proofs are composed by the connectives of the formula language. 2. For an inference rule B=C we have: φ/ψ : A D : ) Proofs are composed by inference rules. 3. For an atom a we have: φ a : A a C; ¯a D¯ B a D; ¯a C¯ : f g f g ) f g Proofs are composed by substitution. Two (relatively) new formalisms Open deduction: composition by (1) connectives and (2) inference rules. I It exists [5] and it can be taken as a definition for deep inference. I It generalises the sequent calculus by removing its restrictions. I Sequents ‘depth-1 inference without full inference composition'. ≈ I Hypersequents ‘depth-2 inference without full inference ≈ composition'. I It compresses proofs by cut, dagness and depth itself (new, exponential speed-up). I I'll show the main ideas and some results. ‘Formalism B': open deduction + (3) substitution. I It almost exists (work with Bruscoli, Gundersen and Parigot). I It further compresses proofs by substitution (conjectured further superpolynomial speed-up, it is equivalent to Frege + substitution). I I'll show some ideas and what I think we can get. 4 PAOLA BRUSCOLI, ALESSIO GUGLIELMI, TOM GUNDERSEN, AND MICHEL PARIGOT 4 PAOLA BRUSCOLI, ALESSIO GUGLIELMI, TOM GUNDERSEN, AND MICHEL PARIGOT ξ γ instances of α and β, respectively, generates an (inference) step ρ { }, for each context ξ ξδ C instances of A and B, respectively, generates an (inference) step ρ{ {} }, for each context ξ .Aderivation, Φ, from α (premiss) to β (conclusion) is a chain ofξ−−−−−−−− inferenceD steps with {} { } ξ .Aderivation, Φ, from A (premiss) to B (conclusion) is aα chain of inference steps with {} α at the top and β at the bottom, and is usually indicated by Φ ! A, where is the name " A at the top and B at the bottom, and is usually indicated by! Φ , where" is the name β B of the proof system or a set of inference rules (we might omit and ); a proof, often of the proof system or a set of inference rules (we might omitΦ and ); a proof, often denoted by , is a derivation with premiss t; besides , we denoteΦ derivations" with . denoted byΠ , is a derivation with premiss t; besidesΦ , we denote derivations withΨ . Sometimes weΠ group n 0 inference steps of the same ruleΦ ρ together into one step, andΨ Sometimes we group!n 0 inference steps of the same rule ρ together into one step, and we label the step with n ρ. we label the step with n ρ. The size α of a formula· α, and the size of a derivation , is the number of unit The size A of a formula· A, and the sizeΦ of a derivationΦ , is the number of unit and atom occurrences| | appearing in it. | | Φ Φ and atom occurrences| | appearing in it. | | By α a /β ,...,a /β , we denote the operation of simultaneously substituting for- By A1 a /1 B ,...,h a /hB , we denote the operation of simultaneously substituting for- mulae β{ ,...,1 β1 intoh all}h the occurrences of the atoms a ,..., a in the formula α, mulae1B{,...,hB into all} the occurrences of the atoms1 a ,...,h a in the formula A, respectively;1 note thath the occurrences of a¯ ,..., a¯ are not automatically1 h substituted. respectively; note that the occurrences of1 a¯ ,...,h a¯ are not automatically substituted. Often, we only substitute certain occurrences of1 atoms,h and these are indicated with su- Often, we only substitute certain occurrences of atoms, and these are indicated with su- perscripts that establish a relation with atomic flows.
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