Bulletin of the Section of Logic Volume 41:1/2 (2012), pp. 89{104 Andrzej Indrzejczak CUT-FREE HYPERSEQUENT CALCULUS FOR S4.3. Abstract Hypersequent calculi (HC) are the generalization of ordinary sequent calculi ob- tained by operating with sets of sequents instead of single sequents. HC, intro- duced independently by Pottinger and Avron, proved to be very useful in the field of nonclassical logics. Nevertheless their application to modal logics was rather limited. We provide a cut-free hypersequent calculus HCS4.3 for modal logic of linear frames based on the idea of hypersequent formalization for G¨odel- Dummett's logic due to Avron. Also a variant of HCS4.3 with analytic cut but with nonbranching logical rules is presented. 1. Introduction G. Pottinger [19], and independently A. Avron [3], introduced a generaliza- tion of Gentzen's sequent calculus called hypersequent calculus where hy- persequents are (multi)sets of ordinary sequents. This simple and straight- forward modification significantly increases the expressive power of or- dinary Gentzen apparatus by allowing additional transfer of information among different sequents. Hypersequents proved to be very useful in build- ing cut-free formalization of many nonclassical logics, including modal, rel- evant, multi-valued and fuzzy logics (e.g. Avron [4, 5], Baaz, Ciabattoni and Ferm¨uller[7] and other papers of these authors). In particular, replac- ing ordinary sequents with hypersequents made possible obtaining different cut-free systems for S5 (Pottinger [19], Avron [4], Poggiolesi [18]) and for one of the most important intermediate (or superintuitionistic) logic G¨odel- Dummett's logic (Avron [4, 6]). 90 Andrzej Indrzejczak The last one is of particular interest for us since it is the logic of intu- itionistic relational frames with linear accessibility relation. In what follows we propose a hypersequent calculus HCS4.3 for the basic modal logic of such frames, namely, S4.3. Our calculus is based on Avron's idea of captur- ing linearity by specific structural rule, however, differences concerning the language (nonmodal versus modal) and the logical basis (intuitionistic ver- sus classical) of both approaches lead to some important consequences. In particular, our specific rule is not purely structural, and the completeness proof of Avron [6] for cut-free hypersequent version of G¨odel-Dummett's logic cannot be simulated directly. Nevertheless, the calculus HCS4.3 is analytic in the sense that all rules satisfy the subformula property since completeness is proved directly for cut-free version. Additionaly, we present briefly a variant of HCS3.4 which has nonbranching logical rules but re- quires analytic (subformula-preserving) cut. The latter system implements in hypersequent framework the idea of KE system of D'Agostino and Mon- dadori [2]. 2. The Logic Let us recall the basic facts concerning S4.3 in standard characterization, i.e., as an axiomatic system adequate to the class of linear Kripke frames. One can axiomatize S4.3 by adding to any system for classical propositional logic the following schemata: K (' ! ) ! (' ! ) T ' ! ' 4 ' ! ' 3 (' ! ) _ ( ! ') Clearly, the system is closed under MP (modus ponens) and GR (G¨odel's rule). It is adequate to the class of Kripke frames where acessibility relation is reflexive, transitive and satisfies the condition of strong connectedness: 8xyz(Rxy ^ Rxz ! Ryz _ Rzy) However, it may be shown that S4.3 is also characterised by the nar- rower class of linear structures, where instead of strong connectedness there holds the condition of dichotomy1: 1In fact, no axiom in standard modal language corresponds to this condition; see Goldblatt [10] for details. Cut-Free Hypersequent Calculus for S4.3. 91 8xy(Rxy _ Ryx) We will prove directly the adequacy of our hypersequent calculus with respect to such a class. 3. Survey of Previous Approaches S4.3, and other kinds of linear modal logics, was formalised by means of other calculi like natural deduction or tableaux. Also sequent calculi of different sorts were used to that aim. Most of the proposed solutions however, are based on the use of several forms of labelling which encodes some elements of relational semantics2. Before we present our solution, a brief survey of proposed approaches may appear helpful. All the systems for linear modal logics may be divided roughly according to two criteria: (a) the shape of rules and (b) the implicit strategy of linearization (of attempted falsifying model). The rules serving to express suitable conditions of connectedness, di- chotomy, trichotomy, and the like, may be divided into nonbranching and branching. The former solution is rather rare, one can mention here la- belled tableau system of Marx, Mikulas, Reynolds [16] and of Baldoni [8], as well as labelled natural deduction system of Indrzejczak [13]. Since con- struction of nonbranching rules for (strong) connectedness is based on the following form of this condition: 8xyz(Rxy ^ Rxz ^ :Ryz ! Rzy) one must have in the system sufficient resources not only for expressing that some worlds are in the relation R but also that some are not, i.e. either the apparatus of labels must be sufficiently strong (like in Baldoni's system) or some forms of cut must be involved (in the remaining proposals). Surprisingly enough, in case of richer language of bimodal temporal logics of linear frames one can define suitable nonbranching rules for natural deduction system even without using labels (see Indrzejczak [12]). Most solutions are based on the application of branching rules. In some systems the number of branches is fixed { two in case of strong connected- ness or dichotomy and three in case of weak connectedness (trichotomy). 2See, in particular, Gor´e[11] and Indrzejczak [14] Chapter 9 for the comprehensive surveys. 92 Andrzej Indrzejczak This is obvious since suitable rules are directly modelled on semantic con- ditions involving disjunction. One may mention here, e.g. display calculus of Wansing [22], labelled sequent calculus of Negri [17], or nested tableau system of Kashima [15]. Both nonbranching and fixed-branching approaches realize the local strategy of linearization of attempted falsifying model. It means that we compare only two worlds at a time and put them in some order, either disjunctively (by means of branching rules) or by choosing one possibility since the other is excluded (nonbranching rules). However, there are formal systems for linear logics which realize global strategy of linearization, i.e. their rules are defined in such a way that all worlds which are generated at the same moment are immediately put in the sequence. This group consists of systems which use rules with the number of branches not fixed in ad- vance. It depends on the number of modal formulae which are responsible for creation of new worlds in the attempted model. One may distinguish here between solutions which make it in the decreasing way (Zeman's se- quent calculus [23], Gor´etableau systems [11] or increasing way (Rescher and Urquhart's tableau system for linear temporal logics [20]). In both approaches we eventually create all possible sequences of worlds hence the number of branches is exponentially dependent on the number of world's creating modal formulae. In this paper we provide a characterization of S4.3 in terms of hyper- sequent calculus which closely follow the lines of Avron's [6] solution for G¨odel-Dummett'slogic. In terms of described solutions the system should be characterised as based on fixed-branching rules realizing local strategy of searching linear model. We do not use any labels, our solution for lin- earization of worlds is quite similar to that of Kashima [15] but instead of nested tableaux we use hypersequents. In some sense, hypersequent calculi may be seen as being the simplest modification of ordinary Gentzen's se- quent calculi, and independent of the kind of underlying semantics. This is the main reason why this approach was successfully applied to several kinds of nonclassical logics characterised by means of different semantic structures (see Avron [5]). 4. Hypersequent Calculus In this paper hypersequents are defined as finite sets of ordinary Gentzen's sequents. We will use the following notation: Cut-Free Hypersequent Calculus for S4.3. 93 • Γ ) ∆ or s for sequents where Γ ; ∆ are finite sets of formulae. • G; H for hypersequents; in particular s j G or Γ ) ∆ j G stand for hypersequents with displayed sequent s (or Γ ) ∆) • Γ = f 2 Γ : := 'g • in the schema of (IW) (internal weakening) Γ ⊆ Γ 0 and ∆ ⊆ ∆0. The calculus HCS4.3 consists of the following rules: Γ ; Π ) ∆ j G Π ; Γ ) Σ j G (AX) ' ) ' (Dich) Γ ) ∆ j Π ) Σ j G G Γ ) ∆ j G (EW) (IW) Γ ) ∆ j G Γ 0 ) ∆0 j G Γ ) ∆;' j G '; Γ ) ∆ j G (:)) ():) :'; Γ ) ∆ j G Γ ) ∆; :' j G '; ; Γ ) ∆ j G Γ ) ∆;' j G Γ ) ∆; j G (^)) ()^) ' ^ ; Γ ) ∆ j G Γ ) ∆;' ^ j G Γ ) ∆; '; j G '; Γ ) ∆ j G ; Γ ) ∆ j G ()_) (_)) Γ ) ∆;' _ j G ' _ ; Γ ) ∆ j G '; Γ ) ∆; j G Γ ) ∆;' j G ; Γ ) ∆ j G ()!) (!)) Γ ) ∆;' ! j G ' ! ; Γ ) ∆ j G '; '; Γ ) ∆ j G Γ ) '1 j ::: j Γ ) 'n j G ()) ()) '; Γ ) ∆ j G Γ ) ∆; '1; :::; 'n j G Note that: 1. All rules satisfy the subformula property. 2. Note that sequents are built from sets, hence in ()) we also need an occurrence of ' in the premise. This is necessary because dur- ing the proof-search performed in reverse order, i.e. from conclusion to premises, boxed formulae may be needed many times. Clearly, if sequents were made of multisets of formulae and (internal) contrac- tion rules were present in the system such an effect would be always obtainable by making a copy of ' by contraction before ordinary ()) is applied.