Tautology Elimination, Cut Elimination and S4?

Infinitary and regular proofs are commonly used in fixed point logics. Being natural intermediate devices between semantics and traditional finitary proof systems, they are commonly found in completeness arguments, automated deduction, verification, etc. However, their proof theory is surprisingly underdeveloped. In particular, very little is known about the computational behavior of such proofs through cut elimination. Taking such aspects into account has unlocked rich developments at the intersection of proof theory and programming language theory. One would hope that extending this to infinitary calculi would lead, e.g., to a better understanding of recursion and corecursion in programming languages. Structural proof theory is notably based on two fundamental properties of a proof system: cut elimination and focalization. The first one is only known to hold for restricted (purely additive) infinitary calculi, thanks to the work of Santocanale and Fortier; the second one has never been studied in infinitary systems. In this paper, we consider the infinitary proof system muMALLi for multiplicative and additive linear logic extended with least and greatest fixed points, and prove these two key results. We thus establish muMALLi as a satisfying computational proof system in itself, rather than just an intermediate device in the study of finitary proof systems.

This dissertation pertains to algebraic proof theory, a research field aimed at solving problems in structural proof theory using results and insights from algebraic logic, universal algebra, duality and representation theory for classes of algebras. The main contributions of this dissertation involve the very recent theory of multi-type calculi on the proof-theoretic side, and the well established theory of heterogeneous algebras on the algebraic side. Given a cut-admissible sequent calculus for a basic logic (e.g. the full Lambek calculus), a core question in structural proof theory concerns the identification of axioms which can be added to the given basic logic so that the resulting axiomatic extensions can be captured by calculi which are again cut-admissible. This question is very hard, since the cut elimination theorem is notoriously a very fragile result. However, algebraic proof theory has given a very satisfactory answer to this question for substructural logics, by identifying a hierarchy (Nn, Pn) of axioms in the language of the full Lambek calculus, referred to as the substructural hierarchy, and guaranteeing that, up to the level N2, these axioms can be effectively transformed into special structural rules (called analytic) which can be safely added to a cut-admissible calculus without destroying cut-admissibility. The research program of algebraic proof theory can be exported to arbitrary signatures of normal lattice expansions, to the study of the systematic connections between algebraic logic and display calculi, and even beyond display calculi, to the study of the systematic connections between the theory of heterogeneous algebras and multi-type calculi, a proof-theoretic format generalizing display calculi, which has proven capable to encompass logics which fall out of the scope of the proof-theoretic hierarchy, and uniformly endow them with calculi enjoying the same excellent properties which (single-type) proper display calculi have. The defining feature of the multi-type calculi format is that it allows entities of different types to coexist and interact on equal ground: each type has its own internal logic (i.e. language and deduction relation), and the interaction between logics of different types is facilitated by special heterogeneous connectives, primitive to the language, and treated on a par with all the others. The fundamental insight justifying such a move is the very natural consideration, stemming from the algebraic viewpoint on (unified) correspondence, that the fundamental properties underlying this theory are purely order-theoretic, and that as long as maps or logical connectives have these fundamental properties, there is very little difference whether these maps have one and the same domain and codomain, or bridge different domains and codomains. This enriched environment is specifically designed to address the problem of expressing the interactions between entities of different types by means of analytic structural rules. In the present dissertation, we extend the semantic cut elimination and finite model property from the signature of residuated lattices to arbitrary signatures of normal lattice expansions, and build or refine the multi-type algebraic proof theory of three logics, each of which arises in close connection with a well known class of algebras (semi De Morgan algebras, bilattices, and Kleene algebras) and is problematic for standard proof-theoretic methods.

Proof theory is a central area of mathematical logic of special interest to philosophy. It has its roots in the foundational debate of the 1920s, in particular, in Hilbert’s program in the philosophy of mathematics, which called for a formalization of mathematics, as well as for a proof, using philosophically unproblematic, “finitary” means, that these systems are free from contradiction. Structural proof theory investigates the structure and properties of proofs in different formal deductive systems, including axiomatic derivations, natural deduction, and the sequent calculus. Central results in structural proof theory are the normalization theorem for natural deduction, proved here for both intuitionistic and classical logic, and the cut-elimination theorem for the sequent calculus. In formal systems of number theory formulated in the sequent calculus, the induction rule plays a central role. It can be eliminated from proofs of sequents of a certain elementary form: every proof of an atomic sequent can be transformed into a “simple” proof. This is Hilbert’s central idea for giving finitary consistency proofs. The proof requires a measure of proof complexity called an ordinal notation. The branch of proof theory dealing with mathematical systems such as arithmetic thus has come to be called ordinal proof theory. The theory of ordinal notations is developed here in purely combinatorial terms, and the consistency proof for arithmetic presented in detail.

Cut elimination in sequent calculus is indispensable in bounding the number of distinct formulas to appear during a backward proof search. A usual approach to prove cut admissibility is permutation of derivation trees. Extra care must be taken, however, when contraction appears as an explicit inference rule. In G1i for example, a simple-minded permutation strategy comes short around contraction interacting directly with cut formulas, which entails irreducibility of the derivation height of Cut instances. One of the practices employed to overcome this issue is the use of MultiCut (the “mix” rule) which takes into account the eject of contraction within. A more recent substructural logic BI inherits the characteristics of the intuitionistic logic but also those of multiplicative linear logic (without exponentials). Following Pym's original work, the cut admissibility in LBI (the original BI sequent calculus) is supposed to hold with the same tweak. However, there is a critical issue in the approach: MultiCut does not take care of the eject of structural contraction that LBI permits. In this paper, we show a proper proof of the LBI cut admissibility based on another derivable rule BI-MultiCut.

Regaining cut admissibility in deduction modulo using abstract completion

In his doctoral dissertation of I935,1 Gentzen introduced two new approaches to the theory of proofs: natural deduction and the sequent calculus. He formulated both classical and intuitionist predicate logics using these approaches. For the resulting sequent calculi, he proved his Hauptsatz, or main theorem, which states that the rule of cut may be dispensed with in all proofs. The significance of this 'cut-elimination theorem' has been much debated over the past fifty years, but no clear consensus has arisen. In this paper, I will examine some of the ways in which this theorem has been interpreted. In particular, I will focus on the idea that the introduction rules for a logical constant in a sequent calculus may be taken as defining the constant or 'giving its meaning'; and I will consider the relation of cut-elimination to this thesis. My discussion will be illustrated by a specific example, drawing on recent work on truth and the semantic paradoxes.

We introduce the logic LRC, designed to describe and reason about agents’ abilities and capabilities in using resources. The proposed framework bridges two—up to now—mutually independent strands of literature: the one on logics of abilities and capabilities, developed within the theory of agency, and the one on logics of resources, motivated by program semantics. The logic LRC is suitable to describe and reason about key aspects of social behaviour in organizations. We prove a number of properties enjoyed by LRC (soundness, completeness, canonicity, and disjunction property) and its associated analytic calculus (conservativity, cut elimination, and subformula property). These results lay at the intersection of the algebraic theory of unified correspondence and the theory of multitype calculi in structural proof theory. Case studies are discussed which showcase several ways in which this framework can be extended and enriched while retaining its basic properties, so as to model an array of issues, both practically and theoretically relevant, spanning from planning problems to the logical foundations of the theory of organizations.

Counterfactual conditions such as “if A were not true, then C would not have been true” have been formally studied by philosophers for causal claims for decades. Counterfactuals are often used informally in practice for diagnosing systems and identifying errors or misconfigurations. This paper develops a proof theory for counterfactual reasoning of Horn clauses, which have applications in domains including security and database and program analysis. The application to security that this paper focuses on is modeling and reasoning about probing attacks in Datalog-based trust management systems, where an attacker can apply counterfactual reasoning to obtain sensitive information embedded in the system. Our work is inspired by a Hilbert style axiomatized system for counterfactual reasoning for Horn clauses, which are hard to use to construct proofs or study properties of the system. To alleviate this difficulty, we develop a sequent calculus from first principles. We show that the sequent calculus has cut elimination and is sound and complete with regard to the corresponding Hilbert style axiomatized system. We also show how to construct proofs that model practical counterfactual reasoning scenarios in trust management systems using our sequent calculus rules.

Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions of such systems from logic to mathematics, and on the connection between the two main forms of structural proof theory - natural deduction and sequent calculus. The authors emphasize the computational content of logical results. A special feature of the volume is a computerized system for developing proofs interactively, downloadable from the web and regularly updated.

We study a fragment of Intuitionistic Linear Logic combined with non-normal modal operators. Focusing on the minimal modal logic, we provide a Gentzen-style sequent calculus as well as a semantics in terms of Kripke resource models. We show that the proof theory is sound and complete with respect to the class of minimal Kripke resource models. We also show that the sequent calculus allows cut elimination. We put the logical framework to use by instantiating it as a logic of agency. In particular, we apply it to reason about the resource-sensitive use of artefacts.

The multiary version of the λ-calculus with generalized applications integrates smoothly both a fragment of sequent calculus and the system of natural deduction of von Plato. It is equipped with reduction rules (corresponding to cut-elimination/normalisation rules) and permutation rules, typical of sequent calculus and of natural deduction with generalised elimination rules. We argue that this system is a suitable tool for doing structural proof theory as rewriting. As an illustration, we investigate combinations of reduction and permutation rules and whether these combinations induce rewriting systems which are confluent and terminating. In some cases, the combination allows the simulation of non-terminating reduction sequences known from explicit substitution calculi. In other cases, we succeed in capturing interesting classes of derivations as the normal forms w.r.t. well-behaved combinations of rules. We identify six of these “combined” normal forms, among which are two classes, due to Herbelin and Mints, in bijection with normal, ordinary natural deductions. A computational explanation for the variety of “combined” normal forms is the existence of three ways of expressing multiple application in the calculus.KeywordsNormal FormProof SystemMultiple ApplicationReduction RuleNatural DeductionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction: Elementary Proof Theory. The Fall of Hilbert's Program. Hilbert's Program. Recursive Functions. The First Incompleteness Theorem. The Second Incompleteness Theorem. Exercises. Annex: Intuitionism. Part I: Sigma 0 1 Proof Theory. The Calculus of Sequents. Definitions. Completeness of the Sequent Calculus. The Cut-Elimination Theorem. The Subformula Property. Intuitionistic Sequent Calculus. Herbrand's Theorem. Generalization. Annex: Natural Deduction. The Church-Rosser Property. Strong Normalization. The Semantics of Sequent Calculus. Completeness of the Cut-Free Rules. Three-Valued Models. Three-Valued Logic. Annex: Takeuti's Conjecture. Limitations of Takeuti's Conjecture. Three-Valued Equivalence. Cut-Free Analysis. Three-Valued Semantics and Generalized Logics. Applications of the ``Hauptsatz''. The Interpolation Lemma. The Reflection Schema of PA. Elementary Consistency Proofs. 1-Consistency. Annex: The Hauptsatz in a Concrete Case. Normalization in HA. Normalization for NL 2 J. Part II: Pi 1 1 Proof Theory. Pi 1 1 Formulas and Well-Foundedness. The Projective Hierarchy. Well-Founded Trees. Well-Orders. Equivalents of (Sigma 0 1 -CA * ). Recursive Well-Orders. Hyperarithmetical Sets. Annex: Kleene's 0. Hierarchies Indexed by 0. Paths Through 0. The Classification Problem. The omega-Rule. omega-Logic. The Cut-Elimination Theorem. Bounds for Cut-Elimination. Equivalents for (Sigma 0 1 -CA * ). Annex: The Calculus Lomega 1 omega. Cut-Elimination in Lomega 1 omega. The Ordinal epsilon o and Arithmetic. Ordinal Analysis of PA. Extensions to other Systems. Ordinals and Theories. Annex: Godel's System T. Functional Interpretation. Spector's Interpretation. No Conterexample Interpretation. An Application. Bibliography. Analytical Index.

Deduction modulo is a framework in which theories are integrated into proof systems such as natural deduction or sequent calculus by presenting them using rewriting rules. When only terms are rewritten, cut admissibility in those systems is equivalent to the confluence of the rewriting system, as shown by Dowek, RTA 2003, LNCS 2706. This is no longer true when considering rewriting rules involving propositions. In this paper, we show that, in the same way that it is possible to recover confluence using Knuth-Bendix completion, one can regain cut admissibility in the general case using standard saturation techniques. This work relies on a view of proposition rewriting rules as oriented clauses, like term rewriting rules can be seen as oriented equations. This also leads us to introduce an extension of deduction modulo with conditional term rewriting rules.KeywordsInference RuleProof SystemNatural DeductionSaturation ProcessSequent CalculusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Focusing and selection are techniques that shrink the proof search space for respectively sequent calculi and resolution. To bring out a link between them, we generalize them both: we introduce a sequent calculus where each occurrence of an atom can have a positive or a negative polarity; and a resolution method where each literal, whatever its sign, can be selected in input clauses. We prove the equivalence between cut-free proofs in this sequent calculus and derivations of the empty clause in that resolution method. Such a generalization is not semi-complete in general, which allows us to consider complete instances that correspond to theories of any logical strength. We present three complete instances: first, our framework allows us to show that ordinary focusing corresponds to hyperresolution and semantic resolution; the second instance is deduction modulo theory and the related framework called superdeduction; and a new setting, not captured by any existing framework, extends deduction modulo theory with rewriting rules having several left-hand sides, which restricts even more the proof search space.

Tautology elimination rule was successfully applied in automated deduction and recently considered in the framework of sequent calculi where it is provably equivalent to cut rule. In this paper we focus on the advantages of proving admissibility of tautology elimination rule instead of cut for sequent calculi. It seems that one may find simpler proofs of admissibility for tautology elimination than for cut admissibility. Moreover, one may prove its admissibility for some calculi where constructive proofs of cut admissibility fail. As an illustration we present a cut-free sequent calculus for S5 based on tableau system of Fitting and prove admissibility of tautology elimination rule for it.