Extended Kripke lemma and decidability for substructural

Revantha Ramanayake Institute of and Computation, Technische Universität Wien (TU Wien) Wolfgang Pauli Institute Austria [email protected]

Abstract are conflated in the classical and intuitionistic setting. While We establish the decidability of every axiomatic extension of the logical connectives of classical (intuitionistic) logic model “truth" (resp. “constructibility"), the logical connectives of the commutative Full Lambek calculus with contraction FLec that has a cut-free hypersequent calculus. The axioms include substructural logics model a much wider range of notions familiar properties such as linearity (fuzzy logics) and the by varying the structural rules that are retained and adding substructural versions of bounded width and weak excluded further axioms in this setting. For example, and its many variants (computational and resource awareness), middle. Kripke famously proved the decidability of FLec by combining structural and combinatorics. Lambek calculus (syntax and syntactic types of natural lan- This work significantly extends both ingredients: height- guage, context-free grammars, linguistics), relevant logics preserving admissibility of contraction by internalising a (refined accounts of the implication connective to avoid the fixed amount of contraction (a Curry’s lemma for hyperse- paradoxes of material implication), fuzzy logics (degrees of quent calculi) and an extended Kripke lemma for hyperse- truth, fuzzy systems modelling), bunched implication log- quents that relies on the componentwise partial order on ics (software program verification and systems modelling). n-tuples being an ω2-well-quasi-order. Along with modal logics, substructural logics play a signifi- cant role in applied and theoretical computer science. CCS Concepts: • Theory of computation → Proof the- Throughout, we identify a logic with the set of its theo- ory; Automated reasoning; Linear logic; Complexity theory rems. A logic is decidable if there is an algorithm to decide and logic; Turing machines; • Mathematics of computing if a given formula is a theorem of the logic or not. Such is → Combinatoric problems; Combinatorial algorithms. the interest in decidability that this question is likely to have Keywords: decidability, substructural logics, proof theory been considered for every logic in the literature! In this work we obtain decidability for the commutative Full Lambek cal- ACM Reference Format: 1 culus with contraction FLec extended with any finite choice Revantha Ramanayake. 2020. Extended Kripke lemma and decid- from infinitely many properties, including many familiar ability for hypersequent substructural logics. In Proceedings of the ones such as linearity (every extension of FL +linearity is a 35th Annual ACM/IEEE Symposium on Logic in Computer Science ec (LICS ’20), July 8–11, 2020, Saarbrücken, Germany. ACM, New York, [11]), and the substructural versions of bounded NY, USA, 12 pages. https://doi.org/10.1145/3373718.3394802 width (ubiquitous) and weak excluded middle (preservation property of rough sets). A striking feature is the expansive 1 Introduction breadth of logics that are covered and the fact that almost all of the decidability results are new. To achieve this, we over- A substructural logic is a logic that lacks some of the struc- come two major challenges: the presence of the contraction tural rules of classical and such as weaken- rule (c) in the absence of the weakening rule (w), and taming ing, contraction, exchange or associativity. As a consequence, the structural language of the hypersequent calculus. a substructural logic will distinguish logical connectives that To explain the nature of these challenges, let us first con- Permission to make digital or hard copies of all or part of this work for sider how we might establish the decidability of intuition- personal or classroom use is granted without fee provided that copies are not istic propositional logic. Its (proof) calculus is ob- made or distributed for profit or commercial advantage and that copies bear tained by adding the of weakening (w) to FLec this notice and the full citation on the first page. Copyrights for components (i.e. FL ). Due to Gentzen’s cut-elimination theorem, it has of this work owned by others than ACM must be honored. Abstracting with ecw credit is permitted. To copy otherwise, or republish, to post on servers or to the subformula property (every formula in a proof is a sub- redistribute to lists, requires prior specific permission and/or a fee. Request formula of the end formula). Additionally, repetitions of a permissions from [email protected]. formula in the antecedent of a sequent are inter-derivable LICS ’20, July 8–11, 2020, Saarbrücken, Germany © 2020 Association for Computing Machinery. 1Also called: intuitionistic negation-free fragment of relevant logic R with- ACM ISBN 978-1-4503-7104-9/20/07...$15.00 out distribution; the positive fragment of Lattice R (denoted LR+ [30]); https://doi.org/10.1145/3373718.3394802 intuitionistic multiplicative additive linear logic with contraction I MALLC. LICS ’20, July 8–11, 2020, Saarbrücken, Germany Revantha Ramanayake and hence need not be distinguished e.g. A ⇒ B is deriv- who construct hypersequent calculi with the subformula able from A, A ⇒ B by (c) and the other direction uses (w). property for infinitely many axiomatic extensions of FLe ′ (For this reason, a set is often used in this calculus as the (acyclic P3 axioms in the substructural hierarchy) via the ad- data structure of the antecedent in place of a list.) These dition of analytic structural rules. These extensions comprise two observations imply that only finitely many different a significant portion of logics in the sense that no further need occur in the proof of a given formula up to axiomatic extensions can be obtained via analytic structural inter-derivability. The length of each branch in the proof rule extension [8] and the hierarchy closes [16] (up to for- search tree can then be made finite and decidability follows. mula equivalence in FLe) at the next level N3. We can restate A more careful counting of branch length can even establish the central contribution of this paper as decidability of every (optimal) PSPACE-complexity. In the absence of weakening, logic that has an analytic structural rule extension of the 2 the number of repetitions matters, so this argument fails hypersequent calculus for FLec. in FLec (A, A ⇒ B is no longer derivable from A ⇒ B). There- There is a price to pay for using the hypersequent calculus. fore the task was to find some other way of bounding the Its structural language is so expressive that it is challeng- branch length in the proof search tree. ing (compared to the ) to use it to prove Enter Kripke [18] who in 1959 gave a famous decidability meta-logical results. As Metcalfe et al. [21] observe: “For argument for FLec later described by Urquhart [30] as a “tour hypersequent calculi, however, [establishing decidability] is de force of combinatorial reasoning". Kripke established the further complicated by the presence of the external contrac- finiteness of the proof search tree by combining (i) Curry’s tion rule (EC) that can duplicate whole sequents." The issue is lemma from structural proof theory which shows how to the interaction of the structural rules with (EC), and it is inde- internalise a fixed amount of contraction into the other rules pendent of the complications due to (c) discussed above. For to obtain height-preserving admissibility of contraction, and example, although MTL (= FLew + prelinearity) was proved (ii) showing that if a branch of the proof search tree is infinite, decidable (see [10]) by specialised semantic arguments, at- then some sequent on the branch must be derivable by con- tempts to find a syntactic proof have been obstructed bythe traction from a later sequent and hence redundant. The latter complex interactions of (EC) with the structural rule (com) is the celebrated Kripke lemma that Meyer (see [28]) subse- which expand the proof search tree unmanageably. The chal- quently identified as being equivalent to Dickson’s lemma lenges posed by contraction and hypersequent calculi are from number theory. From an order-theoretic perspective, also illustrated by the fact that recent decidability results it expresses that the usual componentwise partial order on in the vicinity make significant concessions: decidability n-tuples of natural numbers is a well-quasi-order (wqo). and 2EXPTIME-complexity for analytic structural rule ex- Kripke’s lemma is a ‘sledge hammer’ in that it says some- tensions of FLecm are obtained in [9]. Here m stands for thing about the property of all infinite sequences of sequents the mingle axiom which is a weaker version of weakening, without any direct consideration of the rules of the calcu- but enough to allow a set to be used for the antecedent. lus. Could we be missing a simpler decision argument? The Meanwhile decidability and complexity results are presented tight upper and lower complexity bounds for FLec due to in [29] for a large class of extensions of FLe using semantic Urquhart [30] confirm that we are not: there is no prim- and syntactic means. However, the extensions are confined itive recursive algorithm; it is primitive recursive in the to those that have a cut-free sequent calculus, and the only Ackermann function. Indeed, omit commutativity and the extensions of FLec are by specific mingle-type axioms. logic FLc is undecidable [6]. Also undecidable is its predicate A hypersequent is a multiset of sequents so what we do— version ∀FLec [17]. Kripke’s lemma was subsequently utilised informally speaking—is extend Kripke’s argument over sets to prove decidability for specific logics in the vicinity e.g. of n-tuples of natural numbers. The argument is technically the pure implication fragments of E and R, their implication- involved but the extended Kripke lemma that we obtain gives negation fragments [5], and via translation, lattice R [22]. us a handle on all infinite sequences of , and it However, there is a fundamental proof theoretic limitation is this that supports our general results. In Sec.3 we extend that prevents broader application: the sequent calculus for- Curry’s lemma to all the hypersequent calculi introduced malism (which crucially underpins Kripke’s lemma) is not in [7]. Height-preserving admissibility of the contraction expressive enough to give a with the subfor- rules is a desirable property in many structural proof theo- mula property for most logics of interest. retic investigations so this is a valuable result in its own right. This takes us into the realm of structural proof theory We have written this section independent of the remainder where the limitations of the sequent calculus have motivated so that this generalisation can be lifted directly to other appli- the development of numerous new proof formalisms that cations. Our extended Kripke lemma (Sec.4) is deeper than extend the sequent calculus by adding new structure. These formalisms are then used to define proof calculi with the sub- 2 formula property for the logics of interest. A notable result The distributivity axiom of the relevant logic R is in N3 so it is not covered for substructural logics is the work of Ciabattoni et al. [7] by our result. This is what we should expect given that R is undecidable. Extended Kripke lemma and decidability for hypersequent substructural logics LICS ’20, July 8–11, 2020, Saarbrücken, Germany the original in a formal sense as it relies on the stronger prop- A hypersequent calculus consists of a finite set of hyperse- erty that the usual componentwise partial order on n-tuples quent rule schemas. Each rule schema has a conclusion hyper- of natural numbers is an ω2-wqo. Finally, to combine struc- sequent and some number of premise hypersequent(s). A rule tural proof theory and order theory, a novel partial order schema with no premises is called an initial hypersequent. on hypersequents is formulated (we subsequently identified The conclusion and every premises of the rule schema has this partial order as the Smyth ordering from domain theory) the form s1| ... |sn where each si is either (i) a hypersequent- and the infinite Ramsey theorem is deployed (something that variable (denoted by H), (ii) a sequent-variable (written here is not required in Kripke’s argument). This work reiterates as X ⇒ Π), (iii) ⃗Γ ⇒ Π, or (iv) ⃗Γ ⇒. In the last two cases, Meyer’s insights on Kripke’s lemma and on the potential for ⃗Γ is a finite list comprising of multiset-variables (denoted logical problems to be recast in a way that takes advantage using X,Y, Z), formulas built from formula-variables (A, B) of the mathematical literature. and propositional-variables (p,q,r), and Π is a succedent- multiset-variable. Each si is not a sequent, but it is still re- 2 Preliminaries ferred to as a component. This overloading of the terminol- Let N denote the set {0, 1, 2,...} of natural numbers. For m > ogy with (1) is standard practice. The variables in the rule m schema are collectively referred to as schematic variables. 0, let N denote the set {(a1,..., am )|ai ∈ N} of m-tuples of natural numbers. A rule schema that contains neither formula-variables nor Logical formulas are given by the following grammar. Var propositional-variables is called a structural rule schema. denotes a countably infinite set of propositional variables. The extension of the hypersequent calculus H by the finite set R of rule schemas is the hypersequent calculus H ∪ R F := p ∈ Var|⊤|⊥|1|0|F ∧ F |F ∨ F |F · F |F → F (following standard convention, we write H + R). A rule instance is obtained from a rule schema by uni- The connective · is called fusion. It is also called multiplicative formly instantiating the schematic variables with concrete conjunction to contrast it with the additive conjunction ∧. objects of the corresponding type. I.e. a hypersequent-variable The meaning of the connectives in the context of substruc- with a hypersequent, a sequent-variable with a sequent, a tural logics is conveyed well through its algebraic semantics multiset-variable with a multiset, and so on. In particular, a of residuated lattices [14]: ∧, ∨ are the familiar lattice connec- succedent-multiset-variable is instantiated by a multiset that tives; ⊤ and ⊥ are respectively the top and bottom element of is either empty or consists of a single formula. the lattice; · is a monoidal operation with unit 1, and 0 is an It is worth noting that the standard practice in the field is arbitrary element of the lattice; the implication connective → to not make a rigorous distinction between rule schema and is residuated with ·. rule instance as it can be determined easily from the context. We identify a logic with the set of formulas that are theo- Any component in the premise or conclusion of a rule rems. The axiomatic extension L + F of the logic L by a finite schema that is not a hypersequent-variable, as well as the set F of formulas is defined in the usual way as the closure corresponding component in a rule instance, is called an of L ∪ F under the axioms and rules of its Hilbert calculus. active component. A sequent is a tuple written X ⇒ Π where X (the an- A derivation of the hypersequent h in the hypersequent tecedent) is a multiset of formulas and Π (the succedent) is a calculus H is defined in the usual way as a finite tree whose multiset that is either empty or consists of a single formula. nodes are labelled with hypersequents such that the root A hypersequent is a (possibly empty) multiset of sequents is labelled with h, the leaves are labelled with instances of and it is written explicitly as follows. initial hypersequents, and the labels of an interior node and its child node(s) are the conclusion and premise(s) of an X1 ⇒ Π1| ... |Xn ⇒ Πn (1) instance of some rule schema in H . A branch in a derivation is a path from the root to a leaf. Each Xi ⇒ Πi is called a component of the hypersequent. Of The height of a derivation is the number of nodes on the course, each of these components is a sequent. longest branch. The hypersequent calculus HFLec is given in Fig.1. Note that the cut-rule is eliminable from this calculus so it is not Example 2.1. Consider the rule schema named (∧R). H |X ⇒ AH |X ⇒ B included in the calculus. (∧R) As an aid to the reader unfamiliar with proof theory, we H |X ⇒ A ∧ B provide a brief exposition of the terminology below: It has two premises H |X ⇒ A (the active component is X ⇒ A “hypersequent calculus" is a type of formal proof sys- A) and H |X ⇒ B (active component X ⇒ B), and its conclu- tem (introduced independently in [3, 23, 25]) that can be sion is H |X ⇒ A ∧ B (active component X ⇒ A ∧ B). used to generate proofs (‘derivations’) of an infinite set of A rule instance of this schema is obtained by uniformly hypersequents. It is an extension of the “sequent calculus" instantiating H by a hypersequent, X by a multiset of for- formal proof system introduced by Gentzen [15] in 1935. mulas, and the formula-variables A and B by formulas (the LICS ’20, July 8–11, 2020, Saarbrücken, Germany Revantha Ramanayake instantiation of A ∧ B is fully determined by the latter). Here (coupling) For every component in the conclusion with are some rule instances. succedent Π, there is a multiset-variable Y in its an- ⇒ p ⇒ q r,r ⇒ p ∧ q r,r ⇒ q tecedent such that the pair (Y, Π) always occur to- ⇒ p ∧ q r,r ⇒ (p ∧ q) ∧ q gether in a premise, and Y occurs exactly once there. (strong subformula property) Each multiset-variable ⇒ s|q → p ⇒ p|r ⇒ p ∧ q ⇒ s|q → p ⇒ p|r ⇒ q in the premise occurs in the conclusion. ⇒ s|q → p ⇒ p|r ⇒ (p ∧ q) ∧ q

The active components in the rule instances above appear Let P0 = N0 be the set Var of propositional variables. The as the rightmost component of each hypersequent. substructural hierarchy [7] is defined as follows. Example 2.2. Consider the rule schema named (com). Pn+1 := 1 | ⊥ | Nn | Pn+1 ∨ Pn+ | Pn+1 ·Pn+1 H |X1,Y1 ⇒ Π1 H |X2,Y2 ⇒ Π2 (com) Nn+1 := 0 | ⊤ | Pn | Nn+1 ∧ Nn+ | Pn+1 → Nn+1 H |X1,Y2 ⇒ Π1|X2,Y1 ⇒ Π2 The conclusion has two active components X1,Y2 ⇒ Π1 , ⇒ ′ ′ ′ ′ ′ and X2 Y1 Π2. Meanwhile each premise has a single active P3 := 1 | ⊥ | N2 ∧ 1 | P3 ∨ P3 | P3 ·P3 component. Consider the following instantiations: The constructors in the definition of Pi+1 (i.e. ∨, ·) are those X1 7→ {r,r } Y1 7→ {p} Π1 7→ {p} whose rules in HFLe are invertible in the antecedent; the X2 7→ ∅ Y2 7→ {r,q} Π2 7→ ∅ H 7→ p ⇒ |q ⇒ constructors in the definition of Ni+1 (i.e. ∧, →) are those Here is the corresponding rule instance. Following a slight whose rules in HFLe are invertible in the succedent. Observe ′ (and standard) abuse of notation, when writing a rule in- that Ui ⊂ Vi+1 (U ,V ∈ {P, N}) and P3 ⊂ P3. stance, the curly brackets {} denoting the multiset are dropped, In the presence of weakening, every formula in P3 can be and the comma between multiset-variables in the rule schema effectively transformed into an equivalent analytic structural ′ is interpreted as multiset union. rule schema. In its absence, this holds for formulas in P3 that p ⇒ |q ⇒ |r,r,p ⇒ p p ⇒ |q ⇒ |r,q ⇒ ∅ are acyclic (i.e. those that do not lead the transformation into p ⇒ |q ⇒ |r,r,r,q ⇒ p|p ⇒ ∅ cycles, see [8, Def. 4.11] for further details). This motivates the following definition. Notation. We use lower case д,h to denote hypersequents to make it easy to distinguish them from the hypersequent- Definition 2.4 (amenable). A set F of formulas is amenable variable H. For an index set J = {r1,...,rn }, let H | Xj ⇒ if (i) F ⊆ P3 and left weakening p · q → p ∈ F , or (ii) F ⊆ Π (j ∈ J ) denote the hypersequent ′ j P3 consists of acyclic formulas. H | X ⇒ Π | ... | X ⇒ Π r1 r1 rn rn We say that H is a hypersequent calculus for the logic L if Let An denote A,..., A (n copies). We say that X contains at for every formula B: B ∈ L iff H derives ⇒ B. least n copies of A (or A occurs at least n times in X) if there exists X ′ such that X is the multiset union of {An } and X ′. Theorem 2.5 ([7,8]) . (i) From every finite set F of amenable ′ ′ If additionally X does not contain A (denoted A < X ) then formulas, a finite set R F of analytic structural rule schemas “at least" can be replaced by “exactly". is computable such that HFLe +R F is a calculus for FLe + F with cut-elimination and the subformula property. Hypersequent calculi for substructural logics. Ciabat- (ii) Every analytic structural rule extension HFLe + R is a toni et al. [7, 8] establish equivalence between a large set calculus for an amenable axiomatic extension of FLe. of axiomatic extensions of FLe and the analytic structural rule extensions of HFLe (i.e. HFLec minus the contraction Some amenable formulas and the analytic structural rules rule (c)). The following reformulates [7], [8, Def. 4.16]. computed from them are shown in Fig.2 Definition 2.3. An analytic structural rule schema has the Remark 1. In addition to the above syntactic characterisation following form where each ⃗Γik and ⃗∆l is a list of multiset- of analytic structural rule extensions of HFLe via amenable variables, and satisfies the conditions listed below. formulas, there is also a semantic characterisation in terms of ⃗ ⃗ {H | Yi , Γik ⇒ Πi }i ∈I,k ∈Ki {H | ∆l ⇒}l ∈L closure under hyper-MacNeille algebraic completions (see [8]).

H | Yi , Xi1,..., Xiαi ⇒ Πi (i ∈ I ) | Zj1,..., Zjβj ⇒ (j ∈ J ) (2) 3 Internalising a limited contraction: ⇝ ⇝ (linear conclusion) The multiset-variables that occur from HFLec + R to HFLec + R in the conclusion occur exactly once. A rule schema is hp-admissible (height-preserving admissi- (separation) No multiset-variable occurs in an antecedent ble) if whenever the premises of a rule instance are derivable, and in a succedent. there is a derivation with no greater height of its conclusion. Extended Kripke lemma and decidability for hypersequent substructural logics LICS ’20, July 8–11, 2020, Saarbrücken, Germany

H |X ⇒ Π H |X ⇒ | ⇒ |⊥ ⇒ | ⇒ ⊤ | ⇒ | ⇒ H p p H , X Π H X H 0 H 1 H |1, X ⇒ Π H |X ⇒ 0 H |X, A, B ⇒ Π H |X ⇒ AH |Y ⇒ B H |X, A ⇒ Π H |X, B ⇒ Π H |X ⇒ A (·L) (·R) (∨L) (∨R) H |X, A · B ⇒ Π H |X,Y ⇒ A · B H |X, A ∨ B ⇒ Π H |X ⇒ A ∨ B H |X, A ⇒ Π H |X ⇒ AH |X ⇒ B H |X ⇒ AH |Y, B ⇒ Π H |X, A ⇒ B (∧L) (∧R) (→L) (→R) H |X, A ∧ B ⇒ Π H |X ⇒ A ∧ B H |X,Y, A → B ⇒ Π H |X ⇒ A → B H |X ⇒ Π|X ⇒ Π H H |X, A, A ⇒ Π (EC) (EW) (c) H |X ⇒ Π H |X ⇒ Π H |X, A ⇒ Π

Figure 1. The (cut-free) hypersequent calculus HFLec for FLec

In this section we will construct a new hypersequent The initial hypersequent with conclusion H |⊥, X ⇒ Π ⇝ ⇝ calculus HFLec + R from an analytic structural rule ex- has two components (hence the component multiplicity is 2) tension HFLec + R that derives exactly the same hyperse- and the antecedent of the active component is ⊥, X. There quents but where (EW), (c) and (EC) are hp-admissible. Hp- are 2 elements in this list so the formula multiplicity is 2. admissibility of (EW) even holds in HFLec + R, so the chal- The initial hypersequent with conclusion H | ⇒ 1 has two lenge is (c) and (EC). The idea is to show that these rules can components (hence the component multiplicity is 2) and be permuted with every rule instance above it. The insight of the antecedent of the active component is empty. Hence the Curry’s lemma was that such a permutation is achievable if formula multiplicity is 0. the rules internalise a fixed amount of contraction. Here we The conclusion H |X,Y, A → B ⇒ Π of (→L) has two present the extension of Curry’s lemma to the hypersequent components (hence the component multiplicity is 2) and the calculus. antecedent of the active component is X,Y, A → B. There are 3 elements in this list so the formula multiplicity is 3. 3.1 Formula multiplicity, component multiplicity The conclusion H |X, A ⇒ Π of (c) has two components Consider how we might permute an instance of (c) with a and the antecedent of the active component is X, A. There rule instance above it. are two elements in this list so the formula multiplicity is 2. If the duplicate formulas to be contracted are in the instan- The general form of an analytic structural rule schema is tiation of the same schematic variable in the conclusion of the given in (2). In particular, its conclusion is: rule instance, then contraction is applied in every premise H | Yi , Xi1,..., Xiα ⇒ Πi (i ∈ I ) | Zj1,..., Zjβ ⇒ (j ∈ J ) where this variable occurs and permutation is achieved. The i j obstacle therefore is when each of the duplicate formulas By inspection, its component multiplicity is 1 + |I | + |J |, and belongs to the instantiation of a different schematic variable. its formula multiplicity is   The solution is to internalise a fixed amount of contraction max max({αi |i ∈ I }) + 1 , max({βj |j ∈ J }) into the rule to handle this situation. The more variables Some analytic structural rule schemas and their formula and that are present in the antecedent of an active component in component multiplicities are given in Fig.2. the conclusion, the more contractions we need to internalise until we get to the stage where there are enough duplicate Definition 3.3. Let HFLec + R be any analytic structural formulas in the conclusion to ensure that the instantiation of rule extension of HFLec. some variable must contains multiple copies. This motivates The formula multiplicity of HFLec + R is the maximum of the definition of formula multiplicity (the terminology isdue the formula multiplicities of its rule schemas. to [13], where an alternative definition is given). The component multiplicity of HFLec + R is the maximum of the component multiplicities of its rule schemas. Definition 3.1. The formula multiplicity of a rule schema is the maximum of the numbers of elements in the antecedents A quick inspection of Fig.1 reveals that the formula mul- of each of the active components in its conclusion. tiplicity of HFLec is 3 and its component multiplicity is 2. To permute (EC) with any rule instance above it, we re- quire a similar definition, this time at the level of components ⇝ ⇝ 3.2 (EW), (c), (EC) are hp-admissible in HFLec + R rather than elements in the antecedent of a component. Define these binary relations on hypersequents h and h1. Definition 3.2. The component multiplicity of a rule schema k • h ⇝ h1 iff h1 can be obtained from h by applying is the number of components in its conclusion (including (c) some number of (c) such that every formula is up to k the hypersequent-variable). occurrences fewer in a component in h1 than in the Let us compute some of the values by way of example. corresponding component in h. LICS ’20, July 8–11, 2020, Saarbrücken, Germany Revantha Ramanayake

H |X ,Y ⇒ Π H |X ,Y ⇒ Π H |X,Y ⇒ 1 1 1 2 2 2 (com) (wem) H |X1,Y2 ⇒ Π1|X2,Y1 ⇒ Π2 H |X ⇒ |Y ⇒ (p → q)∧1 ∨ (q → p)∧1 (p → 0)∧1 ∨ ((p → 0) → 0)∧1 (2, 3) (1, 3)

H |Xi , Xj ⇒ Πi (0 ≤ i, j ≤ k;i , j) H |Xi , Xj ⇒ Πi (0 ≤ i ≤ k − 1;i + 1 ≤ j ≤ k) (Bwk) (Bck) H |X0 ⇒ Π0| ... |Xk ⇒ Πk H |X0 ⇒ Π0| ... |Xk−1 ⇒ Πk−1|Xk ⇒ k ∨ → ∨ ∨ ∧ ∧ → ∨i=0 (pi → (∨j,i pj ))∧1 (p0)∧1 (p0 p1)∧1 ... ((p0 ... pk−1) pk )∧1 (1,k + 2) (1,k + 2)

{ | ⇒ { } ⊆ { }} H |Y, X1 ⇒ Π H |Y, X2 ⇒ Π H Y, Xi1 ,..., Xim Π s.t. i1,...,im 1,...,n n mingle knotm H |Y, X1, X2 ⇒ Π H |Y, X1,..., Xn ⇒ Π p · p → p pn → pm (n,m ≥ 0) (3, 2) (n + 1, 2)

Figure 2. Some analytic structural rule schemas, each computed from the amenable formula that appears below it. Below that is its (formula multiplicity, component multiplicity). Note: (A)∧1 is notation for (A ∧ 1).

• k h ⇝(EC) h1 iff h1 can be obtained from h by applying For each instance of r there corresponds a finite number of some number of (EC) such that every component is instances of r ⇝ .A base instance of r ⇝ is a particular instance up to k occurrences fewer in h1 than in h. such that the conclusion is h0. In other words, the conclu- • k ′ k ′ sion is the hypersequent that would have been obtained by h ⇝l h1 iff there exists h such that h ⇝(c) h and ′ l applying r. h ⇝(EC) h1. ⇝ ⇝ Example 3.4. Consider the following hypersequents. Lemma 3.6. HFLec + R and HFLec + R derive the identical 3 2 2 set of hypersequents. h0 := p ,q ⇒ r | p ,q ⇒ r | p,q ⇒ r | ⇒ s 2 2 2 Proof. Every instance of a rule schema r ⇝ ∈ HFL⇝+R⇝ can h1 := p ,q ⇒ r | p ,q ⇒ r | p,q ⇒ r | ⇒ s ec be simulated in HFL + R by a rule instance of r followed by , ⇒ | , ⇒ | , ⇒ | ⇒ ec h2 := p q r p q r p q r s some applications of (c) and (EC). Meanwhile every instance = , ⇒ | ⇒ ⇝ h3 : p q r s of a rule schema r ∈ HFLec + R can be simulated in HFLec + ⇝ ⇝ h4 := ⇒ s R since the latter is a base instance of r . □ 3 2 h5 := p ⇒ r | p ,q ⇒ r | p,q ⇒ r | ⇒ s ⇝ ⇝ ⇝ ⇝ ⇝ Since (EW ), (c ) and (EC ) are in HFLec + R , the 2 2 2 admissibility of (EW), (c), and (EC) is trivial. The significance Then h0 ⇝(c) h1. Also: h0 ⇝(c) h2 and h2 ⇝(EC) h3 and 2 2 of the following lemma is that this admissibility is height- hence h0 ⇝ h3. However it is not the case that h3 ⇝ h4 2 (EC) preserving. since h4 cannot be obtained from h3 by (EC). Nor is it that 1 h h h p,q ⇒ r h ⇝ ⇝ 2 ⇝(EC) 3 since 3 has two fewer than 2 . Nor is Lemma 3.7. (EW), (c), (EC) are hp-admissible in HFLec +R . 2 it that h0 ⇝ h5 since h5 cannot be obtained from h1 by (c). (c) Proof. We prove hp-admissibility of each rule schema in turn. ⇝ ⇝ Example 3.5. The analytic structural rule extension HFLec+ (EW): given a derivation δ in HFLec + R of h, let us (com)+(Bw4) has formula multiplicity 3 and component mul- obtain a derivation of h|X ⇒ Π of no greater height. Proof tiplicity 6 (refer Figs.1,2). By Thm. 2.5 it is a calculus for the by induction on the height of δ. 4 logic FLec + (p → q)∧1 ∨ (q → p)∧1 + ∨i=0 (pi → (∨j,i pj ))∧1. Base case: if h is an instance of an initial hypersequent, For the remainder of this section: fix an arbitrary ana- then so is h|X ⇒ Π. Inductive case: let r⇝ be the last rule instance in δ. Let lytic structural rule extension HFLec + R. Denote its formula multiplicity by fm. Denote its component multiplicity by cm. us illustrate the argument when r⇝ has two premises д|д1 ⇝ ⇝ and д|д2 (the general case of ≥ 1 premise is analogous) and The hypersequent calculus HFLec + R is obtained from conclusion д|д , and д is the hypersequent that instantiates the calculus HFLec + R by replacing each rule schema r (be- 0 low left) in the latter with the rule r⇝ (below right) that the hypersequent-variable of the rule (every rule schema ⇝ ⇝ internalises a fixed amount of contraction in the conclusion. in HFLec + R has exactly one hypersequent-variable). By h1 ... hn h ... hn the induction hypothesis (IH) we obtain derivations of height r 1 ⇝ fm−1 r with h0 ⇝ − д | ⇒ | | ⇒ | ⇝ h0 д cm 1 lesser than δ of д X Π д1 and д X Π д2. Applying r Extended Kripke lemma and decidability for hypersequent substructural logics LICS ’20, July 8–11, 2020, Saarbrücken, Germany to these hypersequents with the multiset-variable now in- rest take values strictly less than fm − 1. I.e. stantiated by д|X ⇒ Π, we obtain д|X ⇒ Π|д0. This is a l components derivation of h|X ⇒ Π with height no greater than δ. z }| { | , , ⇒ ′ k+2 k+2 (c): given a derivation δ of h A A X Π let us obtain a д0| A , X1 ⇒ Π| ... |A , Xl ⇒ Π | derivation of h|A, X ⇒ Π of the same height. Induction on Ak+2+fm−1, X ⇒ Π| ... |Ak+2+fm−1, X ⇒ Π| the height of δ. l+1 l+N k+2+βl+N +1 k+2+βl+1+α Base case: if h|A, A, X ⇒ Π is an instance of an initial A , Xl+N +1 ⇒ Π| ... |A , Xl+1+α ⇒ Π hypersequent, then so is h|A, X ⇒ Π. For every i such that l + 1 ≤ i ≤ l + N , either the com- Inductive case. Let r⇝ be the last rule in δ. Then there ponent Ak+2+fm−1, X ⇒ Π is in the instantiation of the must be a base instance д of r⇝ and some д such that i 0 1 hypersequent-variable, or at least 2 of the copies of A oc- cur in the instantiation of some multiset-variable (since the fm−1 cm−1 | ⇒ д0 ⇝(c) д1 ⇝(EC) h A, A, X Π (3) antecedent of every active component in the conclusion of the rule schema has at most fm elements by the defini- Since X contains exactly k copies of A for some k ≥ 0, write X tion of formula multiplicity). In either case apply the IH to as Ak , X − such that A < X −. Since h contains exactly l identi- those instantiations in the premise(s)3 to make A, A into A. cal components of A, A, X ⇒ Π (i.e. Ak+2, X − ⇒ Π) for some Then apply r⇝ to get the following base instance. Note that l ≥ 0, partition it into the portion h− that does not contain A, A 7→ A does not change anything else because the hyperse- Ak+2, X − ⇒ Π and the remainder. So quent variable (see (2)) and each multiset-variable (by linear conclusion Def. 2.3) occur exactly once in the conclusion.

h|A, A, X ⇒ Π := l components z }| { l components ′ | k+2 ⇒ | | k+2 ⇒ | z }| { д0 A , X1 Π ... A , Xl Π − − − − h | Ak+2, X ⇒ Π| ... |Ak+2, X ⇒ Π |Ak+2, X ⇒ Π A, A 7→ A via IH on premises | {z } z }| { h k+1+fm−1 k+1+fm−1 A , Xl+1 ⇒ Π| ... |A , Xl+N ⇒ Π |

Ak+2+βl+N +1 , X ⇒ Π| ... |Ak+2+βl+1+α , X ⇒ Π cm−1 | k+2, − ⇒ l+N +1 l+1+α Since д1 ⇝(EC) h A X Π by (3), partition д1 into the ′ − ′ − − portion д that externally contracts to h (i.e. д cm 1 h ), Since βi < fm−1 for every i such that l +N +1 ≤ i ≤ l +1+α, 1 1 ⇝(EC) fm−1 and the portion that externally contracts to the remainder. the above is related under ⇝(c) to Specifically, there exists α with 0 ≤ α ≤ cm − 1 such that д1 l components has the following form. Here α is the number of times fewer z }| { k+2 − ⇒ | ⇒ ′ k+2 − k+2 − that A , X Π appears in h A, A, X Π than in д1 (due д1| A , X ⇒ Π| ... |A , X ⇒ Π | cm−1 to ⇝(EC) ). 1 + α components z }| { Ak+1, X − ⇒ Π| ... |Ak+1, X − ⇒ Π l + 1 + α components z }| { ′ cm−1 − ′ k+2 − k+2 − Since 0 ≤ α ≤ cm − 1 and д1 ⇝ h , the above is related д1 := д | A , X ⇒ Π| ... |A , X ⇒ Π (EC) 1 cm−1 under ⇝(EC) to k+2 − For each of the l + 1 +α components of A , X ⇒ Π above, l components the corresponding component in д will have up to fm − 1 z }| { 0 −| k+2, − ⇒ | ... | k+2, − ⇒ | k+1, − ⇒ A fm−1 i h A X Π A X Π A X Π more occurrences of (due to ⇝(c) ). Therefore for each k+1 − (1 ≤ i ≤ l + 1 + α) there exists βi with 0 ≤ βi ≤ fm − 1, Since A , X is A, X, this is exactly h|A, X ⇒ Π. and also a multiset Xi where the number of occurrences of (EC): given a derivation δ of h|s|s let us obtain a derivation each formula is no less and at most fm − 1 greater than in X. of h|s of the same height. Induction on the height of δ. ′ ′ fm−1 ′ Base case: if h|s|s is an instance of an initial hypersequent, Let д0 be such that д0 ⇝(c) д1. Then then so is h|s. ⇝ l + 1 + α components Inductive case. Let r be the last rule in δ. Then there z }| { must be a base instance д0 of r⇝ and some д1 such that д := д′ | Ak+2+β1 , X ⇒ Π| ... |Ak+2+βl+1+α , X ⇒ Π 0 0 1 l+1+α fm−1 cm−1 | | д0 ⇝(c) д1 ⇝(EC) h s s (4)

Without loss of generality, in the list βl+1,..., βl+1+α , we 3The multiset-variable is not required to occur in the premises (e.g. consider can assume that there is some initial segment βl+1,..., βl+N if r was the standard weakening rule). In that case, no IH on the premises (possibly empty) such that each value is fm − 1 and that the is required; simply use A instead of A, A in that variable’s instantiation. LICS ’20, July 8–11, 2020, Saarbrücken, Germany Revantha Ramanayake

Since h contains exactly l (l ≥ 0) identical components of s 4 The extended Kripke lemma − for some l ≥ 0, write h|s|s as follows such that s h . th < We write (ai ) to denote an infinite sequence with i ele- l ment ai , more standardly denoted (ai )i ∈N. For strictly in- z }| { creasing functions r and s on N: (a ) is a subsequence h|s|s := h−| s| ... |s |s|s r (i) of (a ) (denoted (a ) ⊑ (a )); also we drop parentheses and i r (i) i cm−1 | | write (ars (i) ) to mean the subsequence (ar (s (i)) ) of (ar (i) ). Since д1 ⇝(EC) h s s by (4), we can partition д1 into the ′ − ′ cm−1 − portion д1 that externally contracts to h (i.e. д1 ⇝(EC) h ), Lemma 4.1 (Kripke lemma [18]). For every sequence (si ) of and the portion that externally contracts to the remainder. sequents built using a finite set Ω of formulas, there exists i, j Specifically, there exists α with 0 ≤ α ≤ cm − 1 such that д1 such that i < j, and si can be obtained from sj by repeated has the following form. Here α is the number of times fewer applications of (c). that s appears in h|s|s than д due to cm−1. 1 ⇝(EC) Meyer observed (see [28]) that the above is equivalent to l + 2 + α Dickson’s lemma from number theory. Order-theoretically it z }| { expresses that the standard componentwise ordering on Nm д := д′ | s| ... |s 1 1 is a wqo.

< − cm−1 | fm−1 | The main result in this section is the following. If α cm 1 then д1 ⇝(EC) h s, so we have д0 ⇝cm−1 h s as required. So assume that α = cm − 1. In that case д0 has the Lemma 4.2 (extended Kripke lemma). For every sequence (hi ) ′ ′ fm−1 ′ following form for some д0 with д0 ⇝(c) д1, and sequents si of hypersequents built using a finite set Ω of formulas, there fm−1 ≤ ≤ exists i < j such that hi can be obtained from hj by repeated (1 i l + 1 + cm) such that si ⇝(c) s. applications of (c), (EC) and (EW). l + 1 + cm ′ z }| { 4.1 Some order theoretic observations д0 := д0| s1| ... |sl+cm−1|sl+cm|sl+1+cm A partially ordered set (S, ≤) is a set S with a partial order ≤. Since the number of components in the conclusion of a Let x < y (also y > x) denote x ≤ y and x , y. A sequence rule schema is at most cm, and since one of these is the (xi ) from S is a descending chain wrt ≤ if xi+1 < xi for every hypersequent-variable, the number of active components is i ∈ N; it is an antichain wrt ≤ if for all i, j ∈ N, xi ≤ xj at most cm − 1. Therefore we may assume without loss of implies i = j; it is an ascending chain wrt ≤ if xi+1 > xi generality that the last two components sl+cm and sl+1+cm for every i ∈ N. The elements x and y are incomparable if are in the instantiation of the hypersequent-variable. In each neither x ≤ y nor y ≤ x. Thus an antichain (xi ) is a sequence premise, make use of hp-admissibility of (c) (proved above) such that xi and xj (i , j) are incomparable. to replace sl+cm|sl+1+cm with s|s, and then replace s|s with s The following is standard (see e.g. [1]). using the IH. Reapplying r⇝ we can obtain the base instance Lemma 4.3. Let (A, ≤) be a partially ordered set, and let (xi ) l+cm−1 be a sequence from A. Then there is a subsequence (x ) such z }| { r (i) ′ that one of the following holds wrt ≤: д0| s1| ... |sl+cm−1 |s 1. (xr (i) ) is a descending chain Because we had д′ fm−1 д′ and each s fm−1 s, the above 0 ⇝(c) 1 i ⇝(c) 2. (xr (i) ) is an antichain fm−1 is related under ⇝(c) to 3. (xr (i) ) is an ascending chain 4. (xr (i) ) is constant i.e. xri = xrj for all i, j ∈ N l+cm−1 z }| { For a partially ordered set (S, ≤), a subset U ⊆ S is upward д′ | s| ... |s |s 1 closed if x ∈ U and x ≤ y implies y ∈ U . For a subset X ⊆ S,

′ cm−1 − the upward closed set generated by X is defined ↑X := {y ∈ Because we had д1 ⇝ h it follows that the above is (EC) S|∃x.x ∈ X and x ≤ y}. The set U (S, ≤) of upward closed related under cm−1 to ⇝(EC) subsets of S is partially ordered by set inclusion ⊆. l We use the symbol ≤ specifically to denote the standard z }| { componentwise partial order on Nm defined: (a ,..., a ) ≤ h−| s| ... |s |s 1 m (b ,...,b ) iff a ≤ b for every i (1 ≤ i ≤ m). 1 m i i This is exactly h|s. □ Theorem 4.4. Let m > 0. (i) (Nm, ≤) has no descending chain and no antichain. Remark 2. From the above lemma it follows using every- (ii) (U (Nm, ≤), ⊆) has no ascending chain and no antichain. where minimal derivations (see [19]) that (EW), (c), and (EC) ⇝ ⇝ ⇝ ⇝ ⇝ are hp-admissible also in HFLec +R \{(EW ), (c ), (EC )}. Proof. (i) This result is known as Dickson’s lemma [1, 12]. Extended Kripke lemma and decidability for hypersequent substructural logics LICS ’20, July 8–11, 2020, Saarbrücken, Germany

(ii) See [1, 2, 20]. This proof relies on the fact that (Nm, ≤) hence there exists y ∈ Y such that y ≤ x. Similarly y ∈ Y is a ω2-wqo. In particular, it does not hold for every wqo implies y ∈ ↑X and hence there exists x ∈ X such that x ≤ y. (Rado [26] has given a counterexample). □ This implies X ∼s Y , contradicting the incomparability of X and Y . We conclude that ↑X and ↑Y are incomparable. □ Let P(Nm ) denote the powerset of Nm. m 1 P m ∼ For X,Y ∈ P(N ), let us define ≼s as follows. It is easily Lemma 4.6. ( (N )/ s , ≼s ) has no descending chain and seen that this relation is reflexive and transitive. no antichain. 1 P ∼ X ≼s Y iff ∀y ∈ Y ∃x ∈ X (x ≤ y) Proof. If (Xi ) is a descending chain on ( (N)/ s , ≼s ), then (↑X ) is an ascending chain on (U (Nm, ≤), ⊆) by Lem. 4.5(i). However it is not antisymmetric since e.g. when m = 2: i If (X ) is an antichain chain on (P(N)/∼ , ), then (↑X ) is {(1, 2)} 1 {(1, 2), (10, 20)} 1 {(1, 2)}. To obtain antisym- i s ≼s i ≼s ≼s an antichain on (U (Nm, ≤), ⊆) by Lem. 4.5(ii). Each of these metry (and hence a partial order), we will need to treat such consequences contradicts Thm. 4.4(ii). □ elements as equivalent. We therefore define the following equivalence relation: 4.2 Relating to hypersequents 1 1 X ∼s Y iff X ≼s Y and Y ≼s X Let set(M) denote the set of elements that occur a positive m m number of times in multiset M. E.g. if M is multiset {p,p,q,r } Then ∼s partitions P(N ) into the set P(N )/∼s of equiva- lence classes. As usual, let [X] denote the equivalence class then set(M) = {p,q,r }. r containing X. For X,Y ∈ P(Nm ), define Define the following restriction (EW ) of (EW). h h contains a component Y ⇒ A 1 (EWr ) [X] ≼s [Y] iff X ≼s Y h|X ⇒ A such that set(X ) = set(Y ) It is easy to check that the definition of ≼s does not depend Example 4.7. An instance of (EWr ) is given below. on the representative elements that are chosen and hence p5,q3 ⇒ r is well-defined: let X ∼ X ′ and Y ∼ Y ′, and suppose that (EWr ) s s p5,q3 ⇒ r |p2,q4 ⇒ r [X] [Y]. Then X ′ 1 X 1 Y 1 Y ′ so [X ′] [Y ′] as ≼s ≼s ≼s ≼s ≼s 5 3 ⇒ | 4 ⇒ required. However e.g. p ,q r q r is not obtainable from p5,q3 ⇒ r because set({q4}) = q , {p,q} = set({p5,q3}). Clearly ≼s is reflexive, transitive and antisymmetric so ≼s m is a partial order on P(N )/∼s . Let Ω be a set of formulas, and let HΩ denote the set of 1 The definition of ≼s is motivated by the fact that it paral- hypersequents built using formulas from Ω. lels the operations of (EW), (c) and (EC) on hypersequents. For h1,h2 ∈ HΩ, define h1 ∼h h2 iff each hypersequent is This will become clear in the next subsection. In turns out obtainable from the other using repeated applications of (c), 1 r that ≼s is known as the minoring ordering (see [4]), and is (EC) and (EW ). also called the Smyth ordering in the power domain literature. Clearly ∼h is an equivalence relation. Therefore it parti- tions H into the set H /∼ of equivalence classes. Let [h] Lemma 4.5. For X,Y ∈ P(Nm ): (i) [X] ≺ [Y ] iff ↑X ⊃ ↑Y. Ω Ω h s denote the equivalence class containing h. (ii) If X and Y are incomparable wrt ≼s , then ↑X and ↑Y are incomparable wrt ⊆. Example 4.8. Observe that p,q ⇒ r is equivalent under ∼h to p,q ⇒ r |p2,q3 ⇒ r since Proof. (i) Assume [X] ≺s [Y ] (so [X] ≼s [Y] and [X] , [Y ]). For every u ∈ ↑Y there exists y ∈ Y such that y ≤ u. p,q ⇒ r ⇒ | 2 3 ⇒ (EWr ) p,q r p ,q r Since [X] ≼s [Y], there exists x ∈ X such that x ≤ y and p,q ⇒ r |p2,q3 ⇒ r p,q ⇒ r (c),(EC) hence x ≤ u. Thus u ∈ ↑X so ↑X ⊇ ↑Y . Let us show that the For [h1], [h2] ∈ HΩ/∼h, define [h1] h [h2] iff h1 can inequality is strict. Since [X] , [Y] then X ̸∼s Y and hence ≼ be obtained from h2 by repeated applications of (c), (EC) ∀x ∈ X∃y ∈ y(y ≤ x) does not hold. So there exists x0 ∈ X and (EWr ). such that y ≰ x0 for every y ∈ Y . Thus x0 < ↑Y , so ↑X ⊃ ↑Y . Assume ↑X ⊃ ↑Y. For arbitrary y ∈ Y : y ∈ ↑X so there Let us establish that the definition of ≼h does not depend on the representative elements that are chosen and hence exists x ∈ X such that x ≤ y and hence [X] ≼s [Y]. Let ′ ′ is well-defined. Let h1 ∼h h1 and h2 ∼h h2, and suppose us show that the inequality is strict. Let u ∈ ↑X \ ↑Y . Then ′ ∈ ≤ ∼ that [h1] ≼h [h2]. Then h1 is obtainable from h1 using (c), there exists x X such that x u. If [X] = [Y ] then X s Y r and hence there would exist y ∈ Y such that y ≤ x. But (EC), (EW ), similarly the latter from h2, and similarly the h′ h′ h′ then y ≤ u and hence u ∈ ↑Y and this is a contradiction. latter from 2. Thus [ 1] ≼h [ 2]. So [X] ≺ [Y]. Clearly ≼h is reflexive, transitive, and antisymmetric, so s H /∼ (ii) Assume that X and Y are incomparable. ↑X ⊂ ↑Y is it is a partial order on Ω h. not possible because this would imply [X] ≻s [Y ] by (i), Definition 4.9 ((S, Π)-component, hypersequent, sequence). 1 and hence X ≻s Y. Similarly ↑Y ⊂ ↑X is not possible. Fi- Let S be a finite set of formulas, and let Π be either empty or nally, suppose ↑X = ↑Y. Then x ∈ X implies x ∈ ↑Y and consist of a single formula. LICS ’20, July 8–11, 2020, Saarbrücken, Germany Revantha Ramanayake

A (S, Π)-component has the form X ⇒ Π with set(X ) = S. from д1 by using (EC) to remove excess copies so this is no If every component in a hypersequent is a (S, Π)-component, longer the case. Finally, observe that h may contain some call this a (S, Π)-hypersequent. (S, Π)-components that do not occur in д2. Obtain д3 from д2 A sequence of (S, Π)-hypersequents is called a (S, Π)- by applying (EWr ) to insert these missing components. It sequence. follows that д3 is identical to h, and thus [h] ≼h [д]. □ ⊣ ⊣ For a non-empty finite set S, fix an enumeration A1,..., A|S |. From the above, also [h] ≺h [д] iff [h ] ≺s [д ]. Let X1 ⇒ Π| ... |XN ⇒ Π be any (S, Π)-hypersequent h. Then define h⊣ ∈ P(N|S | ): Corollary 4.12. If (hi ) is a (S, Π)-sequence with S finite, [ ([hi ]) is neither a descending chain nor an antichain wrt ≼h. {(a1,..., a |S | )|Ai occurs exactly ai times in Xj } 1≤j ≤N Proof. Immediate if S = ∅ as ([hi ]) would be constant. Sup- pose that S , ∅. If ([hi ]) is a descending chain, then [hi ] ≻h Example 4.10. Let h be the ({p,q,p → q},r )-hypersequent ⊣ ⊣ ⊣ [hi+ ] for every i. Lem. 4.11 implies [h ] ≻s [h ], so ([h ]) below left. Let us enumerate {p,q,p → q} as p,q,p → q. 1 i i+1 i would be a descending chain on (P(N|S | )/∼ , ). If ([h ]) is Then h⊣ ∈ P(N3) is given below right. s ≼s i an antichain, for every i, j neither [hi ] ≼h [hj ] nor [hj ] ≼h → ⇒ | → ⇒ { } ⊣ ⊣ ⊣ ⊣ p,p,p q r p,q,p q r (2, 0, 1) , (1, 1, 1) [hi ]. From Lem. 4.11: neither [hi ] ≼h [hj ] nor [hj ] ≼h [hi ], ⊣ | | ⊣ ⊣ ⊣ P N S ∼ Observe that (д|h) = д ∪ h . so ([hi ]) would be an antichain on ( ( )/ s , ≼s ). Each of these consequences contradicts Lem. 4.6. □ Lemma 4.11. For a non-empty finite set S of formulas, and ⊣ ⊣ (S, Π)-hypersequents д,h: [h] ≼h [д] iff [h ] ≼s [д ]. 4.3 Proof of the extended Kripke lemma 1 We will use the infinite Ramsey theorem (see [1, 24, 27]). Proof. By inspection, the relation ≼s satisfies the following m properties for arbitrary X1, X2,Y1,Y2 ∈ P(N ). Theorem 4.13 (Ramsey [27]). For every function c mapping 1 1 1 (i) X1 ≼s Y1 and X2 ≼s Y2 implies X1 ∪ X2 ≼s Y1 ∪ Y2. each 2-element subset of N to an element of a finite set, there 1 (ii) X1 ≼s Y1 implies X1 ∪ X2 ≼s Y1. exists an infinite set Y ⊆ N such that the restriction of c to the In the following rule instances, suppose that the premise 2-element subsets of Y is a constant. (S, Π) (and hence the conclusion) is a -hypersequent. The (S, Π)-reduct of a hypersequent h—denoted h(S, Π)— is obtained by deleting every non-(S, Π)-component in h. h|X, A, A ⇒ Π h|X ⇒ Π|X ⇒ Π h (EWr ) Clearly h(S, Π) is a (S, Π)-hypersequent. Also, (hi (S, Π)) is h|X, A ⇒ Π h|X ⇒ Π h|X ⇒ Π a (S, Π)-sequence for every sequence (hi ) of hypersequents. We observe—using (i) and (ii)—that in every instance of one Proof of Lemma 4.2. Let (hi ) be a sequence of hypersequents of the above rule schemas, conclusion⊣ 1 premise⊣. I.e. ≼s from HΩ for a finite set Ω of formulas. Since there are only ⊣ ⊣ ⊣ 1 ⊣ ⊣ ⊆ (h|X, A ⇒ Π) = h ∪ (X, A ⇒ Π) ≼s h ∪ (X, A, A ⇒ Π) finitely many choices for S Ω and Π (empty or an ele- ⊑ = (h|X, A, A ⇒ Π)⊣ ment of Ω), it follows that there is a subsequence (hq(i) ) (hi ) such that the signature of hq(i)—define this as the set | ⇒ ⊣ | ⇒ ⊣ Hence [(h X, A Π) ] ≼s [(h X, A, A Π) ]. {(S, Π)|hq(i) has a (S, Π)-component}—is the same set T = (h|X ⇒ Π)⊣ = h⊣ ∪ (X ⇒ Π)⊣ = h⊣ ∪ (X ⇒ Π)⊣ ∪ (X ⇒ Π)⊣ {(S1, Π1),..., (SN , ΠN )} for every i. IfT is empty then (hq(i) ) is a sequence of empty hypersequents so the result follows = (h|X ⇒ Π|X ⇒ Π)⊣ trivially. Therefore take T as non-empty. ⊣ ⊣ Hence [(h|X ⇒ Π) ] ≼s [(h|X ⇒ Π|X ⇒ Π) ]. Let us suppose that ([hr (i)]) ⊑ ([hq(i)]) is a descending ⊣ ⊣ ⊣ 1 ⊣ Also (h|X ⇒ Π) = h ∪ (X ⇒ Π) ≼s h and hence we chain and obtain a contradiction. Then for every i, j ∈ N ⊣ ⊣ have [(h|X ⇒ Π) ] ≼s [h ]. with i < j: [hr (i)] ≻h [hr (j)]. Hence [hr (i)] ⋠ h [hr (j)]. Suppose that [h] ≼h [д]. By definition, there is some Since hr (i) and hr (j) have the same signature, any missing r finite sequence д = д0,...,дN = h such that дi+1 is obtained components of hr (i) can be supplied by (EW ). Therefore r ij ij ij ij from дi by (c), (EC) or (EW ). By the observations above, there must exist some (S , Π )-component ((S , Π ) ∈ T ) ⊣ 1 ⊣ 1 ⊣ 1 ⊣ дi+1 ≼s дi . By transitivity of ≼s it follow that h ≼s д , and in hr (j) that cannot be contracted to any component in hr (i). ⊣ ⊣ ij ij hence [h ] ≼s [д ]. Assign (S , Π ) ∈ T to the 2-element subset {i, j} ⊂ N. By ⊣ ⊣ ⊣ 1 ⊣ Now suppose that [h ] ≼s [д ]. By definition h ≼s д and Ramsey’s theorem there is an infinite Y ⊆ N such that every thus ∀y ∈ д⊣∃x ∈ h⊣ (x ≤ y). Since д and h are both (S, Π)- 2-element subset of Y is assigned the same element (S∗, Π∗) ∈ hypersequents, every component in д can be made identical T . Write Y = {s(0),s(1),s(2),...} with s(0) < s(1) < s(2) < ∗ ∗ ∗ ∗ to some component in h using (c). Let д1 be the hyperse- .... Then [hrs (i) (S , Π )] ≻h [hrs (i+1) (S , Π )], and thus we ∗ ∗ quent obtained from д in this way. Note that a sequent may have that ([hrs (i) (S , Π )]) is a descending chain, contradict- occur as a component more times in д1 than in h. Obtain д2 ing Cor. 4.12. Extended Kripke lemma and decidability for hypersequent substructural logics LICS ’20, July 8–11, 2020, Saarbrücken, Germany

Let us suppose that ([hr (i)]) ⊑ ([hq(i)]) is an antichain is a finite search because there are only finitely many possi- ⇝ ⇝ and obtain a contradiction. Then for every i, j ∈ N with i < j: ble rule instances in HFLec + R with a given hypersequent [hr (i)] ⋠ h [hr (j)] and [hr (j)] ⋠ h [hr (i)]. From the former, conclusion (this was why we internalised a fixed amount ′ because hr (i) and hr (j) have the same signature, there ex- of contraction!). Replace each immediate subderivation d ij ij ij ij ′ ists some (S , Π )-component ((S , Π ) ∈ T ) in hr (j) that of min(d) with min(d ). In the derivation so obtained, re- ′′ ′ cannot be contracted to any component in hr (i). Assign place each immediate subderivation d of those min(d ) (Sij , Πij ) ∈ T to the 2-element subset {i, j} ⊂ N. Apply- with min(d ′′). Proceed in this way until the initial hyper- ing Ramsey’s theorem there is an infinite set Y ⊆ N such sequents are reached. The result is an everywhere minimal that every 2-element subset of Y is assigned the same ele- derivation of h. ment (S∗, Π∗). Write Y = {s(0),s(1),s(2),...} with s(0) < Observation 2. An everywhere minimal derivation is irre- ∗ ∗ ∗ ∗ s(1) < s(2) < . . .. Then [hrs (i) (S , Π )] ⋠ h [hrs (j) (S , Π )] dundant. Indeed, if d were a everywhere minimal derivation for every i < j. We continue: that is not irredundant, then it would contain a hyperse- For every i, j ∈ N with i < j, define c({i, j}) = 1 if quent h1 and a hypersequent h2 above it from which h1 could ∗ ∗ ∗ ∗ [hrs (i) (S , Π )] ≻h [hrs (j) (S , Π )] and 0 otherwise. Once be obtained by repeated applications from (c), (EC), (EW). again by Ramsey’s theorem, there is an infinite set Z ⊆ N Then the subderivation of h1 would not have minimal height such such that every 2-element subset of Z is assigned ei- since a derivation of lesser height is obtainable from h2 by ther 1 or 0. Write Z = {t (0), t (1), t (2),...} with t (0) < hp-admissibility of the latter rules (Lem. 3.7). ∗ ∗ t (1) < t (2) < . . .. If it is 1 then ([hrst (i) (S , Π )]) is a de- We have shown that if ⇒ F is derivable, it has an ir- scending chain so by Cor. 4.12 we obtain a descending chain redundant derivation. The point of irredundancy (unlike |S | in (P(N ), ≼s ) contradicting Thm. 4.6. So it must be 0. everywhere minimality) is that we can enforce it during ∗ ∗ Then for every i, j ∈ N such that i < j: [hrst (i) (S , Π )] ⋠ h backward proof search. Perform irredundant backward proof ∗ ∗ ∗ ∗ [hrst (j) (S , Π )] (previous paragraph) and [hrst (i) (S , Π )] ̸≻h search on ⇒ F by constructing a proof search tree as follows: ∗ ∗ ∗ ∗ [hrst (j) (S , Π )]. It follows that ([hrst (i) (S , Π )]) is an an- place ⇒F at the root. Repeatedly, for each hypersequent in tichain, contradicting Cor. 4.12. the tree, place all the premises of all possible rule instances Since no subsequence of ([hq(i)]) is a decreasing chain or as its children, omitting those rule instances that would in- an antichain, it must have a constant or ascending subse- troduce a premise that violates irredundancy of the tree. The qence ([hu (i)]) (Lem. 4.3). Then [hu (1)] ≼h [hu (2)], so hu (1) proof search tree is finitely branching since there are only r can be obtained from hu (2) using (c), (EC) and (EW ). □ finitely many possible rule instances that apply to agiven hypersequent conclusion. Moreover every branch has finite 5 Decidability via backward proof search length in the limit since, by Lem. 4.2, there is no infinite We are ready to establish the decidability of every amenable sequence (hi ) of hypersequents built using subformulas of F axiomatic extension of FLec. Equivalently, these are the log- that can satisfy irredundancy (i.e. no element is obtainable ics that can be formalised by extending the hypersequent from a later element by (c), (EC), (EW)). The proof search calculus HFLec with analytic structural rules. tree is therefore finite by König’s lemma and thus the con- struction will terminate. Finally, check if the proof search Theorem 5.1 . Every axiomatic extension (Main theorem) tree contains a derivation as a subtree. If it does, then ⇒F is of FL by amenable formulas is decidable. ec derivable, otherwise it is not. □ Proof. Let L be an arbitrary amenable axiomatic extension of FLec. By Thm. 2.5(i) we can compute a finite set R of analytic structural rules such that for every formula F: F ∈ 6 Future work L iff ⇒ F is derivable in HFLec + R. From Lem. 3.6: F ∈ L iff Complexity bounds for the amenable extensions of FLec ⇝ ⇝ ⇒ F is derivable in HFLec + R . Therefore in the following are a challenging question in their own right, as indicated ⇝ ⇝ we solely consider derivations in HFLec + R . by Urquhart’s [30] complexity arguments for FLec. A derivation has minimal height if there is no derivation It is likely that an upper bound could be obtained here of the same hypersequent with lesser height. A derivation along the lines of [30]: the idea is to show that arbitrarily long is everywhere minimal [19] if every subderivation of it has necessarily finite non-constant non-ascending sequences minimal height. A derivation is irredundant if no hyperse- can be replaced by ones with bounded length (‘controlled quent in the derivation can be obtained from a hypersquent bad sequences’) and use an upper bound for controlled bad above it by repeated applications from (c), (EC), (EW). sequences. Controlled bad sequences can be used because the Observation 1. Every derivable hypersequent has an ev- amount of contraction internalised in each rule is fixed and erywhere minimal derivation. We proceed following [19]. hence the number of copies of a formula in the antecedent Given a derivation d of h, obtain a derivation min(d) of h can only increase by a fixed amount from conclusion to of minimal height by exhaustively searching for derivations premise. Balasubramanian [4] gives lower and upper bounds (‘backward proof search’) with height at most that of d. This for controlled bad sequences over the Smyth (minoring) wqo LICS ’20, July 8–11, 2020, Saarbrücken, Germany Revantha Ramanayake of finite sets of Nm that exceed Urquhart’s tight ‘primitive [11] Petr Cintula and Carles Noguera. 2010. Implicational (semilinear) recursive in the Ackermann function’ bound for FLec. logics I: a new hierarchy. Arch. Math. Log. 49, 4 (2010), 417–446. 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