# Two-Variable Separation Logic and Its Inner Circle

@article{Demri2015TwoVariableSL, title={Two-Variable Separation Logic and Its Inner Circle}, author={Stephane Demri and Morgan Deters}, journal={ACM Transactions on Computational Logic (TOCL)}, year={2015}, volume={16}, pages={1 - 36} }

Separation logic is a well-known assertion language for Hoare-style proof systems. We show that first-order separation logic with a unique record field restricted to two quantified variables and no program variables is undecidable. This is among the smallest fragments of separation logic known to be undecidable, and this contrasts with the decidability of two-variable first-order logic. We also investigate its restriction by dropping the magic wand connective, known to be decidable with…

## 16 Citations

### Separation Logic with One Quantified Variable

- MathematicsTheory of Computing Systems
- 2016

It is shown that the satisfiability problem for 1SL1 is PSPACE-complete and the expressive power of the fragment is characterized by showing that every formula is equivalent to a Boolean combination of atomic properties.

### Expressive Completeness of Separation Logic with Two Variables and No Separating Conjunction

- PhilosophyACM Trans. Comput. Log.
- 2016

It is shown that first-order separation logic with one record field restricted to two variables and the separating implication (no separating conjunction) is as expressive as weak second-order logic, substantially sharpening a previous result.

### Semipositivity in Separation Logic with Two Variables

- MathematicsSETTA
- 2016

It is proved that satisfiability of the fragment of SPSL2 where neither separating conjunction nor septraction occurs in the scope of universal quantifiers, is complete, and it is shown that the finite satisfiability problem of first-order logic with two variables and a bounded number of function symbols is \(\textsc {nexptime}\)-complete.

### On the Complexity of Modal Separation Logics

- PhilosophyAdvances in Modal Logic
- 2018

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…

### Extending Propositional Separation Logic for Robustness Properties

- Computer ScienceFSTTCS
- 2018

We study an extension of propositional separation logic that can specify robustness properties, such as acyclicity and garbage freedom, for automatic verification of stateful programs with…

### The power of modal separation logics

- PhilosophyJ. Log. Comput.
- 2019

It is established that the full logic MSL admits an undecidable satisfiability problem and variants of MSL with alternative semantics are investigated and bridges with interval temporal logics and with logics equipped with sabotage operators are built.

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

- Philosophy, Computer ScienceFoSSaCS
- 2020

A set of simple features are described that are sufficient in order to make the satisfiability problem of logics interpreted on trees Tower-hard and it is proved that this logic is captured by four other logics that were independently found to be Tower-complete.

### Decidability of weak logics with deterministic transitive closure

- MathematicsCSL-LICS
- 2014

This paper considers the extension of the two-variable fragment of first-order logic by the deterministic transitive closure of a single binary relation, and proves that the satisfiability and finite satisfiability problems for the obtained logic are decidable and ExpSpace-complete.

### Automated reasoning and randomization in separation logic

- Computer Science
- 2020

A weakest precondition calculus à la Dijkstra is developed, which is a sound and conservative extension of both separation logic and McIver and Morgan’s weakest preexpectations which preserves virtually all properties of classical separation logic.

### Separation logics and modalities: a survey

- Philosophy, Computer ScienceJ. Appl. Non Class. Logics
- 2015

This survey presents similarities between separation logic as an assertion language and modal and temporal logics and proposes a selection of landmark results about decidability, complexity and expressive power.

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It is shown that first-order separation logic with one record field restricted to two variables and the separating implication (no separating conjunction) is as expressive as weak second-order logic, substantially sharpening a previous result.

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It is shown that the purely propositional fragment of separation logic is undecidability, and a number of propositional systems which approximate separation logic are undecidable as well, including both Boolean BI and Classical BI.

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