Bidirectional Decision Procedures for the Intuitionistic Propositional Modal Logic IS4

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1 Bidirectional ecision Procedures for the Intuitionistic Propositional Modal Logic IS4 Samuli Heilala and Brigitte Pientka School of Computer Science, McGill University, Montreal, Canada Technical Report SOCS-TR , May 2007 Abstract. We present a multi-context focused sequent calculus whose derivations are in bijective correspondence with normal natural deductions in the propositional fragment of the intuitionistic modal logic IS4. This calculus, suitable for the enumeration of normal proofs, is the starting point for the development of a sequent calculus-based bidirectional decision procedure for propositional IS4. In this system, relevant derived inference rules are constructed in a forward direction prior to proof search, while derivations constructed using these derived rules are searched over in a backward direction. We also present a variant which searches directly over normal natural deductions. Experimental results show that on most problems, the bidirectional prover is competitive with both conventional backward provers using loop-detection and inverse method provers, significantly outperforming them in a number of cases. 1 Introduction Intuitionistic modal logics are constructive logics incorporating operators of necessity ( ) and possibility ( ). Fitch [8], Prawitz [17], Satre [19], and more recently Simpson [20], Bierman and de Paiva [2], and Pfenning and avies [16] have investigated a broad range of proof-theoretical properties of various logics of this kind. Recently, such logics have also found applications in hardware verification [7] and proposed type systems for staged computation [4] and distributed computing [14]. A logic frequently used in these settings is either the intuitionistic variant of the classical modal logic S4, which we will call IS4, or a logic that can be expressed through IS4, such as Fairtlough and Mendler s lax logic [7] (see for instance [16] for the relationship between IS4 and lax logic). In this light, it is surprising that proof search in IS4 has not received more attention. Howe has investigated proof enumeration and theorem proving in lax logic [13] and, coming closer to our work, has presented a backward decision procedure for the fragment of propositional IS4 without the possibility modality [12]. His system performs loop-detection using a history mechanism, but is encumbered by a large number of rules and related provisos (21 axioms and inference rules). It would only grow with the addition of the possibility modality, which would also require a different loop-detection mechanism. 1

2 Our contributions begin with a sequent calculus for propositional IS4 suitable for the enumeration of normal proofs. This forms the basis for the development of a sequent calculus-based bidirectional IS4 decision procedure, in which derived inference rules relevant to the query are constructed in a forward direction prior to proof search, while derivations constructed using these derived rules are searched over in a backward direction. We also demonstrate that this approach corresponds very closely to an elegant bidirectional decision procedure that searches directly over normal natural deductions. The key to our theoretical justification of both of these decision procedures is a refinement of the well-known subformula property, which we use to restrict nondeterminism in focused proof search in the presence of multiple contexts. To evaluate our approach empirically, we have put together a set of 50 benchmark formulas for IS4. Experimental results show that on most problems, the bidirectional prover is competitive with both conventional backward provers using loop-detection and inverse method provers, significantly outperforming them in a number of cases. Although we concentrate on propositional IS4 in this paper, we believe that the techniques presented are general enough to find applications in other constructive logics, such as the contextual modal logic of Nanevski, Pfenning, and Pientka [15]. This paper is an extended version of [11]. In Sect. 2 we summarize the relevant background and introduce our core natural deduction formalism, while Sect. 3 presents a corresponding Hilbertstyle axiomatization. In Sect. 4 we restrict our natural deduction system to allow only normal natural deductions, and in Sect. 5 we give the corresponding sequent calculus formulation that serves as the basis for our proof search strategies. A more refined sequent calculus for forward proof search is developed in Sect. 6, while Sects. 7 and 8 describe our bidirectional decision procedure in sequent calculus and natural deduction settings, respectively. Experimental results are given in Sect. 9, and Sect. 10 concludes with related and future work. 2 Natural eduction Formulas in the propositional fragment of IS4 are given by the grammar A ::= P A A A A A A A A where P is taken from a countable set of atomic propositional constants and negation and truth are defined notationally in the usual way: A def = A def = Operator precedences are, from highest to lowest,,,,,, while all binary operators are considered right-associative. Our starting point is a multi-context natural deduction formulation for IS4 similar to ones proposed by Pfenning and avies [16] and Bierman and de Paiva [2], featuring hypothetical judgements of the form ; Γ A. Such a judgement asserts the provability of a formula A under hypotheses and Γ, where Γ = A 1,..., A n is a context of true hypotheses 2

3 ; Γ 1, A, Γ 2 A hyp 1 1, A, 2; Γ A hyp 2 ; Γ ; Γ A E ; Γ, A 1 A 2 ; Γ A 1 A 2 I ; Γ A 1 A 2 ; Γ A 1 ; Γ A 2 E ; Γ A 1 ; Γ A 2 ; Γ A 1 A 2 I ; Γ A 1 A 2 ; Γ A j E j ; Γ A j ; Γ A 1 A 2 ; Γ, A 1 C ; Γ, A 2 C I j ; Γ A 1 A 2 ; Γ C ; A ; Γ A I ; Γ A, A; Γ C ; Γ C ; Γ A ; Γ A I ; Γ A ; A C ; Γ C j {1, 2} E E E Fig.1. NJ IS4 and = B 1,..., B m is a modal context of valid hypotheses. Valid hypotheses are hypotheses whose truth does not depend on the truth of other formulas, that is, hypotheses that are in some sense always or necessarily true. We use to denote the empty context. Our base system differs from Pfenning and avies natural deduction formulation for IS4 [16] in that we make no judgemental distinction between truth and possibility. Instead, we internalize the possibility judgement using the operator. Interestingly, this gives rise to a system that is essentially identical to Bierman and de Paiva s multi-context natural deduction formulation for IS4 [2]. This base calculus, which we will call NJ IS4, is shown in Fig. 1. Although the contexts of this system are formally ordered lists, we can afford to be flexible with them, as NJ IS4 has the usual structural properties of weakening, contraction, and exchange for both contexts. We formalize some of these properties below. In the remainder, we will not always formally state structural properties that are not required for the proofs of the main results. For convenience, we will generally think of contexts in NJ IS4 as multisets, and will write A Γ to denote that Γ has the form Γ 1, A, Γ 2 for some (possibly empty) Γ 1 and Γ 2. Lemma 1 (Weakening for NJ IS4 ). If ; Γ 1, Γ 2 C, then ; Γ 1, A, Γ 2 C. Proof. By induction on the structure of. 1 1 Whenever a proof is by a very simple induction, we will omit the cases. 3

4 Lemma 2 (Modal Weakening for NJ IS4 ). If 1, 2 ; Γ C, then 1, A, 2 ; Γ C. Proof. By induction on the structure of. Lemma 3 (Exchange for NJ IS4 ). If ; Γ 1, A 1, Γ 2, A 2, Γ 3 C, then ; Γ 1, A 2, Γ 2, A 1, Γ 3 C. Proof. By induction on the structure of. Lemma 4 (Modal exchange for NJ IS4 ). If 1, A 1, 2, A 2, 3 ; Γ C, then 1, A 2, 2, A 1, 3 ; Γ C. Proof. By induction on the structure of. The system NJ IS4 also has the following important substitution properties. They encapsulate the intuitive notion of replacing uses of a hypothesis by separate derivations of that hypothesis. Lemma 5 (Substitution property for NJ IS4 ). E If ; Γ 1, A, Γ 2 C and ; Γ 1, Γ 2 A, then ; Γ 1, Γ 2 C. Proof. By induction on the structure of, using Lemmas 1 and 2 where needed. Lemma 6 (Modal substitution property for NJ IS4 ). E If 1, A, 2 ; Γ C and 1, 2 ; A, then 1, 2 ; Γ C. Proof. By induction on the structure of, using Lemmas 1 and 2 where needed. The inference rules of NJ IS4 are largely standard, but to glean some intuition about the modal rules and the two contexts, it is useful to think of the modalities as quantifying truth over worlds in some universe, with some reachability relation defined on the worlds. To say that A is true is to say that A is true in all worlds reachable from the current one, while to say that A is true is to say that A is true in some world reachable from the current one. The current world represents the environment in which the provability of the succedent is to be established. Under this interpretation, the hypotheses in the modal context can be used in all reachable worlds, while those in the regular context can only be used in the current world. This notion is formalized by the following results, which highlight the relationship between global validity in the form of the modal context and the local internalization of validity in the form of the necessity operator. Theorem 1 (Globalizing necessity in NJ IS4 ). If 1, 2 ; Γ 1, A, Γ 2 C, then 1, A, 2 ; Γ 1, Γ 2 C. 4

5 Proof. (1) 1, A, 2 ; Γ 1, A, Γ 2 C Lemma 2 () (2) 1, A, 2 ; A hyp 2 (3) 1, A, 2 ; Γ 1, Γ 2 A I (2) (4) 1, A, 2 ; Γ 1, Γ 2 C Lemma 5 (1, 3) Theorem 2 (Localizing validity in NJ IS4 ). If 1, A, 2 ; Γ 1, Γ 2 C, then 1, 2 ; Γ 1, A, Γ 2 C. Proof. (1) 1, A, 2 ; Γ 1, A, Γ 2 C Lemma 1 () (2) 1, 2, A; Γ 1, A, Γ 2 C Lemma 4 (1) (3) 1, 2 ; Γ 1, A, Γ 2 A hyp 1 (4) 1, 2 ; Γ 1, A, Γ 2 C E (3, 2) Theorem 2 suggests that the contextual separation of truth and validity is not strictly necessary, and that we can obtain an embedding of NJ IS4 into a single-context system simply by providing every valid hypothesis with a operator and merging the two contexts. To obtain a single-context system of natural deduction for IS4, however, those inference rules that depend on the separation of the modal and regular contexts (such as I, E, and E) need to be modified. The details of single-context natural deduction systems for IS4 have been explored in more detail by Bierman and de Paiva [2]. To lend support to our claim that NJ IS4 is indeed a formalism for a constructive variant of the classical modal logic S4, we will next present a Hilbert-style axiomatization of IS4 and show that NJ IS4 is equivalent to it. 3 Axiomatization While several proposals for Hilbert-style axiomatic formulations of constructive versions of S4 exist in the literature, we will use a standard system motivated by applications discussed by Alechina et al. [1] and Pfenning and avies [16]. In it, a standard axiomatization of intuitionistic propositional logic (see for instance [21]) is augmented by the six modal axioms K : (A 1 A 2 ) A 1 A 2 K : (A 1 A 2 ) A 1 A 2 T : A A T : A A 4 : A A 4 : A A and the necessitation rule If A is a theorem, then A is a theorem. The complete axiomatization, which we will call A IS4, is shown in Fig. 2. For consistency with NJ IS4, we again consider contexts to be ordered lists, but an interpretation of contexts as sets yields the same results. Note that we show general weakening as an explicit rule w, while an alternative approach, followed by Bierman and de 5

6 Axioms: 1 : A 1 A 2 A 1 2 : (A 1 A 2 A 3) (A 1 A 2) A 1 A 3 1 : A 1 A 2 A 1 A 2 1 : (A 1 A 3) (A 2 A 3) A 1 A 2 A 3 2 : A 1 A 2 A 1 2 : A 1 A 1 A 2 3 : A 1 A 2 A 2 3 : A 2 A 1 A 2 : A K : (A 1 A 2) A 1 A 2 K : (A 1 A 2) A 1 A 2 T : A A T : A A 4 : A A 4 : A A Rules: A A A ass A A ax (A is an axiom) Γ A A 1 A 2 Γ A A 1 Γ A A 2 mp Γ 1, Γ 2 A C Γ 1, A, Γ 2 A C w A A A A nec Fig.2. A IS4 Paiva [2], is to incorporate weakening locally into the assumption, axiom, and necessitation rules, and to prove its general form to be admissible. We will use w to denote zero or more applications of the weakening rule w, and ass and ax to denote applications of the assumption rule ass and the axiom rule ax, respectively, followed by zero or more applications of the weakening rule w. The following common intermediate results hold for our axiomatization. Lemma 7 (Reflexivity of for A IS4 ). Γ A A A. Proof. (1) Γ A (A (A A) A) (A A A) A A ax ( 2 ) (2) Γ A A (A A) A ax ( 1 ) (3) Γ A (A A A) A A mp (1, 2) (4) Γ A A A A ax ( 1 ) (5) Γ A A A mp (3, 4) Lemma 8 (eduction theorem for A IS4 ). If Γ 1, A 1, Γ 2 A A 2, then Γ 1, Γ 2 A A 1 A 2. Proof. By induction on the structure of. = A A A ass (1) A A A Lemma 7 6

7 = Γ 1, Γ 2 A A 2 Γ 1, A 1, Γ 2 A A 2 w (1) Γ 1, Γ 2 A A 2 A 1 A 2 ax ( 1 ) (2) Γ 1, Γ 2 A A 1 A 2 mp (1, ) = Γ 1, Γ 2 A A 2 Γ 1, B, Γ 2 A A 2 w (A 1 Γ 1, Γ 2 ) (1) Γ 1, Γ 2 A A 1 A 2 i.h. ( ) (2) Γ 1, B, Γ 2 A A 1 A 2 w (1) 2 = Γ 1, A 1, Γ 2 A B A 2 Γ 1, A 1, Γ 2 A B mp Γ 1, A 1, Γ 2 A A 2 (1) Γ 1, Γ 2 A (A 1 B A 2 ) (A 1 B) A 1 A 2 ax ( 2 ) (2) Γ 1, Γ 2 A A 1 B A 2 i.h. ( ) (3) Γ 1, Γ 2 A (A 1 B) A 1 A 2 mp (1, 2) (4) Γ 1, Γ 2 A A 1 B i.h. ( 2 ) (5) Γ 1, Γ 2 A A 1 A 2 mp (3, 4) Lemma 9 (Transitivity of for A IS4 ). E If Γ A A 1 A 2 and Γ A A 2 A 3, then Γ A A 1 A 3. Proof. (1) Γ, A 1 A A 2 A 3 w (E) (2) Γ A A 1 A 2 A 3 Lemma 8 (1) (3) Γ A (A 1 A 2 A 3 ) (A 1 A 2 ) A 1 A 3 ax ( 2 ) (4) Γ A (A 1 A 2 ) A 1 A 3 mp (3, 2) (5) Γ A A 1 A 3 mp (4, ) Lemma 10 (Generalized necessitation for A IS4 ). If A 1,..., A n A C, then A 1,..., A n A C. Proof. By induction on n. n = 0. (1) A C nec () 7

8 n > 0. (1) A 1,..., A n 1 A A n C Lemma 8 () (2) A 1,..., A n 1 A ( A n C) i.h. (1) (3) A 1,..., A n 1 A ( A n C) A n C ax (K ) (4) A 1,..., A n 1 A A n C mp (3, 2) (5) A 1,..., A n 1 A A n A n ax (4 ) (6) A 1,..., A n 1 A A n C Lemma 9 (5, 4) (7) A 1,..., A n 1, A n A A n C w (6) (8) A 1,..., A n 1, A n A A n ass (9) A 1,..., A n 1, A n A C mp (7, 8) Armed with these intermediate results, we can now prove the equivalence of NJ IS4 and A IS4. As a shorthand, if = A 1,...,A n, then we will write to denote A 1,..., A n. Theorem 3 (Soundness of NJ IS4 with respect to A IS4 ). If ; Γ A, then, Γ A A. Proof. By induction on the structure of. We show the base cases and the cases involving modalities. = ; Γ1, A, Γ 2 A hyp 1 (1), Γ 1, A, Γ 2 A A ass = 1, A, 2 ; Γ A hyp 2 (1) 1, A, 2, Γ A A A ax (T ) (2) 1, A, 2, Γ A A ass (3) 1, A, 2, Γ A A mp (1, 2) = ; A ; Γ A I (1) A A i.h. ( ) (2) A A Lemma 10 (1) (3), Γ A A w (2) 2 = ; Γ B, B; Γ A E ; Γ A (1), B, Γ A A i.h. ( 2 ) (2), Γ A B A Lemma 8 (1) (3), Γ A B i.h. ( ) (4), Γ A A mp (2, 3) 8

9 = ; Γ A ; Γ A I (1), Γ A A i.h. ( ) (2), Γ A A A ax (T ) (3), Γ A A mp (2, 1) 2 = ; Γ B ; B A E ; Γ A (1), B A A i.h. ( 2 ) (2) A B A Lemma 8 (1) (3) A (B A) Lemma 10 (2) (4) A (B A) B A ax (K ) (5) A B A mp (4, 3) (6), Γ A B A w (5) (7), Γ A B i.h. ( ) (8), Γ A A mp (6, 7) (9), Γ A A A ax (4 ) (10), Γ A A mp (9, 8) Lemma 11 (Completeness of NJ IS4 with respect to the axioms of A IS4 ). If A is an axiom of A IS4, then ; A. Proof. We show the cases involving modalities. The derivations are linearized to save space. K (1) ; (A 1 A 2 ), A 1 (A 1 A 2 ) hyp 1 (2) A 1 A 2 ; (A 1 A 2 ), A 1 A 1 hyp 1 (3) A 1 A 2, A 1 ; A 1 A 2 hyp 2 (4) A 1 A 2, A 1 ; A 1 hyp 2 (5) A 1 A 2, A 1 ; A 2 E (3, 4) (6) A 1 A 2, A 1 ; (A 1 A 2 ), A 1 A 2 I (5) (7) A 1 A 2 ; (A 1 A 2 ), A 1 A 2 E (2, 6) (8) ; (A 1 A 2 ), A 1 A 2 E (1, 7) (9) ; (A 1 A 2 ) A 1 A 2 I (8) (10) ; (A 1 A 2 ) A 1 A 2 I (9) 9

10 K (1) ; (A 1 A 2 ), A 1 (A 1 A 2 ) hyp 1 (2) A 1 A 2 ; (A 1 A 2 ), A 1 A 1 hyp 1 (3) A 1 A 2 ; A 1 A 1 A 2 hyp 2 (4) A 1 A 2 ; A 1 A 1 hyp 1 (5) A 1 A 2 ; A 1 A 2 E (3, 4) (6) A 1 A 2 ; A 1 A 2 I (5) (7) A 1 A 2 ; (A 1 A 2 ), A 1 A 2 E (2, 6) (8) ; (A 1 A 2 ), A 1 A 2 E (1, 7) (9) ; (A 1 A 2 ) A 1 A 2 I (8) (10) ; (A 1 A 2 ) A 1 A 2 I (9) T (1) ; A A hyp 1 (2) A; A A hyp 2 (3) ; A A E (1, 2) (4) ; A A I (3) T (1) ; A A hyp 1 (2) ; A A I (1) (3) ; A A I (2) 4 (1) ; A A hyp 1 (2) A; A hyp 2 (3) A; A I (2) (4) A; A A I (3) (5) ; A A E (1, 4) (6) ; A A I (5) 4 (1) ; A A hyp 1 (2) ; A A hyp 1 (3) ; A A E (1, 2) (4) ; A A I (3) Theorem 4 (Completeness of NJ IS4 with respect to A IS4 ). If Γ A A, then ; Γ A. Proof. By induction on the structure of. = A A A ass (1) ; A A hyp 1 = A A ax (1) ; A Lemma 11 10

11 = Γ 1, Γ 2 A A Γ 1, B, Γ 2 A A w (1) ; Γ 1, Γ 2 A i.h. ( ) (2) ; Γ 1, B, Γ 2 A Lemma 1 (1) 2 = Γ A B A Γ A B mp Γ A A (1) ; Γ B A i.h. ( ) (2) ; Γ B i.h. ( 2 ) (3) ; Γ A E (1, 2) = A A A A nec (1) ; A i.h. ( ) (2) ; A I (1) These results show that the system NJ IS4 is equivalent to the axiomatization A IS4, so it is reasonable to choose NJ IS4 as a foundation on which to develop proof search strategies for IS4. While Girard, Lafont, and Taylor suggest that we should think of natural deductions as the true proof objects [10], natural deduction systems have traditionally not seen much use as formalisms for proof search, mainly as a result of their lack of syntax-directedness. The first step towards being able to perform proof search over natural deductions is to restrict the proof space to those natural deductions that are in normal form. We formalize this restriction in the next section. 4 Normalized Natural eduction NJ IS4 is a possible starting point for proof search strategies, but a better idea is to restrict the proof space described by NJ IS4 in such a way that only natural deductions in normal form can be constructed. This is achieved by annotating judgements with their intended direction of reasoning: ; Γ A A has a normal proof under hypotheses and Γ, ; Γ A A can be extracted from hypotheses in and Γ using only elimination rules. The resulting system, which we will call NJ N IS4, is shown in Fig. 3. As in NJ IS4, the contexts of this system are formally ordered lists, but NJ N IS4 also has the structural properties of weakening, contraction, and exchange for both contexts. Again, we will generally think of contexts in NJ N IS4 as multisets. 11

12 ; Γ 1, A, Γ 2 A hyp 1 1, A, 2; Γ A hyp 2 ; Γ, A 1 A 2 ; Γ A 1 A 2 I ; Γ A 1 A 2 ; Γ A 1 ; Γ A 2 ; Γ A j ; Γ A 1 A 2 I j ; Γ A 1 ; Γ A 2 ; Γ A 1 A 2 I ; Γ A 1 A 2 ; Γ A j ; Γ ; Γ A E E j E ; Γ A 1 A 2 ; Γ, A 1 C ; Γ, A 2 C ; Γ C ; A ; Γ A I ; Γ A, A; Γ C ; Γ C ; Γ A ; Γ A I ; Γ A ; A C ; Γ C ; Γ A (A is atomic) ; Γ A j {1, 2} E E E Fig.3. NJ N IS4 Lemma 12 (Weakening for NJ N IS4 ). 1. If ; Γ 1, Γ 2 C, then ; Γ 1, A, Γ 2 C. E 2. If ; Γ 1, Γ 2 C, then ; Γ 1, A, Γ 2 C. Proof. By simultaneous induction on the structure of the given derivation. Lemma 13 (Modal weakening for NJ N IS4 ). 1. If 1, 2 ; Γ C, then 1, A, 2 ; Γ C. E 2. If 1, 2 ; Γ C, then 1, A, 2 ; Γ C. Proof. By simultaneous induction on the structure of the given derivation. The substitution properties of NJ N IS4 are similar to those of NJ IS4, but take the direction of reasoning into account. Lemma 14 (Substitution property for NJ N IS4 ). E 1. If ; Γ 1, A, Γ 2 C and ; Γ 1, Γ 2 A, then ; Γ 1, Γ 2 C. F G 2. If ; Γ 1, A, Γ 2 C and ; Γ 1, Γ 2 A, then ; Γ 1, Γ 2 C. 12

13 Proof. By simultaneous induction on the structure of the first given derivation, using Lemmas 12 and 13 where needed. Lemma 15 (Modal substitution property for NJ N IS4 ). E 1. If 1, A, 2 ; Γ C and 1, 2 ; A, then 1, 2 ; Γ C. F G 2. If 1, A, 2 ; Γ C and 1, 2 ; A, then 1, 2 ; Γ C. Proof. By simultaneous induction on the structure of the first given derivation, using Lemmas 12 and 13 where needed. Note that while NJ N IS4 requires the principal formula of the coercion rule to be atomic, the following result shows that a corresponding unconstrained rule is admissible. Lemma 16 (Admissibility of nonatomic coercion for NJ N IS4 ). If ; Γ A, then ; Γ A. Proof. By induction on the size of A. We show the base cases and the cases involving modalities. A is atomic. (1) ; Γ A () A =. (1) ; Γ E () A = A 1. (1), A 1 ; A 1 hyp 2 (2), A 1 ; A 1 i.h. (1) (3), A 1 ; Γ A 1 I (2) (4) ; Γ A 1 E (, 3) A = A 1. (1) ; A 1 A 1 hyp 1 (2) ; A 1 A 1 i.h. (1) (3) ; A 1 A 1 I (2) (4) ; Γ A 1 E (, 3) Before continuing, it is important to establish that NJ N IS4 is equivalent, in terms of provability, to NJ IS4. The soundness of NJ N IS4 with respect to NJ IS4 is easy to show. Theorem 5 (Soundness of NJ N IS4 with respect to NJ IS4). 1. If ; Γ A, then ; Γ A. E 2. If ; Γ A, then ; Γ A. 13

14 ; Γ 1, A, Γ 2 A init (A is atomic) ; Γ 1,, Γ 2 A L ; Γ, A 1 A 2 ; Γ A 1 A 2 R 1, A, 2; Γ, A C 1, A, 2; Γ C refl ; Γ1, A1 A2, Γ2 A1 ; Γ1, A1 A2, Γ2, A2 C ; Γ 1, A 1 A 2, Γ 2 C ; Γ A 1 ; Γ A 2 ; Γ A 1 A 2 R ; Γ1, A1 A2, Γ2, Aj C ; Γ 1, A 1 A 2, Γ 2 C Lj ; Γ A j ; Γ 1, A 1 A 2, Γ 2, A 1 C ; Γ 1, A 1 A 2, Γ 2, A 2 C R j ; Γ A 1 A 2 ; Γ 1, A 1 A 2, Γ 2 C ; A ; Γ A R, A;Γ1, A, Γ2 C ; Γ 1, A, Γ 2 C L ; Γ A ; Γ A R ; A C ; Γ 1, A, Γ 2 C L j {1, 2} L L Fig.4. LJ IS4 Proof. By simultaneous induction on the structure of the first given derivation. The proof of the completeness of NJ N IS4 with respect to NJ IS4 is not as immediate, but still routine. We will develop a number of intermediate results, giving the final completeness result as Theorem 10. Our proof is by cut-elimination in an unfocused sequent calculus system designed to be equivalent to NJ IS4. This system, which we call LJ IS4, is shown in Fig. 4 and features sequents of the the form ; Γ A. Note that LJ IS4 is very similar to a sequent calculus presented by Nanevski, Pfenning, and Pientka for contextual modal logic [15], but is not the sequent calculus system that we will use for proof search. It is purely a technical device that we introduce for the purpose of the proof of completeness of NJ N IS4 with respect to NJ IS4. The cut rule ; Γ 1, A, Γ 2 C ; Γ 1, Γ 2 A ; Γ 1, Γ 2 C cut is not part of LJ IS4, but we will refer to the system obtained by augmenting LJ IS4 by the rule cut as LJ + IS4. To distinguish between sequents of LJ IS4 and LJ + IS4, we will decorate the arrows of LJ+ IS4 sequents with a + sign, viz. ; Γ + A. Note that while the init rules of LJ IS4 and LJ + IS4 are required to have atomic principal formulas, unconstrained variants are admissible, as shown by the following result. 14

15 Lemma 17 (Admissibility of nonatomic init for LJ IS4 ). ; Γ 1, A, Γ 2 A. Proof. By induction on the size of A. We show the base cases and the cases involving modalities. A is atomic. (1) ; Γ 1, A, Γ 2 A init A =. (1) ; Γ 1,, Γ 2 L A = A 1. (1), A 1 ; A 1 A 1 i.h. (2), A 1 ; A 1 refl (1) (3), A 1 ; Γ 1, A 1, Γ 2 A 1 R (2) (4) ; Γ 1, A 1, Γ 2 A 1 L (3) A = A 1. (1) ; A 1 A 1 i.h. (2) ; A 1 A 1 R (1) (3) ; Γ 1, A 1, Γ 2 A 1 L (2) Lemma 18 (Admissibility of nonatomic init for LJ + IS4 ). ; Γ 1, A, Γ 2 + A. Proof. By induction on the size of A. The cases are analogous to those in the proof of Lemma 17. Another system we will introduce for this completeness proof is NJ N+ IS4, which is obtained from NJ N IS4 by augmenting it by the rule ; Γ A ; Γ A Unlike the regular coercion rule, the principal formula of this rule need not be atomic. To distinguish between the judgements of NJ N+ IS4 and NJN IS4, we will decorate the turnstiles of NJ N+ IS4 judgements with a + sign, viz. ; Γ + A. As with NJ N IS4, an unconstrained regular coercion rule is admissible for NJN+ Lemma 19 (Admissibility of nonatomic coercion for NJ N+ If ; Γ + A, then ; Γ + A. IS4 ). Proof. By induction on the size of A. The cases are analogous to those in the proof of Lemma 16. We can now prove the first intermediate result needed for the proof of completeness of NJ N IS4 with respect to NJ IS4. Theorem 6 (Completeness of NJ N+ IS4 with respect to NJN IS4 ). IS4. 15

16 1. If ; Γ A, then ; Γ + A. E 2. If ; Γ A, then ; Γ + A. Proof. By simultaneous induction on the structure of the given derivation, using Lemma 19 and the rule where needed. For the next few results, we require the usual structural properties of both LJ IS4 and LJ + IS4. Lemma 20 (Weakening for LJ IS4 ). If ; Γ 1, Γ 2 C, then ; Γ 1, A, Γ 2 C. Proof. By induction on the structure of. Lemma 21 (Modal weakening for LJ IS4 ). If 1, 2 ; Γ C, then 1, A, 2 ; Γ C. Proof. By induction on the structure of. Lemma 22 (Exchange for LJ IS4 ). If ; Γ 1, A 1, Γ 2, A 2, Γ 3 C, then ; Γ 1, A 2, Γ 2, A 1, Γ 3 C. Proof. By induction on the structure of. Lemma 23 (Modal exchange for LJ IS4 ). If 1, A 1, 2, A 2, 3 ; Γ C, then 1, A 2, 2, A 1, 3 ; Γ C. Proof. By induction on the structure of. Lemma 24 (Contraction for LJ IS4 ). If ; Γ 1, A, Γ 2, A, Γ 3 C, then ; Γ 1, A, Γ 2, Γ 3 C. Proof. By induction on the structure of. Lemma 25 (Modal contraction for LJ IS4 ). If 1, A, 2, A, 3 ; Γ C, then 1, A, 2, 3 ; Γ C. Proof. By induction on the structure of. All of the structural properties also hold for LJ + IS4. Lemma 26 (Weakening for LJ + IS4 ). + + If ; Γ 1, Γ 2 C, then ; Γ1, A, Γ 2 C. Proof. By induction on the structure of. 16

17 Lemma 27 (Modal weakening for LJ + IS4 ). If 1, 2 ; Γ + C, then 1, A, 2 ; Γ + C. Proof. By induction on the structure of. Lemma 28 (Exchange for LJ + IS4 ). + + If ; Γ 1, A 1, Γ 2, A 2, Γ 3 C, then ; Γ1, A 2, Γ 2, A 1, Γ 3 C. Proof. By induction on the structure of. Lemma 29 (Modal exchange for LJ + IS4 ). If 1, A 1, 2, A 2, 3 ; Γ + C, then 1, A 2, 2, A 1, 3 ; Γ + C. Proof. By induction on the structure of. Lemma 30 (Contraction for LJ + IS4 ). + + If ; Γ 1, A, Γ 2, A, Γ 3 C, then ; Γ1, A, Γ 2, Γ 3 C. Proof. By induction on the structure of. Lemma 31 (Modal contraction for LJ + IS4 ). If 1, A, 2, A, 3 ; Γ + C, then 1, A, 2, 3 ; Γ + C. Proof. By induction on the structure of. Moreover, the proofs of these structural properties for both LJ IS4 and LJ + IS4 show that applying any form of weakening, exchange, or contraction to a derivation does not affect its structure. This is an important observation that will be used tacitly in the proof of the admissibility of the cut rule for LJ IS4, Lemma 33. First, however, we can prove the next key result on the way to Theorem 10. Theorem 7 (Completeness of LJ + IS4 1. If ; Γ + A, then ; Γ + A. 2. If E ; Γ 1, Γ 2 + A and with respect to NJN+ IS4 ). F ; Γ 1, A, Γ 2 + C, then ; Γ1, Γ 2 + C. Proof. By simultaneous induction on the structure of the first given derivation. We show the base cases and the cases involving modalities. E = ; Γ1, Γ 2 + A hyp 1 (A Γ 1, Γ 2 ) (1) ; Γ 1, Γ 2 + C Lemma 30 (F) 17

18 E = 1, A, 2 ; Γ 1, Γ 2 + A hyp 2 (1) 1, A, 2 ; Γ 1, Γ 2, A + C Lemma 28 (F) + (2) 1, A, 2 ; Γ 1, Γ 2 C refl (1) = ; + A ; Γ + A I (1) ; + A i.h. ( ) (2) ; Γ + A R (1) 2 = ; Γ + B, B; Γ + A E ; Γ + A (1), B; Γ + A i.h. ( 2 ) (2), B; Γ, B + A Lemma 26 (1) (3) ; Γ, B + A L (2) (4) ; Γ + A i.h. (, 3) = ; Γ + A ; Γ + A I (1) ; Γ + A i.h. ( ) (2) ; Γ + A R (1) 2 = ; Γ + B ; B + A E ; Γ + A (1) ; B + A i.h. ( 2 ) (2) ; Γ, B + A L (1) (3) ; Γ + A i.h. (, 2) = ; Γ + A ; Γ + A (1) ; Γ, A + A init (2) ; Γ + A i.h. (, 1) E 1 E = ; Γ + A ; Γ + A 18

19 (1) ; Γ + A i.h. (E 1 ) (2) ; Γ + C cut (F, 1) The following intermediate result will be used in the proof of the admissibility of the cut rule for LJ IS4, Lemma 33. It effectively establishes the admissibility of cut when used in a very particular way. Lemma 32 ( -cutting for LJ IS4 ). If ; Γ, then ; Γ A. Proof. By induction on the structure of. We are now able to prove the fundamental cut-elimination result that we need. Lemma 33 (Admissibility of cut for LJ IS4 ). E 1. (Cut) If ; Γ 1, A, Γ 2 C and ; Γ 1, Γ 2 A, then ; Γ 1, Γ 2 C. F G 2. (Modal cut) If 1, A, 2 ; Γ C and 1, 1 ; A, then 1, 2 ; Γ C. Proof. By simultaneous nested induction. Our induction variables are triples of the form size,case,height 1 + height 2, where size is the size of the cut formula A, computed inductively in the usual way, case is 1 or 2, depending on which induction case (cut or modal cut) we are considering, and height 1 and height 2 are the heights of the two given derivations. We let v 1,...,v n < w 1,...,w n if there is an i {1,...,n} such that v j = w j for j < i and v i < w i. We show the base cases and the cases involving modalities. At the top level, the cases of the proof are in two categories, those for cut and those for modal cut. We first give the cases for cut, where we can only appeal to the induction hypothesis for modal cut if the cut formula decreases in size. Either or E ends with an application of init, L, or refl. Subcase: = ; Γ1, A, Γ 2 C init (A = C) (1) ; Γ 1, Γ 2 C Equality (E) Subcase: = ; Γ1, A, Γ 2 C init (A C, C Γ 1, Γ 2 ) (1) ; Γ 1, Γ 2 C Lemma 17 Subcase: E = ; Γ1, Γ 2 A init (A Γ 1, Γ 2 ) (1) ; Γ 1, Γ 2 C Lemma 24 () 19

20 Subcase: = ; Γ1, A, Γ 2 C L (A = ) (1) ; Γ 1, Γ 2 Equality (E) (2) ; Γ 1, Γ 2 C Lemma 32 (1) Subcase: = ; Γ1, A, Γ 2 C L (A, Γ 1, Γ 2 ) (1) ; Γ 1, Γ 2 C L Subcase: E = ; Γ1, Γ 2 A L ( Γ 1, Γ 2 ) (1) ; Γ 1, Γ 2 C L Subcase: = 1, B, 2 ; Γ 1, A, Γ 2, B C 1, B, 2 ; Γ 1, A, Γ 2 C refl (1) 1, B, 2 ; Γ 1, Γ 2, B A Lemma 20 (E) (2) 1, B, 2 ; Γ 1, Γ 2, B C i.h. (, 1) (3) 1, B, 2 ; Γ 1, Γ 2 C refl (2) Subcase: E = E 1 1, B, 2 ; Γ 1, Γ 2, B A 1, B, 2 ; Γ 1, Γ 2 A (1) 1, B, 2 ; Γ 1, A, Γ 2, B C Lemma 20 () (2) 1, B, 2 ; Γ 1, Γ 2, B C i.h. (1, E 1 ) (3) 1, B, 2 ; Γ 1, Γ 2 C refl (2) The cut formula is the principal formula of the last inference of both and E. Subcase: = E 1, A; Γ 1, A, Γ 2 C ; Γ 1, A, Γ 2 C L and E = ; A ; Γ 1, Γ 2 A R (1), A; Γ 1, Γ 2 A Lemma 21 (E) (2), A; Γ 1, Γ 2 C i.h. (, 1) (3) ; Γ 1, Γ 2 C i.h. (2, E 1 ) Subcase: = E 1 ; A C ; Γ 1, A, Γ 2 C L and E = ; Γ 1, Γ 2 A ; Γ 1, Γ 2 A R (1) ; Γ 1, Γ 2, A C Lemma 20 ( ) (2) ; Γ 1, Γ 2 C i.h. (1, E 1 ) The cut formula is not the principal formula of the last inference of. refl 20

21 Subcase: = ; C ; Γ 1, A, Γ 2 C R (1) ; Γ 1, Γ 2 C R ( ) Subcase: = ; Γ 1, A, Γ 2 C ; Γ 1, A, Γ 2 C R (1) ; Γ 1, Γ 2 C i.h. (, E) (2) ; Γ 1, Γ 2 C R (1) Subcase: =, B; Γ 1, A, Γ 2 C ; Γ 1, A, Γ 2 C L ( B Γ 1, Γ 2 ) (1), B; Γ 1, Γ 2 A Lemma 21 (E) (2), B; Γ 1, Γ 2 C i.h. (, 1) (3) ; Γ 1, Γ 2 C L (3) Subcase: = ; B C ; Γ 1, A, Γ 2 C L ( B Γ 1, Γ 2 ) (1) ; Γ 1, Γ 2 C L ( ) The cut formula is not the principal formula of the last inference of E. Subcase: E 1 E =, B; Γ 1, Γ 2 A ; Γ 1, Γ 2 A L ( B Γ 1, Γ 2 ) (1), B; Γ 1, A, Γ 2 C Lemma 21 () (2), B; Γ 1, Γ 2 C i.h. (1, E 1 ) (3) ; Γ 1, Γ 2 C L Subcase: and E = = ; A C ; Γ 1, A, Γ 2 C L E 1 ; B A ; Γ 1, Γ 2 A L ( B Γ 1, Γ 2 ) Here, we require the last inference of to have the cut formula A as its principal formula. Our cases remain exhaustive since the cut formula not being the principal formula of the last inference of is covered by the previous case. (1) ; A, B C L ( ) (2) ; B C i.h. (1, E 1 ) (3) ; Γ 1, Γ 2 C L (2) 21

22 Next, we give the cases for modal cut. Here, we can appeal to the induction hypothesis for cut even if the size of the cut formula stays the same. F = 1, A, 2 ; Γ 1, C, Γ 2 C init (1) 1, 2 ; Γ 1, C, Γ 2 C init F = 1, A, 2 ; Γ 1,, Γ 2 C L (1) 1, 2 ; Γ 1,, Γ 2 C L F 1 F = 1, A, 2 ; Γ, B C 1, A, 2 ; Γ C refl (B 1, 2 ) (1) 1, 2 ; Γ, B C i.h. (F 1, G) (2) 1, 2 ; Γ C refl (1) F = F 1 1, A, 2 ; Γ, A C 1, A, 2 ; Γ C (1) 1, 2 ; Γ, A C i.h. (F 1, G) (2) 1, 2 ; Γ A Lemma 20 (G) (3) 1, 2 ; Γ C i.h. (1, 2) F = refl F 1 1, A, 2 ; C 1, A, 2 ; Γ C R (1) 1, 2 ; C i.h. (F 1, G) (2) 1, 2 ; Γ C R (1) F = F 1 1, A, 2, B; Γ 1, B, Γ 2 C 1, A, 2 ; Γ 1, B, Γ 2 C (1) 1, 2, B; A Lemma 21 (G) (2) 1, 2, B; Γ 1, B, Γ 2 C i.h. (F 1, 1) (3) 1, 2 ; Γ 1, B, Γ 2 C L (2) F = F 1 1, A, 2 ; Γ C 1, A, 2 ; Γ C R (1) 1, 2 ; Γ C i.h. (F 1, G) (2) 1, 2 ; Γ C R (1) F = L F 1 1, A, 2 ; B C 1, A, 2 ; Γ 1, B, Γ 2 C L (1) 1, 2 ; B C i.h. (F 1, G) (2) 1, 2 ; Γ 1, B, Γ 2 C L (1) 22

23 Theorem 8 (Completeness of LJ IS4 with respect to LJ + IS4 ). If ; Γ + A, then ; Γ A. Proof. By induction on the structure of, using Lemma 33 where needed. Theorem 9 (Completeness of NJ N IS4 with respect to LJ IS4). If ; Γ A, then ; Γ A. Proof. By induction on the structure of. We show the base cases and the cases involving modalities. = ; Γ1, A, Γ 2 A init (1) ; Γ 1, A, Γ 2 A hyp 1 (2) ; Γ 1, A, Γ 2 A (1) = ; Γ1,, Γ 2 A L (1) ; Γ 1,, Γ 2 hyp 1 (2) ; Γ 1,, Γ 2 A E (1) = 1, B, 2 ; Γ, B A 1, B, 2 ; Γ A (1) 1, B, 2 ; Γ, B A i.h. ( ) (2) 1, B, 2 ; Γ B hyp 2 (3) 1, B, 2 ; Γ A Lemma 14 (1, 2) = ; A ; Γ A R (1) ; A i.h. ( ) (2) ; Γ A I (1) = refl, B; Γ 1, B, Γ 2 A ; Γ 1, B, Γ 2 A L (1) ; Γ 1, B, Γ 2 B hyp 1 (2), B; Γ 1, B, Γ 2 A i.h. ( ) (3) ; Γ 1, B, Γ 2 A E (1, 2) = ; Γ A ; Γ A R (1) ; Γ A i.h. ( ) (2) ; Γ A I (1) 23

24 = ; B A ; Γ 1, B, Γ 2 A L (1) ; Γ 1, B, Γ 2 B hyp 1 (2) ; B A i.h. ( ) (3) ; Γ 1, B, Γ 2 A E (1, 2) With this collection of intermediate results, the completeness of NJ N IS4 with respect to NJ IS4 follows in a step-by-step way. Theorem 10 (Completeness of NJ N IS4 with respect to NJ IS4). If ; Γ A, then ; Γ A. Proof. (1) ; Γ + A Theorem 6 () (2) ; Γ + A Theorem 7 (1) (3) ; Γ A Theorem 8 (2) (4) ; Γ A Theorem 9 (3) Theorem 10 is a weak normalization result: if a judgement is derivable in our natural deduction system NJ IS4, then it has a normal natural deduction, where the normal forms of natural deductions are defined by the restrictions imposed by the arrow annotations in NJ N IS4. An alternative normalization proof is to explicitly consider the normalization steps β-conversions and commuting (or permutative) conversions arising from the rules of NJ IS4, and to show that for any derivation, a sequence of normalization steps can be found that terminates in a corresponding derivation in normal form. What then remains to be shown is that a derivation can be constructed in NJ N IS4 if and only if it is in normal form. We will not give the details of such an alternative proof, but the conversions arising from the NJ IS4 rules involving modalities are as follows, while those for the remaing rules are standard (see for instance [10] or [21]). 1. β-conversion for I and E: ; A ; Γ A I 2, A; Γ C ; Γ C E β where E is obtained by applying Lemma 6 to 2 and. 2. β-conversion for I and E: E ; Γ C ; Γ A ; Γ A I 2 ; A C ; Γ C E β E ; Γ C 24

25 where E is obtained by applying Lemma 1 to 2 to obtain ; Γ, A C and then applying Lemma 5 to this and. 3. Commuting conversion for E: 2 ; Γ A, A; Γ C E ; Γ C i E ; Γ 2 1, A; Γ C E i c E ; Γ A, A; Γ E ; Γ where E is any elimination rule, denotes some number of minor premises with derivations i, i = 3, 4,..., and E i is obtained by applying Lemma 2 to i to add A to the modal context of the final judgement. 4. Commuting conversion for E: 2 ; Γ A ; A C E 3 ; Γ C ; C E ; Γ ; A C ; C c E ; Γ A ; A E ; Γ Since the succedent of the conclusion of an application of E is of the form C and the only elimination rule whose major premise has a succedent of the form C is E, this is the only commuting conversion arising from the modality. As a platform for proof search, NJ N IS4 is superior to NJ IS4, since the proof space is restricted to allow only normal natural deductions. However, NJ N IS4 still suffers from a lack of syntax-directedness in the case of the elimination rules, an issue that we will address by using techniques from forward proof search. Although we will ultimately return to NJ N IS4 in our search for bidirectional decision procedures, the relationship between introduction and elimination rules and backward and forward proof search is perhaps most vividly demonstrated in a focused sequent calculus setting, which we turn to next. 5 Sequent Calculi Although we could use LJ IS4, presented in Sect. 4, as a basis for proof search in a backward direction, its refl rule is a source of inconvenient nondeterminism. We will instead follow the approach of yckhoff and Pinto [6] and construct a 25

26 ; Γ A A init (A is atomic) ; Γ A L ; Γ 1, A, Γ 2 A C ; Γ 1, A, Γ 2 C ; Γ, A 1 A 2 ; Γ A 1 A 2 R ch 1 ; Γ A 1 ; Γ A 2 ; Γ A 1 A 2 R 1, A, 2; Γ A C 1, A, 2; Γ C ; Γ A1 ; Γ A2 C ; Γ A 1 A 2 C ch 2 L ; Γ A j C ; Γ A 1 A 2 C Lj ; Γ A j ; Γ, A 1 C ; Γ, A 2 C R j ; Γ A 1 A 2 ; Γ A 1 A 2 C ; A ; Γ A R, A; Γ C ; Γ A C L ; Γ A ; Γ A R ; A C ; Γ A C L j {1, 2} L Fig.5. MJ IS4 focused sequent calculus for propositional IS4 whose derivations are in bijective correspondence with normal natural deductions. This system, which we will call MJ IS4, is shown in Fig. 5 and involves two forms of sequents: ; Γ C C can be proved from assumptions, Γ, ; Γ A C C can be proved from assumptions, Γ, A, focusing on the assumption A. If a sequent is focused on a formula A, then the only applicable rules are those with A as a principal formula. Following Girard [9], we will call the position of the focused formula the stoup. As in the natural deduction formulations, contexts in MJ IS4 are technically ordered lists, but the usual structural properties of weakening, contraction, and exchange hold here as well, so an interpretation of contexts as multisets is reasonable. Lemma 34 (Weakening for MJ IS4 ). 1. If ; Γ 1, Γ 2 C, then ; Γ 1, A, Γ 2 C. E 2. If ; Γ 1, Γ 2 B C, then ; Γ 1, A, Γ 2 B C. Proof. By simultaneous induction on the structure of the given derivation. Lemma 35 (Modal weakening for MJ IS4 ). 1. If 1, 2 ; Γ C, then 1, A, 2 ; Γ C. 26

27 E 2. If 1, 2 ; Γ B C, then 1, A, 2 ; Γ B C. Proof. By simultaneous induction on the structure of the given derivation. Lemma 36 (Contraction for MJ IS4 ). 1. If ; Γ 1, A, Γ 2, A, Γ 3 C, then ; Γ 1, A, Γ 2, Γ 3 C. E 2. If ; Γ 1, A, Γ 2, A, Γ 3 B C, then ; Γ 1, A, Γ 2, Γ 3 B C. Proof. By simultaneous induction on the structure of the given derivation. Lemma 37 (Modal contraction for MJ IS4 ). 1. If 1, A, 2, A, 3 ; Γ C, then 1, A, 2, 3 ; Γ C. E 2. If 1, A, 2, A, 3 ; Γ B C, then 1, A, 2, 3 ; Γ B C. Proof. By simultaneous induction on the structure of the given derivation. Lemma 38 (Spanning contraction for MJ IS4 ). 1. If 1, A, 2 ; Γ 1, A, Γ 2 C, then 1, A, 2 ; Γ 1, Γ 2 C. E 2. If 1, A, 2 ; Γ 1, A, Γ 2 B C, then 1, A, 2 ; Γ 1, Γ 2 B C. Proof. By simultaneous induction on the structure of the given derivation, replacing applications of ch 1 by applications of ch 2 where needed. The key result of this section is that there is a very close correspondence between MJ IS4 and NJ N IS4. In fact, derivations in one correspond bijectively to derivations in the other. To show this, we will define functions that map MJ IS4 derivations to corresponding NJ N IS4 derivations and vice versa. More formally, we define a function f that takes an MJ IS4 derivation of ; Γ C and returns a corresponding NJ N IS4 derivation of ; Γ C. It requires an auxiliary function f which takes two arguments, an MJ IS4 derivation of ; Γ A C and an NJ N IS4 derivation of ; Γ A. Given these arguments, f returns an NJ N IS4 derivation of ; Γ C. The operational interpretation is that f is the main generator function, traversing the unfocused segments of the given MJ IS4 derivation in an upward direction, producing corresponding segments of judgements of a partial NJ N IS4 derivation, also constructed in an upward direction. When a formula is raised into the stoup in the MJ IS4 derivation being traversed, the auxiliary function f is called upon to continue traversing the MJ IS4 derivation (the first argument to f ) in an upward direction, while the corresponding segment of judgements in the partial NJ N IS4 derivation (the second argument to f ) is constructed in a downward direction. When the focus in the original MJ IS4 derivation blurs, the downward-constructed partial NJ N IS4 derivation meets the previously upward-constructed partial NJ N IS4 derivation, 27

28 and the main function f is possibly called upon again to continue the traversal of the next unfocused segment of the MJ IS4 derivation. The function f is defined as follows. f ( ) f ; Γ 1, A, Γ 2 A C ; Γ 1, A, Γ 2 C ch 1 ( ) f 1, A, 2 ; Γ A C 1, A, 2 ; Γ C ch 2 ( f ( ; Γ, A 1 A 2 ; Γ A 1 A 2 R 2 ; Γ A 1 ; Γ A 2 R ; Γ A 1 A 2 f ( ; Γ A j R j ; Γ A 1 A 2 f f ( ( ; A ; Γ A R ; Γ A ; Γ A R ) ) ) ) ) ( def = f, ; Γ1, A, Γ 2 A hyp ) 1 ( def = f, 1, A, 2 ; Γ A hyp ) 2 def = def = def = def = def = The function f is defined as follows. f ( ) ; Γ, A 1 A 2 ; Γ A 1 A 2 I f ( ) f ( 2 ) ; Γ A 2 ; Γ A 1 ; Γ A 1 A 2 f ( ) ; Γ A j ; Γ A 1 A 2 I j f ( ) ; A ; Γ A I f ( ) ; Γ A ; Γ A I j {1, 2} ( f ; Γ A A init ) F def, F = ; Γ A ; Γ A ( f ; Γ A L ) F def, F = ; Γ ; Γ A E E 1, A; Γ C ; Γ A C L, F ( E 1 f ; A C ; Γ A C L, F f ( ) ) def = def = F ; Γ A ; Γ C F ; Γ A ; Γ C I f (E 1 ), A; Γ C f (E 1 ) ; A C E E ( E 1 E 2 f ; Γ A 1 ; Γ A 2 C ; Γ A 1 A 2 C L, F ) 28

29 def = f E 2, ( E ) 1 f ; Γ A j C ; Γ A 1 A 2 C L, F j def = f (E 1, F f (E 1 ) ; Γ A 1 A 2 ; Γ A 1 E ; Γ A 2 F ; Γ A 1 A 2 ; Γ A j E j ) ( ) E 1 E 2 f ; Γ, A 1 C ; Γ, A 2 C L, F ; Γ A 1 A 2 C F f (E 1 ) def = ; Γ A 1 A 2 ; Γ, A 1 C ; Γ C j {1, 2} f (E 2 ) ; Γ, A 2 C The functions g and g are similar to f and f, but convert NJ N IS4 derivations into MJ IS4 derivations. Formally, g takes an NJ N IS4 derivation of ; Γ C and returns a corresponding MJ IS4 derivation of ; Γ C. It requires an auxiliary function g which takes two arguments, an NJ N IS4 derivation of ; Γ A and an MJ IS4 derivation of ; Γ A C. Given these arguments, g returns an MJ IS4 derivation of ; Γ C. The interactions between g and g are very similar to those between f and f. g is the main generator function that proceeds upward along the sequents of the given NJ N IS4 derivation. The corresponding MJ IS4 derivation is also constructed in an upward direction. If the direction of reasoning in the original NJ N IS4 derivation reverses, g is called upon. Since the corresponding segment of the derivation in MJ IS4 is focused, the construction of this focused MJ IS4 segment (the second argument to g ) proceeds in a downward direction, while g continues to traverse the original NJ N IS4 derivation (the first argument to g ) in an upward direction. The partial MJ IS4 derivations meet when the leaves of the NJ N IS4 derivation are reached. The function g is defined as follows. ( ) g ; Γ A ( def ; Γ A = g, init ) ; Γ A A g ( g ( ; Γ ; Γ A E ; Γ, A 1 A 2 ; Γ A 1 A 2 I ) ) 29 ( def = g, L ) ; Γ A def = g ( ) ; Γ, A 1 A 2 ; Γ A 1 A 2 R E

30 ( ) 2 def g ; Γ A 1 ; Γ A 2 = I ; Γ A 1 A 2 g ; Γ A j ; Γ A 1 A 2 I def = 1 ( ) g ; A ; Γ A I def = g ( ) ; Γ A 1 g ( 2 ) ; Γ A 2 ; Γ A 1 A 2 R g ( ) ; Γ A j ; Γ A 1 A 2 R j g ( ) ; A ; Γ A R ( ) 2 def g ; Γ A, A; Γ C = g E 1, ; Γ C ( ) def g ; Γ A ; Γ A I = ( ) 2 def g ; Γ A ; A C = g E 1, ; Γ C ( g ; Γ A 1 A 2 g ( ) ; Γ A ; Γ A R g ( 2 ), A; Γ C ; Γ A C L g ( 2 ) ; A C ; Γ A C L ) 2 3 ; Γ, A 1 C ; Γ, A 2 C E ; Γ C g ( 2 ) g ( 3 ) def = g, ; Γ, A 1 C ; Γ, A 2 C L ; Γ A 1 A 2 C j {1, 2} The function g is defined as follows. ( g ; Γ 1, A, Γ 2 A hyp ) F def 1, F = ; Γ 1, A, Γ 2 A C ; Γ 1, A, Γ 2 C ch 1 ( g 1, A, 2 ; Γ A hyp ) F def 2, F = 1, A, 2 ; Γ A C 1, A, 2 ; Γ C ch 2 g E 1 ; Γ A 1 A 2 E, F j ; Γ A j def = g (E 1, F ; Γ A j C ; Γ A 1 A 2 C L j ( ) E 1 E 2 g ; Γ A 1 A 2 ; Γ A 1 E, F ; Γ A 2 g (E 2 ) F def = g E 1, ; Γ A 1 ; Γ A 2 C L ; Γ A 1 A 2 C 30 )

31 j {1, 2} Note that operationally, each call to any one of the functions f, f, g, or g terminates, since the size of the first argument decreases in every recursive call. Finally, to establish a bijection between MJ IS4 and NJ N IS4 derivations, it is sufficient to show that f and g are both invertible. In fact, it is perhaps not surprising that g is the inverse of f and vice versa. The following two results formalize this notion. Lemma 39 (Invertibility of f and g (1)). 1. If ; Γ C, then f(g()) =. E F 2. If ; Γ A and ; Γ A C, then f(g (E, F)) = f (F, E). Proof. By simultaneous induction on the structure of the first given derivation. We show the base cases and the cases involving modalities. f f ( ( ( ) g ; Γ 1, A, Γ 2 A hyp )) F 1, F = f ; Γ 1, A, Γ 2 A C ; Γ 1, A, Γ 2 C ch 1 ( = f F, ; Γ1, A, Γ 2 A hyp ) 1 ( ( ( ) g 1, A, 2 ; Γ A hyp )) F 2, F = f 1, A, 2 ; Γ A C 1, A, 2 ; Γ C ch 2 ( = f F, 1, A, 2 ; Γ A hyp ) 2 ( ( )) f g ; Γ ; Γ A E = f (g (, L )) ; Γ A ( i.h = f ; Γ A L ), = ; Γ ; Γ A E ( ( )) f g ; A ; Γ A I = f g ( 1) ; A ; Γ A R 31

32 f (g ( )) = ; A ; Γ A I i.h. = ; A ; Γ A I f ( g ( ; Γ A ; Γ C 2, A; Γ C = f )) E g, i.h. = f i.h. = g ( 2 ), A; Γ C ; Γ A C L g ( 2 ), A; Γ C ; Γ A C L, ; Γ A ; Γ C f (g ( 2 )) = ; Γ A, A; Γ C E ; Γ C 2, A; Γ C E ( ( )) f g ; Γ A ; Γ A I = f g ( 1) ; Γ A ; Γ A R = i.h. = f (g ( )) ; Γ A ; Γ A I ; Γ A ; Γ A I ( ( )) 2 f g ; Γ A ; A C E ; Γ C 32

33 g ( 2 ) = f g, ; A C ; Γ A C L g ( 2 ) i.h. = f ; A C ; Γ A C L, 1 f (g ( 2 )) = ; Γ A ; A C E ; Γ C 2 ; A C i.h. = ; Γ A ; Γ C E ( ( )) f g ; Γ A ; Γ A = f (g (, init )) ; Γ A A ( i.h. = f ; Γ A A init ), = ; Γ A ; Γ A Lemma 40 (Invertibility of f and g (2)). 1. If ; Γ C, then g(f()) =. E F 2. If ; Γ A C and ; Γ A, then g(f (E, F)) = g (F, E). Proof. By simultaneous induction on the structure of the first given derivation. We show the base cases and the cases involving modalities. ( ( ( ) g f ; Γ A A init )) F, F = g ; Γ A ; Γ A = g ( F, ; Γ A A init ) ( ( ( ) g f ; Γ A L )) F, F = g ; Γ ; Γ A E ( = g F, L ) ; Γ A 33

34 ( ( )) g f ; Γ 1, A, Γ 2 A C ; Γ 1, A, Γ 2 C ch 1 ( ( )) g f 1, A, 2 ; Γ A C 1, A, 2 ; Γ C ch 2 ( ( = g f, ; Γ1, A, Γ 2 A hyp )) 1 ( i.h. = g ; Γ 1, A, Γ 2 A hyp ) 1, = ; Γ 1, A, Γ 2 A C ; Γ 1, A, Γ 2 C ch 1 ( ( = g f, 1, A, 2 ; Γ A hyp )) 2 ( i.h. = g 1, A, 2 ; Γ A hyp ) 2, = 1, A, 2 ; Γ A C 1, A, 2 ; Γ C ch 2 ( ( )) g f ; A ; Γ A R = g f ( 1) ; A ; Γ A I = i.h. = g (f ( )) ; A ; Γ A R ; A ; Γ A R ( )) E 1 g (f, A; Γ C ; Γ A C L, F F f (E 1 ) = g ; Γ A, A; Γ C E ; Γ C g (f (E 1 )) = g F,, A; Γ C ; Γ A C L ) E 1 i.h. = g (F,, A; Γ C ; Γ A C L ( ( )) g f ; Γ A ; Γ A R = g f ( 1) ; Γ A ; Γ A I 34

35 g (f ( )) = ; Γ A ; Γ A R i.h. = ; Γ A ; Γ A R ( )) E 1 g (f ; A C ; Γ A C L, F F = g ; Γ A ; Γ C = g F, i.h. = g (F, f (E 1 ) ; A C g (f (E 1 )) ; A C ; Γ A C L E 1 ; A C ; Γ A C L E A function is bijective if and only if it is both left- and right-invertible. As a result, the bijectivity of both f and g follows immediately from Lemmas 39 and 40. The soundness and completeness of MJ IS4 with respect to NJ N IS4 follow directly from this result. Theorem 11 (Bijection between MJ IS4 and NJ N IS4 derivations). erivations of unfocused sequents in MJ IS4 correspond bijectively to derivations of judgements in NJ N IS4. Proof. Functions f and g, defined above, map derivations of unfocused sequents in MJ IS4 to derivations of judgements in NJ N IS4 and vice versa, respectively. The fact that f and g are bijections follows immediate from Lemmas 39 and 40. Although MJ IS4 is suitable for proof search in a backward direction, a naive approach still requires loop-detection to achieve a decision procedure. We will not pursue this direction further here, but instead concentrate on forward proof search, and on how we can combine ideas from backward and forward search to perform bidirectional proof search. ) 6 Forward Proof Search Constructing MJ IS4 proofs in a forward direction from the top down is complicated by the presence of multiple contexts, making MJ IS4 less than ideal for forward proof search. All MJ IS4 derivations begin, at the leaves, with focused sequents of the form ; Γ A A, with A atomic. After a sequence of (possibly zero) left-rule applications, the stoup formula is dropped from the stoup into one 35

36 of the contexts by an application of ch 1 or ch 2. In a focused forward calculus used as the basis for the inverse method [5], we would proceed in a similar way, but it is not clear which context a stoup formula should be dropped into. To address this uncertainty, we refine the idea of focusing and develop the system MJ F IS4, which is suitable for forward proof search and features sequents of three kinds, involving both modal and nonmodal stoups: ; Γ C C can be proved using all assumptions in, Γ, ; Γ A C C can be proved using all assumptions in, Γ, A, with A assumed true, ; Γ A C C can be proved using all assumptions in, Γ, A, with A assumed valid. Note that the forms of the focused sequents reveal which context the stoup formula will drop into. For brevity, we write ; Γ i A C, i {1, 2} for either form of focused sequent. The inference rules of MJ F IS4, shown in Fig. 6, are obtained by reinterpreting the rules of MJ IS4 in a forward fashion and by defining the ch i rules to behave as sketched above. The contexts of MJ F IS4, however, are interpreted differently, in that sequents ; Γ C and ; Γ i A C, i {1, 2} assert that all assumptions in and Γ, as well as A if the sequent is focused, are needed to prove C. General weakening, which holds in MJ IS4, is thus disallowed, but local weakening is incorporated in the rule R 2. Contexts in MJ F IS4 are treated as sets rather than multisets, and we write Γ 1, Γ 2 and Γ, A for Γ 1 Γ 2 and Γ {A}, respectively. Note that a rule such as 1 ; Γ 1, A 1 C 2 ; Γ 2, A 2 C 1, 2 ; Γ 1, Γ 2 i A 1 A 2 C L i could more pedantically be written 1 ; Γ 1 C 2 ; Γ 2 C 1 2 ; (Γ 1 \ {A 1 }) (Γ 2 \ {A 2 }) i A 1 A 2 C L i (A 1 Γ 1, A 2 Γ 2 ) but we use the former notation for readability and consistency with our previous systems. No essential ambiguity is introduced by this convention. In the following soundness and completeness results, we tacitly convert contexts from lists to sets and vice versa. More precisely, if and Γ are MJ IS4 contexts, then when used in MJ F IS4 sequents ; Γ C or ; Γ i A C, they are converted into the sets that contain all distinct formulas in the contexts and Γ. Similarly, if and Γ are sets of formulas, that is, MJ F IS4 contexts, then when used in MJ IS4 sequents ; Γ C or ; Γ A C, they are converted into lists containing all elements in the sets and Γ. As a shorthand, if Γ is a list and Γ is a set, then we write Γ Γ to mean that every element of Γ is contained in Γ. Theorem 12 (Soundness of MJ F IS4 with respect to MJ IS4). 36

37 ; i A A init i (A is atomic) ; i A Li ; Γ A C ; Γ A C ; Γ, A C ch1, A;Γ C ch2 ; Γ, A 1 A 2 ; Γ A 1 A 2 R 1 ; Γ A 2 ; Γ A 1 A 2 R 2 1; Γ 1 A 1 2; Γ 2 i A 2 C 1, 2; Γ 1, Γ 2 i A 1 A 2 C Li 1; Γ 1 A 1 2; Γ 2 A 2 1, 2; Γ 1, Γ 2 A 1 A 2 R ; Γ i A j C ; Γ i A 1 A 2 C L i,j ; Γ A j 1; Γ 1, A 1 C 2; Γ 2, A 2 C R j ; Γ A 1 A 2 1, 2; Γ 1, Γ 2 i A 1 A 2 C ; A ; A R, A; Γ C ; Γ A ; Γ i A C Li ; Γ A R ; A C ; i A C Li i, j {1, 2} Li Fig.6. MJ F IS4 1. If ; Γ C, then ; Γ C. E 2. If ; Γ i A C, i {1, 2}, then ; Γ A C. Proof. By simultaneous induction on the structure of the given derivation. We show the base cases and the cases involving modalities. E = ; i A A init i (1) ; A A init E = ; i A L i (1) ; A L = ; Γ A C ; Γ, A C ch (A Γ) 1 (1) ; Γ A C i.h. ( ) (2) ; Γ C ch 1 (1) = ; Γ A C ; Γ, A C ch 1 37 (A / Γ)