arXiv subjects

Tim S. Lyon

Publications and source records attributed to Tim S. Lyon.

At least 19 recordsLinked to original sources

arXiv2026

Non-Wellfounded and Cyclic Proofs for LTL: A Syntactic Correspondence with Linear Nested Sequents

We introduce and investigate non-wellfounded and cyclic linear nested sequent calculi, and, as a case study, develop such systems for linear temporal logic (LTL). The paper addresses two central problems, which we call 'cycle recognition' and 'unraveling.' Cycle recognition concerns identifying cycles in non-wellfounded proofs in order to extract corresponding cyclic proofs, while unraveling studies the converse transformation, from cyclic proofs to non-wellfounded ones. Although these processes are well understood for Gentzen sequents, they have received little attention for more expressive sequent formalisms and become more challenging in the linear nested sequent setting. To address cycle recognition, we show the completeness of non-wellfounded proofs relative to a particular normal form exhibiting a property we call 'saturation recurrence,' which enables the systematic extraction of cyclic proofs. To address unraveling, we introduce a specialized procedure that shifts rule applications forward along linear nested sequents, allowing non-wellfounded proofs to be reconstructed from cyclic ones. Overall, our work provides new proof-theoretic techniques for cycle recognition and unraveling in expressive multisequent formalisms.

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arXiv2026

Optimizing Proof-Search via Linearization for G\"odel-L\"ob Logic with Tree-Hypersequents

We answer a question posed by Poggiolesi concerning a syntactic decidability proof for GL in the tree-hypersequent system CSGL, and resolve a challenge identified by Maggesi and Perini Brogi, who sought a PSPACE proof-search algorithm for GL in expressive sequent-based formalisms. We work with a notational variant of CSGL formulated in terms of (labeled) tree sequents. Our answer is complexity-optimal: we present a proof-search algorithm that decides the (in)validity of formulae and runs in PSPACE, matching the known PSPACE-completeness of GL. To achieve this, we introduce a "linearization method," which constructs only a single branch of a derivation and of a tree sequent at a time, avoiding the exponential blowup typical of naive proof-search in sequent formalisms. We show how to systematically combine fragments of tree sequents generated during proof-search to extract finite counter-models, which serves as a theoretical device for establishing the correctness of the algorithm when proof-search fails. Finally, we show that every valid formula admits a proof consisting solely of line sequents, which correspond to linear nested sequents. This establishes a connection between depth-first proof-search and linear nested sequent calculi. Our results not only answer the aforementioned questions, but also provide new insights into proof-search and correctness arguments in tree sequent systems for modal logics.

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arXiv2026

Nested Sequents for Horn-Characterizable Quantified Modal Logics with Equality via Reachability Rules

We introduce cut-free nested sequent systems for a broad class of quantified modal logics (QMLs). The QMLs we consider are semantically defined using relational models that assign both an inner and outer domain to each world. This rich model structure enables the specification of various QMLs by enforcing different frame conditions, including increasing, decreasing, constant, and empty domains, as well as general path conditions and seriality. We extend the usual notion of nested sequent to include signatures, i.e., multisets of terms, which let us naturally define rules capturing the aforementioned domain conditions. A distinctive feature of our nested sequent systems is the use of reachability rules--inference rules parameterized by formal grammars (viz., semi-Thue systems). These rules operate by propagating or consuming formulae or terms along certain paths within a nested sequent, where paths are encoded as strings generated by a parameterizing grammar. This paper is the first to provide sound and complete nested systems for QMLs semantically characterized by models using both inner and outer domains. We analyze the proof-theoretic properties of these systems, identify a number of admissible structural rules, establish the invertibility of all rules, and prove a non-trivial syntactic cut-elimination theorem. We also observe that the standard universal quantifier rule used in nested systems subsumes the Extended Barcan Rule, which forces nested systems to capture QMLs with constant outer domains.

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arXiv2026

The Varieties of Ought-Implies-Can and Deontic STIT Logic

STIT logic is a prominent framework for the analysis of multi-agent choice-making. In the available deontic extensions of STIT, the principle of Ought-implies-Can (OiC) fulfills a central role. However, in the philosophical literature a variety of alternative OiC interpretations have been proposed and discussed. This paper provides a modular framework for deontic STIT that accounts for a multitude of OiC readings. In particular, we discuss, compare, and formalize ten such readings. We provide sound and complete sequent-style calculi for all of the various STIT logics accommodating these OiC principles. We formally analyze the resulting logics and discuss how the different OiC principles are logically related. In particular, we propose an endorsement principle describing which OiC readings logically commit one to other OiC readings.

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arXiv2026

Loop-Checking and Counter-Model Extraction for Intuitionistic Tense Logics via Nested Sequents

This paper develops a novel nested sequent proof-search methodology for intuitionistic tense logics (ITLs), supporting finite counter-model extraction. We introduce a new loop-checking method that detects repeating nested sequents using homomorphisms, thereby bounding the height of derivations during proof-search. Due to the non-invertibility of some inference rules, the algorithm does not construct a single derivation, but a generalized structure we call a 'computation tree.' We show how proofs and counter-models can be extracted from computation trees when proof-search succeeds or fails, respectively. This establishes the finite model property for each ITL of the form IKt + A with A a subset of {T,B,D}.

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arXiv2025

Nested Sequents for Intuitionistic Multi-Modal Logics: Modularity, Cut-Elimination, and Undecidability

We introduce and study single-conclusioned nested sequent calculi for a broad class of intuitionistic multi-modal logics known as "intuitionistic grammar logics (IGLs)." These logics serve as the intuitionistic counterparts of classical grammar logics, and subsume standard intuitionistic modal and tense logics, including IK and IKt extended with combinations of the T, B, 4, 5, and D axioms. We analyze fundamental invertibility and admissibility properties of our calculi and introduce a novel structural rule, called the "shift rule," which unifies standard structural rules arising from modal frame conditions into a single rule. This rule enables a purely syntactic proof of cut-admissibility that is uniform over all IGLs, and yields completeness of our nested calculi as a corollary. Finally, we define a negative translation that constitutes a faithful embedding of classical grammar logics (CGLs) into IGLs, witnessed by proof transformations between multi-conclusioned and single-conclusioned nested sequent proofs for CGLs and IGLs, respectively. This reduces the general validity problem for CGLs to that of IGLs. The general validity problem over a class C of logics asks: given a logic L in C and a formula A, is A valid in L? As this problem is known to be undecidable for CGLs, our reduction implies its undecidability for IGLs as well.

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arXiv2024

On Explicit Solutions to Fixed-Point Equations in Propositional Dynamic Logic

Propositional dynamic logic (PDL) is an important modal logic used to specify and reason about the behavior of software. A challenging problem in the context of PDL is solving fixed-point equations, i.e., formulae of the form $x \equiv \phi(x)$ such that $x$ is a propositional variable and $\phi(x)$ is a formula containing $x$. A solution to such an equation is a formula $\psi$ that omits $x$ and satisfies $\psi \equiv \phi(\psi)$, where $\phi(\psi)$ is obtained by replacing all occurrences of $x$ with $\psi$ in $\phi(x)$. In this paper, we identify a novel class of PDL formulae arranged in two dual hierarchies for which every fixed-point equation $x \equiv \phi(x)$ has a solution. Moreover, we not only prove the existence of solutions for all such equations, but also provide an explicit solution $\psi$ for each fixed-point equation.

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arXiv2024

Unifying Sequent Systems for G\"odel-L\"ob Provability Logic via Syntactic Transformations

We demonstrate the inter-translatability of proofs between the most prominent sequent-based formalisms for G\"odel-L\"ob provability logic. In particular, we consider Sambin and Valentini's sequent system GLseq, Shamkanov's non-wellfounded and cyclic sequent systems GL$\infty$ and GLcirc, Poggiolesi's tree-hypersequent system CSGL$^{*}$, and Negri's labeled sequent system G3KGL. Shamkanov provided proof-theoretic correspondences between GLseq, GL$\infty$, and GLcirc, and Gor\'e and Ramanayake showed how proofs could be transformed between CSGL$^{*}$ and G3KGL; however, the exact nature of proof transformations between the former three systems and the latter two systems has remained an open problem. We solve this open problem by showing how to restructure tree-hypersequent proofs into an end-active form and introduce a novel 'linearization technique' that transforms such proofs into linear nested sequent proofs. As a result, we obtain a new proof-theoretic tool for extracting linear nested sequent systems from tree-hypersequent systems, which yields the first cut-free linear nested sequent calculus LNGL for G\"odel-L\"ob provability logic. We show how to transform proofs in LNGL into a certain normal form, where proofs repeat in stages of modal and local rule applications, and which are translatable into GLseq and G3KGL proofs. These new syntactic transformations, together with those mentioned above, establish full proof-theoretic correspondences between GLseq, GL$\infty$, GLcirc, CSGL$^{*}$, G3KGL, and LNGL while also giving (to the best of the author's knowledge) the first constructive proof mappings between structural (viz. labeled, tree-hypersequent, and linear nested sequent) systems and a cyclic sequent system.

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arXiv2024

Realizing the Maximal Analytic Display Fragment of Labeled Sequent Calculi for Tense Logics

We define and study translations between the maximal class of analytic display calculi for tense logics and labeled sequent calculi, thus solving an open problem about the translatability of proofs between the two formalisms. In particular, we provide PTIME translations that map cut-free display proofs to and from special cut-free labeled proofs, which we dub 'strict' labeled proofs. This identifies the space of cut-free display proofs with a polynomially equivalent subspace of labeled proofs, showing how calculi within the two formalisms polynomially simulate one another. We analyze the relative sizes of proofs under this translation, finding that display proofs become polynomially shorter when translated to strict labeled proofs, though with a potential increase in the length of sequents; in the reverse translation, strict labeled proofs may become polynomially larger when translated into display proofs. In order to achieve our results, we formulate labeled sequent calculi in a new way that views rules as 'templates', which are instantiated with substitutions to obtain rule applications; we also provide the first definition of primitive tense structural rules within the labeled sequent formalism. Therefore, our formulation of labeled calculi more closely resembles how display calculi are defined for tense logics, which permits a more fine-grained analysis of rules, substitutions, and translations. This work establishes that every analytic display calculus for a tense logic can be viewed as a labeled sequent calculus, showing conclusively that the labeled formalism subsumes and extends the display formalism in the setting of primitive tense logics.

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arXiv2024

Decidability of Quasi-Dense Modal Logics

The decidability of axiomatic extensions of the modal logic K with modal reduction principles, i.e. axioms of the form $\Diamond^{k} p \rightarrow \Diamond^{n} p$, has remained a long-standing open problem. In this paper, we make significant progress toward solving this problem and show that decidability holds for a large subclass of these logics, namely, for 'quasi-dense logics.' Such logics are extensions of K with with modal reduction axioms such that $0 < k < n$ (dubbed 'quasi-density axioms'). To prove decidability, we define novel proof systems for quasi-dense logics consisting of disjunctive existential rules, which are first-order formulae typically used to specify ontologies in the context of database theory. We show that such proof systems can be used to generate proofs and models of modal formulae, and provide an intricate model-theoretic argument showing that such generated models can be encoded as finite objects called 'templates.' By enumerating templates of bound size, we obtain an EXPSPACE decision procedure as a consequence.

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arXiv2024

Taking Bi-Intuitionistic Logic First-Order: A Proof-Theoretic Investigation via Polytree Sequents

It is well-known that extending the Hilbert axiomatic system for first-order intuitionistic logic with an exclusion operator, that is dual to implication, collapses the domains of models into a constant domain. This makes it an interesting problem to find a sound and complete proof system for first-order bi-intuitionistic logic with non-constant domains that is also conservative over first-order intuitionistic logic. We solve this problem by presenting the first sound and complete proof system for first-order bi-intuitionistic logic with increasing domains. We formalize our proof system as a polytree sequent calculus (a notational variant of nested sequents), and prove that it enjoys cut-elimination and is conservative over first-order intuitionistic logic. A key feature of our calculus is an explicit eigenvariable context, which allows us to control precisely the scope of free variables in a polytree structure. Semantically this context can be seen as encoding a notion of Scott's existence predicate for intuitionistic logic. This turns out to be crucial to avoid the collapse of domains and to prove the completeness of our proof system. The explicit consideration of the variable context in a formula sheds light on a previously overlooked dependency between the residuation principle and the existence predicate in the first-order setting, which may help to explain the difficulty in designing a sound and complete proof system for first-order bi-intuitionistic logic.

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arXiv2024

Constructive Interpolation and Concept-Based Beth Definability for Description Logics via Sequents

We introduce a constructive method applicable to a large number of description logics (DLs) for establishing the concept-based Beth definability property (CBP) based on sequent systems. Using the highly expressive DL RIQ as a case study, we introduce novel sequent calculi for RIQ-ontologies and show how certain interpolants can be computed from sequent calculus proofs, which permit the extraction of explicit definitions of implicitly definable concepts. To the best of our knowledge, this is the first sequent-based approach to computing interpolants and definitions within the context of DLs, as well as the first proof that RIQ enjoys the CBP. Moreover, due to the modularity of our sequent systems, our results hold for restrictions of RIQ, and are applicable to other DLs by suitable modifications.

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arXiv2024

Proof Theory and Decision Procedures for Deontic STIT Logics

This paper provides a set of cut-free complete sequent-style calculi for deontic STIT ('See To It That') logics used to formally reason about choice-making, obligations, and norms in a multi-agent setting. We leverage these calculi to write a proof-search algorithm deciding deontic, multi-agent STIT logics with (un)limited choice and introduce a loop-checking mechanism to ensure the termination of the algorithm. Despite the acknowledged potential for deontic reasoning in the context of autonomous, multi-agent scenarios, this work is the first to provide a syntactic decision procedure for this class of logics. Our proof-search procedure is designed to provide verifiable witnesses/certificates of the (in)validity of formulae, which permits an analysis of the (non)theoremhood of formulae and act as explanations thereof. We show how the proof system and decision algorithm can be used to automate normative reasoning tasks such as duty checking (viz. determining an agent's obligations relative to a given knowledge base), compliance checking (viz. determining if a choice, considered by an agent as potential conduct, complies with the given knowledge base), and joint fulfillment checking (viz. determining whether under a specified factual context an agent can jointly fulfill all their duties).

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arXiv2023

Internal and External Calculi: Ordering the Jungle without Being Lost in Translations

This paper gives a broad account of the various sequent-based proof formalisms in the proof-theoretic literature. We consider formalisms for various modal and tense logics, intuitionistic logic, conditional logics, and bunched logics. After providing an overview of the logics and proof formalisms under consideration, we show how these sequent-based formalisms can be placed in a hierarchy in terms of the underlying data structure of the sequents. We then discuss how this hierarchy can be traversed using translations. Translating proofs up this hierarchy is found to be relatively straightforward while translating proofs down the hierarchy is substantially more difficult. Finally, we inspect the prevalent distinction in structural proof theory between 'internal calculi' and 'external calculi.' We discuss the ambiguities involved in the informal definitions of these categories, and we critically assess the properties that (calculi from) these classes are purported to possess.

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arXiv2023

Derivation-Graph-Based Characterizations of Decidable Existential Rule Sets

This paper establishes alternative characterizations of very expressive classes of existential rule sets with decidable query entailment. We consider the notable class of greedy bounded-treewidth sets (gbts) and a new, generalized variant, called weakly gbts (wgbts). Revisiting and building on the notion of derivation graphs, we define (weakly) cycle-free derivation graph sets ((w)cdgs) and employ elaborate proof-theoretic arguments to obtain that gbts and cdgs coincide, as do wgbts and wcdgs. These novel characterizations advance our analytic proof-theoretic understanding of existential rules and will likely be instrumental in practice.

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arXiv2023

Nested Sequents for Quantified Modal Logics

This paper studies nested sequents for quantified modal logics. In particular, it considers extensions of the propositional modal logics definable by the axioms D, T, B, 4, and 5 with varying, increasing, decreasing, and constant domains. Each calculus is proved to have good structural properties: weakening and contraction are height-preserving admissible and cut is (syntactically) admissible. Each calculus is shown to be equivalent to the corresponding axiomatic system and, thus, to be sound and complete. Finally, it is argued that the calculi are internal -- i.e., each sequent has a formula interpretation -- whenever the existence predicate is expressible in the language.

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arXiv2023

Nested Sequents for Intermediate Logics: The Case of G\"odel-Dummett Logics

We present nested sequent systems for propositional G\"odel-Dummett logic and its first-order extensions with non-constant and constant domains, built atop nested calculi for intuitionistic logics. To obtain nested systems for these G\"odel-Dummett logics, we introduce a new structural rule, called the "linearity rule," which (bottom-up) operates by linearizing branching structure in a given nested sequent. In addition, an interesting feature of our calculi is the inclusion of reachability rules, which are special logical rules that operate by propagating data and/or checking if data exists along certain paths within a nested sequent. Such rules require us to generalize our nested sequents to include signatures (i.e. finite collections of variables) in the first-order cases, thus giving rise to a generalization of the usual nested sequent formalism. Our calculi exhibit favorable properties, admitting the height-preserving invertibility of every logical rule and the (height-preserving) admissibility of a large collection of structural and reachability rules. We prove all of our systems sound and cut-free complete, and show that syntactic cut-elimination obtains for the intuitionistic systems. We conclude the paper by discussing possible extensions and modifications, putting forth an array of structural rules that could be used to provide a sizable class of intermediate logics with cut-free nested sequent systems.

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arXiv2023

Connecting Proof Theory and Knowledge Representation: Sequent Calculi and the Chase with Existential Rules

Chase algorithms are indispensable in the domain of knowledge base querying, which enable the extraction of implicit knowledge from a given database via applications of rules from a given ontology. Such algorithms have proved beneficial in identifying logical languages which admit decidable query entailment. Within the discipline of proof theory, sequent calculi have been used to write and design proof-search algorithms to identify decidable classes of logics. In this paper, we show that the chase mechanism in the context of existential rules is in essence the same as proof-search in an extension of Gentzen's sequent calculus for first-order logic. Moreover, we show that proof-search generates universal models of knowledge bases, a feature also exhibited by the chase. Thus, we formally connect a central tool for establishing decidability proof-theoretically with a central decidability tool in the context of knowledge representation.

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