5 Search Results for "Ghilardi, Silvio"

Document
DOI: 10.4230/LIPIcs.FSCD.2025.8
The Unification Type of an Equational Theory May Depend on the Instantiation Preorder

Authors: Franz Baader and Oliver Fernández Gil

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract
The unification type of an equational theory is defined using a preorder on substitutions, called the instantiation preorder, whose scope is either restricted to the variables occurring in the unification problem, or unrestricted such that all variables are considered. It has been known for more than three decades that the unification type of an equational theory may vary, depending on which instantiation preorder is used. More precisely, it was shown in 1991 that the theory ACUI of an associative, commutative, and idempotent binary function symbol with a unit is unitary w.r.t. the restricted instantiation preorder, but not unitary w.r.t. the unrestricted one. In 2016 this result was strengthened by showing that the unrestricted type of this theory also cannot be finitary. Here, we considerably improve on this result by proving that ACUI is infinitary w.r.t. the unrestricted instantiation preorder, thus precluding type zero. We also show that, w.r.t. this preorder, the unification type of ACU (where idempotency is removed from the axioms) and of AC (where additionally the unit is removed) is infinitary, though it is respectively unitary and finitary in the restricted case. In the other direction, we prove (using the example of unification in the description logic EL) that the unification type may actually improve from type zero to infinitary when switching from the restricted instantiation preorder to the unrestricted one. In addition, we establish some general results on the relationship between the two instantiation preorders.
Cite as

Franz Baader and Oliver Fernández Gil. The Unification Type of an Equational Theory May Depend on the Instantiation Preorder. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 8:1-8:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{baader_et_al:LIPIcs.FSCD.2025.8,
  author =	{Baader, Franz and Fern\'{a}ndez Gil, Oliver},
  title =	{{The Unification Type of an Equational Theory May Depend on the Instantiation Preorder}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{8:1--8:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.8},
  URN =		{urn:nbn:de:0030-drops-236230},
  doi =		{10.4230/LIPIcs.FSCD.2025.8},
  annote =	{Keywords: Unification type, Instantiation preorder, Equational theories, Modal and Description Logics}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.42
Unifying Sequent Systems for Gödel-Löb Provability Logic via Syntactic Transformations

Authors: Tim S. Lyon

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
We demonstrate the inter-translatability of proofs between the most prominent sequent-based formalisms for Gödel-Löb provability logic. In particular, we consider Sambin and Valentini’s sequent system GL_{seq}, Shamkanov’s non-wellfounded and cyclic sequent systems GL_∞ and GL_{circ}, Poggiolesi’s tree-hypersequent system CSGL, and Negri’s labeled sequent system G3GL. Shamkanov provided proof-theoretic correspondences between GL_{seq}, GL_∞, and GL_{circ}, and Goré and Ramanayake showed how to transform proofs between CSGL and G3GL, 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 Gödel-Löb 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 GL_{seq} and G3GL proofs. These new syntactic transformations, together with those mentioned above, establish full proof-theoretic correspondences between GL_{seq}, GL_∞, GL_{circ}, CSGL, G3GL, 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.
Cite as

Tim S. Lyon. Unifying Sequent Systems for Gödel-Löb Provability Logic via Syntactic Transformations. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 42:1-42:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lyon:LIPIcs.CSL.2025.42,
  author =	{Lyon, Tim S.},
  title =	{{Unifying Sequent Systems for G\"{o}del-L\"{o}b Provability Logic via Syntactic Transformations}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{42:1--42:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.42},
  URN =		{urn:nbn:de:0030-drops-227992},
  doi =		{10.4230/LIPIcs.CSL.2025.42},
  annote =	{Keywords: Cyclic proof, G\"{o}del-L\"{o}b logic, Labeled sequent, Linear nested sequent, Modal logic, Non-wellfounded proof, Proof theory, Proof transformation, Tree-hypersequent}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.41
Taking Bi-Intuitionistic Logic First-Order: A Proof-Theoretic Investigation via Polytree Sequents

Authors: Tim S. Lyon, Ian Shillito, and Alwen Tiu

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
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.
Cite as

Tim S. Lyon, Ian Shillito, and Alwen Tiu. Taking Bi-Intuitionistic Logic First-Order: A Proof-Theoretic Investigation via Polytree Sequents. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 41:1-41:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lyon_et_al:LIPIcs.CSL.2025.41,
  author =	{Lyon, Tim S. and Shillito, Ian and Tiu, Alwen},
  title =	{{Taking Bi-Intuitionistic Logic First-Order: A Proof-Theoretic Investigation via Polytree Sequents}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{41:1--41:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.41},
  URN =		{urn:nbn:de:0030-drops-227987},
  doi =		{10.4230/LIPIcs.CSL.2025.41},
  annote =	{Keywords: Bi-intuitionistic, Cut-elimination, Conservativity, Domain, First-order, Polytree, Proof theory, Reachability, Sequent}
}
Document
DOI: 10.4230/LIPIcs.RTA.2011.171
Rewriting-based Quantifier-free Interpolation for a Theory of Arrays

Authors: Roberto Bruttomesso, Silvio Ghilardi, and Silvio Ranise

Published in: LIPIcs, Volume 10, 22nd International Conference on Rewriting Techniques and Applications (RTA'11) (2011)

Abstract
The use of interpolants in model checking is becoming an enabling technology to allow fast and robust verification of hardware and software. The application of encodings based on the theory of arrays, however, is limited by the impossibility of deriving quantifier-free interpolants in general. In this paper, we show that, with a minor extension to the theory of arrays, it is possible to obtain quantifier-free interpolants. We prove this by designing an interpolating procedure, based on solving equations between array updates. Rewriting techniques are used in the key steps of the solver and its proof of correctness. To the best of our knowledge, this is the first successful attempt of computing quantifier-free interpolants for a theory of arrays.
Cite as

Roberto Bruttomesso, Silvio Ghilardi, and Silvio Ranise. Rewriting-based Quantifier-free Interpolation for a Theory of Arrays. In 22nd International Conference on Rewriting Techniques and Applications (RTA'11). Leibniz International Proceedings in Informatics (LIPIcs), Volume 10, pp. 171-186, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{bruttomesso_et_al:LIPIcs.RTA.2011.171,
  author =	{Bruttomesso, Roberto and Ghilardi, Silvio and Ranise, Silvio},
  title =	{{Rewriting-based Quantifier-free Interpolation for a Theory of Arrays}},
  booktitle =	{22nd International Conference on Rewriting Techniques and Applications (RTA'11)},
  pages =	{171--186},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-30-9},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{10},
  editor =	{Schmidt-Schauss, Manfred},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2011.171},
  URN =		{urn:nbn:de:0030-drops-31155},
  doi =		{10.4230/LIPIcs.RTA.2011.171},
  annote =	{Keywords: rewriting, interpolation, arrays, model-checking}
}
Document
DOI: 10.4230/DagSemProc.07401.4
From Non-Disjoint Combination to Satisfiability and Model-Checking of Infinite State Systems

Authors: Silvio Ghilardi, Silvio Ranise, Enrica Nicolini, and Daniele Zucchelli

Published in: Dagstuhl Seminar Proceedings, Volume 7401, Deduction and Decision Procedures (2007)

Abstract
In the first part of our contribution, we review recent results on combined constraint satisfiability for first order theories in the non-disjoint signatures case: this is done mainly in view of the applications to temporal satisfiability and model-checking covered by the second part of our talk, but we also illustrate in more detail some case-study where non-disjoint combination arises. The first case deals with extensions of the theory of arrays where indexes are endowed with a Presburger arithmetic structure and a length expressing `dimension' is added; the second case deals with the algebraic counterparts of fusion in modal logics. We then recall the basic features of the Nelson-Oppen method and investigate sufficient conditions for it to be complete and terminating in the non-disjoint signatures case: for completeness we rely on a model-theoretic $T_0$-compatibility condition (generalizing stable infiniteness) and for termination we impose a noetherianity requirement on positive constraints chains. We finally supply examples of theories matching these combinability hypotheses. In the second part of our contribution, we develop a framework for integrating first-order logic (FOL) and discrete Linear time Temporal Logic (LTL). Manna and Pnueli have extensively shown how a mixture of FOL and LTL is sufficient to precisely state verification problems for the class of reactive systems: theories in FOL model the (possibly infinite) data structures used by a reactive system while LTL specifies its (dynamic) behavior. Our framework for the integration is the following: we fix a theory $T$ in a first-order signature $Sigma$ and consider as a temporal model a sequence $cM_1, cM_2, dots$ of standard (first-order) models of $T$ and assume such models to share the same carrier (or, equivalently, the domain of the temporal model to be `constant'). Following Plaisted, we consider symbols from a subsignature $Sigma_r$ of $Sigma$ to be emph{rigid}, i.e. in a temporal model $cM_1, cM_2, dots$, the $Sigma_r$-restrictions of the $cM_i$'s must coincide. The symbols in $Sigmasetminus Sigma_r$ are called `flexible' and their interpretation is allowed to change over time (free variables are similarly divided into `rigid' and `flexible'). For model-checking, the emph{initial states} and the emph{transition relation} are represented by first-order formulae, whose role is that of (non-deterministically) restricting the temporal evolution of the model. In the quantifier-free case, we obtain sufficient conditions for %undecidability and decidability for both satisfiability and model-checking of safety properties emph{by lifting combination methods} for emph{non-disjoint} theories in FOL: noetherianity and $T_0$-compatibility (where $T_0$ is the theory axiomatizing the rigid subtheory) gives decidability of satisfiability, whereas $T_0$-compatibility and local finiteness give safety model-checking decidability. The proofs of these decidability results suggest how decision procedures for the constraint satisfiability problem of theories in FOL and algorithms for checking the satisfiability of propositional LTL formulae can be integrated. This paves the way to employ efficient Satisfiability Modulo Theories solvers in the model-checking of infinite state systems. We illustrate our techniques on some examples and discuss further work in the area.
Cite as

Silvio Ghilardi, Silvio Ranise, Enrica Nicolini, and Daniele Zucchelli. From Non-Disjoint Combination to Satisfiability and Model-Checking of Infinite State Systems. In Deduction and Decision Procedures. Dagstuhl Seminar Proceedings, Volume 7401, pp. 1-2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{ghilardi_et_al:DagSemProc.07401.4,
  author =	{Ghilardi, Silvio and Ranise, Silvio and Nicolini, Enrica and Zucchelli, Daniele},
  title =	{{From Non-Disjoint Combination to Satisfiability and Model-Checking of Infinite State Systems}},
  booktitle =	{Deduction and Decision Procedures},
  pages =	{1--2},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7401},
  editor =	{Franz Baader and Byron Cook and J\"{u}rgen Giesl and Robert Nieuwenhuis},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07401.4},
  URN =		{urn:nbn:de:0030-drops-12479},
  doi =		{10.4230/DagSemProc.07401.4},
  annote =	{Keywords: Non disjoint combination, linear temporal logic, model checking}
}
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