An approach to complexity theory which offers a means of analysing algorithms in terms of their tractability. The authors consider the problem in terms of parameterized languages and taking "k-slices" of the language, thus introducing readers to new classes of algorithms which may be analysed more precisely than was the case until now. The book is as self-contained as possible and includes a great deal of background material. As a result, computer scientists, mathematicians, and graduate students interested in the design and analysis of algorithms will find much of interest.

Parameterized Complexity

A type system is a syntactic method for automatically checking the absence of certain erroneous behaviors by classifying program phrases according to the kinds of values they compute. The study of type systems -- and of programming languages from a type-theoretic perspective -- has important applications in software engineering, language design, high-performance compilers, and security.This text provides a comprehensive introduction both to type systems in computer science and to the basic theory of programming languages. The approach is pragmatic and operational; each new concept is motivated by programming examples and the more theoretical sections are driven by the needs of implementations. Each chapter is accompanied by numerous exercises and solutions, as well as a running implementation, available via the Web. Dependencies between chapters are explicitly identified, allowing readers to choose a variety of paths through the material.The core topics include the untyped lambda-calculus, simple type systems, type reconstruction, universal and existential polymorphism, subtyping, bounded quantification, recursive types, kinds, and type operators. Extended case studies develop a variety of approaches to modeling the features of object-oriented languages.

https://www.seas.upenn.edu/~bcpierce/tapl/contents.pdf

Types and Programming Languages

This paper starts with the project of finding a large subclass of NP which exhibits a dichotomy. The approach is to find this subclass via syntactic prescriptions. While the paper does not achieve this goal, it does isolate a class (of problems specified by) "monotone monadic SNP without inequality" which may exhibit this dichotomy. We justify the placing of all these restrictions by showing, essentially using Ladner's theorem, that classes obtained by using only two of the above three restrictions do not show this dichotomy. We then explore the structure of this class. We show that all problems in this class reduce to the seemingly simpler class CSP. We divide CSP into subclasses and try to unify the collection of all known polytime algorithms for CSP problems and extract properties that make CSP problems NP-hard. This is where the second part of the title, "a study through Datalog and group theory," comes in. We present conjectures about this class which would end in showing the dichotomy.

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The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory

This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.

http://types10.mimuw.edu.pl/content/slides/day1/HenkBarendregt.pdf

Lambda Calculus with Types

This article surveys various complexity and expressiveness results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results on plain logic programming (pure Horn clause programs), more recent results on various important extensions of logic programming are surveyed. These include logic programming with different forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming.

/pdf/complexity-and-expressive-power-of-logic-programming-334cjkp9dg.pdf

Complexity and expressive power of logic programming

Datalog is the language of logic programs without function symbols. It is used as a database query language. If it is possible to eliminate recursion from a Datalog program F’, then t’ is said to be bounded. It is shown that the problem of deciding whether a given Datalog program is bounded is undecidable, even for linear programs (i.e., programs in which each rule contains at most one occurrence of a recursive predicate). It is then shown that every semantic property of Datalog programs is undecidable if it is stable, is strongly nontrivial, and contains An earlier version of this work appeared under the same title in the Proceedings of the 2nd IEEE Symposium on Logic i~z Computer Science (Ithaca, N.Y.). IEEE, New York, 1987, pp. 106-115. Most of the research reported here was done while H. Gaifman was visiting the AI Center of SRI International whose support he wishes to acknowledge. He also wishes to thank IBM Watson Research Center and IBM Almaden Research Center for support in the summer of 1989, when the concluding work on this paper was done. The research reported here was done partly while H. Mairson was at the Computer Science Department of Stanford University and was supported by the Office of Naval Research (ONR) contract NOO014-85-C-0731 and partly while he was at the Programming Research Group of Oxford University. The research reported here was done while Y. Sagiv was visiting the Computer Science Department of Stanford University and was supported by a grant of AT & T Foundation, a grant of IBM Corporation and the National Science Foundation (NSF) grant 1ST 84-12791. Authors’ addresses: H. Gaifman and Y. Sagiv, Hebrew University, Jerusalem 91904, Israel; H. Mairson, Department of Computer Science, Brandeis University, Waltham, MA 02254; M. Y. Vardi, IBM Almaden Research Center, K53-802, 650 Harry Road, San Jose, CA 95120-6099. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage. the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 01993 ACM 0004-5411/93/0700-0683 $01.50 Journal of the Awxmt]on for Computing Machinery, VO1 40, No 3. July 1993. PP 683-713 684 H. GAIFMAN ET AL. boundedness. In particular, the property of being first-order 1s undecidable and (assuming that PTIME 1s different from LOGSPACE and from NC) the same holds for the property of being equivalent to a linear program and for the properties of being in LOGSPACE and of being in NC

Undecidable Optimization Problems for Database Logic Programs

A well known but incorrect piece of functional programming folklore is that ML expressions can be efficiently typed in polynomial time. In probing the truth of that folklore, various researchers, including Wand, Buneman, Kanellakis, and Mitchell, constructed simple counterexamples consisting of typable ML programs having length n, with principal types having O(2cn) distinct type variables and length O(22cn). When the types associated with these ML constructions were represented as directed acyclic graphs, their sizes grew as O(2cn). The folklore was even more strongly contradicted by the recent result of Kanellakis and Mitchell that simply deciding whether or not an ML expression is typable is PSPACE-hard.We improve the latter result, showing that deciding ML typability is DEXPTIME-hard. As Kanellakis and Mitchell have shown containment in DEXPTIME, the problem is DEXPTIME-complete. The proof of DEXPTIME-hardness is carried out via a generic reduction: it consists of a very straightforward simulation of any deterministic one-tape Turing machine M with input k running in O(c|k|) time by a polynomial-sized ML formula PM,k, such that M accepts k iff PM,k is typable. The simulation of the transition function d of the Turing Machine is realized uniquely through terms in the lambda calculus without the use of the polymorphic let construct. We use let for two purposes only: to generate an exponential amount of blank tape for the Turing Machine simulation to begin, and to compose an exponential number of applications of the ML formula simulating state transition.It is purely the expressive power of ML polymorphism to succinctly express function composition which results in a proof of DEXPTIME-hardness. We conjecture that lower bounds on deciding typability for extensions to the typed lambda calculus can be regarded precisely in terms of this expressive capacity for succinct function composition.To further understand this lower bound, we relate it to the problem of proving equality of type variables in a system of type equations generated from an ML expression with let-polymorphism. We show that given an oracle for solving this problem, deciding typability would be in PSPACE, as would be the actual computation of the principal type of the expression, were it indeed typable.

Deciding ML typability is complete for deterministic exponential time

We consider the problem of computing all intersections between two sets S and T of line segments in the plane, where no two segments in S (similarly, T) intersect. We present an asymptotically optimal algorithm which reports all those intersections in O(n log n + k) time and O(n) space, where n is the total number of line segments, and k is the number of intersections. Our algorithm works also for general arcs of single-valued curves, within the same time bounds. Applications include the intersection of two polygons and the “merge” of two planar maps in time O(n log n + m) and space O(n + m), where n and m are the number of input and output edges, respectively. In the case of straight line segments, a simple modification allows us to count the number of intersections (without reporting them) in time O((n +√ nk)log n) = O(n 1.5log n).

Reporting and Counting Intersections Between Two Sets of Line Segments

Undecidable optimization problems for database logic programs

A given Datalog program is bounded if its depth of recursion is independent of the input database. Deciding boundedness is a basic task for the analysis of database logic programs. The undecidability of Datalog boundedness was first demonstrated by Gaifman et al. [7]. We introduce new techniques for proving the undecidability of (various kinds of) boundedness, which allow us to considerably strengthen the results of Gaifman et al. [7]. In particular, (1) we use a new generic reduction technique to show that program boundedness is undecidable for arity 2 predicates, even with linear rules; (2) we use the mortality problem of Turing machines to show that uniform boundedness is undecidable for arity 3 predicates and for arity 1 predicates when ≠ is also allowed; (3) by encoding all possible transitions of a two-counter machine in a single rule, we show that program (resp., predicate) boundedness is undecidable for two linear rules (resp., one rule and a projection) and one initialization rule, where all predicates have small arities (6 or 7).

Harry G. Mairson

Papers

Undecidable Optimization Problems for Database Logic Programs

Deciding ML typability is complete for deterministic exponential time

Reporting and Counting Intersections Between Two Sets of Line Segments

Undecidable optimization problems for database logic programs

Undecidable boundedness problems for datalog programs