Mircea Marin

A survey of the Theorema project

The Theorems project aims at extending current computer algebra systems by facilities jor supporting mathematical proving. The present early-prototype version of the Theorems software system is implemented in Mathetnatica 3.0. The system consists of a general higher-order predicate logic prover and a collection of special provers that call each other depending on the particular proof situations. The individual provers imitate the proof style of human mathematicians and aim at producing human-readable proofs in natuml language presented in nested cells that facilitate studying the computer-generated proofs at various levels of detail. The special provers are intimately connected with the junctors that build up the various mathematical domains. 1 The Objectives of the Theorems Project The Tlaeorema project aims at providing a uniform (logic and software) frame for computing, solving, and proving. In a simplified view, given a “knowledge base” K of formulae (and a logical / computational derivation mechanism L),

Matching with regular constraints

We describe a sound, terminating, and complete matching algorithm for terms built over flexible arity function symbols and context, function, sequence, and individual variables. Context and sequence variables allow matching to move in term trees to arbitrary depth and breadth, respectively. The values of variables can be constrained by regular expressions which are not necessarily linear. We describe heuristics for optimization, and discuss applications.

Foundations of the rule-based system ρLog

We describe the foundations of a system for rule-based programming which integrates two powerful mechanisms: (1) matching with context variables, sequence variables, and regular constraints for their matching values; and (2) strategic programming with labeled rules. The system is called ρLog, and is built on top of the pattern matching and rule-based programming capabilities of Mathematica.

Computational construction of a maximum equilateral triangle inscribed in an origami

We present an origami construction of a maximum equilateral triangle inscribed in an origami, and an automated proof of the correctness of the construction. The construction and the correctness proof are achieved by a computational origami system called Eos (E-origami system). In the construction we apply the techniques of geometrical constraint solving, and in the automated proof we apply Grobner bases theory and the cylindrical algebraic decomposition method. The cylindrical algebraic decomposition is indispensable to the automated proof of the maximality since the specification of this property involves the notion of inequalities. The interplay of construction and proof by Grobner bases method and the cylindrical algebraic decomposition supported by Eos is the feature of our work.

Programming with Sequence Variables: the Sequentica Package

Sequence variables are an advanced feature of modern programming languages. They enhance the support for writing programs in a declarative and easily understood way. To our knowledge, Mathematica provides the best support for programming with sequence variables, but it requires a good understanding of how the interpreter chooses the matcher. This is so because matching against patterns with sequence variables is in general not unitary. We claim that there is room to improve the programming style with sequence variables. We propose a number of new programming constructs which impose certain strategies on the pattern matching process. Our constructs enable to control the selection of a matcher by annotating sequence variables with binding priorities and ranges for their lengths, and to compute optimal values characterized by a score function to be optimized. To this end we have developed the package Sequentica. With Sequentica the Mathematica programmers and users get additional support for defining functions and transformation rules in an easy and convenient way. We outline the algorithmic difficulties to support these extensions and describe how they are implemented in Sequentica. The usefulness of these extensions is illustrated with various examples. We regard these extensions as a first step towards identifying a new programming style: programming with sequence variables. Such a programming style is useful to solve problems based on sequence analysis such as bio-informatics, cryptography or data mining.

Logical and algebraic view of Huzita's origami axioms with applications to computational origami

We describe Huzitas origami axioms from the logical and algebraic points of view. Observing that Huzitas axioms are statements about the existence of certain origami constructions, we can generate basic origami constructions from those axioms. Origami construction is performed by repeated application of Huzitas axioms. We give the logical specification of Huzitas axioms as constraints among geometric objects of origami in the language of the first-order predicate logic. The logical specification is then translated into logical combinations of algebraic forms, i.e. polynomial equalities, disequalities and inequalities, and further into polynomial ideals (if inequalities are not involved). By constraint solving, we obtain solutions that satisfy the logical specification of the origami construction problem. The solutions include fold lines along which origami paper has to be folded. The obtained solutions both in numeric and symbolic forms make origami computationally tractable for further treatments, such as visualization and automated theorem proving of the correctness of the origami construction.

On Reducing the Search Space of Higher-Order Lazy Narrowing

Higher-order lazy narrowing is a general method for solving E-unification problems in theories presented as sets of rewrite rules. In this paper we study the possibility of improving the search for normalized solutions of a higher-order lazy narrowing calculus LN. We introduce a new calculus, LNff, obtained by extending LN and define an equation selection strategy S n such that LNff with strategy S n is complete. The main advantages of using LNff with strategy S n instead of LN include the possibility of restricting the application of outermost narrowing at variable position, and the computation of more specific solutions because of additional inference rules for solving flex-flex equations. We also show that for orthogonal pattern rewrite systems we can adopt an eager variable elimination strategy that makes the calculus LNff with strategy S n even more deterministic.

Cooperative constraint functional logic programming

Describes the current status of the development of CFLP (constraint functional logic programming), a system which aims at the integration of the best features of functional logic programming (FLP), cooperative constraint solving (CCS) and distributed constraint solving. FLP provides support for defining ones own abstractions (user-defined functions and predicates) over a constraint domain in an easy and comfortable way, whereas CCS is employed to solve systems of mixed constraints by iterating specialized constraint-solving methods in accordance with a well-defined strategy. CFLP is a distributed implementation of a cooperative FLP scheme obtained from the integration of higher-order lazy narrowing for FLP with CCS. The implementation takes advantage of the existence of several constraint-solving resources located in a distributed environment, which communicate asynchronously via message passing.

New completeness results for lazy conditional narrowing

We show the completeness of the lazy conditional narrowing calculus (LCNC) with leftmost selection for the class of deterministic conditional rewrite systems (CTRSs). Deterministic CTRSs permit extra variables in the right-hand sides and conditions of their rewrite rules. From the completeness proof we obtain several insights to make the calculus more deterministic. Furthermore, and similar to the refinements developed for the unconditional case, we succeeded in removing all nondeterminism due to the choice of the inference rule of LCNC by imposing further syntactic conditions on the participating CTRSs and restricting the set of solutions for which completeness needs to be established.

Solving, Reasoning, and Programming in Common Logic

Common Logic (CL) is a recent ISO standard for exchanging logic-based information between disparate computer systems. Sharing and reasoning upon knowledge represented in CL require equation solving over terms of this language. We study computationally well-behaved fragments of such solving problems and show how they can influence reasoning in CL and transformations of CL expressions.