Jerzy Tyszkiewicz

Mathematical Foundations of Computer Science 2008

This book constitutes the refereed proceedings of the 33rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2008, held in Torun, Poland, in August 2008. The 45 revised full papers presented together with 5 invited lectures were carefully reviewed and selected from 119 submissions. All current aspects in theoretical computer science and its mathematical foundations are addressed, ranging from algorithmic game theory, algorithms and data structures, artificial intelligence, automata and formal languages, bioinformatics, complexity, concurrency and petrinets, cryptography and security, logic and formal specifications, models of computations, parallel and distributed computing, semantics and verification.

Statistical properties of simple types

We consider types and typed lambda calculus over a finite number of ground types. We are going to investigate the size of the fraction of inhabited types of the given length n against the number of all types of length n. The plan of this paper is to find the limit of that fraction when n → ∞. The answer to this question is equivalent to finding the ‘density’ of inhabited types in the set of all types, or the so-called asymptotic probability of finding an inhabited type in the set of all types. Under the Curry–Howard isomorphism this means finding the density or asymptotic probability of provable intuitionistic propositional formulas in the set of all formulas. For types with one ground type (formulas with one propositional variable), we prove that the limit exists and is equal to 1s2 + √5s10, which is approximately 72.36%. This means that a long random type (formula) has this probability of being inhabited (tautology). We also prove that for every finite number k of ground-type variables, the density of inhabited types is always positive and lies between (4k + 1)s(2k + 1)2 and (3k + 1)s(k + 1)2. Therefore we can easily see that the density is decreasing to 0 with k going to infinity. From the lower and upper bounds presented we can deduce that at least 1s3 of classical tautologies are intuitionistic.

Spreadsheet as a relational database engine

Spreadsheets are among the most commonly used applications for data management and analysis. Perhaps they are even among the most widely used computer applications of all kinds. However, the spreadsheet paradigm of computation still lacks sufficient analysis. In this paper we demonstrate that a spreadsheet can play the role of a relational database engine, without any use of macros or built-in programming languages, merely by utilizing spreadsheet formulas. We achieve that by implementing all operators of relational algebra by means of spreadsheet functions. Given a definition of a database in SQL, it is therefore possible to construct a spreadsheet workbook with empty worksheets for data tables and worksheets filled with formulas for queries. From then on, when the user enters, alters or deletes data in the data worksheets, the formulas in query worksheets automatically compute the actual results of the queries. Thus, the spreadsheet serves as data storage and executes SQL queries, and therefore acts as a relational database engine. The paper is based on Microsoft Excel (TM), but our constructions work in other spreadsheet systems, too. We present a number of performance tests conducted in the beta version of Excel 2010. Their conclusion is that the performance is sufficient for a desktop database with a couple thousand rows.

The Semijoin Algebra and the Guarded Fragment

In the 1970s Codd introduced the relational algebra, with operators selection, projection, union, difference and product, and showed that it is equivalent to first-order logic. In this paper, we show that if we replace in Codd’s relational algebra the product operator by the “semijoin” operator, then the resulting “semijoin algebra” is equivalent to the guarded fragment of first-order logic. We also define a fixed point extension of the semijoin algebra that corresponds to μGF.

Petri net + nested relational calculus = dataflow

In this paper we propose a formal, graphical workflow language for dataflows, i.e., workflows where large amounts of complex data are manipulated and the structure of the manipulated data is reflected in the structure of the workflow. It is a common extension of Petri nets, which are responsible for the organization of the processing tasks, and Nested relational calculus, which is a database query language over complex objects, and is responsible for handling collections of data items (in particular, for iteration) and for the typing system. We demonstrate that dataflows constructed in hierarchical manner, according to a set of refinement rules we propose, are sound: initiated with a single token (which may represent a complex scientific data collection) in the input node, terminate with a single token in the output node (which represents the output data collection). In particular they always process all of the input data, leave no ”debris data” behind and the output is always eventually computed.

Database Query Processing Using Finite Cursor Machines

We introduce a new abstract model of database query processing, finite cursor machines, that incorporates certain data streaming aspects. The model describes quite faithfully what happens in so-called “one-pass” and “two-pass query processing”. Technically, the model is described in the framework of abstract state machines. Our main results are upper and lower bounds for processing relational algebra queries in this model, specifically, queries of the semijoin fragment of the relational algebra.

Distributed computation of web queries using automata

We introduce and investigate a distributed computation model for querying the Web. Web queries are computed by interacting automata running at different nodes in the Web. The automata which we are concerned with can be viewed as register automata equipped with an additional communication component. We identify conditions necessary and sufficient for systems of automata to compute Web queries, and investigate the computational power of such systems.

Infinitary queries and their asymptotic probabilities. II. Properties definable in least fixed point logic

We develop an almost complete theory for existence of asymptotic probabilities of least fixed point and partial fixed pint definable properties, and for the complexity of the associated almost sure theory. Our method works for any randomized class of finite structures.

A formal model of dataflow repositories

Dataflow repositories are databases containing dataflows and their different runs. We propose a formal conceptual data model for such repositories. Our model includes careful formalisations of such features as complex data manipulation, external service calls, subdataflows, and the provenance of output values.

On the expressive power of semijoin queries

The semijoin algebra is the variant of the relational algebra obtained by replacing the join operator by the semijoin operator. We provide an Ehrenfeucht-Fraisse game, characterizing the discerning power of the semijoin algebra. This game gives a method for showing that queries are not expressible in the semijoin algebra.