# Introduction

Introduction of Syntax Definition for Language Prototyping

Language prototyping is the activity of designing and testing definitions of new or existing computer languages. An important aspect of a language definition is the definition of its syntax. The subject of this thesis are new formalisms and techniques that support the development and prototyping of syntax definitions. There are four main subjects: (1) Techniques for parsing and disambiguation of context-free languages. (2) Design and implementation of a new syntax definition formalism. (3) Design of a multi-level algebraic specification formalism. (4) Study of polymorphic syntax definition. This chapter sketches the background and motivation of this work and gives an overview of the thesis.

## General

### Language Prototyping

Computer languages are used to instruct computers or to encode data processed by computers. According to their application area languages are classified as programming language, domain specific language, specification language or data format.

Language design is a recurrent activity in computer science and software engineering. \cite{langlist} lists about 2350 languages that have been designed since computers were first developed in the 1940s. Bearing in mind that this list contains only a fraction of all languages it is probably an underestimation to say that a new language appears every week. New general purpose languages are designed as new technology becomes available that poses new requirements or provides new opportunities. A recent example is the Java programming language that addresses problems posed by exchanging programs over networks. New domain specific languages are developed to encapsulate domain knowledge previously expressed in a general purpose language.

The design of a new computer language requires a considerable investment. By developing a prototype of the new language that contains its essential features at an early stage, its design can be tested and adjusted if necessary. Design tools can considerably speed up the design process by generating components of a prototype.

### Language Definitions

The core of the design of a language is a language definition consisting of the description of its syntax and semantics. The syntax describes the form and structure of its sentences. The semantics describes the meaning of the syntactic constructs, which can vary from the interpretation of expressions as computer programs to the translation of expressions to another language. A \emph{formal} language definition is a definition of syntax and semantics in a formal language specification formalism, consisting of a syntax definition formalism and a logical or computational formalism for the expressing its semantics.

Language prototyping involves testing a language definition by executing it as a computer program. Execution of a language definition comprises the syntactic analysis of expressions in the language according to the syntax definition of the language and the computation of the semantics of syntactically correct expressions. A specification formalism is called executable if language definitions can be tested directly on a computer or if tools exist that construct executable computer programs from language definitions. A set of tools that supports the development of programs or specifications in some language could be called a programming environment. A programming environment for developing languages and their programming environments is called a meta-environment. Other requirements for language definition formalisms besides executability are that they support the description of existing languages, that language definitions are extensible and can be combined with other language definitions, and that the formalism is not overly restrictive.

### Syntax Definition Formalisms

A syntax definition formalism is a formal language for the specification of the syntactic rules of a language. The syntax describes the sentences of the languages and assigns a structure to these sentences. An example of a syntax definition formalism is the context-free grammar formalism introduced by \cite{Cho56}. A context-free production $A_1\ldots{}A_n\to{}A_0$ determines that a sentence of type $A_0$ can be composed by concatenating sentences of type $A_1$ to $A_n$ in that order. It can also be considered as a composition rule for trees: a tree of type $A_0$ can be composed by creating a new tree node labeled $A_0$ with trees of type $A_1$ to $A_n$ as direct descendants.

In general, a syntax definition formalism can be characterized as follows. A syntax definition (also called grammar) is a set of rules that describe how to generate a set of trees. The concatenation of all leafs nodes of a tree (the \emph{yield}) gives a sentence. The language defined by a grammar is the set of yields of the trees it generates. A parser is a function that assigns to a sentence a tree that represents its structure. If more than one tree has a sentence as its yield, the sentence is ambiguous. To solve ambiguities the syntax definition formalism may provide a disambiguation method that allows the formulation of disambiguation rules for selecting the intended tree for a sentence.

This abstract approach to syntax is also applicable to the type systems of programming languages. A signature describes the valid typed expressions (the trees), untyped expressions (the sentences) can be derived from the typed expressions, a type checker analyses untyped expresssions and assigns them a type. In particular, it is applicable to algebraic signatures.

### Algebraic Specification of Languages

An algebra is a set of data with operations on those data. In case of many-sorted algebras the data can be divided over a collection of sets. An algebraic specification is the description of an algebra. It consists of a signature declaring the types of the algebraic operations and a logical formula describing properties of these operations. Several algebraic specification formalisms have been developed that use conditional equational logic as defining logic. Equations can be executed by interpreting them as rewrite rules.

\cite{Rus72} and later also \cite{GTWW77} showed that context-free grammars correspond to algebraic signatures (see also \cite{HR76}). The composition of a tree, i.e., the construction of a new tree from subtrees, corresponds to the application of a function. In this view, languages correspond to algebras. A number of algebraic specification formalisms (for instance, OBJ, Pluss, ASF+SDF, Elan) exploit this property by using signatures with mixfix operators or even arbitrary context-free grammars instead of a prefix signature. A definition can be viewed as a context-free grammar or as an algebraic signature. The grammar view is used to generate parsers from a definition. The signature view describes the abstract syntax trees that are used by semantic tools. A mapping from parse trees to abstract syntax trees is used as interface between parser and semantic tool. In \cite{HHKR89} these views are made explicit by translating an \Sdf{} definition to a contex-free grammar (\BNF) and to a first-order algebraic signature and by providing a translation from parse trees to abstract syntax trees.

### ASF+SDF

ASF+SDF is an algebraic specification formalism designed for the specification and prototyping of programming language tools. It uses the syntax definition formalism SDF for the definition of the syntax of a language. This enables an expressive notation in specifications, since functions can be prefix, postfix, infix and mix-fix. Furthermore, the syntax of the programming language under consideration is also expressed in these signatures. The semantics of a language can be defined by means of operations on the language specified by means of conditional equations.

Language prototyping is supported by the ASF+SDF Meta-Environment \cite[]{Kli93.meta,DHK96}. It is an interactive development environment for modular ASF+SDF specifications. Given a specification of a language, a programming environment for that language is generated automatically. The use of incremental generation techniques makes that changes to the specification are immediately applied to the generated environment. This makes it possible to experiment with alternative designs.

Although the formalism in combination with the Meta-Environment provides a powerful system for language prototyping, there are several shortcomings. For instance, it is not possible to generate stand-alone environments and the evaluation of equations by means of term rewriting is interpreted instead of compiled. Currently much research is invested in overcoming these shortcomings.

### Syntax Definition for Language Prototyping

This thesis is concerned with the design and implementation of methods to enhance the expressive power and usability of the syntactic aspects of language definition formalisms. The main theme is the development of techniques for providing an \emph{expressive} syntax definition formalism. The point of departure is the syntax definition formalism SDF of \cite{HHKR89} that is used in combination with the algebraic specification formalism ASF of \cite{BHK89.asf}. This setting provides the direct background and motivation for this work, but the techniques developed are applicable in other syntax definition settings as well. There are four main results:

• Scannerless generalized-LR parsing is a new approach to parsing without scanners that solves a number of problems of conventional parsing techniques, by combining the following techniques: parsing without scanner, generalized-LR parsing, static disambiguation with priority and associativity declarations, lexical disambiguation with follow restrictions and reject productions.
• SDF2 is an expressive syntax definition formalism for context-free syntax defintion. It is a redesign of SDF as a family of orthogonally defined features for syntax definition.
• The multi-level algebraic specification formalism MLS extends first-order many-sorted algebraic specification by making the sorts used in a signature a user-definable algebraic data type. This provides a simple and uniform framework for the specification of advanced type constructs including polymorphism and higher-order functions.
• Polymorphic syntax definition is the combination of the flexible notation facilities of SDF with the flexible typing facilities of MLS.

Each of these subjects brings its own technical problems that are addressed in this thesis. In the rest of this chapter we give an overview of this development and indicate the connections.

## Part I: Context-free Parsing Techniques

Part~I describes techniques for parsing and disambiguation of context-free languages.

### Scannerless Generalized-LR Parsing

Current deterministic parsing techniques have a number of problems. These include the limitations of parser generators for deterministic languages and the complex interface between scanner and parser. Scannerless parsing is a parsing technique in which lexical and context-free syntax are integrated into one grammar and are all handled by a single context-free analysis phase. This approach has a number of advantages including discarding of the scanner and lexical disambiguation by means of the context in which a lexical token occurs. Scannerless parsing generates a number of interesting problems as well. Integrated grammars do not fit the requirements of the conventional deterministic parsing techniques. A plain context-free grammar formalism leads to unwieldy grammars, if all lexical information is included. Lexical disambiguation needs to be reformulated for use in context-free parsing.

The scannerless generalized-LR parsing approach presented in \ChapRef{Vis97.sglr} solves these problems. Grammar normalization is used to support an expressive grammar formalism without complicating the underlying machinery. Follow restrictions are used to express longest match lexical disambiguation. Reject productions are used to express the prefer keywords rule for lexical disambiguation. An adaptation of the SLR(1) parser generation algorithm is used to implement disambiguation by general priority and associativity declarations and interprets follow restrictions. Generalized-LR parsing is used to provide dynamic lookahead and to support parsing of arbitrary context-free grammars including ambiguous ones. An adaptation of the GLR algorithm supports the interpretation of grammars with reject productions.

### Disambiguation Filters

An ambiguous context-free grammar defines a language in which some sentences have multiple interpretations. For conciseness, ambiguous context-free grammars are frequently used to define even completely unambiguous languages and numerous disambiguation methods exist for specifying which interpretation is the intended one for each sentence. The existing methods can be divided in parser specific’ methods that describe how some parsing technique deals with ambiguous sentences and logical’ methods that describe the intended interpretation without reference to a specific parsing technique.

\ChapRef{KV94} proposes a framework of {\em filters\/} to describe and compare a wide range of disambiguation problems in a parser-independent way. A filter is a function that selects from a set of parse trees (the canonical representation of the interpretations of a sentence) the intended trees. A number of general properties of disambiguation filters is defined and several case studies are discussed including disambiguation by means of priorities.

### Optimizing Parsing Schemata by Disambiguation Filters

Although disambiguation filters give an abstract account of disambiguation, implementation of disambiguation by means of a filter applied to the parse forest after parsing can be too inefficient for a number of disambiguation methods. Therefore, it would be attractive if a declaratively defined disambiguation filter could be efficiently implemented by applying it during parsing or even during parser generation.

In \ChapRef{Vis95.acc} a study into the optimization of the composition of parsing algorithms and disambiguation filters is started, by considering two filters based on priorities. The first filters a set of parse trees and selects trees without priority conflict. The second selects the trees which are lowest in the multi-set ordering on parse trees induced by the priority relation on productions.

The theory of parsing schemata of \cite{Sik93} gives an abstract account of parsing algorithms. In \ChapRef{Vis95.acc} the parsing schema for Earley’s parsing algorithm is optimized by applying the two priority filters. For the priority conflict filter this results in an optimized LR(0) parser generator that yields parsers that do not produce parse trees with a priority conflict. This provides the formal derivation of the imlementation rules presented in \ChapRef{Vis97.sglr}. For a restricted case of the multi-set filter an optimization of Earley’s algorithm is derived.

## Part II: A Family of Syntax Definition Formalisms

The formalism SDF is a syntax definition formalism for specification of lexical and context-free syntax of programming languages. The design of the formalism is rather monolithic, which makes it difficult to extend with new features or experiment with the implementation. In \PartRef{Vis97.sdf} SDF is redesigned and specified as a modular and extensible family of syntax definition formalisms. Each feature is specified as an extension of a kernel formalism, orthogonal with respect to other features. The meaning of most features is expressed in terms of the primitives of the kernel formalism by means of normalization functions. One of the members of this family is SDF2, the successor of SDF.

The syntax definition formalism SDF2 is a formalism for the concise definition of context-free syntax. The semantic core of the formalism is formed by context-free grammars extended with character clases, priorities, follow restrictions and reject productions. Grammars in this basic format describe a set of parse trees to which strings are associated that form the language of the grammar. In connection with semantics specification formalisms such as ASF, a grammar is interpreted as a signature and the parse trees it generates as terms in the term algebra generated by the signature. The implementation of SDF2 consists of a grammar normalizer, a parser generator and a generic parser. It supports arbitrary context-free grammars using the GLR parsing algorithm.

One of the main contributions of SDF2 is the complete integration of lexical and context-free syntax. The formalism supports the definition of lexical and context-free syntax providing a separate name space for symbols such that interference is prevented. The grammar normalizer integrates lexical and context-free syntax into a single context-free grammar. The scannerless parser generated for such a grammar reads input characters directly and combines lexical analysis with context-free analysis in a single parsing phase.

Ambiguous grammars can be disambiguated by means of three disambiguation facilities. Priority and associativity declarations can be used to disambiguate mixfix expression grammars in a very general way. Disambiguation by means of priorities is implemented in the parser generator. For the disambiguation of lexical ambiguities two features are introduced. With follow restrictions the follow-set of grammar symbols can be restricted, which enables the expression of the prefer longest match’ disambiguation rule. With reject productions one can express the prefer keywords’ rule. Follow restrictions are interpreted during parser generation, reject productions are interpreted during parsing.

Other disambiguation methods can be defined as filters on parse forests (compact representations of sets of parse trees). Due to the open design of the SDF2 implementation such filters can be easily attached to the parser. A number of case studies of disambiguation filters are discussed in \ChapRef{KV94}.

Other features of SDF2 are literals, an expressive set of regular expressions, and symbol aliases that serve to abbreviate complicated regular expressions. Furthermore, the formalism supports modular syntax definitions and flexible reuse of modules by means of symbol and production renamings.

Many of the features are defined in terms of the core features by means of a normalization function on syntax definitions. The formalism can be coupled to any semantics specification language based on first-order many-sorted signatures providing user-definable syntax. The modular design of the formalism supports experiments with new features.

\Chapter{Family} gives an introduction to SDF2 and discusses the approach of designing it as a family of syntax definition formalisms. \Chapter{CFG} defines the kernel of the family consisting of context-free grammars with sorts, character classes and literals. The semantics of the formalism is defined by means of a well-formedness predicate on parse trees characterizing the trees generated by a grammar. \Chapter{DisAbr} defines disambiguation by means of priorities, follow restrictions and reject productions. Regular expressions are defined to abbreviate several common patterns in syntax definitions such as lists, optional constructs and tuples. The integration of lexical and context-free syntax is defined. \Chapter{RenMod} introduces a renaming operator on grammars that can be used to rename sorts and productions. Renamings are then used to define symbol aliases. A module mechanism is defined that supports the modularization of syntax definitions. Modules can be parameterized with a list of symbols and renamings can be applied to imports. \Chapter{SDF2} discusses the assembly of SDF2 from the features defined in the previous chapters and discusses possible improvements.

## Part III: Multi-Level Algebraic Specification

Polymorphic, higher-order functions in functional programming languages provide a powerful abstraction method to construct reusable software. The first-order signatures provided by conventional many-sorted first-order algebraic specification formalisms (such as ASF+SDF) do not support polymorphic or higher-order functions. In Part~II a multi-level algebraic specification formalism is designed and specified in order to study the extension of first-order formalisms with polymorphism and higher-order functions.

The multi-level specification formalism MLS extends first-order many-sorted algebraic specification by making the algebra of types a user-definable data type. The structure of the types used in the signature of a specification is specified by means of an algebraic specification itself. This process is formalized in a multi-level setting. The terms over a signature at level $i+1$ can be used as type expressions at level $i$. Variables in type expressions are interpreted as universally quantified type parameters. Function declarations with such a universally quantified type are interpreted as declaration schemata for functions with closed type expressions and thus represent polymorphic functions and constants. Functions can also be overloaded, i.e., have more than one type. The term structure is applicative, enabling higher-order functions. The formalism supports modular specifications.

The formalism MLS is defined by means of a specification in ASF+SDF. This specification also forms the basis for a prototype environment for MLS. The environment consists of a typechecker that is defined in terms of a well-formedness checker and a type assignment procedure. Type assignment is an extension of the Hindley/Milner algorithm to many-sorted types, multi-levels and overloading of functions. Furthermore, the environment contains a term rewrite interpreter for equations in specifications.

Applications of multi-level specifications include all functional programs expressible in a Hindley/Milner system. Due to the many-sortedness of the signatures of types and kinds (as opposed to the single-sorted types of functional languages) more distinction can be made in type assignment. This enables the definition of data types such as stratified stacks and tuples. By means of equations over types still more advanced typing constructs can be modeled. An example is the type of the {\tt zip} function that maps a list of tuples to a tuple of lists. Other applications of type equations are type abbreviations, recursive types, record types, the polytypic functions of \cite{JJ97}, the type classes of Haskell and the constructor classes of \cite{Jon93}. The specification of type assignment presented here only deals with syntactic equality and not with equality modulo type equations.

\ChapRef{IntroMLS} gives an introduction to the formalism and discusses related work. \ChapRef{DefOLS} handles the case of specifications consisting of a single level. This corresponds to first-order algebraic specification. First an untyped equational specification formalism with equational logic and term rewriting is defined. This is extended with a first-order monomorphic applicative type system. In \ChapRef{XmplMLS} the possibilities of multi-level specifications are explained by means of a number of example specifications. The formalism is defined in \Chapter{DefMLS}, building on the language of \ChapRef{DefOLS}. Appendix~\absref{Chap:AppMLS} defines several auxiliary tools such as substitution, matching and unification.

## Part IV: Polymorphic Syntax Definition

The signatures of multi-level specifications only support prefix and infix functions. \ChapRef{Vis97.psd} develops theory to combine the type flexibility of multi-level specifications with the notational flexibility of context-free grammars.

The combination of the idea of grammars as signatures with multi-level algebraic specification leads to a multi-level grammar formalism. In a multi-level grammar the set of non-terminals becomes a user-definable data type in the same way as the types in multi-level specifications. Moreover, types and object level data are specified by means of a context-free grammar instead of with a signature, leading to flexible notation.

The combination provides a formalism for polymorphic syntax definition, in which common language constructs can be described generically and reused in many specifications. It turns out that while both formalisms have a decidable type-assignment/parsing problem, the combination in its full generality has an undecidable parsing problem. However, a subset of such multi-level grammars can be characterized that have a decidable parsing problem, while not being too restrictive for use in abstract data type specification. For this class of grammars that satisfy the finite-chain property, a parsing algorithm is presented.

When restricted to two levels we have a formalism that is similar to Van Wijngaarden grammars. The difference is that VWGs use derived strings with variables (sentential forms) as types at level~0, while our two-level grammars use parse trees with variables. This restriction ensures that syntactic unification is decidable, which it is not in VWGs. The further restriction of two-level grammars to grammars that satisfy the finite-chain property results in grammars with a decidable parsing problem. Van Wijngaarden grammars were not succesful in executable definition of programming languages. The discovery that $\epsilon$-productions could be used to encode the static and even dynamic semantics of a language led to a formalism with a very difficult parsing problem. This sparked developments in the usage of VWGs as a programming language instead of a grammar formalism. In \ChapRef{Vis97.psd} it is shown that this development has hidden the very useful application of two-level grammars to polymorphic syntax definition, opening the flexibility of polymorphism to grammar development.

## Origins of the Chapters

Most of the chapters in this thesis were published before as separate papers. We list their origin.

• Chapter 3 on the scannerless generalized-LR parsing approach is a new paper that gives an overview of the design and implementation of the SDF2 normalizer, parser generator and parser. It appeared as a technical report P9707.

• Chapter 4 on disambiguation filters is joint work with Paul Klint. It was presented under the title using filters for the disambiguation of context-free grammars at the ASMICS Workshop on Parsing Theory in Milan and appeared in the proceedings.

• Chapter 5 was presented at the Accolade{95 conference on logic in Amsterdam and appeared in the proceedings. It has been accepted for presentation at the International Workshop on Parsing Technology (IWPT’97) in Boston and for publication in the proceedings.

• Part II on the specification of SDF2 as a family of syntax definition formalisms is an update and extension of a paper that was presented at the ASF+SDF’95 workshop on Generating Tools from Algebraic Specifications and appeared in the proceedings. In its current form it appeared as technical report P9706

• Part III appeared as a single chapter in the book Language Prototyping. An Algebraic Specification Approach. The version in this thesis has been split up in five chapters and several example specifications have been added. Furthermore, the specification has been improved in a few places.

• An extended abstract of Chapter 15 was presented at the AMAST workshop on Algebraic Methods in Language Processing (AMiLP[^?^](/bin/edit/EelcoVisser/AMiLP?topicparent=EelcoVisser.SyntaxDefinitionForLanguagePrototypingIntro)’95) in Enschede and was published in the proceedings. The current version is accepted for publication in a special issue of Theoretical Computer Science dedicated to the workshop.