Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

Word equations are a crucial element in the theoretical foundation of constraint solving over strings, which have received a lot of attention in recent years. A word equation relates two words over string variables and constants. Its solution amounts to a function mapping variables to constant strings that equate the left and right hand sides of the equation. While the problem of solving word equations is decidable, the decidability of the problem of solving a word equation with a length constraint (i.e., a constraint relating the lengths of words in the word equation) has remained a long-standing open problem. In this paper, we focus on the subclass of quadratic word equations, i.e., in which each variable occurs at most twice. We first show that the length abstractions of solutions to quadratic word equations are in general not Presburger-definable. We then describe a class of counter systems with Presburger transition relations which capture the length abstraction of a quadratic word equation with regular constraints. We provide an encoding of the effect of a simple loop of the counter systems in the theory of existential Presburger Arithmetic with divisibility (PAD). Since PAD is decidable, we get a decision procedure for quadratic words equations with length constraints for which the associated counter system is \emph{flat} (i.e., all nodes belong to at most one cycle). We show a decidability result (in fact, also an NP algorithm with a PAD oracle) for a recently proposed NP-complete fragment of word equations called regular-oriented word equations, together with length constraints. Decidability holds when the constraints are additionally extended with regular constraints with a 1-weak control structure.Comment: 18 page

(Un)Decidability Results for Word Equations with Length and Regular Expression Constraints

We prove several decidability and undecidability results for the satisfiability and validity problems for languages that can express solutions to word equations with length constraints. The atomic formulas over this language are equality over string terms (word equations), linear inequality over the length function (length constraints), and membership in regular sets. These questions are important in logic, program analysis, and formal verification. Variants of these questions have been studied for many decades by mathematicians. More recently, practical satisfiability procedures (aka SMT solvers) for these formulas have become increasingly important in the context of security analysis for string-manipulating programs such as web applications. We prove three main theorems. First, we give a new proof of undecidability for the validity problem for the set of sentences written as a forall-exists quantifier alternation applied to positive word equations. A corollary of this undecidability result is that this set is undecidable even with sentences with at most two occurrences of a string variable. Second, we consider Boolean combinations of quantifier-free formulas constructed out of word equations and length constraints. We show that if word equations can be converted to a solved form, a form relevant in practice, then the satisfiability problem for Boolean combinations of word equations and length constraints is decidable. Third, we show that the satisfiability problem for quantifier-free formulas over word equations in regular solved form, length constraints, and the membership predicate over regular expressions is also decidable.Comment: Invited Paper at ADDCT Workshop 2013 (co-located with CADE 2013

An Optimal Bound on the Solution Sets of One-Variable Word Equations and its Consequences

We solve two long-standing open problems on word equations. Firstly, we prove that a one-variable word equation with constants has either at most three or an infinite number of solutions. The existence of such a bound had been conjectured, and the bound three is optimal. Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system of equations is at most of size 17. Although probably not optimal, this is the first finite bound found. However, the conjecture of that bound being actually two still remains open

What is Decidable about Strings?

We prove several decidability and undecidability results for the satisfiability/validity problem of formulas over a language of finite-length strings and integers (interpreted as lengths of strings). The atomic formulas over this language are equality over string terms (word equations), linear inequality over length function (length constraints), and membership predicate over regularexpressions (r.e.). These decidability questions are important in logic, program analysis and formal verification. Logicians have been attempting to resolve some of these questions for many decades, while practical satisfiability procedures for these formulas are increasingly important in the analysis of string-manipulating programs such as web applications and scripts. We prove three main theorems. First, we consider Boolean combination of quantifier-free formulas constructed out of word equations and length constraints. We show that if word equations can be converted to a solved form, a form relevant in practice, then the satisfiability problem for Boolean combination of word equations and length constraints is decidable. Second, we show that the satisfiability problem for word equations in solved form that areregular, length constraints and r.e. membership predicate is also decidable. Third, we show that the validity problem for the set of sentences written as a forall-exists quantifier alternation applied to positive word equations is undecidable. A corollary of this undecidability result is that this set is undecidable even with sentences with at most two occurrences of a string variable

An Optimal Bound on the Solution Sets of One-Variable Word Equations and its Consequences

We solve two long-standing open problems on word equations. Firstly, we prove that a one-variable word equation with constants has either at most three or an infinite number of solutions. The existence of such a bound had been conjectured, and the bound three is optimal. Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system has at most 17 equations. Although probably not optimal, this is the first finite bound found. However, the conjecture of that bound being actually two still remains open

One-Variable Word Equations and Three-Variable Constant-Free Word Equations

We prove connections between one-variable word equations and three-variable constant-free word equations, and use them to prove that the number of equations in an independent system of three-variable constant-free equations is at most logarithmic with respect to the length of the shortest equation in the system. We also study two well-known conjectures. The first conjecture claims that there is a constant c such that every one-variable equation has either infinitely many solutions or at most c. The second conjecture claims that there is a constant c such that every independent system of three-variable constant-free equations with a nonperiodic solution is of size at most c. We prove that the first conjecture implies the second one, possibly for a different constant

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We solve two long-standing open problems on word equations. Firstly, we prove that a onevariable word equation with constants has either at most three or an infinite number of solutions. The existence of such a bound had been conjectured, and the bound three is optimal. Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system of equations is at most of size 17. Although probably not optimal, this is the first finite bound found. However, the conjecture of that bound being actually two still remains open.</p

Polynomial-time complexity for instances of the endomorphism problem in free groups

We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F, there is an endomorphism Φ of F sending W to U. This work analyzes an approach due to C. Edmunds and improved by C. Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a two- generator word. We show that when W is a two-generator word this algorithm solves the problem in time polynomial in the length of U. This result gives a polynomial-time algorithm for solving, in free groups, two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side

Analisis Kesulitan Dalam Menyelesaikan Soal Cerita Sistem Persamaan Linear Dua Variabel Siswa Kelas X Akuntansi Di SMK Negeri 1 Tenga

Mathematics is a very important science because human activities generally use mathematics. Mathematics itself is a subject that is taught from elementary school to university. Based on observations made by researchers, most students were unable to understand mathematical concepts, this caused students to experience many difficulties and make mistakes in working on math problems. Besides that, many students experience difficulties when memorizing formulas in learning mathematics. The purpose of this study was to find out and analyze the difficulties and factors that cause students' difficulties in working on word problems on a system of two-variable linear equations. The type of research used is qualitative research. The approach to be used in this research is a descriptive approach. And the results of his research, the average student still has difficulty understanding the basic concepts of mathematics, difficulty understanding concepts, difficulties in solving and calculating steps. Because students still need more guidance and encouragement with students who still encounter many obstacles in solving story problems of a system of two-variable linear equations