Sequent Calculus Research Articles

Reasoning about Context Information in Cloud Computing Environments

The notion of context provides flexibility and adaptation to cloud computing services. Location, time identity and activity of users are examples of primary context types. The motivation of this paper is to formalize reasoning about context information in cloud computing environments. To formalize such context-aware reasoning, the logic LCM of context-mixture is introduced based on a Gentzen-type sequent calculus for an extended resource-sensitive logic. LCM has a specific inference rule called the context-mixture rule, which can naturally represent a mechanism for merging formulas with context information. Moreover, LCM has a specific modal operator called the sequence modal operator, which can suitably represent context information. The cut-elimination and embedding theorems for LCM are proved, and a fragment of LCM is shown to be decidable. These theoretical results are intended to provide a logical justification of context-aware cloud computing service models such as a flowable service model.

FORMAL REASONING IN LOGIC DESIGN OF RECONFIGURABLE CONTROLLERS

The rigorous digital design of embedded Reconfigurable Logic Controller starts from hierarchical concurrent state machine model (HCSM), which has been formally derived from modular control interpreted Petri net. The colored tokens, arcs, places and transitions distinguish nested State Machine Modules. Colored coordination places, called doubles facilitate effective and flexible Petri net state encoding. The rule based on a textual logic description of the Petri net is accepted by the hardware description language VHDL and easily mapped into Field Programmable Gate Array macrocells. Several combinatorial procedures in formal digital design of logic controller are supported by formal reasoning in the monotone Gentzen calculus.

A cut-free sequent system for two-dimensional modal logic, and why it matters

The two-dimensional modal logic of Davies and Humberstone (1980) [3] is an important aid to our understanding the relationship between actuality, necessity and a priori knowability. I show how a cut-free hypersequent calculus for 2D modal logic not only captures the logic precisely, but may be used to address issues in the epistemology and metaphysics of our modal concepts. I will explain how the use of our concepts motivates the inference rules of the sequent calculus, and then show that the completeness of the calculus for Davies–Humberstone models explains why those concepts have the structure described by those models. The result is yet another application of the completeness theorem.

Formula omitted], [formula omitted], and Non-Permutability of [formula omitted]-Steps

Using a human-oriented formal example proof of the lim+-theorem (that the sum of limits is the limit of the sum), we exhibit a non-permutability of β-steps and δ+-steps (according to Smullyan’s classification), which is not visible with non-liberalized δ-rules and dissolves into a problem of mere inefficiency with further liberalized δ-rules, such as the δ++-rules. Beside a careful presentation of the human-oriented search for a formal proof of (lim+), our main intention is to show where sequent and tableau calculi are in conflict with human-oriented proof construction.

A sequent calculus for propositional temporal logic with time gaps

A sequent calculus with Kripke semantics internalization for a propositional temporal logic with time gaps is introduced. All rules of the calculus are context-free andheight-preserving invertible. Structural rules are admissible. The calculus is cut free and is proved to be complete.

A specialization of definitions in common knowledge logic

It is known that one of main aims of specializations of derivations in nonclassical logics is the various tools which allow us to simplify the searching of termination of derivations. The traditional techniques used to ensure termination of derivations in various non-classical logics are based on loop-checking. In this paper the reflexive common knowledge logic based on modal logic K2 is considered. For considered logic a sequent calculus with specialized non-logical (loop-type) axioms is presented.

Contraction-free calculi for modal logics S5 and KD45

It is known that termination and backtracking are among the most important problems in constructing derivations in non-classical logics. In this paper contractionfree and backtracking-free sequent calculi for modal logics S5 and KD45 are presented and founded. These are index-style sequent calculi with some specific indexed axioms. The main new ideas in considered calculi are to use metavariables (along with natural numbers) as elements of indexes and to numerate by different natural number all the positive occurrences of modality 2.

Sequent calculus for logic of correlated knowledge

Sound and complete sequent calculi for general epistemic logic and logic of correlated knowledge are presented in this paper.

A logic of sequences

The notion of “sequences” is fundamental to practical reasoning in computer science, because it can appropriately represent “data (information) sequences”, “program (execution) sequences”, “action sequences”, “time sequences”, “trees”, “orders” etc. The aim of this paper is thus to provide a basic logic for reasoning with sequences. A propositional modal logic LS of sequences is introduced as a Gentzen-type sequent calculus by extending Gentzen’s LK for classical propositional logic. The completeness theorem with respect to a sequence-indexed semantics for LS is proved, and the cut-elimination theorem for LS is shown. Moreover, a first-order modal logic FLS of sequences, which is a first-order extension of LS, is introduced. The completeness theorem with respect to a first-order sequence-indexed semantics for FLS is proved, and the cut-elimination theorem for FLS is shown. LS and the monadic fragment of FLS are shown to be decidable.

Cut, invariant rule, and loop-check free sequent calculus for PLTL

In this paper, some loop-check free saturation-like decision procedure is proposed for propositional linear temporal logic (PLTL) with temporal operators “next” and “always”. This saturation procedure terminates when special type sequents are obtained. Properties of PLTL allows us to construct backtracking-free decision procedure without histories and loop-check.

Partial cut elimination for combinations of propositional multi-modal logics with past time

We consider combinations of nine propositional multi-modal logics with propositional discrete linear time temporal logic with past time. For these combinations, we present sound and complete Gentzen-type sequent calculi with a restricted cut rule.

Proof analysis in intermediate logics

Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Godel---McKinsey---Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the explanation of the rules.

Algebraic proof theory for substructural logics: Cut-elimination and completions

We carry out a unified investigation of two prominent topics in proof theory and order algebra: cut-elimination and completion, in the setting of substructural logics and residuated lattices. We introduce the substructural hierarchy — a new classification of logical axioms (algebraic equations) over full Lambek calculus FL, and show that a stronger form of cut-elimination for extensions of FL and the MacNeille completion for subvarieties of pointed residuated lattices coincide up to the level N 2 in the hierarchy. Negative results, which indicate limitations of cut-elimination and the MacNeille completion, as well as of the expressive power of structural sequent calculus rules, are also provided. Our arguments interweave proof theory and algebra, leading to an integrated discipline which we call algebraic proof theory.

Visible acyclic differential nets, Part I: Semantics

We give a geometric condition that characterizes the differential nets having a finitary interpretation in finiteness spaces: visible acyclicity. This is based on visible paths, an extension to differential nets of a class of paths we introduced in the framework of linear logic nets. The characterization is then carried out as follows: the differential nets having no visible cycles are exactly those whose interpretation is a finitary relation. Visible acyclicity discloses a new kind of correctness for the promotion rule of linear logic, which goes beyond sequent calculus correctness.

A modal logic internalizing normal proofs

In the proof-theoretic study of logic, the notion of normal proof has been understood and investigated as a metalogical property. Usually we formulate a system of logic, identify a class of proofs as normal proofs, and show that every proof in the system reduces to a corresponding normal proof. This paper develops a system of modal logic that is capable of expressing the notion of normal proof within the system itself, thereby making normal proofs an inherent property of the logic. Using a modality △ to express the existence of a normal proof, the system provides a means for both recognizing and manipulating its own normal proofs. We develop the system as a sequent calculus with the implication connective ⊃ and the modality △, and prove the cut elimination theorem. From the sequent calculus, we derive two equivalent natural deduction systems.

Constructive linear-time temporal logic: Proof systems and Kripke semantics

In this paper we study a version of constructive linear-time temporal logic (LTL) with the “next” temporal operator. The logic is originally due to Davies, who has shown that the proof system of the logic corresponds to a type system for binding-time analysis via the Curry–Howard isomorphism. However, he did not investigate the logic itself in detail; he has proved only that the logic augmented with negation and classical reasoning is equivalent to (the “next” fragment of) the standard formulation of classical linear-time temporal logic. We give natural deduction, sequent calculus and Hilbert-style proof systems for constructive LTL with conjunction, disjunction and falsehood, and show that the sequent calculus enjoys cut elimination. Moreover, we also consider Kripke semantics and prove soundness and completeness. One distinguishing feature of this logic is that distributivity of the “next” operator over disjunction “ ◯ ( A ∨ B ) ⊃ ◯ A ∨ ◯ B ” is rejected in view of a type-theoretic interpretation.

Sequent calculi and decidability for intuitionistic hybrid logic

In this paper we study the proof theory of the first constructive version of hybrid logic called Intuitionistic Hybrid Logic ( IHL) in order to prove its decidability. In this perspective we propose a sequent-style natural deduction system and then the first sequent calculus for this logic. We prove its main properties like soundness, completeness and also the cut-elimination property. Finally we provide, from our calculus, the first decision procedure for IHL and then prove its decidability.

Cut-free Gentzen calculus for multimodal [formula omitted

This paper extends previous work on the modal logic CK as a reference system, both proof-theoretically and model-theoretically, for a correspondence theory of constructive modal logics. First, the fundamental nature of CK is discussed and compared with the intuitionistic modal logic IK which is traditionally taken to be the base line. Then, it is shown, that CK admits of a cut-free Gentzen sequent calculus G - CK which has (i) a local interpretation in constructive Kripke models and (ii) does not require explicit world labels. Finally, the paper demonstrates how non-classical modal logics such as IK , C S 4 , CL , or Masiniʼs deontic system of 2-sequents arise as theories of CK , presented both as special rules and as frame classes.

An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic

Craig interpolation has become a versatile tool in formal verification, used for instance to generate program assertions that serve as candidates for loop invariants. In this paper, we consider Craig interpolation for quantifier-free Presburger arithmetic (QFPA). Until recently, quantifier elimination was the only available interpolation method for this theory, which is, however, known to be potentially costly and inflexible. We introduce an interpolation approach based on a sequent calculus for QFPA that determines interpolants by annotating the steps of an unsatisfiability proof with partial interpolants. We prove our calculus to be sound and complete. We have extended the Princess theorem prover to generate interpolating proofs, and applied it to a large number of publicly available Presburger arithmetic benchmarks. The results document the robustness and efficiency of our interpolation procedure. Finally, we compare the procedure against alternative interpolation methods, both for QFPA and linear rational arithmetic.

Synchronized Linear-Time Temporal Logic

A new combined temporal logic called synchronized linear-time temporal logic (SLTL) is introduced as a Gentzen-type sequent calculus. SLTL can represent the n-Cartesian product of the set of natural numbers. The cut-elimination and completeness theorems for SLTL are proved. Moreover, a display sequent calculus ?SLTL is defined.