Sequent Systems Research Articles

Extended Natural Deduction Images of Conversions from the System of Sequents

A modification of natural deduction system for intuitionistic predicate logic, the extension of natural deduction system, will be defined. The main result of such modification will be that each conversion from the set of conversions of a cut-elimination procedure in the system of sequents has the corresponding conversion in the set of conversions of a normalization procedure in that extension of natural deduction.

Algebraic Aspects of Cut Elimination

We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasi-completion of these Gentzen structures. It is shown that the quasi-completion is a generalization of the MacNeille completion. Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and Okada-Terui [17].

Identity of Proofs Based on Normalization and Generality

AbstractSome thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal formin natural deduction. Lambek worked on a normalization proposal analogous to Prawitz's, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to a renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic.The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well.The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to low-dimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.

Two Restrictions on Contraction

I show two limitations of multiplicative (or context-independent) sequent calculi: contraction can neither be restricted to atoms nor to the bottom of a proof tree. There is a a set of rules for classical propositional logic, system SKS [1], with two interesting properties: 1. Contraction can be restricted to atoms. As a consequence, no rule in system SKS requires the duplication of formulae of unbounded size. 2. Contraction can be restricted to the bottom of a proof. As a consequence, a proof in SKS can be separated into two phases (seen bottom-up): In the first phase atoms are duplicated and in the second phase all other rules are applied and the size of the frontier of the proof (in terms of number of atoms) does not increase. System SKS is not a sequent calculus but is presented in a more general formalism, the calculus of structures [2]. While the sequent calculus restricts the application of rules to the main connective of a formula, the calculus of structures is more expressive by admitting rules that can be applied anywhere deep inside formulae. This leads to the question whether the extra expressive power of the calculus of structures is needed to observe these properties, or whether they can be observed in sequent systems as well. This question seems quickly answered by sequent system G3cp [3] in which contraction is admissible and can thus trivially be restricted to atoms or to the bottom of a proof. However, the R∧-rule in G3cp is additive (or context-sharing), so none of the above mentioned interesting consequences for system SKS follow for G3cp. Even though contraction is admissible, formulae of unbounded size still have to be duplicated in the context of an additive R∧. The separation of a proof into two phases as in system SKS also is not observable in G3cp, simply because there is no rule that just duplicates formulae. To answer the question whether similar properties to those of system SKS can be achieved in sequent systems, I thus consider systems with a multiplicative R∧-rule. For the specific case of system GS1p [3], a Gentzen-Schutte formulation of classical logic shown in Fig. 1, I now prove by counterexample that it does not admit the properties of SKS. Atoms can be either positive or negative and are denoted by a or b. The negation of an atom a is denoted by ā and is again an atom. In a proof, contraction is considered to be at the bottom if there is no application of a rule different from contraction such that one of its premises is the conclusion of a contraction. An application of the contraction rule is considered atomic if its principal formula is an atom. Theorem 1. There is a sequent that is valid but for which there is no cut-free proof in GS1p with multiplicative context treatment in which all contractions are at the bottom. Ax A, Ā Γ,A ∆, Ā Cut Γ,∆ Γ,A,B R∨ Γ,A ∨B Γ,A ∆,B R∧ Γ,∆,A ∧B Γ,A,A RC Γ,A Γ RW Γ,A Fig. 1. GS1p with multiplicative context treatment Proof. Consider the following sequent: a ∧ a, ā ∧ ā . It suffices to show that a ∧ a, . . . , a ∧ a, ā ∧ ā, . . . , ā ∧ ā (for any number of occurrences of the formulae a ∧ a and ā ∧ ā) is not provable in GS1p without contraction and cut. The connective ∨ does not occur in the conclusion. Thus, the only rules that can appear in the proof are RW ,Ax and R∧. A proof has to have all branches closed with axioms A, Ā. Since no rule that may occur introduces new formulae, the only formula that can take the place of A in an axiom is the atom a. There can be no such proof since there is always one branch in the derivation tree in which there is at most one single atom, as is shown by induction on the derivation tree: Base Case There is only one branch in a ∧ a, . . . , a ∧ a, ā ∧ ā, . . . , ā ∧ ā and it contains at most one (that is, no) single atom. Inductive Case Weakening does not increase the number of single atoms. When a R∧ rule is applied, the context is split, so the only single atom that may occur in the conclusion goes to one branch. Choose the other branch, which has at most one (that is, exactly one) atom. Theorem 2. There is a sequent that is valid but for which there is no cut-free proof in GS1p with multiplicative context treatment in which all contractions are atomic.

Sequent systems for compact bilinear logic

AbstractCompact Bilinear Logic (CBL), introduced by Lambek [14], arises from the multiplicative fragment of Noncommutative Linear Logic of Abrusci [1] (also called Bilinear Logic in [13]) by identifying times with par and 0 with 1. In this paper, we present two sequent systems for CBL and prove the cut‐elimination theorem for them. We also discuss a connection between cut‐elimination for CBL and the Switching Lemma from [14].

G-dinaturality

An extension of the notion of dinatural transformation is introduced in order to give a criterion for preservation of dinaturality under composition. An example of an application is given by proving that all bicartesian closed canonical transformations are dinatural. An alternative sequent system for intuitionistic propositional logic is introduced as a device, and a cut-elimination procedure is established for this system.

A New Machine-checked Proof of Strong Normalisation for Display Logic

We use a deep embedding of the display calculus for relation algebras δRA in the logical framework Isabelle/HOL to formalise a new, machine-checked, proof of strong normalisation and cut-elimination for δRA which does not use measures on the size of derivations. Our formalisation generalises easily to other display calculi and can serve as a basis for formalised proofs of strong normalisation for the classical and intuitionistic versions of a vast range of substructural logics like the Lambek calculus, linear logic, relevant logic, BCK-logic, and their modal extensions. We believe this is the first full formalisation of a strong normalisation result for a sequent system using a logical framework.

Minimum segments in sequent derivations

In a system of sequents for intuitionistic predicate logic derivations without a special kind of cuts (maximum cuts) will be considered. The following be shown: in a derivation without maximum cuts there are paths of the same form as paths in a normal derivation of natural deduction, i.e., these paths have the E-part, the I-part, and one minimum part which corresponds to a minimum segment in a normal derivation.

On different proof-search strategies for orthologic

In this paper, we consider three different search strategies for a cut-free sequent system formalizing orthologic, and estimate the respective search spaces. Applying backward search, there are classes of formulae for which both the minimal proof length and the search space are exponential. In a combined forward and backward approach, all proofs are polynomial, but the potential search space remains exponential. Using a forward strategy, the potential search space becomes polynomial yielding a polynomial decision procedure for orthologic and the word problem for free ortholattices.

A Cut-Free Sequent System for the Smallest Interpretability Logic

The idea of interpretability logics arose in Visser [Vis90]. He introduced the logics as extensions of the provability logic GLwith a binary modality ▸. The arithmetic realization of A ▸ B in a theory T will be that T plus the realization of B is interpretable in T plus the realization of A (T + A interprets T + B). More precisely, there exists a function f (the relative interpretation) on the formulas of the language of T such that T + B ⊢ C implies T + A ⊢ f(C). The interpretability logics were considered in several papers. An arithmetic completeness of the interpretability logic ILM, obtained by adding Montagna's axiom to the smallest interpretability logic IL, was proved in Berarducci [Ber90] and Shavrukov [Sha88] (see also Hajek and Montagna [HM90] and Hajek and Montagna [HM92]). [Vis90] proved that the interpretability logic ILP, an extension of IL, is also complete for another arithmetic interpretation. The completeness with respect to Kripke semantics due to Veltman was, for IL, ILMand ILP, proved in de Jongh and Veltman [JV90]. The fixed point theorem of GLcan be extended to ILand hence ILMand ILP(cf. de Jongh and Visser [JV91]). The unary pendant "T interprets T + A" is much less expressive and was studied in de Rijke [Rij92]. For an overview of interpretability logic, see Visser [Vis97], and Japaridze and de Jongh [JJ98]. In this paper, we give a cut-free sequent system for IL. To begin with, we give a cut-free system for the sublogic IL4of IL, whose ▸-free fragment is the modal logic K4. A cut-elimination theorem for ILis proved using the system for IK4and a property of Lob's axiom.

Sequent Calculi for Visser's Propositional Logics

This paper introduces sequent systems for Visser's two propositional logics: Basic Propositional Logic (BPL) and Formal Propositional Logic (FPL). It is shown through semantical completeness that the cut rule is admissible in each system. The relationships with Hilbert-style axiomatizations and with other sequent formulations are discussed. The cut-elimination theorems are also demonstrated by syntactical methods.

Sequent Calculi for Visser's Propositional Logics

This paper introduces sequent systems for Visser's two propositional logics: Basic Propositional Logic (BPL) and Formal Propositional Logic (FPL). It is shown through semantical completeness that the cut rule is admissible in each system. The relationships with Hilbert-style axiomatizations and with other sequent formulations are discussed. The cut-elimination theorems are also demonstrated by syntactical methods.

On Different Intuitionistic Calculi and Embeddings from Int to S4

In this paper, we compare several cut-free sequent systems for propositional intuitionistic logic Intwith respect to polynomial simulations. Such calculi can be divided into two classes, namely single-succedent calculi (like Gentzen's LJ) and multi-succedent calculi. We show that the latter allow for more compact proofs than the former. Moreover, for some classes of formulae, the same is true if proofs in single-succedent calculi are directed acyclic graphs (dags) instead of trees. Additionally, we investigate the effect of weakening rules on the structure and length of dag proofs. The second topic of this paper is the effect of different embeddings from Int to S4. We select two different embeddings from the literature and show that translated (propositional) intuitionistic formulae have sometimes exponentially shorter minimal proofs in a cut-free Gentzen system for S4than the original formula in a cut-free single-succedent Gentzen system for Int. Moreover, the length and the structure of proofs of translated formulae crucially depend on the chosen embedding.

Focussing and proof construction

This paper proposes a synthetic presentation of the proof construction paradigm, which underlies most of the research and development in the so-called “logic programming” area. Two essential aspects of this paradigm are discussed here: true non-determinism and partial information. A new formulation of Focussing, the basic property used to deal with non-determinism in proof construction, is presented. This formulation is then used to introduce a general constraint-based technique capable of dealing with partial information in proof construction. One of the baselines of the paper is to avoid to rely on syntax to describe the key mechanisms of the paradigm. In fact, the bipolar decomposition of formulas captures their main structure, which can then be directly mapped into a sequent system that uses only atoms. This system thus completely “dissolves” the syntax of the formulas and retains only their behavioural content as far as proof construction is concerned. One step further is taken with the so-called “abstract” proofs, which dissolves in a similar way the specific tree-like syntax of the proofs themselves and retains only what is relevant to proof construction.

A Uniform Procedure for Converting Matrix Proofs into Sequent-Style Systems

We present a uniform algorithm for transforming machine-found matrix proofs in classical, constructive, and modal logics into sequent proofs. It is based on unified representations of matrix characterizations, of sequent calculi, and of prefixed sequent systems for various logics. The peculiarities of an individual logic are described by certain parameters of these representations, which are summarized in tables to be consulted by the conversion algorithm.

Noncharacteristic line harmonics of AC/DC converters with high DC current ripple

To investigate the basic mechanisms generating noncharacteristic line current harmonics of AC/DC converters with high DC ripple, the assumption of negligible small commutation is used. Thus, simple analytical models can be derived describing the effect of voltage unbalance in case of equidistant phase angle control and control angle asymmetry, respectively. By means of superposition of both components, the according behavior applying individual phase angle control is described. Finally, it is shown that the effect of resonance excitation in diode rectifier circuits can also be estimated using the proposed modeling. Comparing simulations confirm the validity of the approximations. However, the strong influence of the commutation especially on the negative sequent systems requires further investigations.

Sequent calculi for finite-valued Lukasiewicz logics via Boolean decompositions

In this paper we define internal cut-free sequent calculi for any n-valued Lukasiewicz logic Ln. These calculi are based on a representation of formulas of Ln, by n - 1 many {0, 1}-valued formulas of Ln. They enjoy the usual properties of sequent systems like symmetry, subformula property and invertibility of the rules. Upon dualizing our calculi one obtains Hähnle's tableau systems. Then they provide a reformulation of Hähnle's approach to theorem proving that makes no use of nonlogical elements.

CATEGORIAL GRAPH GRAMMAR: A DIRECT APPROACH TO FUNCTOR-ARGUMENTOR-STRUCTURE

Systems of categorial grammar usually define functor-argumentor-structure (FAS, for short) in terms of derivational history. Here a more direct approach is proposed. FASs of linguistic items are conceived of as graphs of a certain kind. Category assignment statements are used to classify these graphs into categories and grammatical rules are formulated as sequents relating such type assignment statements. A base grammar G is a natural deduction system whose rules are given by sequents of the kind described. These rules may also be interpreted as specifying the mapping behaviour of the basic functors of G. In order to endow G with its full combinatorial power, the meta-rules of a so-called framework F are used for the construction of more complex functors from the basic ones. From a logical point of view, these complex functors are just the derived rules of the system G. A full grammar F(G) is a sequent system for the production of these derived rules. Index constructors (like / and \), though theoretically suspensable in F(G), may be characterized within this systems by special rules. Given this definitional rules, the usual cancellation laws of categorial grammar may be derived.

An O(n log n)-Space Decision Procedure for the Relevance Logic B+

In previous work we gave a new proof-theoretical method for establishing upper-bounds on the space complexity of the provability problem of modal and other propositional non-classical logics. Here we extend and refine these results to give an O(n log n)-space decision procedure for the basic positive relevance logic B+. We compute this upper-bound by first giving a sound and complete, cut-free, labelled sequent system for B+, and then establishing bounds on the application of the rules of this system.

A cut-elimination proof in intuitionistic predicate logic

In this paper we give a new proof of cut elimination in Gentzen's sequent system for intuitionistic first-order predicate logic. The point of this proof is that the elimination procedure eliminates the cut rule itself, rather than the mix rule.