scholarly journals Beyond Symbolic Heaps: Deciding Separation Logic With Inductive Definitions

10.29007/vkmj ◽  
2020 ◽  
Author(s):  
Jens Pagel ◽  
Florian Zuleger
Keyword(s):  
Recent Work ◽  
Upper Bound ◽  
Separation Logic ◽  
Satisfiability Problem ◽  
Magic Wand ◽  
Inductive Definitions ◽  
Direct Model

Symbolic-heap separation logic with inductive definitions is a popular formalism for reasoning about heap-manipulating programs. The fragment SLIDbtw introduced by Iosif, Rogalewicz and Simacek, is one of the most expressive fragments with a decidable entailment problem. In recent work, we improved on the original decidability proof by providing a direct model-theoretic construction, obtaining a 2-Exptime upper bound. In this paper, we investigate separation logics built on top of the inductive definitions from SLIDbtw, i.e., logics that feature the standard Boolean and separation-logic operators. We give an almost tight delineation between decidability and undecidabilty. We establish the decidability of the satisfiability problem (in 2-Exptime) of a separation logic with conjunction, disjunction, separating conjunction and guarded forms of negation, magic wand, and septraction. We show that any further generalization leads to undecidabilty (under mild assumptions).

On the Modal μ-Calculus Over Finite Symmetric Graphs

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
Giovanna D’Agostino ◽  
Giacomo Lenzi
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Satisfiability Problem ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Symmetric Graphs ◽  
Trivial Consequence

AbstractIn this paper we consider the alternation hierarchy of the modal μ-calculus over finite symmetric graphs and show that in this class the hierarchy is infinite. The μ-calculus over the symmetric class does not enjoy the finite model property, hence this result is not a trivial consequence of the strictness of the hierarchy over symmetric graphs. We also find a lower bound and an upper bound for the satisfiability problem of the μ-calculus over finite symmetric graphs.

scholarly journals The power of modal separation logics

2019 ◽  
Vol 29 (8) ◽  
pp. 1139-1184 ◽  
Author(s):  
Stéphane Demri ◽  
Raul Fervari
Keyword(s):  
Separation Logic ◽  
Satisfiability Problem ◽  
Alternative Semantics ◽  
Temporal Logics ◽  
Original Language ◽  
Modal Operators ◽  
Logical Connectives ◽  
Np Complete ◽  
Memory States ◽  
The Difference

Abstract We introduce a modal separation logic MSL whose models are memory states from separation logic and the logical connectives include modal operators as well as separating conjunction and implication from separation logic. With such a combination of operators, some fragments of MSL can be seen as genuine modal logics whereas some others capture standard separation logics, leading to an original language to speak about memory states. We analyse the decidability status and the computational complexity of several fragments of MSL, obtaining surprising results by design of proof methods that take into account the modal and separation features of MSL. For example, the satisfiability problem for the fragment of MSL with $\Diamond $, the difference modality $\langle \neq \rangle $ and separating conjunction $\ast $ is shown Tower-complete whereas the restriction either to $\Diamond $ and $\ast $ or to $\langle \neq \rangle $ and $\ast $ is only NP-complete. We establish that the full logic MSL admits an undecidable satisfiability problem. Furthermore, we investigate variants of MSL with alternative semantics and we build bridges with interval temporal logics and with logics equipped with sabotage operators.

scholarly journals An approximative inference method for solving ∃∀SO satisfiability problems

2012 ◽  
Vol 45 ◽  
pp. 79-124 ◽  
Author(s):  
H. Vlaeminck ◽  
J. Vennekens ◽  
M. Denecker ◽  
M. Bruynooghe
Keyword(s):  
Constraint Propagation ◽  
Second Order ◽  
Order Logic ◽  
Satisfiability Problem ◽  
Finite Domain ◽  
Satisfiability Problems ◽  
Conformant Planning ◽  
Inductive Definitions ◽  
Powerful Knowledge ◽  
Second Order Logic

This paper considers the fragment ∃∀SO of second-order logic. Many interesting problems, such as conformant planning, can be naturally expressed as finite domain satisfiability problems of this logic. Such satisfiability problems are computationally hard (ΣP2) and many of these problems are often solved approximately. In this paper, we develop a general approximative method, i.e., a sound but incomplete method, for solving ∃∀SO satisfiability problems. We use a syntactic representation of a constraint propagation method for first-order logic to transform such an ∃∀SO satisfiability problem to an ∃SO(ID) satisfiability problem (second-order logic, extended with inductive definitions). The finite domain satisfiability problem for the latter language is in NP and can be handled by several existing solvers. Inductive definitions are a powerful knowledge representation tool, and this moti- vates us to also approximate ∃∀SO(ID) problems. In order to do this, we first show how to perform propagation on such inductive definitions. Next, we use this to approximate ∃∀SO(ID) satisfiability problems. All this provides a general theoretical framework for a number of approximative methods in the literature. Moreover, we also show how we can use this framework for solving practical useful problems, such as conformant planning, in an effective way.

scholarly journals Sur les solutions friables de l'équation a+b=c

2013 ◽  
Vol 154 (3) ◽  
pp. 439-463 ◽  
Author(s):  
SARY DRAPPEAU
Keyword(s):  
Lower Bound ◽  
Asymptotic Behaviour ◽  
Recent Work ◽  
Upper Bound ◽  
Smooth Function ◽  
Riemann Hypothesis ◽  
Smooth Solutions ◽  
Compactly Supported ◽  
The One

AbstractIn a recent paper [5], Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Conditionally under the Generalised Riemann Hypothesis, they obtain an estimate for the number of those solutions weighted by a compactly supported smooth function, as well as a lower bound for the number of bounded unweighted solutions. In this paper, we prove a more precise conditional estimate for the number of weighted solutions that is valid when y is relatively large with respect to x, so as to connect our estimate with the one obtained by La Bretèche and Granville in a recent work [2]. We also prove, conditionally under the Generalised Riemann Hypothesis, the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour.

scholarly journals An Auxiliary Logic on Trees: on the Tower-Hardness of Logics Featuring Reachability and Submodel Reasoning

2020 ◽  
pp. 462-481
Author(s):  
Alessio Mansutti
Keyword(s):  
Modal Logic ◽  
Separation Logic ◽  
Satisfiability Problem ◽  
Computation Tree Logic ◽  
Computation Tree ◽  
Simple Features ◽  
Fixed Node

AbstractWe describe a set of simple features that are sufficient in order to make the satisfiability problem of logics interpreted on trees Tower-hard. We exhibit these features through an Auxiliary Logic on Trees (), a modal logic that essentially deals with reachability of a fixed node inside a forest and features modalities from sabotage modal logic to reason on submodels. After showing that admits a Tower-complete satisfiability problem, we prove that this logic is captured by four other logics that were independently found to be Tower-complete: two-variables separation logic, quantified computation tree logic, modal logic of heaps and modal separation logic. As a by-product of establishing these connections, we discover strict fragments of these logics that are still non-elementary.

A proof system for separation logic with magic wand

2014 ◽  
Vol 49 (1) ◽  
pp. 477-490 ◽  
Author(s):  
Wonyeol Lee ◽  
Sungwoo Park
Keyword(s):  
Separation Logic ◽  
Proof System ◽  
Magic Wand
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