# On the Computational Complexity of the Numerically Definite Syllogistic and Related Logics

@article{PrattHartmann2008OnTC,
title={On the Computational Complexity of the Numerically Definite Syllogistic and Related Logics},
author={I. Pratt-Hartmann},
journal={Bulletin of Symbolic Logic},
year={2008},
volume={14},
pages={1 - 28}
}
• I. Pratt-Hartmann
• Published 2008
• Computer Science, Mathematics
• Bulletin of Symbolic Logic
Abstract The numerically definite syllogistic is the fragment of English obtained by extending the language of the classical syllogism with numerical quantifiers. The numerically definite relational syllogistic is the fragment of English obtained by extending the numerically definite syllogistic with predicates involving transitive verbs. This paper investigates the computational complexity of the satisfiability problem for these fragments. We show that the satisfiability problem (= finite… Expand
A Note on the Complexity of the Satisfiability Problem for Graded Modal Logics
• Computer Science, Mathematics
• 2009 24th Annual IEEE Symposium on Logic In Computer Science
• 2009
Tight complexity bounds are obtained for the problem of determining the satisfiability of a given graded modal logic formula over the classes of frames characterized by any combination of reflexivity, seriality, symmetry, transitivity and the Euclidean property. Expand
Decidable first-order modal logics with counting quantifiers
This paper shows that the decision (validity) problem for the one- variable fragment of the minimal first-order modal logic QK with counting quantifiers is coNExpTime-complete and provides optimal upper bounds on the complexity of the decision problem for several one-variable fragments, by establishing the finite model property. Expand
No Syllogisms for the Numerical Syllogistic
This paper answers the question in the negative: no finite collection of syllogism-like rules, broadly conceived, is sound and complete for the numerical syllogistic. Expand
On the Complexity of Graded Modal Logics with Converse
• Mathematics, Computer Science
• JELIA
• 2019
This paper fills the gaps remaining in an analogous classification of the graded modal language with graded converse modalities over traditional classes of frames and shows its NExpTime-completeness over the class of Euclidean frames, demonstrating this way that over this class the considered language is harder than the language without graded modalities or without conversemodalities. Expand
An algebraic analysis of categorical syllogisms by using Carroll’s diagrams
• Mathematics
• 2019
In this paper, we analyze the algebraic properties of categorical syllogisms by constructing a logical calculus system called Syllogistic Logic with Carroll Diagrams (SLCD). We prove that anyExpand
A new algorithmic decision for categorical syllogisms via Carroll’s diagrams
• Computer Science
• Soft Comput.
• 2020
The empirical contributions of this paper are to design a polynomial-time algorithm at the first time in the literature to conduce to researchers getting into the act in different areas of science used categorical syllogisms such as artificial intelligence, engineering, computer science and etc. Expand
Easy Solutions for a Hard Problem? The Computational Complexity of Reciprocals with Quantificational Antecedents
• Computer Science
• ESSLLI Logic & Cognition Workshop
• 2012
The results from the picture completion experiment suggest that intractable readings occur in language comprehension and it is argued that during verification, guessing strategies are used to reduce computational complexity. Expand
On the Complexity and Expressiveness of Description Logics with Counting
Simple counting quantifiers that can be used to compare the number of role successors of an individual or the cardinality of a concept with a fixed natural number have been employed in DescriptionExpand
Algorithms for Deciding Counting Quantifiers over Unary Predicates
• Computer Science
• AAAI
• 2017
An algebraic formulation of the CQU satisfiability problem in terms of Integer Linear Programming is provided based on which two algorithms are proposed, a direct reduction to SAT instances and an Integer Linear programming version extended with a column generation mechanism. Expand
On the Expressive Power of Description Logics with Cardinality Constraints on Finite and Infinite Sets
• Mathematics, Computer Science
• FroCos
• 2019
The main contribution of this paper is to give a characterization of the first-order fragment of $$\mathcal {ALCSCC} ^\infty$$, a notion of bisimulation that characterizes this fragment. Expand

#### References

SHOWING 1-10 OF 27 REFERENCES
Complexly fractionated syllogistic quantifiers
Building on Peterson (1985), the coherence of such a syllogistic can, however, be demonstrated with an algebra which provides its semantics; e.g., “almost 1/2 the S are P” is represented as “−(3(SP)≫SP)”. Expand
PSPACE Reasoning for Graded Modal Logics
• S. Tobies
• Computer Science, Mathematics
• J. Log. Comput.
• 2001
A PSPACE algorithm that decides satisfiability of the graded modal lo gic Gr(KR), a natural extension of propositional modal logic KR by counting expressions, is presented, which is the first known algorithm which meets the lower bound for the complexity of the problem. Expand
Towards Efficient Satisfiability Checking for Boolean Algebra with Presburger Arithmetic
• Mathematics, Computer Science
• 2007
This paper presents an algorithm for checking satisfiability of QFBAPA formulas by reducing them to formulas of quantifier-free PA, with only O(n log(n) increase in formula size, and proves the correctness of the algorithm using a theorem about sparse solutions of integer linear programming problems. Expand
Formal Logic: Or, The Calculus of Inference, Necessary and Probable
Preface 1. First notions 2. On objects, ideas, and names 3. On the abstract form of the proposition 4. On propositions 5. On the syllogism 6. On the syllogism (cont.) 7. On the Aristotelian syllogismExpand
Complexity Results for First-Order Two-Variable Logic with Counting
• Computer Science, Mathematics
• SIAM J. Comput.
• 2000
It is proved that the satisfiability problem for first-order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) $i$" for $i\leq p$ is NEXPTIME-complete. Expand
A Polynomial Translation from the Two-Variable Guarded Fragment with Number Restrictions to the Guarded Fragment
A satisfiability preserving transformation of formulas in this fragment to the three-variable guarded fragment is given and can be computed in polynomial time and produces a formula that is linear in the size of the initial formula even for the binary coding of number restrictions. Expand
Logics in Artificial Intelligence 9th European Conference, Jelia 2004, Lisbon, Portugal, September 27-30, 2004 : Proceedings
• Mathematics, Computer Science
• 2004
Invited Talks.- Representing and Reasoning with Preferences.- Engineering of Logics for the Content-Based Representation of Information.- Formal Methods in Robotics.- Multi-agent Systems.- Games forExpand
More Fragments of Language
• Computer Science
• Notre Dame J. Formal Log.
• 2006
The present paper considers various fragments of English involving ditransitive verbs and determines their semantic complexity. Expand
Complexity of the Two-Variable Fragment with Counting Quantifiers
The satisfiability and finite satisfiability problems for the two-variable fragment of first-order logic with counting quantifiers are both in NEXPTIME, even when counting quantifiers are codedExpand
Pure Logic and Other Minor Works
• Psychology
• 1991
This collection of Jevons' papers falls naturally into two groups. There are papers developing Jevons' positive conception of logic as a purely abstract, formal discipline; and there is a group ofExpand