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Permutability of Proofs in Intuitionistic Sequent Calculi

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Permutability of Proofs in Intuitionistic Sequent Calculi Permutabilityofproofsinintuitionistic sequent calculi Roy Dyckho Scho ol of Mathematical Computational Sciences St Andrews University St Andrews Scotland y Lus Pinto Departamento de Matematica Universidade do Minho Braga Portugal Abstract Weprove a folklore theorem that two derivations in a cutfree se quent calculus for intuitioni sti c prop ositional logic based on Kleenes G are interp ermutable using a set of basic p ermutation reduction rules derived from Kleenes work in i they determine the same natu ral deduction The basic rules form a conuentandweakly normalisin g rewriting system We refer to Schwichtenb ergs pro of elsewhere that a mo dication of this system is strongly normalising Key words intuitionistic logic pro of theory natural deduction sequent calcu lus Intro duction There is a folklore theorem that twointuitionistic sequent calculus derivations are really the same i they are interp ermutable using p ermutations as de scrib ed by Kleene in Our purp ose here is to make precise and provesuch a p ermutability theorem Prawitz showed howintuitionistic sequent calculus derivations determine LJ to NJ here we consider only natural deductions via a mapping from the cutfree derivations and normal natural deductions resp ectively and in eect that this mapping is surjectiveby constructing a rightinverse of from NJ to LJ Zucker showed that in the negative fragment of the calculus c LJ ie LJ including cut two derivations have the same image under i they are interconvertible using a sequence of p ermutativeconversions eg p ermutations of logical rules with the cut rule In the presentpaperwe provea Supp orted by the Europ ean Commission via the ESPRIT BRA GENTZEN y Supp orted by the Centro de Matematica da Universidade do Minho Braga Portugal similar result for a cutfree system making precise the idea referred to ab ove In fact we showhow certain p ermutation reduction rules can b e used to reduce an arbitrary derivation to normal form and that the set of such reductions is conuent With minor changes this system is strongly normalising wepointto Schwichtenb ergs for a pro of of this Our interest in these problems arises from the theory of logic programming regarded as in as based on pro of search in a cutfree system if one asks not just What problems are solvable but What solutions do these problems have and Howmany times is each solution obtained one is led to analyse the manyone relationship b etween sequent calculus derivations suitable for pro of search and natural deductions suitable for presenting solutions In fact Herb elins sequent calculus describ ed b elow is a much b etter basis for pro of search than LJ so the original problem disapp ears nevertheless in view of the historical imp ortance of Gentzens calculus and Kleenes variantofit G the p ermutability theorem is of indep endentinterest Mints pap er on the same topic came to our attention in Octob er when an early version of this pap er was b eing distributed we discuss the re lationship b etween his work and our own in x We thank Herb elin Mints Schwichtenberg and Tro elstra for advance copies of their resp ec tivelyWe are pleased to acknowledge that Herb elins pap ers lled the gap b etween the usual denition of normal lamb daterms representing natural deductions and Prawitz denition of our name for the rightinverse of Background Herb elins calculus M Herb elin gives a nonstandard description with origins in of terms representing normal natural deductions Consider rst a standard description of normal terms of the untyp ed lamb da calculus A apA N j vrV N V N j anA where V is a set of variables N is the set of normal terms and A is the set of application terms We use explicit constructors an and vr to ensure consistency with our typ echecked implementations The head variable of such a term is for a large term buried deep inside Herb elins representation brings it to the wemake the following surface So following Herb elin who calls the calculus Denition The set M of untypeddeduction terms and the set Ms of lists of such terms are dened simultaneously as fol lows M V Ms j V M Ms j M Ms Note the use again of the same symbol The notation M M abbreviates n the term M M The suggestion that such terms are lists is adequate n while we deal with implication alone but not when we add the other connectives Terms are equal i they are alphaconvertible we use the symbol for this relation Adding typ e restrictions gives us a description of the typable deduction terms We call the asso ciated typ ed system MJasitisintermediate b etween LJ and NJ rather than use Herb elins name LJT already used in There is a bijective translation b etween M and N mentioned but not de tailed in xM M translates into the term apapx N N n n usually written as xN N where N is the translation of M and abstraction n i i y The bijection extends to the typable terms terms translate in the obvious wa elsewhere wehave called such sequent calculus permutationfree meaning that there are no p ermutations ie that the map from MJ to NJ is Further details of this calculus covering all the connectives and several pro ofs of admissibility of cut can b e found in We shall im plicitly use the bijectiveness of the corresp ondences with N and NJ and not trouble to give pro ofs that eg a result shown for M translates correctly to a result claimed without pro of for N The calculus LI LI is a cutfree sequent calculus for intuitionistic implicational logic First formulae A are built up from proposition variables p q using just for implication Second contexts are nite sets of variable formula pairs as so ciating at most one formula to eachtermvariable in V Third there are terms dened as in Denition The set L of terms in cutfree LI derivations is dened as fol lows L varV j appV L VL j V L The notions of free and bound variable and of alphaconversion are as usual there are two binding mechanisms those at the o ccurrences of VL in the ab ove denition Two terms are said to b e equal i they are alphaconvertible again we shall use for this relation Note again the overloaded use of We write x L for x is not free in L similarly x L for x is free in L Fourth there are judgments L A Fifth there are typing rules inductively dening the derivations of the calculus Axiom x A varxA L A y B L C y A L B L R y L A B appx L yL C with the provisos x A B b elongs to in L and y is new iedoesnot app ear in the context in b oth L and R From the term and context parts of the endsequent of a derivation one can recover the entire derivation the terms mo dulo alpha conversion are really just a convenient notation for derivations The rules ab out new variables imply for example that b ound variables are chosen so that the variable y in appx L yL diers from the variable x and do es not o ccur freely in L Wemake no distinction b etween the judgment L A and the assertion of its derivability Weakening is an admissible rule of LIany derivation can b e transformed to aweaker derivation by adding an assumption x A to eachantecedent for new x The two derivations will b e represented by the same term also if a derivation do es not use an assumption x A then it can b e strengthened byremoving x A b oth from the endsequents antecedent and inductively with descendants from the premisses In the following we use b oth the strengthening and the weakening techniques without comment The corresp ondence from L to M Prawitz description see also x of the function from sequent calculus derivations to natural deductions uses the ordinary notion of subs titution recursively dened on the structure of the term b eing substituted into Using Herb elins denition of terms we need a dierentversion of the subs titution function This should b e based on his cut rules as in x for ease of exp osition wenow just intro duce it in an ad hoc wayWedoitjustinthe untyp ed case typing is not necessary for the functions to b e welldened Denition The functions of substitution of a variable x and a term M for a variable y in a term resp terms aredened as fol lows subst V M V M M substx M y y Ms xM substsx M y M s def z substsx M y M s if z y substx M y z Ms def substx M y z M z substx M y M def substs V M V Ms Ms substsx M y def substsx M y M Ms substx M y M substsx M y M s def Care is taken as usual to avoid variable capture ie in line of the denition for subst z x z y and z M L M is dened as fol lows Denition The function varx x def appx L yL substx L yL def xL xL def Our denition is for untyp ed terms we can easily extend it to typ ed terms and consider it as a map from cutfree sequent calculus derivations to normal natural deductions in Herb elins notation WesaythatL determines the term L and similarly for the derivation represented by L and the deduction represented by LWe reserve the name as in for the corresp onding function intro duced but not named in p REMARK from L to N dened by varx vrx def appx L yL apx L y L def xL xL def Note that is just the comp osite of with the bijection from M to N Details are in Denition Anequation L L is trivial i L L similarly for trivial and similarly for permutations and transformations The corresp ondence from M to L Denition The function M L is denedbyrecursion on the size of terms of M as fol lows x varx def xM Ms appx M z z Ms z new def xM xM def P n where siz exM M siz eM and siz exM siz eM n i
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