(Last modified Thu Feb 14 16:20 2008)

The Alloy logic

The Alloy logic is a first-order logic in which the domain is the set of all relations, and terms include relational expressions such as joins. 

Everything in Alloy is a relation! 

• A relation is a set of tuples of the same (positive) arity.  Each tuple lists entities that are related to each other.  The size of the relation is the number of tuples; the arity of the relation is the arity of the tuples. 
• Sets are represented by unary relations.  Each 1-tuple in the unary relation contains an element of the set. 
• Scalars are represented by singleton sets.  Since a set is a unary relation, an scalar is thus represented as a singleton (size 1) unary relation. 

As a result, the operators apply to relations, sets, and scalars, and there are very few cases that produce no result. 

Page numbers refer to Daniel Jackson, Software Abstractions, MIT Press 2006. 

How to update the book for Alloy 4

Quick reference

Set examples

else example

Relation and logic examples

Signatures and relations

(Parts of this subsection describe the Alloy language.)

Each set of atoms is defined by a signature, with keyword sig. 

A signature can contain zero or more relation declarations, separated by commas. Each declaration names a (binary) relation between the set defined by the signature and a set or relation.

  //  Simple example
  abstract sig Person {     // Signature
    father: lone Man,       //   A declaration
    mother: lone Woman      //   Another declaration
  }
  sig Man extends Person {
    wife: lone Woman
  }
  sig Woman extends Person {
    husband: lone Man
  }

Constraining a declaration

There are two ways: 

1. with set or relation multiplicity constraints in the signature.  These are a quick shorthand.  The example above has several of these (all are lone).
2. with a fact 117 that states a constraint on the set or relation.  The constraint is expressed in the Alloy logic. 

   (The fact keyword may be omitted if the fact is only about the relations of a single signature, and it immediately follows that signature — then it is a signature fact, and is implicitly universally quantified over the signature's set, and may use this as if it were the variable of this implied quantification.) 

A more extensive example of signatures, declarations, and constraints

Multiplicity constraints in declarations

Facts 117

A fact contains a formula in the Alloy logic that is assumed to always be true. See the Alloy language for more details. 

Disjointness 71

disj before a list of variables restricts their bindings to be disjoint. 

Disjointness example

Cardinality constraints 80

The prefix operator # (cardinality) on a relation produces the relation's size.  The result can be operated on with + - = < > =< >=.  Positive integer literals can appear in cardinality expressions. 

sum x: e | ie sums the value of ie for each x in set e. 

The Alloy language

The Alloy language uses the Alloy logic plus some other constructs to make models.  In Alloy, a model is "a description of a software abstraction" 4. 

(Recall that in FOL a model means something different.) 

Language constructs

The Alloy language adds these constructs to the Alloy logic: 

1. A module line gives the relative pathname of the model's file (minus the ".als" suffix).  The pathname is relative to the directory that imported module pathnames are going to be relative to.  (Obviously, the module line is mostly redundant with the file's full pathname.) 
2. A sig (signature) declares one or more sets of atoms, and their relations to other sets. 
3. A fun (function) defines a way of getting a relation (or set, or atom).  It can take parameters that are used in getting its result.  It can define a relation (usually using ->) and make use of it to produce its result.  It is a FOL function for the Alloy logic, in which expressions are relations. 
4. A pred (predicate) defines a formula (true or false).  It can take parameters that are used in getting its result.  It is a FOL predicate for the Alloy logic. 
5. A fact defines a formula that you assume is valid (always true, for any world).  The Alloy analyzer uses facts as axioms in constructing its examples and counterexamples. 
6. You run a predicate in order to see the examples (if any) the Alloy analyzer finds for which the predicate is true. 

   You define the scope that the analyzer checks by saying things like "run for 3" or "run for 3 but 4 Dog".  The analyzer will then check only possible examples that contain no more than that many of atoms from each set. 

   If it finds an example, then the predicate is satisfiable. 

   If it finds no examples, the predicate may be either invalid (false for all possible examples); or it may be satisfiable but not within the scope you used. 

7. An assert (assertion) defines a formula that you claim will always be true.  An assertion differs from a fact in that the Alloy analyzer will check an assertion to see if it is true for all the examples in a scope, whereas the analyzer assumes each fact is true and uses them to constrain which examples it looks at. 
8. You check an assertion in order to see whether the Alloy analyzer finds any counterexamples. 

   You define the scope as for a run command. 

   If it finds a counterexample, then the predicate is unsatifiable. 

   If it finds no counterexamples, the predicate may be either valid (true for all possible examples); or it may be unsatifiable but not within the scope you used. 

Which construct to use where?

1. Writing a model (Alloy file) that might need to import other models?  Use module. 
2. Need a set of atoms?  Use a sig. 
3. Need an expression, whose value is a function (or set, or scalar)?  Use a fun (function). 
4. Need a formula, whose value is true or false?  Use a pred (predicate). 
5. Need to state an axiom that you want to be true always?  Use a fact (function). 
6. Need an example for which a pred is true?  run the predicate to see if one exists.  It's like using an existential quantifier over all the predicate's parameters. 
7. Want to claim something is always true?  Use an assert (assertion). 
8. Want to see if an assert is unsatisfiable?  check the assertion to see if any counterexample can be found. 

Signatures

Functions

Predicates

Facts

Assertions

Modules and polymorphism 130 — section under construction

Thomas A. Alspaugh
Assistant Professor, Informatics Dept.
School of Information and Computer Sciences