Sequent Calculus Research Articles

Two complete finitary sequent calculi for reflexive common knowledge

This paper discusses the use of complete sequent calculi for reflexive common knowledge logic. Description of language and complete infinitary calculus for RCL is presented. Then finitary calculi RCLI and RCLL are introduced and completeness of finitary calculi RCLI and RCLL is proven.

Sequent systems for PLTL

We consider three sequent calculi for propositional linear temporal logic (PLTL) which allow us to formalize the properties of operator “always”. The main new results presented in the paper are: (1) introduction of the calculus withlooping axioms; (2) the direct proof that the presented calculi are equivalent; (3) the proof of completeness of the calculi with looping axioms and with invariant-like rule based on completeness of the calculus with the infinitary $\omega$-type rule.

Prehistoric Graph in Modal Derivations and Self-Referentiality

By terms-allowed-in-formulas capacity, Artemov's Logic of Proofs LP Artemov includes self-referential formulas of the form t:?(t) that play a crucial role in the realization of modal logic S4 in LP, and in the Brouwer---Heyting---Kolmogorov semantics of intuitionistic logic via LP. In an earlier work appeared in the Proceedings of CSR 2010 the author defined prehistoric loop in a sequent calculus of S4, and verified its necessity to self-referentiality in S4?LP realization. In this extended version we generalize results there to T and K4, two modal logics smaller than S4 that yet call for self-referentiality in their realizations into corresponding justification logics JT and J4.

Algebra, proof theory and applications for an intuitionistic logic of propositions, actions and adjoint modal operators

We develop a cut-free nested sequent calculus as basis for a proof search procedure for an intuitionistic modal logic of actions and propositions. The actions act on propositions via a dynamic modality (the weakest precondition of program logics), whose left adjoint we refer to as “update” (the strongest postcondition ). The logic has agent-indexed adjoint pairs of epistemic modalities: the left adjoints encode agents' uncertainties and the right adjoints encode their beliefs. The rules for the “update” modality encode learning as a result of discarding uncertainty. We prove admissibility of Cut , and hence the soundness and completeness of the logic with respect to an algebraic semantics. We interpret the logic on epistemic scenarios that consist of honest and dishonest communication actions, add assumption rules to encode them, and prove that the calculus with the assumption rules still has the admissibility results. We apply the calculus to encode (and allow reasoning about) the classic epistemic puzzles of dirty children (a.k.a. “muddy children”) and drinking logicians and some versions with dishonesty or noise; we also give an application where the actions are movements of a robot rather than announcements.

Judgmental subtyping systems with intersection types and modal types

We study how to extend modal type systems based on intuitionistic modal logic S4 or S5 with a subtyping system based on intersection types. In the presence of four type constructors $${\!}\!\rightarrow \!{\!},\,{\!}\wedge {\!},\,\square {}$$ ? , ? , ? , and $$\Diamond {}$$ ? , the traditional approach using a binary subtyping relation does not work well because of lack of orthogonality in subtyping rules and presence of a transitivity rule. We adopt the idea from the judgmental formulation of modal logic (Pfenning and Davies in Math Struct Comput Sci 11(4):511---540, 2001) and use subtyping judgments whose definitions express those notions internalized into type constructors directly at the level of judgments. The resultant judgmental subtyping systems admit cut rules similarly to a sequent calculus for intuitionistic logic and play a key role in designing and verifying the relational subtyping systems based on the binary subtyping relation. We use the proof assistant Coq to prove the admissibility of the cut rules and the equivalence between the two kinds of subtyping systems. The lesson from our study is that by using subtyping judgments instead of the binary subtyping relation, we can overcome the limitation usually associated with the syntactic approach to formulating subtyping systems.

On the proof-theory of a first-order extension of GL

We introduce a first order extension of GL, called ML3, and develop its proof theory via a proxy cut-free sequent calculus GLTS. We prove the highly nontrivial result that cut is a derived rule in GLTS, a result that is unavailable in other known first-order extensions of GL. This leads to proofs of weak reflection and the related conservation result for ML3, as well as proofs for Craig’s interpolation theorem for GLTS. Turning to semantics we prove that ML3 is sound with respect to arithmetical interpretations and that it is also sound and complete with respect to converse well-founded and transitive finite Kripke models. This leads us to expect that a Solovay-like proof of arithmetical completeness of ML3 is possible.

Decision procedures for some strong hybrid logics

Hybrid logics are extensions of standard modal logics, which significantly increase the expressive power of the latter. Since most of hybrid logics are known to be decidable, decision procedures for them is a widely investigated field of research. So far, several tableau calculi for hybrid logics have been presented in the literature. In this paper we introduce a sound, complete and terminating tableau calculus T H(@,E,D, ♦ −) for hybrid logics with the satisfaction operators, the universal modality, the difference modality and the inverse modality as well as the corresponding sequent calculus S H(@,E,D, ♦ −) . They not only uniformly cover relatively wide range of various hybrid logics but they are also conceptually simple and enable effective search for a minimal model for a satisfiable formula. The main novelty is the exploitation of the unrestricted blocking mechanism introduced as an explicit, sound tableau rule.

Labeled sequent calculi for modal logics and implicit contractions

The paper settles an open question concerning Negri-style labeled sequent calculi for modal logics and also, indirectly, other proof systems which make (more or less) explicit use of semantic parameters in the syntax and are thus subsumed by labeled calculi, like Brunnler's deep sequent calculi, Poggiolesi's tree-hypersequent calculi and Fitting's prefixed tableau systems. Specifically, the main result we prove (through a semantic argument) is that labeled calculi for the modal logics K and D remain complete w.r.t. valid sequents whose relational part encodes a tree-like structure, when the unique rule which contains an harmful implicit contraction--by which the condition that the premises be less complex than the conclusion is violated--is modified into a contraction-free one respecting the latter condition, thus making the proof-search space finite.

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.

Logic in pragmatics

Relevance Theory assumes that the deductive device for spontaneous inferences never uses the introduction rules for logical connectives, such as & introduction. Two motivations for this are (I) the overgeneration of propositions expressed and (II) non-terminating inferences, both of which are claimed to be caused by logical introduction rules. However, having both logical introduction rules and elimination rules is essential for the completeness of the inference system. Although a sound and complete inference system is not as strongly preferred in the theory of human reasoning as it is in proof theoretic logic, missing one of the two essential rules for the logical connectives leaves too many holes in the data coverage. Naturally, Relevance Theory has additional rules for covering some of these holes. However, without introduction rules, these additional rules are ⿿non-logical⿿ and therefore create other holes which would then require even more non-logical rules with no principled limit. Also, banning introduction rules to solve particular data problems without independent motivations is stipulative. Based on such considerations, this paper discusses a theory that deals with problems (I) and (II) above without banning introduction rules. After explaining the problems with the ban on introduction rules, I provide a pragmatic solution for the overgeneration problem. I then show that non-terminating inferences arise independently of introduction rules and therefore, removing introduction rules does not provide the right solution. Finally, I substantiate my arguments in the earlier sections by briefly showing that a sound and complete sequent calculus can be used as the deductive system with its performance pragmatically controlled in the way that has been discussed in the previous sections.

BOLZANO’S CONCEPT OF GROUNDING (ABFOLGE) AGAINST THE BACKGROUND OF NORMAL PROOFS

AbstractIn this paper, I provide a thorough discussion and reconstruction of Bernard Bolzano’s theory of grounding and a detailed investigation into the parallels between his concept of grounding and current notions of normal proofs. Grounding (Abfolge) is an objective ground-consequence relation among true propositions that is explanatory in nature. The grounding relation plays a crucial role in Bolzano’s proof-theory, and it is essential for his views on the ideal buildup of scientific theories. Occasionally, similarities have been pointed out between Bolzano’s ideas on grounding and cut-free proofs in Gentzen’s sequent calculus. My thesis is, however, that they bear an even stronger resemblance to the normal natural deduction proofs employed in proof-theoretic semantics in the tradition of Dummett and Prawitz.

Representing any-time and program-iteration by infinitary conjunction

Two new infinitary modal logics are simply obtained from a Gentzen-type sequent calculus for infinitary logic by adding a next-time operator, and a program operator, respectively. It is shown that an any-time operator and a program-iteration operator can respectively be expressed using infinitary conjunction in these logics. The cut-elimination and completeness theorems for these logics are proved using some theorems for embedding these logics into (classical) infinitary logic.

On Gentzen Relations Associated with Finite-valued Logics Preserving Degrees of Truth

When considering m-sequents, it is always possible to obtain an m-sequent calculus VL for every m-valued logic (defined from an arbitrary finite algebra L of cardinality m) following for instance the works of the Vienna Group for Multiple-valued Logics. The Gentzen relations associated with the calculi VL are always finitely equivalential but might not be algebraizable. In this paper we associate an algebraizable 2-Gentzen relation with every sequent calculus VL in a uniform way, provided the original algebra L has a reduct that is a distributive lattice or a pseudocomplemented distributive lattice. We also show that the sentential logic naturally associated with the provable sequents of this algebraizable Gentzen relation is the logic that preserves degrees of truth with respect to the original algebra (in contrast with the more common logic that merely preserves truth). Finally, for some particular logics we obtain 2-sequent calculi that axiomatize the algebraizable Gentzen relations obtained so far.

Admissibility of cut in congruent modal logics

We present a detailed proof of the admissibility of cut in sequent calculus for some congruent modal logics. The result was announced much earlier during the Trends in Logic Conference, Torun 2006 and the proof for monotonic modal logics was provided already in Indrzejczak [5]. Also some tableau and natural deduction formalizations presented in Indrzejczak [6] and Indrzejczak [7] were based on this result but the proof itself was not published so far. In this paper we are going to fill this gap. The delay was partly due to the fact that the author from time to time was trying to improve the result and extend it to some additional logics by testing other methods of proving cut elimination. Unfortunately all these attempts failed and cut elimination holds only for these logics which were proved to satisfy this property already in 2005.

A sequent calculus for a logic of contingencies

We introduce a sequent calculus that is sound and complete with respect to propositional contingencies, i.e., formulas which are neither provable nor refutable. Like many other sequent and natural deduction proof systems, this calculus possesses cut elimination and the subformula property and has a simple proof search mechanism.

A proof-search procedure for intuitionistic propositional logic

A sequent root-first proof-search procedure for intuitionistic propositional logic is presented. The procedure is obtained from modified intuitionistic multi-succedent and classical sequent calculi, making use of Glivenko's Theorem. We prove that a sequent is derivable in a standard intuitionistic multi-succedent calculus if and only if the corresponding prefixed-sequent is derivable in the procedure.

Temporal BI: Proof system, semantics and translations

The logic BI of bunched implications is a combination of intuitionistic logic and multiplicative intuitionistic linear logic. In this paper, a temporal extension tBI of BI is introduced and studied. A Gentzen-type sequent calculus GtBI for tBI is introduced, and the cut-elimination and decidability theorems for GtBI are proved using a theorem for syntactically embedding GtBI into a sequent calculus GBI for BI. A semantics for GtBI is introduced extending the Grothendieck topological semantics for BI, and the completeness theorem with respect to this semantics is proved using a theorem for semantically embedding tBI into BI. A semantics for GtBI without additive falsity constant is introduced extending the Kripke resource semantics for BI without additive falsity constant, and the completeness theorem with respect to this semantics is proved in a similar way. Moreover, an intuitionistic temporal linear logic, ITLL, is introduced as a Gentzen-type sequent calculus, and a theorem for embedding GtBI into ITLL is proved using a temporal extension of the Girard translation of intuitionistic logic into intuitionistic linear logic.

On the Dynamic Logic of Agency and Action

We present a Hilbert style axiomatization and an equational theory for reasoning about actions and capabilities. We introduce two novel features in the language of propositional dynamic logic, converse as backwards modality and abstract processes specified by preconditions and effects, written as $${\varphi \Rightarrow \psi}$$ and first explored in our recent paper (Hartonas, Log J IGPL Oxf Univ Press, 2012), where a Gentzen-style sequent calculus was introduced. The system has two very natural interpretations, one based on the familiar relational semantics and the other based on type semantics, where action terms are interpreted as types of actions (sets of binary relations). We show that the proof systems do not distinguish between the two kinds of semantics, by completeness arguments. Converse as backwards modality together with action types allow us to produce a new purely equational axiomatization of Dynamic Algebras, where iteration is axiomatized independently of box and where the fixpoint and Segerberg induction axioms are derivable. The system also includes capabilities operators and our results provide then a finitary Hilbert-style axiomatization and a decidable system for reasoning about agent capabilities, missing in the KARO framework.

Lifting lower bounds for tree-like proofs

It is known that constant-depth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider tree-like sequent calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to be of constant depth. Under a plausible hardness assumption concerning small-depth Boolean circuits, we prove exponential lower bounds for such proofs. We prove these lower bounds directly from the computational hardness assumption. We start with a lower bound for cut-free proofs and "lift" it so it applies to proofs with constant-depth cuts. By using the same approach, we obtain the following additional results. We provide a much simpler proof of a known unconditional lower bound in the case where modular connectives are not used. We establish a conditional exponential separation between the power of constant-depth proofs that use different modular connectives. We show that these tree-like proofs with constant-depth cuts cannot polynomially simulate similar dag-like proofs, even when the dag-like proofs are cut-free. We present a new proof of the non-finite axiomatizability of the theory of bounded arithmetic I Δ0(R). Finally, under a plausible hardness assumption concerning the polynomial-time hierarchy, we show that the hierarchy $${G_i^*}$$ of quantified propositional proof systems does not collapse.

Syntax and semantics of simple paracomplete logics

For an arbitrary fixed element $\beta$ in $\{1; 2; 3; ...; \omega\}$ both a sequent calculus and a natural deduction calculus which axiomatise simple paracomplete logic $I_{2;\beta}$ are built. Additionally, a valuation semantic which is adequate to logic $I_{2;\beta}$ is constructed. For an arbitrary fixed element $\gamma$ in $\{1; 2; 3;...\}$ a cortege semantic which is adequate to logic $I_{2;\gamma}$ is described. A number of results obtainable with the axiomatisations and semantics in question are formulated.