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A Course in Universal Algebra

Lattice theory today reflects the general Status of current mathematics: there is a rich production of theoretical concepts, results, and developments, many of which are reached by elaborate mental gymnastics; on the other hand, the connections of the theory to its surroundings are getting weaker and weaker, with the result that the theory and even many of its parts become more isolated. Restructuring lattice theory is an attempt to reinvigorate connections with our general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattice theory.

Restructuring lattice theory: an approach based on hierarchies of concepts

Semantic analysis of programs is essential in optimizing compilers and program verification systems. It encompasses data flow analysis, data type determination, generation of approximate invariant assertions, etc. Several recent papers (among others Cousot & Cousot[77a], Graham & Wegman[76], Kam & Ullman[76], Kildall[73], Rosen[78], Tarjan[76], Wegbreit[75]) have introduced abstract approaches to program analysis which are tantamount to the use of a program analysis framework (A,t,a) where A is a lattice of (approximate) assertions, t is an (approximate) predicate transformer and a is an often implicit function specifying the meaning of the elements of A. This paper is devoted to the systematic and correct design of program analysis frameworks with respect to a formal semantics. Preliminary definitions are given in Section 2 concerning the merge over all paths and (least) fixpoint program-wide analysis methods. In Section 3 we briefly define the (forward and backward) deductive semantics of programs which is later used as a formal basis in order to prove the correctness of the approximate program analysis frameworks. Section 4 very shortly recall the main elements of the lattice theoretic approach to approximate semantic analysis of programs. The design of a space of approximate assertions A is studied in Section 5. We first justify the very reasonable assumption that A must be chosen such that the exact invariant assertions of any program must have an upper approximation in A and that the approximate analysis of any program must be performed using a deterministic process. These assumptions are shown to imply that A is a Moore family, that the approximation operator (wich defines the least upper approximation of any assertion) is an upper closure operator and that A is necessarily a complete lattice. We next show that the connection between a space of approximate assertions and a computer representation is naturally made using a pair of isotone adjoined functions. This type of connection between two complete lattices is related to Galois connections thus making available classical mathematical results. Additional results are proved, they hold when no two approximate assertions have the same meaning. In Section 6 we study and examplify various methods which can be used in order to define a space of approximate assertions or equivalently an approximation function. They include the characterization of the least Moore family containing an arbitrary set of assertions, the construction of the least closure operator greater than or equal to an arbitrary approximation function, the definition of closure operators by composition, the definition of a space of approximate assertions by means of a complete join congruence relation or by means of a family of principal ideals. Section 7 is dedicated to the design of the approximate predicate transformer induced by a space of approximate assertions. First we look for a reasonable definition of the correctness of approximate predicate transformers and show that a local correctness condition can be given which has to be verified for every type of elementary statement. This local correctness condition ensures that the (merge over all paths or fixpoint) global analysis of any program is correct. Since isotony is not required for approximate predicate transformers to be correct it is shown that non-isotone program analysis frameworks are manageable although it is later argued that the isotony hypothesis is natural. We next show that among all possible approximate predicate transformers which can be used with a given space of approximate assertions there exists a best one which provides the maximum information relative to a program-wide analysis method. The best approximate predicate transformer induced by a space of approximate assertions turns out to be isotone. Some interesting consequences of the existence of a best predicate transformer are examined. One is that we have in hand a formal specification of the programs which have to be written in order to implement a program analysis framework once a representation of the space of approximate assertions has been chosen. Examples are given, including ones where the semantics of programs is formalized using Hoare[78]'s sets of traces. In Section 8 we show that a hierarchy of approximate analyses can be defined according to the fineness of the approximations specified by a program analysis framework. Some elements of the hierarchy are shortly exhibited and related to the relevant literature. In Section 9 we consider global program analysis methods. The distinction between "distributive" and "non-distributive" program analysis frameworks is studied. It is shown that when the best approximate predicate transformer is considered the coincidence or not of the merge over all paths and least fixpoint global analyses of programs is a consequence of the choice of the space of approximate assertions. It is shown that the space of approximate assertions can always be refined so that the merge over all paths analysis of a program can be defined by means of a least fixpoint of isotone equations. Section 10 is devoted to the combination of program analysis frameworks. We study and examplify how to perform the "sum", "product" and "power" of program analysis frameworks. It is shown that combined analyses lead to more accurate information than the conjunction of the corresponding separate analyses but this can only be achieved by a new design of the approximate predicate transformer induced by the combined program analysis frameworks.

Systematic design of program analysis frameworks

The diamond lemma for ring theory

There is a population explosion among the logical systems used in computing science. Examples include first-order logic, equational logic, Horn-clause logic, higher-order logic, infinitary logic, dynamic logic, intuitionistic logic, order-sorted logic, and temporal logic; moreover, there is a tendency for each theorem prover to have its own idiosyncratic logical system. The concept of institution is introduced to formalize the informal notion of “logical system.” The major requirement is that there is a satisfaction relation between models and sentences that is consistent under change of notation. Institutions enable abstracting away from syntactic and semantic detail when working on language structure “in-the-large”; for example, we can define language features for building large logical system. This applies to both specification languages and programming languages. Institutions also have applications to such areas as database theory and the semantics of artificial and natural languages. A first main result of this paper says that any institution such that signatures (which define notation) can be glued together, also allows gluing together theories (which are just collections of sentences over a fixed signature). A second main result considers when theory structuring is preserved by institution morphisms. A third main result gives conditions under which it is sound to use a theorem prover for one institution on theories from another. A fourth main result shows how to extend institutions so that their theories may include, in addition to the original sentences, various kinds of constraint that are useful for defining abstract data types, including both “data” and “hierarchy” constraints. Further results show how to define institutions that allow sentences and constraints from two or more institutions. All our general results apply to such “duplex” and “multiplex” institutions.

Institutions: abstract model theory for specification and programming

To prove the statement given in the title take a set Σ 1 of identities characterizing distributive lattices 〈L; ∨, ∧, 0, 1〉 with 0 and 1, and let Then is Σ redundant set of identities characterizing Stone algebras = 〈 L ; ∨, ∧, *, 0, 1〉. To show that we only have to verify that for a ∈ L , a * is the pseudo-complement of a . Indeed, a ∧ a * 0; now, if a ∧ x = 0, then a * ∨ x * 0* = 1, and a ** ∧ = 1* = 0; since a ** is the complement of a *, the last identity implies x ** ≦ a *, thus x ≦ x ** ≦ a *, which was to be proved.

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Stone algebras form an equational class: (Remarks on Lattice Theory III)

In a recent paper, G. Czedli and E. T. Schmidt present a structure theorem for planar semimodular lattices. In this note, we present an alternative proof.

Notes on planar semimodular lattices. VI. On the structure theorem of planar semimodular lattices

On the endomorphism semigroup of simple algebras

Let L be a lattice and let L1, L2 be sublattices of L. Let be a congruence relation of L1. We extend to L by taking the smallest congruence of L containing . Then we restrict to L2, obtaining the congruence L2 of L2. Thus we have de ned a map ConL1 ! ConL2. Obviously, this is an isotone 0-preservingmap of the nite distributive lattice ConL1 into the nite distributive lattice ConL2. The main result of this paper is the converse. Let D1 and D2 be nite distributive lattices, and let : D1 ! D2 be an isotone map that preserves 0. Then there is a nite lattice L with sublattices L1 and L2 such that ConL1 represents D1 and ConL2 represents D2, and the map ConL1 ! ConL2 obtained by rst extending each congruence relation of L1 to L by minimal extension and then restricting the resulting congruence relation to L2 represents .

Isotone Maps as Maps of Congruences I. Abstract Maps

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George Grätzer

Papers

Stone algebras form an equational class: (Remarks on Lattice Theory III)

Notes on planar semimodular lattices. VI. On the structure theorem of planar semimodular lattices

On the endomorphism semigroup of simple algebras

Isotone Maps as Maps of Congruences I. Abstract Maps

On the number of polynomials of a universal algebra. II