Characterising Choiceless Polynomial Time with First-Order Interpretations

Grädel, Erich; Pakusa, Wied; Schalthöfer, Svenja; Kaiser, Łukasz

doi:10.1109/lics.2015.68

Cited by 9 publications

(9 citation statements)

References 16 publications

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“…We generalise this result to the case of counting and show that CPT-programs which decide the CFI query over complete graphs have to use set-like objects. In particular, this yields an interesting lower bound for a fragment of Interpretation Logic, a characterisation of Choiceless Polynomial Time which was presented in [12]: Interpretation Logic without congruences cannot decide the Cai-Fürer-Immerman query over complete graphs, and is thus strictly less expressive than the fragment of CPT using only sets of bounded rank.…”

Section: :4mentioning

confidence: 97%

“…Interpretation Logic, which was introduced in [12], characterises CPT and some of its fragments in terms of first-order interpretations (with the Härtig quantifier for equicardinality testing). This framework allows to iterate a first-order interpretation until a first-order halting condition is satisfied, and evaluates a first-order sentence on the resulting structure (for details, see [12]), with polynomial bounds on the number of iterations and the size of the intermediate structures. The fragment of interpretation logic that arises when only interpretations without congruences are allowed, i.e.…”

Section: :16mentioning

confidence: 99%

“…Without counting (respectively equicardinality quantifiers), this fragment corresponds to the database query language while new , which permits construction of new elements for definable tuples. For the fragment with counting, it could be shown [12] that, on structures with bounded colour class size, only CPT computations with sets of bounded rank can be simulated. Theorem 23 now implies that the fragment not only fails to define sets of unbounded rank, but also any kind of set-like objects.…”

Section: :16mentioning

confidence: 99%

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Definability of Cai-Fürer-Immerman Problems in Choiceless Polynomial Time

Pakusa

Schalthöfer

Selman

2018

ACM Trans. Comput. Logic

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Choiceless Polynomial Time (CPT) is one of the most promising candidates in the search for a logic capturing P time . The question whether there is a logic that expresses exactly the polynomial-time computable properties of finite structures, which has been open for more than 30 years, is one of the most important and challenging problems in finite model theory. The strength of Choiceless Polynomial Time is its ability to perform isomorphism-invariant computations over structures, using hereditarily finite sets as data structures. But, because of isomorphism-invariance, it is choiceless in the sense that it cannot select an arbitrary element of a set—an operation that is crucial for many classical algorithms. CPT can define many interesting P time queries, including (a certain version of) the Cai-Fürer-Immerman (CFI) query. The CFI-query is particularly interesting, because it separates fixed-point logic with counting from P time and has since remained the main benchmark for the expressibility of logics within P time . The CFI-construction associates with each connected graph a set of CFI-graphs that can be partitioned into exactly two isomorphism classes called odd and even CFI-graphs. The problem is to decide, given a CFI-graph, whether it is odd or even. For the case where the CFI-graphs arise from ordered graphs, Dawar, Richerby, and Rossman proved that the CFI-query is CPT-definable. However, definability of the CFI-query over general graphs remains open. Our first contribution generalises the result by Dawar, Richerby, and Rossman to the variant of the CFI-query derived from graphs with colour classes of logarithmic size, instead of colour class size one. Second, we consider the CFI-query over graph classes where the maximal degree is linear in the size of the graphs. For the latter, we establish CPT-definability using only sets of small, constant rank, which is known to be impossible for the general case. In our CFI-recognising procedures we strongly make use of the ability of CPT to create sets, rather than tuples only, and we further prove that, if CPT worked over tuples instead, then no such procedure would be definable. We introduce a notion of “sequencelike objects” based on the structure of the graphs’ symmetry groups, and we show that no CPT-program that only uses sequencelike objects can decide the CFI-query over complete graphs, which have linear maximal degree. From a broader perspective, this generalises a result by Blass, Gurevich, and van den Bussche about the power of isomorphism-invariant machine models (for polynomial time) to a setting with counting.

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Section: :4mentioning

confidence: 97%

Section: :16mentioning

confidence: 99%

Section: :16mentioning

confidence: 99%

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Definability of Cai-Fürer-Immerman Problems in Choiceless Polynomial Time

Pakusa

Schalthöfer

Selman

2018

ACM Trans. Comput. Logic

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“…In this section we translate a CPT+WSC formula into an equivalent polynomial time DeepWL+WSC-algorithm. The following translation is based on the translation of CPT into interpretation logic in [14] and the translation of interpretation logic in DeepWL in [18]. However, we avoid the route through interpretation logic by directly implementing the ideas of [14] in DeepWL+WSC.…”

Section: From Cpt+wsc To Deepwl+wscmentioning

confidence: 99%

Choiceless Polynomial Time with Witnessed Symmetric Choice

Lichter¹,

Schweitzer²

2022

Preprint

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We extend Choiceless Polynomial Time (CPT), the currently only remaining promising candidate in the quest for a logic capturing Ptime, so that this extended logic has the following property: for every class of structures for which isomorphism is definable, the logic automatically captures Ptime.For the construction of this logic we extend CPT by a witnessed symmetric choice operator. This operator allows for choices from definable orbits. But, to ensure polynomial time evaluation, automorphisms have to be provided to certify that the choice set is indeed an orbit.We argue that, in this logic, definable isomorphism implies definable canonization. Thereby, our construction removes the non-trivial step of extending isomorphism definability results to canonization. This step was a part of proofs that show that CPT or other logics capture Ptime on a particular class of structures. The step typically required substantial extra effort.

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“…For the original definition of CPT by Blass, Gurevich and Shelah, see [2]. More usable formulations of CPT are for example BGS-logic [17] and polynomial interpretation logic (see [9], [18]).…”

Section: Choiceless Polynomial Timementioning

confidence: 99%

Choiceless Polynomial Time, Symmetric Circuits and Cai-Fürer-Immerman Graphs

Pago¹

2021

Preprint

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Choiceless Polynomial Time (CPT) is currently the only candidate logic for capturing PTIME (that is, it is contained in PTIME and has not been separated from it). A prominent example of a decision problem in PTIME that is not known to be CPT-definable is the isomorphism problem on unordered Cai-Fürer-Immerman graphs (the CFI-query). We study the expressive power of CPT with respect to this problem and develop a partial characterisation of solvable instances in terms of properties of symmetric XOR-circuits over the CFI-graphs: The CFI-query is CPT-definable on a given class of graphs only if: For each graph G, there exists an XOR-circuit C, whose input gates are labelled with edges of G, such that C is sufficiently symmetric with respect to the automorphisms of G and satisfies certain other circuit properties. We also give a sufficient condition for CFI being solvable in CPT and develop a new CPT-algorithm for the CFI-query. It takes as input structures which contain, along with the CFI-graph, an XOR-circuit with suitable properties. The strongest known CPT-algorithm for this problem can solve instances equipped with a preorder with colour classes of logarithmic size. Our result implicitly extends this to preorders with colour classes of polylogarithmic size (plus some unordered additional structure). Finally, our work provides new insights regarding a much more general problem: The existence of a solution to an unordered linear equation system A • x = b over a finite field is CPT-definable if the matrix A has at most logarithmic rank (with respect to the size of the structure that encodes the equation system). This is another example that separates CPT from fixed-point logic with counting.

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Characterising Choiceless Polynomial Time with First-Order Interpretations

Cited by 9 publications

References 16 publications

Definability of Cai-Fürer-Immerman Problems in Choiceless Polynomial Time

Definability of Cai-Fürer-Immerman Problems in Choiceless Polynomial Time

Choiceless Polynomial Time with Witnessed Symmetric Choice

Choiceless Polynomial Time, Symmetric Circuits and Cai-Fürer-Immerman Graphs

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