Gentzen's Sequent Calculus Research Articles

Merging Intuitionistic and De Morgan Logics

We introduce De Morgan Heyting logic for Heyting algebras with De Morgan negation (DH-algebras). The variety DH of all DH-algebras is congruence distributive. The lattice of all subvarieties of DH is distributive. We show the discrete dualities between De Morgan frames and DH-algebras. The Kripke completeness and finite approximability of some DH-logics are proven. Some conservativity of DH expansion of a Kripke complete superintuitionistic logic is shown by the construction of frame expansion. Finally, a cut-free terminating Gentzen sequent calculus for the DH-logic of De Morgan Boolean algebras is developed.

Cut-Free Gentzen Sequent Calculi for Tense Logics

The cut-free single-succedent Gentzen sequent calculus GKt for the minimal tense logic Kt is introduced. This sequent calculus satisfies the displaying property. The proof proceeds in terms of a Kolmogorov translation and three intermediate sequent systems. Finally, we show that tense logics axiomatized by strictly positive implication have cut-free Gentzen sequent calculi uniformly.

From axioms to synthetic inference rules via focusing

An important application of focused variants of Gentzen's sequent calculus proof rules is the construction of (possibly) large synthetic inference rules. This paper examines the synthetic inference rules that arise when using theories composed of bipolars, and we do this in both classical and intuitionistic logics. A key step in transforming a formula into synthetic inference rules involves attaching a polarity to atomic formulas and some logical connectives. Since there are different choices in how polarity is assigned, it is possible to produce different synthetic inference rules for the same formula. We show that this flexibility allows for the generalization of different approaches for transforming axioms into sequent rules present in the literature. We finish the paper showing how to apply these results to organize the proof theory of labeled sequent systems for several propositional modal logics.

Expander construction in VNC1

We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson [44], and show that this analysis can be formalized in the bounded arithmetic system VNC1 (corresponding to the “NC1 reasoning”). As a corollary, we prove the assumption made by Jeřábek [28] that a construction of certain bipartite expander graphs can be formalized in VNC1. This in turn implies that every proof in Gentzen's sequent calculus LK of a monotone sequent can be simulated in the monotone version of LK (MLK) with only polynomial blowup in proof size, strengthening the quasipolynomial simulation result of Atserias, Galesi, and Pudlák [9].

Herbrand's theorem as higher order recursion

This article examines the computational content of the classical Gentzen sequent calculus. There are a number of well-known methods that extract computational content from first-order logic but applying these to the sequent calculus involves first translating proofs into other formalisms, Hilbert calculi or Natural Deduction for example. A direct approach which mirrors the symmetry inherent in sequent calculus has potential merits in relation to proof-theoretic considerations such as the (non-)confluence of cut elimination, the problem of cut introduction, proof compression and proof equivalence. Motivated by such applications, we provide a representation of sequent calculus proofs as higher order recursion schemes. Our approach associates to an LK proof π of ⇒∃vF, where F is quantifier free, an acyclic higher order recursion scheme H with a finite language yielding a Herbrand disjunction for ∃vF. More generally, we show that the language of H contains all Herbrand disjunctions computable from π via a broad range of cut elimination strategies.

Quantum-like behavior without quantum physics III : Logic and memory.

We employ some of the machinery developed in previous work to investigate the inferential and memory functions of quantum-like neural networks. We set up a logical apparatus to implement this in the form of a Gentzen sequent calculus which codifies some of the combinatory rules for the state spaces of the neuronal networks introduced earlier. We discuss memory storage in this context and along the way find formal proof that synchronicity promotes binding and storage. These results lead to an algorithmic fragment in calculus that simulates the memory function known as pattern completion. This claim is tested by noting that the failure of certain steps in the algorithm leads to memory deficits essentially identical to those found in such pathologies as Alzheimer's dementia, schizophrenia, and certain forms of autism. Moreover, a specific "power-of-two" wiring architecture and computational logic, which have been postulated and observed across many brain circuits, emerge spontaneously from our model. We draw conclusions concerning the possible nature of such mental processes qua computations.

Gentzen sequent calculi for some intuitionistic modal logics

Abstract Intuitionistic modal logics are extensions of intuitionistic propositional logic with modal axioms. We treat with two modal languages ${\mathscr{L}}_\Diamond $ and $\mathscr{L}_{\Diamond ,\Box }$ which extend the intuitionistic propositional language with $\Diamond $ and $\Diamond ,\Box $, respectively. Gentzen sequent calculi are established for several intuitionistic modal logics. In particular, we introduce a Gentzen sequent calculus for the well-known intuitionistic modal logic $\textsf{MIPC}$. These sequent calculi admit cut elimination and subformula property. They are decidable.

A Graphical Deep Inference System for Intuitionistic Logic

A graphical approach to intuitionistic propositional logic is presented. The system GrIn is a deep inference system and is formulated in terms of Peirce’s existential graphs. GrIn is shown to be sound and complete with respect to the class of all Heyting algebras. The system GrIn is shown to be equivalent to the Gentzen sequent calculus G3ip via translations between intuitionistic graphs and intuitionistic formulae. We also discuss the intuitionistic interpretation of logical constants proposed by the graphical approach.

Elimination of network intrusions via a resource oriented BDI architecture

Abstract We propose a resource-oriented architecture of a rational agent for a network intrusion detection system. This architecture describes the behavior of a rational agent after detection of unwanted network activities. We describe the creation of countermeasures to ward off detected threats. Examples are created based on the proposed architecture, describing the process during a rational agent detection. We have described these examples by linear BDI logic behavioral formulæ, that have been proven by Gentzen sequent calculus.

A cut-free proof system for a predicate extension of the logic of provability

In this paper, we introduce a proof system $\mathsf{NQGL}$ for a Kripke complete predicate extension of the logic $\mathbf{GL}$, that is, the logic of provability, which is defned by $\mathbf{K}$ and the L\"{o}b formula $\Box(\Box p\supset p)\supset\Box p$. $\mathsf{NQGL}$ is a modal extension of Gentzen's sequent calculus $\mathsf{LK}$. Although the propositional fragment of $\mathsf{NQGL}$ axiomatizes $\mathbf{GL}$, it does not have the L\"{o}b formula as its axiom. Instead, it has a non-compact rule, that is, a derivation rule with countably many premises. We show that $\mathsf{NQGL}$ enjoys cut admissibility and is complete with respect to the class of Kripke frames such that for each world, the supremum of the length of the paths from the world is finite.

Simulation of Natural Deduction and Gentzen Sequent Calculus

We consider four natural deduction systems: Fitch-style systems, Gentzen-style systems (in the form of dags), general deduction Frege systems and nested deduction Frege systems, as well as dag-like Gentzen-style sequent calculi. All these calculi soundly and completely formalize classical propositional logic. We show that general deduction Frege systems and Gentzen-style natural calculi provide at most quadratic speedup over nested deduction Frege systems and Fitch-style natural calculi and at most cubic speedup over Gentzen-style sequent calculi.

Proof Analysis of Peirce’s Alpha System of Graphs

Charles Peirce's alpha system $$\mathfrak {S}_\alpha $$S? is reformulated into a deep inference system where the rules are given in terms of deep graphical structures and each rule has its symmetrical rule in the system. The proof analysis of $$\mathfrak {S}_\alpha $$S? is given in terms of two embedding theorems: the system $$\mathfrak {S}_\alpha $$S? and Brunnler's deep inference system for classical propositional logic can be embedded into each other; and the system $$\mathfrak {S}_\alpha $$S? and Gentzen sequent calculus $$\mathbf {G3c}^*$$G3c? can be embedded into each other.

Proof Generalization in $$\mathrm {LK}$$ LK by Second Order Unifier Minimization

We devise a method for generalizing proofs in Gentzen's sequent calculus $$\mathrm {LK}$$LK, presented in a typed $$\lambda $$ź-calculus flavor. A constrained version $$\mathrm {LK}^{{{\mathrm {c}}}}$$LKc of the calculus is introduced, aiming at collecting a second order constraint ensuring that all the inference steps occurring in a proof are syntactically correct. A semantics is provided for $$\mathrm {LK}^{{{\mathrm {c}}}}$$LKc, extending the standard semantics of $$\mathrm {LK}$$LK. It is then established that $$\mathrm {LK}$$LK-proofs correspond to $$\mathrm {LK}^{{{\mathrm {c}}}}$$LKc-proofs with valid constraint thanks to the use of eigenterms replacing $$\mathrm {LK}$$LK's eigenvariables. Next, a lifting theorem shows how a valid $$\mathrm {LK}^{{{\mathrm {c}}}}$$LKc-proof can be lifted to a most general proof, yielding a non-trivial constraint together with a solution. An algorithm is then provided that minimizes this solution of the constraint. The result, applied to the most general proof, yields a valid proof that translates to an $$\mathrm {LK}$$LK-proof more general than the initial one. Finally, clues are given for extending this method to other logics with due care on proof lifting.

Non-commutative classical arithmetical sequent calculi are intuitionistic

A non-commutative first-order sequent calculus NCLK equivalent to Gentzen's sequent calculus LJ for intuitionistic logic was recently presented such that the addition of exchange rules to NCLK yields a calculus equivalent to Gentzen's sequent calculus LK for classical logic. In Gentzen's original treatment, the intuitionistic calculus LJ was obtained from LK by restricting the sequents to at most one formula in the succedent. Distinguishing between a classical and intuitionistic calculus solely by the presence or absence of the exchange rules thus provides a new perspective on these calculi. Unfortunately, the sequent calculus NCLK is highly non-standard for several reasons. In addition, the original proof relies on a technical intermediate step utilizing a complicated induction measure. Here we present a direct proof of this result using a calculus that is almost identical to Gentzen's LK. Recent work by Tatsuta and Berardi suggests that the logic obtained by deleting the exchange rules from a sequent calculus for infinitary Peano arithmetic is not infinitary Heyting arithmetic but instead an extension of the latter by a restricted excluded middle principle. Here we show how to recover infinitary Heyting arithmetic, thus developing further the ‘classical calculus minus exchange rules is the intuitionistic calculus’ paradigm. Finally, we resolve an open problem posed in that work by showing that the sequent calculus for Peano arithmetic minus the exchange rules is Heyting arithmetic. Results are obtained using a uniform method of proof.

Expanding the Universe

In (Béziau 2001), Béziau provides a means by which Gentzen's sequent calculus can be combined with the general semantic theory of bivaluations. In do- ing so, according to Béziau, it is possible to construe the abstract "core" of logics in general, where logical syntax and semantics are "two sides of the same coin". The central suggestion there is that, by way of a modification of the notion of maximal consistency, it is possible to prove the soundness and completeness for any normal logic (without invoking the role of classical negation in the completeness proof). However, the reduction to bivaluation may be a side effect of the architecture of ordinary sequents, which is both overly restrictive, and entails certain expressive restrictions over the language. This paper provides an expansion of Béziau's completeness results for logics, by showing that there is a natural extension of that line of thinking to n-sided sequent constructions. Through analogical techniques to Béziau's construction, it is possible, in this setting, to construct abstract soundness and completeness results for n-valued logics.

Cut Rule for the Resolution Method

In this work, we extend the resolution method with a new rule similar to the cut rule in Gentzen sequent calculi. We obtain upper and lower bounds on the complexity of refutations with this counterpart of the cut rule and without it. The complexity of refutations is also compared to the complexity of sequent proofs with cuts by formulas that do not contain implications, conjunctions, and ∃ quantifiers and in which negation may occur only immediately before elementary formulas. Bibliography: 3 titles.

On Hilbert's Axiomatics of Propositional Logic

In this paper I will consider the axioms for propositional logic which were presented by Hilbert in his conferences during the year 1922, and those which were presented by Hilbert and Bernays in the book Grundlagen der Mathematik, I (1934). I will describe a general procedure in order to translate Hilbert's axioms into rules on sequents and I will show that, following this procedure, Hilbert's axioms become particular cases of (derived or primitive) rules of Gentzen's Sequent Calculus and contain ideas which will be focused and developed in Gentzen's Sequent Calculus and also in more recent logical investigations.

Intuitionistic N-Graphs

The geometric system of deduction called N-Graphs was introduced by de Oliveira in 2001. The proofs in this system are represented by means of digraphs and, while its derivations are mostly based on Gentzen's sequent calculus, the system gets its inspiration from geometrically based systems, such as the Kneales' tables of development, Statman's proofs-as-graphs, Buss' logical flow graphs and Girard's proof-nets. Given that all these geometric systems appeal to the classical symmetry between premises and conclusions, providing an intuitionistic version of any of these is an interesting exercise in extending the range of applicability of the geometric system in question. In this article we produce an intuitionistic version of N-Graphs, based on Maehara's LJ' system, as described by Takeuti. Recall that LJ' has multiple conclusions in all but the essential intuitionistic rules, e.g., implication right and negation right. We show soundness and completeness of our intuitionistic N-Graphs with respect to LJ'. We also discuss how we expect to extend this work to a version of N-Graphs corresponding to the intuitionistic logic system FIL (Full Intuitionistic Logic) of de Paiva and Pereira and sketch future developments.

Completeness and Soundness Results for 𝒳 with Intersection and Union Types

With the eye on defining a type-based semantics, this paper defines intersection and union type assignment for the sequent calculus X, a substitution-free language that enjoys the Curry-Howard correspondence with respect to the implicative fragment of Gentzen's sequent calculus for classical logic. We investigate the minimal requirements for such a system to be complete i.e. closed under redex expansion, and show that the non-logical nature of both intersection and union types disturbs the soundness i.e. closed uder reduction properties. This implies that this notion of intersection-union type assignment needs to be restricted to satisfy soundness as well, making it unsuitable to define a semantics. We will look at two confluent notions of reduction, called Call-by-Name and Call-by-Value, and prove soundness results for those.

Prefixed tableaus and nested sequents

Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational variants of each other, roughly in the same way that Gentzen sequent calculi and tableaus are notational variants. This immediately gives rise to new modal nested sequent systems which may be of independent interest. We discuss some of these, including those for some justification logics that include standard modal operators.