3,626 research outputs found
Expressive Completeness of Separation Logic With Two Variables and No Separating Conjunction ∗
We show 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. Capturing weak secondorder logic with such a restricted form of separation logic requires substantial updates to known proof techniques. We develop these, and as a by-product identify the smallest fragment of separation logic known to be undecidable: first-order separation logic with one record field, two variables, and no separating conjunction
On the Expressive Power of Multiple Heads in CHR
Constraint Handling Rules (CHR) is a committed-choice declarative language
which has been originally designed for writing constraint solvers and which is
nowadays a general purpose language. CHR programs consist of multi-headed
guarded rules which allow to rewrite constraints into simpler ones until a
solved form is reached. Many empirical evidences suggest that multiple heads
augment the expressive power of the language, however no formal result in this
direction has been proved, so far.
In the first part of this paper we analyze the Turing completeness of CHR
with respect to the underneath constraint theory. We prove that if the
constraint theory is powerful enough then restricting to single head rules does
not affect the Turing completeness of the language. On the other hand,
differently from the case of the multi-headed language, the single head CHR
language is not Turing powerful when the underlying signature (for the
constraint theory) does not contain function symbols.
In the second part we prove that, no matter which constraint theory is
considered, under some reasonable assumptions it is not possible to encode the
CHR language (with multi-headed rules) into a single headed language while
preserving the semantics of the programs. We also show that, under some
stronger assumptions, considering an increasing number of atoms in the head of
a rule augments the expressive power of the language.
These results provide a formal proof for the claim that multiple heads
augment the expressive power of the CHR language.Comment: v.6 Minor changes, new formulation of definitions, changed some
details in the proof
Internal Calculi for Separation Logics
We present a general approach to axiomatise separation logics with heaplet semantics with no external features such as nominals/labels. To start with, we design the first (internal) Hilbert-style axiomatisation for the quantifier-free separation logic SL(?, -*). We instantiate the method by introducing a new separation logic with essential features: it is equipped with the separating conjunction, the predicate ls, and a natural guarded form of first-order quantification. We apply our approach for its axiomatisation. As a by-product of our method, we also establish the exact expressive power of this new logic and we show PSpace-completeness of its satisfiability problem
Axiomatising logics with separating conjunctions and modalities
International audienceModal separation logics are formalisms that combine modal operators to reason locally, with separating connectives that allow to perform global updates on the models. In this work, we design Hilbert-style proof systems for the modal separation logics MSL(⇤, h6 =i) and MSL(⇤, 3), where ⇤ is the separating conjunction, 3 is the standard modal operator and h6 =i is the di↵erence modality. The calculi only use the logical languages at hand (no external features such as labels) and take advantage of new normal forms and of their axiomatisation
The Complexity of Prenex Separation Logic with One Selector
We first show that infinite satisfiability can be reduced to finite
satisfiability for all prenex formulas of Separation Logic with
selector fields (\seplogk{k}). Second, we show that this entails the
decidability of the finite and infinite satisfiability problem for the class of
prenex formulas of \seplogk{1}, by reduction to the first-order theory of one
unary function symbol and unary predicate symbols. We also prove that the
complexity is not elementary, by reduction from the first-order theory of one
unary function symbol. Finally, we prove that the Bernays-Sch\"onfinkel-Ramsey
fragment of prenex \seplogk{1} formulae with quantifier prefix in the
language is \pspace-complete. The definition of a complete
(hierarchical) classification of the complexity of prenex \seplogk{1},
according to the quantifier alternation depth is left as an open problem
Stone-Type Dualities for Separation Logics
Stone-type duality theorems, which relate algebraic and
relational/topological models, are important tools in logic because -- in
addition to elegant abstraction -- they strengthen soundness and completeness
to a categorical equivalence, yielding a framework through which both algebraic
and topological methods can be brought to bear on a logic. We give a systematic
treatment of Stone-type duality for the structures that interpret bunched
logics, starting with the weakest systems, recovering the familiar BI and
Boolean BI (BBI), and extending to both classical and intuitionistic Separation
Logic. We demonstrate the uniformity and modularity of this analysis by
additionally capturing the bunched logics obtained by extending BI and BBI with
modalities and multiplicative connectives corresponding to disjunction,
negation and falsum. This includes the logic of separating modalities (LSM), De
Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics
extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as
corollaries soundness and completeness theorems for the specific Kripke-style
models of these logics as presented in the literature: for DMBI, the
sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene
BI (connecting our work to Concurrent Separation Logic), this is the first time
soundness and completeness theorems have been proved. We thus obtain a
comprehensive semantic account of the multiplicative variants of all standard
propositional connectives in the bunched logic setting. This approach
synthesises a variety of techniques from modal, substructural and categorical
logic and contextualizes the "resource semantics" interpretation underpinning
Separation Logic amongst them
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