Artur Korniłowicz

A SAT Based Approach for Solving Formulas over Boolean and Linear Mathematical Propositions

The availability of decision procedures for combinations of boolean and linear mathematical propositions opens the ability to solve problems arising from real-world domains such as verification of timed systems and planning with resources. In this paper we present a general and efficient approach to the problem, based on two main ingredients. The first is a DPLL-based SAT procedure, for dealing efficiently with the propositional component of the problem. The second is a tight integration, within the DPLL architecture, of a set of mathematical deciders for theories of increasing expressive power. A preliminary experimental evaluation shows the potential of the approach.

Bounded Model Checking for Timed Systems

Enormous progress has been achieved in the last decade in the verification of timed systems, making it possible to analyze significant real-world protocols. An open challenge is the identification of fully symbolic verification techniques, able to deal effectively with the finite state component as well as with the timing aspects. In this paper we propose a new, symbolic verification technique that extends the Bounded Model Checking (BMC) approach for the verification of timed systems. The approach is based on the following ingredients. First, a BMC problem for timed systems is reduced to the satisfiability of a math-formula, i.e., a boolean combination of propositional variables and linear mathematical relations over real variables (used to represent clocks). Then, an appropriate solver, called MATHSAT, is used to check the satisfiability of the math-formula. The solver is based on the integration of SAT techniques with some specialized decision procedures for linear mathematical constraints, and requires polynomial memory. Our methods allow for handling expressive properties in a fully-symbolic way. A preliminary experimental evaluation confirms the potential of the approach.

Mizar in a Nutshell

This paper is intended to be a practical reference manual for basic Mizar terminology which may be helpful to get started using the system. The paper describes most important aspects of the Mizar language as well as some features of the verification software.

A Brief Overview of Mizar

Mizar is the name of a formal language derived from informal mathematics and computer software that enables proof-checking of texts written in that language. The system has been actively developed since 1970s, growing into a popular proof assistant accompanied with a huge repository of formalized mathematical knowledge. In this short overview, we give an outline of the key features of the Mizar language, the ideas and theory behind the system, its main applications, and current development.

Mizar: State-of-the-art and Beyond

Mizari¾?is one of the pioneering systems for mathematics formalization, which still has ani¾?active user community. The project has been in constant development since 1973, when Andrzej Trybulec designed the fundamentals of ai¾?language capable of rigorously encoding mathematical knowledge in ai¾?computerized environment which guarantees its full logical correctness. Since then, the system with its feature-rich language devised to approximate mathematics writing has influenced other formalization projects and has given rise to ai¾?number of Mizari¾?modes implemented on top of other systems. However, the information about the system as ai¾?whole is not readily available to developers of other systems. Various papers describing Mizari¾?features have been rather incremental and focused only on particular newly implemented Mizari¾?aspects. The objective of the current paper is to give ai¾?survey of the most important Mizari¾?features that distinguish it from other popular proof checkers. We also go ai¾?step further and describe most important current trends and lines of development that go beyond the state-of-the-art system.

Four Decades of Mizar

This special issue is dedicated to works related to Mizar,xa0the theorem proving project started by Andrzej Trybulec in the 1970s, and other automated proof checking systems used for formalizing mathematics.

On Rewriting Rules in Mizar

This paper presents some tentative experiments in using axa0special case of rewriting rules in Mizar (Mizar homepage: http://www.mizar.org/): rewriting axa0term as its subterm. Axa0similar technique, but based on another Mizar mechanism called functor identification (Korniłowicz 2009) was used by Caminati, in his paper on basic first-order model theory in Mizar (Caminati, J Form Reason 3(1):49–77, 2010, Form Math 19(3):157–169, 2011). However for this purpose he was obligated to introduce some artificial functors. The mechanism presented in the present paper looks promising and fits the Mizar paradigm.

Integrating Boolean and Mathematical Solving: Foundations, Basic Algorithms, and Requirements

In the last years we have witnessed an impressive advance in the efficiency of boolean solving techniques, which has brought large previously intractable problems at the reach of state-of-the-art solvers. Unfortunately, simple boolean expressions are not expressive enough for representing many real-world problems, which require handling also integer or real values and operators. On the other hand, mathematical solvers, like computer-algebra systems or constraint solvers, cannot handle efficiently problems involving heavy boolean search, or do not handle them at all. In this paper we present the foundations and the basic algorithms for a new class of procedures for solving boolean combinations of mathematical propositions, which combine boolean and mathematical solvers, and we highlight the main requirements that boolean and mathematical solvers must fulfill in order to achieve the maximum benefits from their integration. Finally we show how existing systems are captured by our framework.

Formal Mathematics for Mathematicians

The collection of works for this special issue was inspired by the presentations given at the 2011 AMS Special Session on Formal Mathematics for Mathematicians: Developing Large Repositories of Advanced Mathematics. The issue features a collection of articles by practitioners of formalizing proofs who share a deep interest in making computerized mathematics widely available.

Flexary connectives in Mizar

One of the main components of the Mizar project is the Mizar language, a computer language invented to reflect the natural language of mathematics. From the very beginning various linguistic constructions and grammar rules which enable us to write texts which resemble classical mathematical papers have been developed and implemented in the language.The Mizar Mathematical Library is a repository of computer-verified mathematical texts written in the Mizar language. Besides well-known and important theorems, the library contains series of some quite technical lemmas describing some properties formulated for different values of numbers. For example the sequence of lemmas fornbeingNatstn <=1holdsn=0orn=1;fornbeingNatstn <=2holdsn=0orn=1orn=2;fornbeingNatstn <=3holdsn=0orn=1orn=2orn=3;which for a long time contained 13 such formulae.In this paper, we present an extension of the Mizar language - an ellipsis that is used to define flexary logical connectives. We define flexary conjunction and flexary disjunction, which can be understood as generalization of classical conjunction and classical disjunction, respectively. The proposed extension enables us to get rid of lists of such lemmas and to formulate them as single theorems, e.g.form,nbeingNatstn <=mholdsn=0or ... orn=m;covering all cases between the bounds 0 and m in this case.Moreover, a specific inference rule to process flexary formulae, formulae using flexary connectives, is introduced. We describe how ellipses are incorporated into the Mizar language and how they are verified by the Mizar proof checker. HighlightsTwo new logical connectives are introduced to the Mizar language.Inference rules to deal with the connectives are introduced.Processing of the connectives by the Mizar proof checker is described.