Modal Logics with Composition on Finite Forests

Bednarczyk, Bartosz; Demri, Stéphane; Fervari, Raul; Mansutti, Alessio

doi:10.1145/3373718.3394787

Cited by 6 publications

(12 citation statements)

References 46 publications

(89 reference statements)

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“…In particular, our proof technique for Tower-hardness of SAT(QCTL t X ) (and therefore for QK t on finite trees) is simple enough so that it could be further reused or adapted, see e.g. a recent refinement of the proof in [BDFM20].…”

Section: Discussionmentioning

confidence: 99%

Why Does Propositional Quantification Make Modal and Temporal Logics on Trees Robustly Hard?

Bednarczyk¹,

Demri²

2022

Logical Methods in Computer Science

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Adding propositional quantification to the modal logics K, T or S4 is known to lead to undecidability but CTL with propositional quantification under the tree semantics (tQCTL) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of tQCTL as well as of the modal logic K with propositional quantification under the tree semantics. More specifically, we show that tQCTL restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When tQCTL restricted to EX is interpreted on N-bounded trees for some N >= 2, we prove that the satisfiability problem is AExpPol-complete; AExpPol-hardness is established by reduction from a recently introduced tiling problem, instrumental for studying the model-checking problem for interval temporal logics. As consequences of our proof method, we prove Tower-hardness of tQCTL restricted to EF or to EXEF and of the well-known modal logics such as K, KD, GL, K4 and S4 with propositional quantification under a semantics based on classes of trees.

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Section: Discussionmentioning

confidence: 99%

Why Does Propositional Quantification Make Modal and Temporal Logics on Trees Robustly Hard?

Bednarczyk¹,

Demri²

2022

Logical Methods in Computer Science

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Section: Discussionmentioning

confidence: 99%

Why Propositional Quantification Makes Modal Logics on Trees Robustly Hard?

Bednarczyk¹,

Demri²

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Adding propositional quantification to the modal logics K, T or S4 is known to lead to undecidability but CTL with propositional quantification under the tree semantics (QCTL t) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of QCTL t as well as of the modal logic K with propositional quantification under the tree semantics. More specifically, we show that QCTL t restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When QCTL t restricted to EX is interpreted on N-bounded trees for some N ≥ 2, we prove that the satisfiability problem is AExp polcomplete; AExp pol-hardness is established by reduction from a recently introduced tiling problem, instrumental for studying the model-checking problem for interval temporal logics. As consequences of our proof method, we prove Tower-hardness of QCTL t restricted to EF or to EXEF and of the well-known modal logics K, KD, GL, S4, K4 and D4, with propositional quantification under a semantics based on classes of trees.

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“…Moving beyond MFO(#) and MSO(#), what evidence is there for fundamental numerical representations involving polyadicity or multiplication? Of course, our running example of 'many' (like its antonym 'few') is exceedingly common, also appearing early in development, though there is still significant debate about how these expressions should be analyzed [111], 10 and how closely they should be unified with their mass counterparts like 'much' and 'little' ( [113]; cf. our discussion in Section 8.3).…”

Section: Dynamic Modalitiesmentioning

confidence: 99%

“…More recently, some researchers have probed the precise counting capacity of such systems, employing notions of count-bisimulations as well (see, e.g., [3]). There has also been study of related logical systems that are expressively equivalent to, but more complex than, graded modal logic [10], as well as natural expressive extensions that remain of relatively low complexity [30]. Emerging connections between graded modal logic and classes of graph neural networks [6] promise yet further dimensions to our subject.…”

Section: A3 Syllogistic and Propositional Counting Logicmentioning

confidence: 99%

Interleaving Logic and Counting

VAN BENTHEM,

ICARD

2023

Bull. symb. log

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Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus 'grassroots mathematics'.We begin with a brief review of FO(#), first-order logic with counting operators and cardinality comparisons. This system is known to be of very high complexity, and drowns out finer aspects of the combination of logic and counting. We therefore move to a small fragment that can represent numerical syllogisms and basic reasoning about comparative size: monadic first-order logic with counting, MFO(#). We provide normal forms that allow for axiomatization, determine which arithmetical notions can be defined on finite and on infinite models, and conversely, we discuss which logical notions can be defined out of purely arithmetical ones, and what sort of (non-)classical logics can be induced.Next, we investigate a series of strengthenings of MFO(#), again using normal form methods. The monadic second-order version is close, in a precise sense, to additive Presburger Arithmetic, while versions with the natural device of tuple counting take us to Diophantine equations, making the logic undecidable. We also define a system ML(#) that combines basic modal logic over binary accessibility relations with counting, needed to formulate ubiquitous reasoning patterns such as the Pigeonhole Principle. We prove decidability of ML(#), and provide a new kind of bisimulation matching the expressive power of the language.As a complement to the fragment approach pursued here, we also discuss two other ways of lowering the complexity of FO(#) by changing the semantics of counting in natural ways. A first approach replaces cardinalities by abstract but well-motivated values of 'mass' or other mereological aggregating notions. A second approach keeps the cardinalities but generalizes the meaning of counting to work in models that allow dependencies between variables.Finally, we return to our starting point in natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary and syntax, as well as with natural reasoning modules such as the monotonicity calculus. In addition to these encounters with formal semantics, we discuss the role of counting in semantic evaluation procedures for quantifier expressions and determine, for instance, which binary quantifiers are computable by finite 'semantic automata'. We conclude with some general thoughts on yet further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on connecting our analysis to empirical findings in cognitive science.

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Modal Logics with Composition on Finite Forests

Cited by 6 publications

References 46 publications

Why Does Propositional Quantification Make Modal and Temporal Logics on Trees Robustly Hard?

Why Does Propositional Quantification Make Modal and Temporal Logics on Trees Robustly Hard?

Why Propositional Quantification Makes Modal Logics on Trees Robustly Hard?

Interleaving Logic and Counting

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