The Mechanized Semantic Library is a collection of mathematical techniques
useful for developing semantic models of program logics and type systems:
**separation algebras**, **permission share models**,
**indirection theory** (a clean axiomatization of step indexing),
and **modal logics**.

The library has a special emphasis on techniques which integrate smoothly into a
mechanical theorem prover; at the moment the library is only available for use
in Coq.

The current version of MSL is distributed as a component of the Verified Software Toolchain. The MSL component will build by itself.

Release versions.

Bleeding-edge in the Subversion repository.

Older versions: 0.1, 0.2, 0.3.

MSL v0.3 has the following enhancements over v0.2:

- Isolation of separation and approximation
- Design of knot shims to enable different uses for the knot
- Better material on cross-split, join homomorphisms
- Extraction for share model
- New example: separation logic for a simple imperative language
- Sundry engineering improvements

MSL v0.2 has the following enhancements over v0.1:

- Fully-worked example using the MSL: mechanized type soundness of the polymorphic lambda calculus with references.
- Recursive predicates: both regular and higher-order recursion, plus utility lemmas for proving contractiveness.
- Fancier knots: we support knots which provide monotonic predicates and allow for bifunctors.
- A proof that squash and unsquash form Galois connection.
- Minor mathematical improvements: we define a more general "fashionable" and weaken the definition of "unage_identity".
- Engineering upgrades: redo build system; new "msl.v" file to import the entire library for clients; streamline path to use indirection theory without separation algebras; a few definition tweeks for engineering purposes; and various utility lemmas, including a theory of fashionable implication.

The Mechanized Semantic Library is made available under a BSD-style license.

# | Name | Description | Paper(s) |

1. | Base | Axioms for computation; custom tactics; basic theorems | |

2. | Separation Algebras | Compositional semantic models for separation logic | [2], [3], [4] |

3. | Shares | Share accounting to model fractional ownership in separation logic | [2] |

4. | Indirection Theory | Semantic models for approximating contravariant circularities | [3], [4] |

5. | Logics | Definitions of substructural and modal logics | [1], [2], [3], [4] |

# | Title | Authors | Read | Cite |

[1] | Multimodal Separation Logic for Reasoning About Operational Semantics |
R. Dockins, A. W. Appel, A. Hobor | BibTeX | |

[2] | A Fresh Look at Separation Algebras and Share Accounting | R. Dockins, A. Hobor, A. W. Appel | BibTeX | |

[3] | A Theory of Indirection via Approximation | A. Hobor, R. Dockins, A. W. Appel | BibTeX | |

[4] | A Logical Mix of Approximation and Separation | A. Hobor, R. Dockins, A. W. Appel | BibTeX |

List order is by publication date. For a mapping of files to papers please see the file list.

# | Title | Authors | Read | Cite |

[1] | Developing and Mechanizing Semantic Models for Program Logics |
A. Hobor, R. Dockins | - |

The MSL includes some associated examples using various components of the library. The first example, released with v0.2, is a mechanized type soundness proof for the polymorphic lambda calculus with references. The second example, released with v0.3, is a separation logic for a simple imperative language with loops and functions. We plan to include more examples as we continue to build the MSL. These examples are released under the same BSD-style license as the MSL.

# | Example | Components |

1 | Type Soundness of the Polymorphic Lambda Calculus with References | Indirection Theory, Logics |

2 | Separation Logic for While Programs with Functions | Separation Algebras, Logics |

The following projects use the MSL. Although the MSL is utilized in them, and in some cases was developed in conjunction with them, they are independent projects, with their own developers, goals, codebase, and associated license schemes.

The goal of the Verified Software Toolchain Project is to connect machine-verified source programs to machine-verified optimizing compilers. This project grew out of the Concurrent C Minor project, which involved the design and soundness proof of a concurrent separation logic with first-class locks and threads for a variant of Xavier Leroy's C minor language.

Design and soundness proof for a concurrent separation logic for Pthreads-style barriers.

Heap-Hop is a prover for concurrent heap-manipulating programs that use Hoare monitors and message-passing synchronization. Programs are annotated with pre and post-conditions and loop invariants, written in a fragment of separation logic. Communications are governed by a form of session types called contracts. Heap-Hop can prove safety and race-freedom and, thanks to contracts, absence of memory leaks and safety of communications. [Text from Heap-Hop website]

Axioms.v

This file contains the axiom base for the development. We assume dependent
functional extensionality and
propositional extensionality. These axioms are compatible with the law of
excluded middle, dependent unique choice, relational choice, etc.

Extensionality.v

Consequences of the extensionality axioms.

base.v

This exports the parts of the Coq standard library used throughout the
development, our axioms and extensionality facts, and a few custom convenience tactics.

sepalg.v

This file defines our relational form of separation algebras with the
disjointness axiom. We also define the join_sub relation and the joins relation.
Additionally, elementary lemmas about the definitions are proved.

cjoins.v

This file defines a constructive version of some of the (existential) join
definitions, such as join_sub and joins.

sepalg_generators.v

We define SA operators in this file. All the operators mentioned in
[2] (except for the boolean algebra generator, located in
the shares component), [3], and [4] appear here, along with
a few others.

cross_split.v

We define and prove properties about the cross_split axoim discussed in
[2], including the relationship to distributive separation
algebras and how to pull the property through many of the generators.

join_hom_lemmas.v

A series of lemmas that can be used to prove that functions are join
homomorphisms as defined in [3] and [4],
although the lemmas themselves are not covered in any paper.

boolean_alg.v

This file defines boolean algebras from an order-theoretic perspective. We also
define axioms relating to properties we desire of share models, including
relativization, splitting and token factory axioms.

tree_shares.v

Here we construct the boolean-labeled tree share model as discussed in
[2]. Note, however, that the proof of the token counting
axioms follow a slightly different path than the proof in that paper. This
is mostly because reasoning about sets in Coq is inconvenient.

shares.v

This file simply repackages the construction from tree_shares.v into a nicer
interface for downstream users. We also define the notion of a "positive" share;
that is a nonunit share.

extract.v

This file will compile the share model into OCaml code for program analysis
tools. Read the instructions to
see how.

knot.v

This file contains the central "knot" development used to model approximations
to contravariant circularities. Included are both the axiomitization and the
model construction. It follows section 8 of [3] quite
closely.

knot_hered.v

Enhances knot.v by threading a monotonicity property through the construction.
The result is that all predicates extracted from a knot are automatically
hereditary. Although the axiomatization is not much more complex than in
the basic case, the construction is significantly harder. The
axiomatization is explained in section 2.3 of [4]; the
construction will
be explained in a future paper.

knot_full.v

Enhances knot_hered.v to allow the input to the knot to be a bifunctor, and
allow the user to supply a relation that further restricts the predicates that
can be stored in the knot. The axiomatization is slightly more complicated
than in the knot_hered case, but the construction is again more complicated.
This file will be explained in a future paper.

knot_lemmas.v

Easy lemmas that follow from the theory of indirection, including a Galois
connection.

knot_sa.v

The definition of a separation algebra on top of knots as coved in section 7 of
[3] and in more detail in sections 6 and 7 of
[4].

knot_sa_trivial.v

Used when you want to use the logics module, but do not have interesting
separation behavior.

knot_shims.v

Defines knot interfaces for common cases (e.g., when we do not need the
bifunctor) and provides shims that translate to knot_full.

knot_prop.v

This file specializes the knot, knot_lemmas, and knot_sa files to use "Prop" as truth
values.

sepalg_functors.v

A technical development of "unmapping" needed to get the extra unage*
properties in knot_sa as explained in section 7 of [4].

knot_setoid.v

An alternate, axiom-free, development of the knot. We avoid the need for the
extensionality axiom by working explicitly up to equivalence relations and
weakening the axioms of the theory accordingly. Not mainline due to lack
of use and long build time.

knot_unique.v

The development of the uniqueness proof for knots. We prove that any two
implementations of the theory of indirection are isomorphic.

ageable.v

Define a notion of approximation as in section 2
of
[4] as the "ageable" typeclass.

age_sepalg.v

We enhance separation algebras with as in section 6 of
[4]. The
"ASA" typeclass presents an interface sufficient to define the Kripke model
below. The interface is implemented using the properties
obtained from knot_sa.

predicates_hered.v

Definition of higher-order modal logic as discussed in sections
2, 4, and 5 of [1]; briefly in section 4
of [2]; sections 5 and 6 of [3];
and sections 2 and 3 of [4].

predicates_sl.v

Definition of higher-order separation logic as discussed in section 6 of [1], section 4
of [2], section 7 of [3], and
section 6 of [4].

subtypes.v

Lemmas about fashionable implication. This is partially explained in section 4
of [4].

predicates_rec.v

Definition and proofs about equirecursion. We include both the standard
"mu" recursion as well as a more powerful higher-order variant. This file
is partially explained in section 5 of [1] and section 4 of [4] and will be explained in
more detail in a future paper.

contractive.v

A collection of utility lemmas for proving that various predicate functions are
contractive or nonexpansive (for both the regular recursion and the higher-order
variety). This file is partially explained in section 4 of
[4] and will be explained in more detail in a future paper.

msl_standard.v

A file which re-exports the major components of the MSL library for users.

The Mechanized Semantic Library was developed by Andrew W. Appel, Robert Dockins, and Aquinas Hobor.

Copyright (c) 2009-2010, Andrew Appel, Robert Dockins and Aquinas Hobor.

All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

- Redistribution of source code must retain the above copyright notice, this list of conditions and the following disclaimer.

- Redistribution in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.

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