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Tadeusz Litak

FAU Erlangen-Nuernberg, Computer Science, Post-Doc

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Chapman University

Tomasz Kowalski

Jagiellonian University

Francesco Paoli

University of Cagliari

Shunsuke Yatabe

National Institute of Advanced Industrial Science and Technology

Constantine Tsinakis

Ricardo Oscar Rodriguez

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Papers by Tadeusz Litak

An Algebraic Glimpse at Bunched Implications and Separation Logic

We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective i... more We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective inspired by Hiroakira Ono's algebraic approach to substructural logics. We propose generalized BI algebras (GBI-algebras) as a common framework for algebras arising via " declarative resource reading " , intuitionistic generalizations of relation algebras and arrow logics and the distributive Lambek calculus with intuitionistic implication. Apart from existing models of BI (in particular, heap models and effect algebras), we also cover models arising from weakening relations, formal languages or more fine-grained treatment of labelled trees and semistructured data. After briefly discussing the lattice of subvarieties of GBI, we present a suitable duality for GBI along the lines of Esakia and Priestley and an algebraic proof of cut elimination in the setting of residuated frames of Galatos and Jipsen. We also show how the algebraic approach allows generic results on decidability, both positive and negative ones. In the final part of the paper, we gently introduce the substructural audience to some theory behind state-of-art tools, culminating with an algebraic and proof-theoretic presentation of (bi-) abduction.

Constructive Modalities with Provability Smack

Outstanding Contributions to Logic, 2014

Completions of GBL-algebras and acyclic modal algebras: negative results (数学基礎論とその応用--RIMS研究集会報告集)

数理解析研究所講究録, Nov 1, 2006

Guard Your Daggers and Traces: On The Equational Properties of Guarded (Co-)recursion

Electronic Proceedings in Theoretical Computer Science, 2013

Motivated by the recent interest in models of guarded (co-)recursion we study its equational prop... more Motivated by the recent interest in models of guarded (co-)recursion we study its equational properties. We formulate axioms for guarded fixpoint operators generalizing the axioms of iteration theories of Bloom andÉsik. Models of these axioms include both standard (e.g., cpo-based) models of iteration theories and models of guarded recursion such as complete metric spaces or the topos of trees studied by Birkedal et al. We show that the standard result on the satisfaction of all Conway axioms by a unique dagger operation generalizes to the guarded setting. We also introduce the notion of guarded trace operator on a category, and we prove that guarded trace and guarded fixpoint operators are in one-to-one correspondence. Our results are intended as first steps leading to the description of classifying theories for guarded recursion and hence completeness results involving our axioms of guarded fixpoint operators in future work.

Acyclic modal logic: an exercise in irreflexivization

Complete axiomatizations for XPath fragments

Journal of Applied Logic, 2010

We provide complete axiomatizations for several fragments of XPath: sets of equivalences from whi... more We provide complete axiomatizations for several fragments of XPath: sets of equivalences from which every other valid equivalence is derivable. Specically, we axiomatize downward single axis fragments of Core XPath (that is, Core XPath(↓) and Core XPath(↓+)) as well as the full Core XPath. We make use of techniques from modal logic.

An Algebraic Approach to Incompleteness in Modal Logic

$Research paper thumbnail of Some Notes on the Superintuitionistic Logic of Chequered Subsets of $R^\infty$$

Some Notes on the Superintuitionistic Logic of Chequered Subsets of $R^\infty$

The Non-reflexive Counterpart of Grz

Bulletin of the Section of Logic, 2007

The importance of being discrete

The paper discusses discrete frames as an attractive semantics for modal logic. We study question... more The paper discusses discrete frames as an attractive semantics for modal logic. We study questions of completeness, persistence, duality and definability. Notions of completeness, strong global completeness and complexity of dual varieties coincide for discrete frames; moreover, they are equivalent to conservativity of minimal hybrid extensions. The paper also provides some criteria of di-persistence and a Goldblatt-Thomason theorem for discrete frames.

A neighbourhood complete K4 logic not complete with respect to any class of subdirectly irreducible complete modal algebras

submitted to the Tbilisi Conference 2003 I have answered an old question posed by Dziobak [Dz78]

A Complete Axiomatization for Core XPath 1.0

This paper provides a complete algebraic axiomatization of node and path equivalences in Core XPa... more

COMPLETIONS OF GBL-ALGEBRAS AND ACYCLIC MODAL ALGEBRAS : NEGATIVE RESULTS(Mathematical Logic and its Applications)

On Notions of Completeness Weaker than Kripke Completeness

We are going to show that the standard notion of Kripke completeness is the strongest one among m... more We are going to show that the standard notion of Kripke completeness is the strongest one among many provably distinct algebraically motivated completeness properties, some of which seem to be of intrinsic interest. More specically, we are going to investigate notions of completeness with respect to algebras which are either atomic, complete, completely additive or admit residuals (the last notion of completeness coincides with conservativity of minimal tense extensions); we will be also interested in combinations of these properties. 1 Motivation It is known that Kripke frames correspond to complete, atomic and completely additive Boolean algebras with operators (baos). This fact became the basis of duality theory for Kripke frames, developed in the 1970's by Thomason (13), Goldblatt (4) and others. In this paper, we are going to investigate notions of completeness and con- sequence weaker than those associated with standard Kripke frames from an algebraic perspective. Our star...

A Continuum of Incomplete Intermediate Logics

This paper generalizes the 1977 paper of V.B. Shehtman, which constructed the first Kripke incomp... more

様相論理における不完全性への代数的アプロ－チ

Supervisor:小野寛晰情報科学研究科博士 An algebraic approach to incompleteness in modal logic Tadeusz Litak

A Van Benthem/Rosen theorem for coalgebraic predicate logic

Journal of Logic and Computation, 2015

Stone Duality for Nominal Boolean Algebras with И

Lecture Notes in Computer Science, 2011

ABSTRACT We define Boolean algebras over nominal sets with a function-symbol N mirroring the N &a... more

Coalgebraic Predicate Logic: Equipollence Results and Proof Theory

Lecture Notes in Computer Science, 2013

Algebraization of Hybrid Logic with Binders

Lecture Notes in Computer Science, 2006

... Algebraic operators formalizing substitutions in first-order logic have been studied since Ha... more

An Algebraic Glimpse at Bunched Implications and Separation Logic

We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective i... more We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective inspired by Hiroakira Ono's algebraic approach to substructural logics. We propose generalized BI algebras (GBI-algebras) as a common framework for algebras arising via " declarative resource reading " , intuitionistic generalizations of relation algebras and arrow logics and the distributive Lambek calculus with intuitionistic implication. Apart from existing models of BI (in particular, heap models and effect algebras), we also cover models arising from weakening relations, formal languages or more fine-grained treatment of labelled trees and semistructured data. After briefly discussing the lattice of subvarieties of GBI, we present a suitable duality for GBI along the lines of Esakia and Priestley and an algebraic proof of cut elimination in the setting of residuated frames of Galatos and Jipsen. We also show how the algebraic approach allows generic results on decidability, both positive and negative ones. In the final part of the paper, we gently introduce the substructural audience to some theory behind state-of-art tools, culminating with an algebraic and proof-theoretic presentation of (bi-) abduction.

Constructive Modalities with Provability Smack

Outstanding Contributions to Logic, 2014

Completions of GBL-algebras and acyclic modal algebras: negative results (数学基礎論とその応用--RIMS研究集会報告集)

数理解析研究所講究録, Nov 1, 2006

Guard Your Daggers and Traces: On The Equational Properties of Guarded (Co-)recursion

Electronic Proceedings in Theoretical Computer Science, 2013

Motivated by the recent interest in models of guarded (co-)recursion we study its equational prop... more Motivated by the recent interest in models of guarded (co-)recursion we study its equational properties. We formulate axioms for guarded fixpoint operators generalizing the axioms of iteration theories of Bloom andÉsik. Models of these axioms include both standard (e.g., cpo-based) models of iteration theories and models of guarded recursion such as complete metric spaces or the topos of trees studied by Birkedal et al. We show that the standard result on the satisfaction of all Conway axioms by a unique dagger operation generalizes to the guarded setting. We also introduce the notion of guarded trace operator on a category, and we prove that guarded trace and guarded fixpoint operators are in one-to-one correspondence. Our results are intended as first steps leading to the description of classifying theories for guarded recursion and hence completeness results involving our axioms of guarded fixpoint operators in future work.

Acyclic modal logic: an exercise in irreflexivization

Complete axiomatizations for XPath fragments

Journal of Applied Logic, 2010

We provide complete axiomatizations for several fragments of XPath: sets of equivalences from whi... more We provide complete axiomatizations for several fragments of XPath: sets of equivalences from which every other valid equivalence is derivable. Specically, we axiomatize downward single axis fragments of Core XPath (that is, Core XPath(↓) and Core XPath(↓+)) as well as the full Core XPath. We make use of techniques from modal logic.

An Algebraic Approach to Incompleteness in Modal Logic

$Research paper thumbnail of Some Notes on the Superintuitionistic Logic of Chequered Subsets of $R^\infty$$

Some Notes on the Superintuitionistic Logic of Chequered Subsets of $R^\infty$

The Non-reflexive Counterpart of Grz

Bulletin of the Section of Logic, 2007

The importance of being discrete

The paper discusses discrete frames as an attractive semantics for modal logic. We study question... more The paper discusses discrete frames as an attractive semantics for modal logic. We study questions of completeness, persistence, duality and definability. Notions of completeness, strong global completeness and complexity of dual varieties coincide for discrete frames; moreover, they are equivalent to conservativity of minimal hybrid extensions. The paper also provides some criteria of di-persistence and a Goldblatt-Thomason theorem for discrete frames.

A neighbourhood complete K4 logic not complete with respect to any class of subdirectly irreducible complete modal algebras

submitted to the Tbilisi Conference 2003 I have answered an old question posed by Dziobak [Dz78]

A Complete Axiomatization for Core XPath 1.0

This paper provides a complete algebraic axiomatization of node and path equivalences in Core XPa... more

COMPLETIONS OF GBL-ALGEBRAS AND ACYCLIC MODAL ALGEBRAS : NEGATIVE RESULTS(Mathematical Logic and its Applications)

On Notions of Completeness Weaker than Kripke Completeness

We are going to show that the standard notion of Kripke completeness is the strongest one among m... more We are going to show that the standard notion of Kripke completeness is the strongest one among many provably distinct algebraically motivated completeness properties, some of which seem to be of intrinsic interest. More specically, we are going to investigate notions of completeness with respect to algebras which are either atomic, complete, completely additive or admit residuals (the last notion of completeness coincides with conservativity of minimal tense extensions); we will be also interested in combinations of these properties. 1 Motivation It is known that Kripke frames correspond to complete, atomic and completely additive Boolean algebras with operators (baos). This fact became the basis of duality theory for Kripke frames, developed in the 1970's by Thomason (13), Goldblatt (4) and others. In this paper, we are going to investigate notions of completeness and con- sequence weaker than those associated with standard Kripke frames from an algebraic perspective. Our star...

A Continuum of Incomplete Intermediate Logics

This paper generalizes the 1977 paper of V.B. Shehtman, which constructed the first Kripke incomp... more

様相論理における不完全性への代数的アプロ－チ

Supervisor:小野寛晰情報科学研究科博士 An algebraic approach to incompleteness in modal logic Tadeusz Litak

A Van Benthem/Rosen theorem for coalgebraic predicate logic

Journal of Logic and Computation, 2015

Stone Duality for Nominal Boolean Algebras with И

Lecture Notes in Computer Science, 2011

ABSTRACT We define Boolean algebras over nominal sets with a function-symbol N mirroring the N &a... more

Coalgebraic Predicate Logic: Equipollence Results and Proof Theory

Lecture Notes in Computer Science, 2013

Algebraization of Hybrid Logic with Binders

Lecture Notes in Computer Science, 2006

... Algebraic operators formalizing substitutions in first-order logic have been studied since Ha... more

Negative Translations and Normal Modality

One Modal Logic to Rule Them All?

by Tadeusz Litak and Wesley Holliday

An Algebraic Glimpse at Bunched Implications and Separation Logic

We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective i... more We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective inspired by Hiroakira Ono's algebraic approach to substructural logics. We propose generalized BI algebras (GBI-algebras) as a common framework for algebras arising via "declarative resource reading", intuitionistic generalizations of relation algebras and arrow logics and the distributive Lambek calculus with intuitionistic implication. Apart from existing models of BI (in particular, heap models and effect algebras), we also cover models arising from weakening relations, formal languages or more fine-grained treatment of labelled trees and semistructured data. After briefly discussing the lattice of subvarieties of GBI, we present a suitable duality for GBI along the lines of Esakia and Priestley and an algebraic proof of cut elimination in the setting of residuated frames of Galatos and Jipsen. We also show how the algebraic approach allows generic results on decidability, both positive and negative ones. In the final part of the paper, we gently introduce the substructural audience to some theory behind state-of-art tools, culminating with an algebraic and proof-theoretic presentation of (bi-)abduction.

Guard Your Daggers and Traces: {P}roperties of Guarded (Co-)recursion

Motivated by the recent interest in models of guarded (co-)recursion, we study their equational p... more Motivated by the recent interest in models of guarded (co-)recursion, we study their equational properties. We formulate axioms for guarded fixpoint operators generalizing the axioms of iteration theories of Bloom and \'Esik. Models of these axioms include both standard (e.g., cpo-based) models of iteration theories and models of guarded recursion such as complete metric spaces or the topos of trees studied by Birkedal et al. We show that the standard result on the satisfaction of all Conway axioms by a unique dagger operation generalizes to the guarded setting. We also introduce the notion of guarded trace operator on a category, and we prove that guarded trace and guarded fixpoint operators are in one-to-one correspondence. Our results are intended as first steps leading, hopefully, towards future description of classifying theories for guarded recursion.

Lewis meets {B}rouwer: {C}onstructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical p... more C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.

Model Theory and Proof Theory of Coalgebraic Predicate Logic

by Tadeusz Litak, Lutz Schröder, and Katsuhiko Sano

We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a... more We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and completeness results for several natural classes of such logics. Moreover, we show that an entirely general completeness result is not possible. We study the expressive power of our language, both in comparison with coalgebraic hybrid logics and with existing first-order proposals for special classes of Set-coalgebras (apart from relational structures, also neighbourhood frames and topological spaces). Basic model-theoretic constructions and results, in particular ultraproducts, obtain for the two classes that allow completeness---and in some cases beyond that. Finally, we discuss a basic sequent system, for which we establish a syntactic cut-elimination result.

Infinite Populations, Choice and Determinacy

Studia Logica

This paper criticizes non-constructive uses of set theory in formal economics. The main focus is ... more This paper criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow's theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the Hildenbrand criterion and argue that results based on unrestricted Axiom of Choice (AC) do not meet this criterion. The technically novel part of this paper is a proposal to use a set-theoretic principle known as the Axiom of Determinacy (AD), not as a replacement for Choice, but simply to eliminate applications of set theory violating the Hildenbrand criterion. A particularly appealing aspect of AD from the point of view of the research area in question is its game-theoretic character.

Lewis arrow fell off the wall: {D}ecompositions of constructive strict implication