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A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS
, 2008
"... Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent ..."
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Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church’s Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing’s Thesis, characterizing the effective string functions, and—in particular—the effectivelycomputable functions on string representations of numbers.
Logics with rank operators
 In Proceedings of the 24th IEEE Symposium on Logic in Computer Science
, 2009
"... Abstract—We introduce extensions of firstorder logic (FO) and fixedpoint logic (FP) with operators that compute the rank of a definable matrix. These operators are generalizations of the counting operations in FP+C (i.e. fixedpoint logic with counting) that allow us to count the dimension of a de ..."
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Abstract—We introduce extensions of firstorder logic (FO) and fixedpoint logic (FP) with operators that compute the rank of a definable matrix. These operators are generalizations of the counting operations in FP+C (i.e. fixedpoint logic with counting) that allow us to count the dimension of a definable vector space, rather than just count the cardinality of a definable set. The logics we define have data complexity contained in polynomial time and all known examples of polynomial time queries that are not definable in FP+C are definable in FP+rk, the extension of FP with rank operators. For each prime number p and each positive integer n, we have rank operators rkp for determining the rank of a matrix over the finite field GFp defined by a formula over ntuples. We compare the expressive power of the logics obtained by varying the values p and n can take. In particular, we show that increasing the arity of the operators yields an infinite hierarchy of expressive power. The rank operators are surprisingly expressive, even in the absence of fixedpoint operators. We show that FO+rkp can define deterministic and symmetric transitive closure. This allows us to show that, on ordered structures, FO+rkp captures the complexity class MODp L, for all prime values of p. I.
FixedPoint Definability and Polynomial Time
"... Abstract. My talk will be a survey of recent results about the quest for a logic capturing polynomial time. In a fundamental study of database query languages, Chandra and Harel [4] first raised the question of whether there exists a logic that captures polynomial time. Actually, Chandra and Harel p ..."
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Abstract. My talk will be a survey of recent results about the quest for a logic capturing polynomial time. In a fundamental study of database query languages, Chandra and Harel [4] first raised the question of whether there exists a logic that captures polynomial time. Actually, Chandra and Harel phrased the question in a somewhat disguised form; the version that we use today goes back to Gurevich [15]. Briefly, but slightly imprecisely, 1 a logic L captures a complexity class K if exactly those properties of finite structures that are decidable in K are definable in L. The existence of a logic capturing PTIME is still wide open, and it is viewed as one of the main open problems in finite model theory and database theory. One reason the question is interesting is that we know from Fagin’s Theorem [9] that existential secondorder logic captures NP, and we also know that there are logics capturing most natural complexity classes above NP. Gurevich conjectured that there is no logic capturing PTIME. If this conjecture was true, this would not only imply that PTIME ̸ = NP, but it would also show that NP and the complexity
Choiceless Computation and Symmetry
, 2010
"... Many natural problems in computer science concern structures like graphs where elements are not inherently ordered. In contrast, Turing machines and other common models of computation operate on strings. While graphs may be encoded as strings (via an adjacency matrix), the encoding imposes a linear ..."
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Many natural problems in computer science concern structures like graphs where elements are not inherently ordered. In contrast, Turing machines and other common models of computation operate on strings. While graphs may be encoded as strings (via an adjacency matrix), the encoding imposes a linear order on vertices. This enables a Turing machine operating on encodings of graphs to choose an arbitrary element from any nonempty set of vertices at low cost (the Augmenting Paths algorithm for Bipartite Matching being an example of the power of choice). However, the outcome of a computation is liable to depend on the external linear order (i.e., the choice of encoding). Moreover, isomorphisminvariance/encodingindependence is an undecidable property of Turing machines. This trouble with encodings led Blass, Gurevich and Shelah [3] to propose a model of computation known as BGS machines that operate directly on structures. BGS machines preserve symmetry at every step in a computation, sacrificing the ability to make arbitrary choices between indistinguishable elements of the input structure (hence “choiceless computation”). Blass et al. also introduced a complexity class CPT+C (Choiceless Polynomial Time with Counting) defined in terms of polynomially bounded BGS machines. While every property finite structures in CPT+C is polynomialtime computable in the usual sense, it is open whether conversely every isomorphisminvariant property in P belongs to CPT+C. In this paper we give evidence that CPT+C = P by proving the separation of the corresponding classes of function problems. Specifically, we show that there is an isomorphisminvariant polynomialtime computable function problem on finite vector spaces (“given a finite vector space V, output the set of hyperplanes in V ”) that is not computable by any CPT+C program. In addition, we give a new simplified proof of the Support Theorem, which is a key step in the result of [3] that a weak version of CPT+C absent counting cannot decide the parity of sets. 1
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, that the only numeric functions that can be calculated by effective means are the recursive ones (which are the same, extensionally, as the Turingcomputable numeric functions). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
FixedPoint Definability and Polynomial Time on Chordal Graphs and Line Graphs
, 2009
"... The question of whether there is a logic that captures polynomial time was formulated by Yuri Gurevich in 1988. It is still wide open and regarded as one of the main open problems in finite model theory and database theory. Partial results have been obtained for specific classes of structures. In pa ..."
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The question of whether there is a logic that captures polynomial time was formulated by Yuri Gurevich in 1988. It is still wide open and regarded as one of the main open problems in finite model theory and database theory. Partial results have been obtained for specific classes of structures. In particular, it is known that fixedpoint logic with counting captures polynomial time on all classes of graphs with excluded minors. The introductory part of this paper is a short survey of the stateoftheart in the quest for a logic capturing polynomial time. The main part of the paper is concerned with classes of graphs defined by excluding induced subgraphs. Two of the most fundamental such classes are the class of chordal graphs and the class of line graphs. We prove that capturing polynomial time on either of these classes is as hard as capturing it on the class of all graphs. In particular, this implies that fixedpoint logic with counting does not capture polynomial time on these classes. Then we prove that fixedpoint logic with counting does capture polynomial time on the class of all graphs that are both chordal and line graphs.
Choiceless Polynomial Time on structures with small Abelian colour classes
"... Choiceless Polynomial Time (CPT) is one of the candidates in the quest for a logic for polynomial time. It is a strict extension of fixedpoint logic with counting (FPC) but to date it is unknown whether it expresses all polynomialtime properties of finite structures. We study the CPTdefinability ..."
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Choiceless Polynomial Time (CPT) is one of the candidates in the quest for a logic for polynomial time. It is a strict extension of fixedpoint logic with counting (FPC) but to date it is unknown whether it expresses all polynomialtime properties of finite structures. We study the CPTdefinability of the isomorphism problem for relational structures of bounded colour class size q (for short, qbounded structures). Our main result gives a positive answer, and even CPTdefinable canonisation procedures, for classes of qbounded structures with small Abelian groups on the colour classes. Such classes of qbounded structures with Abelian colours naturally arise in many contexts. For instance, 2bounded structures have Abelian colours which shows that CPT captures Ptime on 2bounded structures. In particular, this shows that the isomorphism problem of multipedes is definable in CPT, an open question posed by Blass, Gurevich, and Shelah.
Structure and Specification as Sources of Complexity
 LIPICS LEIBNIZ INTERNATIONAL PROCEEDINGS IN INFORMATICS
, 2009
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