Periodic Reporting for period 4 - ARiAT (Advanced Reasoning in Arithmetic Theories)
Periodo di rendicontazione: 2023-09-01 al 2025-08-31
At school, many of us have encountered questions like this: "A grandmother is 60 years old; her grandson is 4. In how many years will the grandmother be 5 times as old as her grandson?" With some effort, we find the answer to be 10 years. Instead of trial-and-error, this type of problem-solving can be systematized by translating it into a system of equations and then solved algorithmically. This is a technique that has been employed for thousands of years and forms the basis of many of today's modern advancements.
Just as we can turn word problems into equations, we can also turn the behavior of computer programs into logical formulas. These formulas are like more sophisticated kinds of systems of equations: they enable us to unambiguously describe the operations of a computer program, and in turn to mathematically prove that a computer program or system does what it is intended to do. The latter is of crucial importance in the ever-growing areas where computer programs take direct responsibility over the life of humans, e.g. in healthcare (think of a program controlling a pacemaker), or in aviation (think of the software for an autopilot of an aircraft). Without automation, solving logical formulas that model computer programs or systems, or even answering more complex questions about such formulas, would be impossible.
The goal of the ERC-funded "Advanced Reasoning in Arithmetic Theories" (ARiAT) project is to develop and lay the foundations for novel algorithms for automatically reasoning in logical formalisms that can express numerical properties of computer programs and systems. The project aims at pushing the state-of-the-art in efficient algorithms that can automatically prove sophisticated properties of computer systems. It also aims at understanding the algorithmic barriers of automated reasoning in the logical formalisms studied in this project. This amounts to mathematically proving that some existing algorithms cannot be improved, thus sparing other researchers resources who may try to improve algorithms solving certain tasks. Finally, the project aims at developing prototype implementations of those algorithms discovered in the course of this project which have properties that promise to dramatically outperform existing ones.
All program code developed as part of this project will be made freely available and open source.
Just as we can turn word problems into equations, we can also turn the behavior of computer programs into logical formulas. These formulas are like more sophisticated kinds of systems of equations: they enable us to unambiguously describe the operations of a computer program, and in turn to mathematically prove that a computer program or system does what it is intended to do. The latter is of crucial importance in the ever-growing areas where computer programs take direct responsibility over the life of humans, e.g. in healthcare (think of a program controlling a pacemaker), or in aviation (think of the software for an autopilot of an aircraft). Without automation, solving logical formulas that model computer programs or systems, or even answering more complex questions about such formulas, would be impossible.
The goal of the ERC-funded "Advanced Reasoning in Arithmetic Theories" (ARiAT) project is to develop and lay the foundations for novel algorithms for automatically reasoning in logical formalisms that can express numerical properties of computer programs and systems. The project aims at pushing the state-of-the-art in efficient algorithms that can automatically prove sophisticated properties of computer systems. It also aims at understanding the algorithmic barriers of automated reasoning in the logical formalisms studied in this project. This amounts to mathematically proving that some existing algorithms cannot be improved, thus sparing other researchers resources who may try to improve algorithms solving certain tasks. Finally, the project aims at developing prototype implementations of those algorithms discovered in the course of this project which have properties that promise to dramatically outperform existing ones.
All program code developed as part of this project will be made freely available and open source.
The project aimed not only to develop new algorithms and methodologies that make arithmetic reasoning practically more efficient, but also to understand the geometric structure and computational limits of arithmetic theories. The project's research activities focused on three main themes: quantified linear arithmetic, non-linear arithmetic, and counting extensions of arithmetic.
In quantified linear arithmetic, the project developed a framework for identifying efficiently decidable fragments and corresponding algorithms for their decision problems. It produced new methods for representing and manipulating semi-linear sets, and developed a novel quantifier elimination procedure for Presburger arithmetic.
Work on non-linear arithmetic established precise complexity bounds for key variants, including arithmetic over the p-adic numbers and systems with divisibility constraints. The project also explored extensions involving exponentiation, deriving algorithms that can efficiently solve formulas incorporating them.
In the area of counting, the project team developed algorithms for reasoning about arithmetic with counting quantifiers and investigated connections between arithmetic and reachability problems in computational models such as vector addition systems.
Several methodological frameworks were developed to support these advances. These include a framework for fixed-parameter tractable reasoning, a hybrid symbolic–geometric approach to quantifier elimination, a translation linking string constraints to non-linear arithmetic, and a quantitative extension of the Chinese Remainder Theorem.
Alongside theoretical research, the project conducted knowledge-transfer and training activities, including summer schools, tutorials, and the organisation of two international workshops Trends in Arithmetic Theories (2022 and 2024). These events promoted interaction between researchers and practitioners from academia and industry.
The project also produced software prototypes. A notable output is MPL, a multi-precision arithmetic library optimised for variable fixed bit lengths, which provides full integration to serve as a backend for the cvc5 SMT solver.
Results were disseminated through publications in proceedings of major international conferences such as LICS, ICALP, SODA, STACS, and TACAS, as well as through invited talks and industrial seminars.
In quantified linear arithmetic, the project developed a framework for identifying efficiently decidable fragments and corresponding algorithms for their decision problems. It produced new methods for representing and manipulating semi-linear sets, and developed a novel quantifier elimination procedure for Presburger arithmetic.
Work on non-linear arithmetic established precise complexity bounds for key variants, including arithmetic over the p-adic numbers and systems with divisibility constraints. The project also explored extensions involving exponentiation, deriving algorithms that can efficiently solve formulas incorporating them.
In the area of counting, the project team developed algorithms for reasoning about arithmetic with counting quantifiers and investigated connections between arithmetic and reachability problems in computational models such as vector addition systems.
Several methodological frameworks were developed to support these advances. These include a framework for fixed-parameter tractable reasoning, a hybrid symbolic–geometric approach to quantifier elimination, a translation linking string constraints to non-linear arithmetic, and a quantitative extension of the Chinese Remainder Theorem.
Alongside theoretical research, the project conducted knowledge-transfer and training activities, including summer schools, tutorials, and the organisation of two international workshops Trends in Arithmetic Theories (2022 and 2024). These events promoted interaction between researchers and practitioners from academia and industry.
The project also produced software prototypes. A notable output is MPL, a multi-precision arithmetic library optimised for variable fixed bit lengths, which provides full integration to serve as a backend for the cvc5 SMT solver.
Results were disseminated through publications in proceedings of major international conferences such as LICS, ICALP, SODA, STACS, and TACAS, as well as through invited talks and industrial seminars.
At the beginning of the project, the algorithmic foundations of reasoning in arithmetic theories were fragmented and marked by long-standing open problems. Many questions concerning the complexity and decidability of key variants, particularly those involving divisibility, counting, or non-linear operations, had remained unresolved for decades. Existing algorithms were often of high complexity and therefore of limited practical use, while the structural understanding of efficiently solvable fragments was incomplete.
The project advanced significantly beyond this state of the art, by developing new algorithms, improving known complexity bounds, and establishing general theoretical frameworks that substantially broaden our knowledge on how arithmetic reasoning can be made efficient, and where provably no further improvements can be made. It resolved a 30-year-old open problem by establishing the NP-completeness of existential arithmetic over the p-adic numbers and achieved the first improvement in over forty years for deciding classes systems of Diophantine equations with divisibility constraints. The project also produced new decision procedures for extensions of arithmetic involving exponentiation and counting quantifiers, yielding sharper upper bounds and refined classifications of decidable fragments.
In addition to these breakthroughs, the project introduced new conceptual and methodological paradigms that connect logic, geometry, and arithmetic. These include a framework for fixed-parameter tractable reasoning in logical theories, a hybrid symbolic–geometric approach to quantifier elimination, a reduction technique linking string constraints to decidable non-linear arithmetic, and a quantitative generalisation of the Chinese Remainder Theorem.
Together, the project’s results have redefined the algorithmic landscape of arithmetic reasoning. They unify insights from logic, geometry, and number theory, giving faster decision procedures, sharper complexity bounds, and generic frameworks for tractable reasoning about arithmetical constraints. These advances will serve as lasting foundations for next-generation reasoning and verification tools, and future research at the interface of mathematics, logic, and computer science.
The project advanced significantly beyond this state of the art, by developing new algorithms, improving known complexity bounds, and establishing general theoretical frameworks that substantially broaden our knowledge on how arithmetic reasoning can be made efficient, and where provably no further improvements can be made. It resolved a 30-year-old open problem by establishing the NP-completeness of existential arithmetic over the p-adic numbers and achieved the first improvement in over forty years for deciding classes systems of Diophantine equations with divisibility constraints. The project also produced new decision procedures for extensions of arithmetic involving exponentiation and counting quantifiers, yielding sharper upper bounds and refined classifications of decidable fragments.
In addition to these breakthroughs, the project introduced new conceptual and methodological paradigms that connect logic, geometry, and arithmetic. These include a framework for fixed-parameter tractable reasoning in logical theories, a hybrid symbolic–geometric approach to quantifier elimination, a reduction technique linking string constraints to decidable non-linear arithmetic, and a quantitative generalisation of the Chinese Remainder Theorem.
Together, the project’s results have redefined the algorithmic landscape of arithmetic reasoning. They unify insights from logic, geometry, and number theory, giving faster decision procedures, sharper complexity bounds, and generic frameworks for tractable reasoning about arithmetical constraints. These advances will serve as lasting foundations for next-generation reasoning and verification tools, and future research at the interface of mathematics, logic, and computer science.