Fourth Part: The Interplay of Algebra and Logic

Transcription

1 Fourth Part: Th Intrplay of Algbra and Logic Francsco Paoli TACL 2013 Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

2 Adissibility of cut Cut liination, in proof-thortic parlanc, is constructiv: it iplis giving an algorith for roving cuts fro drivations. Th adissibility of cut for th cut-fr syst, howvr, can b attaind by non-constructiv algbraic thods, originating in th work of Mahara and Okada. Th cut-fr syst, in othr words, is shown to b coplt with rspct to so class of algbras. Such rsults suffi c for stablishing dcidability or intrpolation. Th algbraic approach to copltnss has ad squnt calculi or attractiv to algbraists, and proiss to lad to insights and proofs unobtainabl solly by syntactic ans. For xapl, Ciabattoni, Galatos, and Trui hav cobind algbraic and syntactic tchniqus to obtain an algorith that convrts Hilbrt-styl axios of a crtain for into structural ruls for squnt and hyprsqunt calculi that prsrv cut liination. Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

3 Adissibility of cut for GRL (1) Lt M = M,, 1 b a onoid, and for ach X, Y M, dfin: X Y = {x y : x X and y Y }; X \Y = {y M : X {y} Y }; Y /X = {y M : {y} X Y }. (M) = (M),,,, \, /, {1} is a rsiduatd lattic. A nuclus on th powrst (M) is a ap γ : (M) (M) satisfying X γ(x ), γ(γ(x )) γ(x ), X Y iplis γ(x ) γ(y ), and γ(x ) γ(y ) γ(x Y ). La If M is a onoid and γ is a nuclus on (M), thn (M) γ = γ( (M)),, γ, γ, \, /, γ({1}) is a coplt rsiduatd lattic with X γ Y = γ(x Y ) and X γ Y = γ(x Y ). Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

4 Adissibility of cut for GRL (2) W construct a rsiduatd lattic such that validity in this algbra corrsponds to cut-fr drivability in GRL. Lt F b th fr onoid gnratd by th forulas of GRL: th lnts of F ar finit squncs of forulas, ultiplication is concatnation, and th unit lnt is th pty squnc. Intuitivly, w build our algbra fro sts of squncs of forulas that play th sa rol in cut-fr drivations in GRL. W dfin: [Γ 1 _Γ 2 α] = {Γ F : Γ 1, Γ, Γ 2 α is cut-fr drivabl in GRL}; D = {[Γ 1 _Γ 2 α] : Γ 1, Γ 2 F and α F}; γ(x ) = {Y (F ) : X Y D}. Thn γ is a nuclus on (F ) and hnc th algbra (F ) γ is a rsiduatd lattic. Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

5 Adissibility of cut for GRL (3) W dfin an valuation for this algbra by (p) = γ ({p}) and prov by induction on forula coplxity that for ach α F: α (α) [_ α]. Now, lt α 1,..., α n β b such that RL α 1... α n β. So in particular (F ) γ α 1... α n β, whnc (α 1 )... (α n ) (β). Howvr, sinc α i (α i ) for i n and (β) [_ β], α 1,..., α n [_ β] which ans that α 1,..., α n β is cut-fr drivabl in GRL. Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

6 Dcision probls Syntactic thods Santic thods Validity probl cut liination FMP (Dcidability of th (prhaps for display quational thory) or hyprsqunt calculi) Consqunc probl (Dcidability of th quasiquational thory) SFMP Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

7 SFMP and FEP Givn an algbra A = A, fi A : i I of any typ and B A, a partial subalgbra B of A is th partial algbra B, fi B : i I whr for i I, k-ary f i, and b 1,..., b k B, { fi B f A (b 1,..., b k ) = i (b 1,..., b k ) if fi A (b 1,..., b k ) B undfind, othrwis. An bdding of a partial algbra B into an algbra A of th sa typ is a 1-1 ap ϕ : B A such that ϕ(fi B (b 1,..., b k )) = fi A (ϕ(b 1 ),..., ϕ(a k )) whnvr fi B (b 1,..., b k ) is dfind. A class K of algbras of th sa typ has th finit bddability proprty (FEP for short) if vry finit partial subalgbra of so br of K can b bddd into so finit br of K. For quasivaritis of finit typ such as (quasi)varitis of rsiduatd lattics, th FEP is quivalnt to th SFMP. Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

8 Th FEP for rsiduatd lattics (1) Hyting algbras: lt B b a finit partial subalgbra of so A HA. Th lattic D gnratd by B {0, 1} is a finitly gnratd distributiv lattic, hnc finit, vn though this ight not b tru of th Hyting algbra finitly gnratd by B. Sinc t is rsiduatd in any finit distributiv lattic, D can b ad into a Hyting algbra. Morovr, rsiduals coincid and thus B can b bddd into this algbra. A or coplicatd construction by Blok and Van Altn stablishs th FEP for nurous subvaritis of F L obying intgrality or idpotncy. In particular, this construction was usd by Ono to stablish th dcidability of various silinar varitis corrsponding to fuzzy logics. Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

9 Th FEP for rsiduatd lattics (2) For varitis of rsiduatd lattics such as RL and CRL that lack intgrality and idptncy, (vrsions of) th following algbra basd on th intgrs provids a good candidat for a countrxapl: Z = Z, in, ax, +,, 0, whr x y = x + y. Th quasi-quation 1 x & x y 1 x 1. holds in all finit rsiduatd lattics, but fails in Z. So th SFMP and th FEP fail for RL and CRL. Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

10 Dcidability: A synopsis Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

11 Aalgaation and intrpolation A varity V has th aalgaation proprty AP if for all A, B, C V and bddings i and j of A into B and C, rspctivly, thr xist D V and bddings h, k of B and C, rspctivly, into D such that h i = k j. Lt us writ var(k ) for th variabls occurring in so xprssion (forula, quation, st of quations, tc.) K. A varity V is said to hav th dductiv intrpolation proprty DIP if whnvr Σ Eq(V) ε, thr xists a st of quations Π with var(π) var(σ) var(ε) such that Σ Eq(V) Π and Π Eq(V) ε. For varitis with at last on nullary opration and all oprations of finit arity, th AP and th DIP ar quivalnt in th prsnc of th congrunc xtnsion proprty. So, this quivalnc holds for CRL and its subvaritis. Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

12 Th Craig intrpolation proprty A logic L has th Craig intrpolation proprty CIP if, whnvr L α β, thr xists a forula γ with var (γ) var (α) var (β) such that L α γ and L γ β. Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

13 Rlationships aong ths notions Thor Suppos that V CRL is an quivalnt algbraic santics for a logic L. Thn: 1 Γ {α} L β iff Γ L (α 1) n β for so n; 2 If L has th CIP, thn V has th DIP and hnc th AP. Proof. (1) Induction on th hight of drivations. (2) It suffi cs by algbraizability to prov th logical countrpart of th DIP for L. Suppos that Γ L α. Thn γ 1... γ n L α for so {γ 1,..., γ n } Γ and by (1), L (γ 1... γ n 1) n α for so n. If L has th CIP, thn L (γ 1... γ n 1) n γ and L γ α for so forula γ with var(γ) var(γ 1... γ n ) var(α). But thn, again by (1), {γ 1... γ n } L γ and {γ} L α as rquird. Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

14 Aalgaation for coutativ FL algbras (1) Thor FL has th CIP. Proof. W considr a ultist vrsion GFL of th squnt calculus GFL and prov th following statnt: If GFL Γ, α, thn thr is β with var (β) var (Γ) var (, α) such that GFL Γ β and GFL, β α. W procd by induction on th higth of a cut-fr drivation of Γ, α in GFL. If Γ, α is an instanc of ( ), thn: 1 if Γ = (α), = (), lt β = α; 2 if Γ = (), = (α), lt β = 1. Cass of th axios for 0 and 1: siilar. Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

15 Aalgaation for coutativ FL algbras (2) Proof. For th inductiv stp, w ust considr th last application of a rul in d. Exapl: iplication. Suppos first that α is α 1 α 2 and d nds with: By IH thr is β with var (β) var (Γ) var (, α 1 α 2 ) such that GFL Γ β and GFL, β, α 1 α 2. So GFL, β α 1 α 2 by ( ). Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

16 Aalgaation for coutativ FL algbras (3) Proof. If d nds with: thr ar two subcass: 1 If Γ is Γ 1, Γ 2 and is γ 1 γ 2, 1, 2, thn by IH (twic) thr ar β 1, β 2 s.t. GFL Γ i β i (i 2), GFL 1, β 1 γ 1 and GFL 2, γ 2, β 2 α (variabls OK). So, for β = β 1 β 2, Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

17 Aalgaation for coutativ FL algbras (4) Proof. 1 If Γ is Γ 1, Γ 2, γ 1 γ 2 and is 1, 2, thn w considr th drivabl squnt Γ 1, 1 γ 1 and by IH w gt a forula β 1 with var (β 1 ) var (Γ) var (, α) s.t. GFL 1 β 1, GFL Γ 1, β 1 γ 1. Arguing siilarly for Γ 2, 2, γ 2 α, w gt β 2 with var (β 2 ) var (Γ) var (, α) s.t. GFL Γ 2, γ 2 β 2 and GFL 2, β 2 α. So, for β = β 1 β 2, Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18

18 Aalgaation and intrpolation: A synopsis Francsco Paoli (Univ. of Cagliari) Tutorial on algbraic logic TACL / 18