Presburger arithmetic
Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely. The theory is computably axiomatizable; the axioms include a schema of induction.
Presburger arithmetic is much weaker than Peano arithmetic, which includes both addition and multiplication operations. Unlike Peano arithmetic, Presburger arithmetic is a decidable theory. This means it is possible to algorithmically determine, for any sentence in the language of Presburger arithmetic, whether that sentence is provable from the axioms of Presburger arithmetic. The asymptotic running-time computational complexity of this algorithm is at least doubly exponential, however, as shown by.
Overview
The language of Presburger arithmetic contains constants 0 and 1 and a binary function +, interpreted as addition.In this language, the axioms of Presburger arithmetic are the universal closures of the following:
- ¬
- x + 1 = y + 1 → x = y
- x + 0 = x
- x + = + 1
- Let P be a first-order formula in the language of Presburger arithmetic with a free variable x. Then the following formula is an axiom:.
Presburger arithmetic can be viewed as a first-order theory with equality containing precisely all consequences of the above axioms. Alternatively, it can be defined as the set of those sentences that are true in the intended interpretation: the structure of non-negative integers with constants 0, 1, and the addition of non-negative integers.
Presburger arithmetic is designed to be complete and decidable. Therefore, it cannot formalize concepts such as divisibility or primality, or, more generally, any number concept leading to multiplication of variables. However, it can formulate individual instances of divisibility; for example, it proves "for all x, there exists y : ∨ ". This states that every number is either even or odd.
Properties
proved Presburger arithmetic to be:- consistent: There is no statement in Presburger arithmetic that can be deduced from the axioms such that its negation can also be deduced.
- complete: For each statement in the language of Presburger arithmetic, either it is possible to deduce it from the axioms or it is possible to deduce its negation.
- decidable: There exists an algorithm that decides whether any given statement in Presburger arithmetic is a theorem or a nontheorem - note that a "nontheorem" is a formula that cannot be proved, not in general necessarily one whose negation can be proved, but in the case of a complete theory like here the two definitions are equivalent.
In contrast, Peano arithmetic, which is Presburger arithmetic augmented with multiplication, is not decidable, as proved by Church alongside the negative answer to the Entscheidungsproblem. By Gödel's incompleteness theorem, Peano arithmetic is incomplete and its consistency is not internally provable.
Computational complexity
The decision problem for Presburger arithmetic is an interesting example in computational complexity theory and computation. Let n be the length of a statement in Presburger arithmetic. Then proved that, in the worst case, the proof of the statement in first-order logic has length at least, for some constant c>0. Hence, their decision algorithm for Presburger arithmetic has runtime at least exponential. Fischer and Rabin also proved that for any reasonable axiomatization, there exist theorems of length n that have doubly exponential length proofs. Fischer and Rabin's work also implies that Presburger arithmetic can be used to define formulas that correctly calculate any algorithm as long as the inputs are less than relatively large bounds. The bounds can be increased, but only by using new formulas. On the other hand, a triply exponential upper bound on a decision procedure for Presburger arithmetic was proved by Oppen.A more tight complexity bound was shown using alternating complexity classes by. The set of true statements in Presburger arithmetic is shown complete for TimeAlternations. Thus, its complexity is between double exponential nondeterministic time and double exponential space. Completeness is under Karp reductions.
For a more fine-grained result, let PA be the set of true Σi PA statements, and PA the set of true Σi PA statements with each quantifier block limited to j variables. '<' is considered to be quantifier-free; here, bounded quantifiers are counted as quantifiers.
PA is in P, while PA is NP-complete.
For i > 0 and j > 2, PA is ΣiP-complete. The hardness result only needs j>2 in the last quantifier block.
For i>0, PA is ΣiEXP-complete.
Short Presburger Arithmetic is complete. Here, 'short' requires bounded sentence size except that integer constants are unbounded. Also, two variable PA is NP-complete. Short PA is in P, and this extends to fixed-dimensional parametric integer linear programming.
Applications
Because Presburger arithmetic is decidable, automatic theorem provers for Presburger arithmetic exist. For example, the Rocq and Lean proof assistant systems feature the tactic omega for Presburger arithmetic and the Isabelle proof assistant contains a verified quantifier elimination procedure by. The double exponential complexity of the theory makes it infeasible to use the theorem provers on complicated formulas, but this behavior occurs only in the presence of nested quantifiers: describe an automatic theorem prover that uses the simplex algorithm on an extended Presburger arithmetic without nested quantifiers to prove some of the instances of quantifier-free Presburger arithmetic formulas. More recent satisfiability modulo theories solvers use complete integer programming techniques to handle quantifier-free fragment of Presburger arithmetic theory.Presburger arithmetic can be extended to include multiplication by constants, since multiplication is repeated addition. Most array subscript calculations then fall within the region of decidable problems. This approach is the basis of at least five proof-of-correctness systems for computer programs, beginning with the Stanford Pascal Verifier in the late 1970s and continuing through to Microsoft's Spec# system of 2005.
Presburger-definable integer relation
Some properties are now given about integer relations definable in Presburger Arithmetic. For the sake of simplicity, all relations considered in this section are over non-negative integers.A relation is Presburger-definable if and only if it is a semilinear set.
A unary integer relation, that is, a set of non-negative integers, is Presburger-definable if and only if it is ultimately periodic. That is, if there exists a threshold and a positive period such that, for all integer such that, if and only if.
By the Cobham–Semenov theorem, a relation is Presburger-definable if and only if it is definable in Büchi arithmetic of base for all. A relation definable in Büchi arithmetic of base and for and being multiplicatively independent integers is Presburger definable.
An integer relation is Presburger-definable if and only if all sets of integers that are definable in first-order logic with addition and are Presburger-definable. Equivalently, for each relation that is not Presburger-definable, there exists a first-order formula with addition and that defines a set of integers that is not definable using only addition.
Muchnik's characterization
Presburger-definable relations admit another characterization: by Muchnik's theorem. It is more complicated to state, but led to the proof of the two former characterizations. Before Muchnik's theorem can be stated, some additional definitions must be introduced.Let be a set, the section of, for and is defined as
Given two sets and a of integers, the set is called -periodic in if, for all such that then if and only if. For, the set is said to be in if it is for some such that
Finally, for let
denote the cube of size whose lesser corner is.
Intuitively, the integer represents the length of a shift, the integer is the size of the cubes and is the threshold before the periodicity. This result remains true when the condition
is replaced either by or by.
This characterization led to the so-called "definable criterion for definability in Presburger arithmetic", that is: there exists a first-order formula with addition and a predicate that holds if and only if is interpreted by a Presburger-definable relation. Muchnik's theorem also allows one to prove that it is decidable whether an automatic sequence accepts a Presburger-definable set.