%Nr: DS-1995-19
%Author: Erik Aarts
%Title: Investigations in Logic, Language and Computation

As the title says, this PhD thesis is on the interface between logic, 
language and computation. The main perspective of this thesis is that 
of complexity theory. The merit of applying complexity theory to certain 
problems is that complexity analyses provide different kinds of information. 
Complexity analysis does not only show how difficult a problem is, but also 
why it is difficult, and possibly how we can change the problem such that 
it becomes less difficult. 
The thesis is about the complexity of two grammar formalisms and a program­
ming language. The grammar formalisms are categorial grammar based on Lambek 
calculi and a restricted form of the context­sensitive rewrite grammars: the 
acyclic context­sensitive grammars. The programming language is Prolog. 
Complexity theory distinguishes two types of complexity: time complexity and 
space complexity. Time complexity is defined as the number of steps that 
some machine needs to solve a problem. Space complexity is the amount of 
memory the machine needs. For an introduction in complexity theory see
Garey and Johnson (1979). The problems we are going to analyze (i.e. 
estimating how much space and time it costs to solve these problems) are the 
following. 
For the grammar formalisms we have some generator G that generates 
sentences. In categorial grammar, the generator consists of a lexicon of 
words associated with types, and some (Lambek based) calculus. In acyclic 
context­sensitive grammars, the generator consists of a context­sensitive 
grammar. Besides the generator we also have an input sentence. The problem 
we try to solve is the following: "Is the input sentence generated by the 
generator?". 
A Prolog program consists of a number of facts and rules. The rules say how 
we can derive new conclusions from the given facts. The problem we will 
discuss in this thesis is: "Given a program with rules and facts, is 
conclusion C derivable?". 
At first sight there seems to be no relation between the recognition problem 
for grammar formalisms and the derivability problem in Prolog. But there are 
similarities. The Prolog rules can be seen as a representation of infinitely 
many context­free rewrite rules. Every substitution for the variables gives 
another rewrite rule. The facts can be seen as infinitely many so­called 
epsilon­rules, i.e. they rewrite to the empty string. Suppose we want to 
know whether a conclusion C is derivable from a program. We transform the 
program to an "infinite context­free grammar" with start symbol C. Then the 
derivability problem is equivalent to the question whether the empty string 
is grammatical. We have reduced the derivability problem to a recognition 
problem.
The fact that the grammar is infinite makes that we can not simply use 
parsing theory for Prolog theorem provers. But we can use elements of 
parsing theory to obtain theorem provers. The relationship between Prolog 
and grammars has been worked out in (Deransart and Maluszynski 1993). They 
show that Prolog programs are equivalent to W­grammars. 


Organization of the thesis:

The thesis consists of two parts that can be read separately: 
"Grammar Formalisms", and "Programming in Logic".

Part I consists of the following chapters:   

  - Chapter 1 Introduction   
  - Chapter 2 Non­associative Lambek calculus NL
  - Chapter 3 The second order fragment of Lambek calculus
  - Chapter 4 Acyclic Context­Sensitive Grammars

Part II consists of the following chapters
  - Chapter 5 Complexity of Prolog programs
  - Chapter 6 Proof of the time complexity result
  - Chapter 7 The link between the first part and the second part. It shows 
    how we can use the method from chapter 5 for proving that membership for 
    fragments of L can be computed in polynomial time. 

Appendix A contains an implementation in Prolog of the efficient meta­
interpreter described in chapter 6. Appendix B contains a reduction from 
3SAT to ACSG.