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PTIME Computation of Transitive Closures of Octagonal Relations [article]

Filip Konecny
2014 arXiv   pre-print
In this paper, we study difference bounds and octagonal relations and prove that their transitive closure is a PTIME-computable formula in the existential fragment of Presburger arithmetic.  ...  Computing transitive closures of integer relations is the key to finding precise invariants of integer programs.  ...  The running time of each iteration is of the order O(N 3 ) and hence the total running time is of the order O(N 7 ).  ... 
arXiv:1402.2102v1 fatcat:tiw7hxslx5byxc6xjk4atpviqm

PTIME Computation of Transitive Closures of Octagonal Relations [chapter]

Filip Konečný
2016 Lecture Notes in Computer Science  
In this paper, we study difference bounds and octagonal relations and prove that their transitive closure is a PTIMEcomputable formula in the existential fragment of Presburger arithmetic.  ...  Computing transitive closures of integer relations is the key to finding precise invariants of integer programs.  ...  The running time of each iteration is of the order O(N 3 ) and hence the total running time is of the order O(N 7 ).  ... 
doi:10.1007/978-3-662-49674-9_42 fatcat:gctacip2kvc3pnwcsyz6zpweka

Bisimulation-invariant PTIME and higher-dimensional μ-calculus

Martin Otto
1999 Theoretical Computer Science  
Bisimulation-invariant PTIME, or the modal fragment of PTIME, thus proves to be one of the very rare cases in which a logical characterization is known in a setting of unordered structures.  ...  It is a characteristic feature in these examples that they either concern complexity classes beyond PTIME or else concern classes of linearly ordered structures. Indeed, no *  ...  l,a) to its isomorphic standard representation over an initial segment of the natural numbers, naturally ordered by the PTIME computable global ordering + (according to Proposition 2.8).  ... 
doi:10.1016/s0304-3975(98)00314-4 fatcat:klwctsikb5fhpm45567db6vn6y

On the theory of the PTIME degrees of the recursive sets

Juichi Shinoda, Theodore A. Slaman
1990 Journal of computer and system sciences (Print)  
There is an interpretation of first-order arithmetic in the theory of the PTIME degrees of the recursive sets.  ...  There is an interpretation of second-order arithmetic in the first-order theory of the PTIME degrees. These results characterize the Turing degrees of the first order theories of these structures.  ...  (1) There is an interpretation of the first-order theory of arithmetic in the first-order theory of (REC, <p >. (2) There is an interpretation of the theory of second-order arithmetic in the first-order  ... 
doi:10.1016/0022-0000(90)90024-f fatcat:x6rmsefnxreepkfhrvyveejcnq

Automorphisms in the PTIME-Turing degrees of recursive sets

Christine Ann Haught, Theodore A. Slaman
1997 Annals of Pure and Applied Logic  
an automorphism of the ideal of PTIME-degrees below A.  ...  We prove that there is a nontrivial automorphism of an ideal of 3. This can be rephrased in terms of partial relativizations.  ...  We use conditions as approximations, or initial segments of oracle sets.  ... 
doi:10.1016/s0168-0072(95)00065-8 fatcat:63si4h43srcwrnqtp56dju5hum

On the theory of the PTIME degrees of the recursive sets

J. Shinoda, T.A. Slaman
1988 [1988] Proceedings. Structure in Complexity Theory Third Annual Conference  
There is an interpretation of first order arithmetic in the theory of the PTIME degrees of the recursive sets.  ...  There is an interpretation of second order arithmetic in the first order theory of the PTIME degrees. These results characterize the Turing degrees of the first order theories of these structures.  ...  There is an interpretation of first order arithmetic in the theory of the PTIME degrees of the recursive sets.  ... 
doi:10.1109/sct.1988.5285 dblp:conf/coco/ShinodaS88 fatcat:n5hbeedjmbfnzgnfoxrw7wdq64

On polynomial time computation over unordered structures

Andreas Blass, Yuri Gurevich, Saharon Shelah
2002 Journal of Symbolic Logic (JSL)  
We consider several algorithmic problems near the border of the known, logically defined complexity classes contained in polynomial time.  ...  Similar results hold for the multipede examples of Gurevich and Shelah, except that their final version of multipedes is, in a sense, already suitably padded.  ...  In the case of 3-and 4-multipedes, it is further required that S of the shoe is the first segment in the order ≤.  ... 
doi:10.2178/jsl/1190150152 fatcat:xojjlyg65rbplcz4gc7gbhmxr4

On Polynomial Time Computation Over Unordered Structures [article]

Andreas Blass and Yuri Gurevich and Saharon Shelah
2001 arXiv   pre-print
We consider several algorithmic problems near the border of the known, logically defined complexity classes contained in polynomial time.  ...  Similar results hold for the multipede examples of Gurevich and Shelah, except that their final version of multipedes is, in a sense, already suitably padded.  ...  We point out that, although we have formulated these questions for CPT+Card, the logic in which we are primarily interested, the first two of them are open also forCPT, and the last two are open also for  ... 
arXiv:math/0102059v1 fatcat:io6j7jst3zfbvjlmrqj4hqpuzi

XPath Query Satisfiability is in PTIME for Real-World DTDs [chapter]

Manizheh Montazerian, Peter T. Wood, Seyed R. Mousavi
Lecture Notes in Computer Science  
However, in this paper we show that the satisfiability problem is in PTIME for most DTDs used in real-world applications.  ...  The problem of XPath query satisfiability under DTDs (Document Type Definitions) is to decide, given an XPath query p and a DTD D, whether or not there is some document valid with respect to D on which  ...  Therefore, we only need to check whether L(R root ) contains a word w c which includes all of the labels in C (with an arbitrary ordering), where C is the set, as opposed to the multiset, of labels of  ... 
doi:10.1007/978-3-540-75288-2_3 fatcat:yd6j6vfpxvcjlnw7ylkpx6wncu

An analysis of the nonemptiness problem for classes of reversal-bounded multicounter machines

Rodney R. Howell, Louis E. Rosier
1987 Journal of computer and system sciences (Print)  
Furthermore, we show that in most cases, the complexity of the nonemptiness problem does not change signiticantly when the reversal bound is dropped for one of the counters.  ...  In this paper, we present an efficient nondeterministic algorithm to decide nonemptiness for reversal-bounded multicounter machines.  ...  Hence, within each segment, the behavior of each counter is "nice" enough that the exact order of the transitions in the computation is not necessary to know.  ... 
doi:10.1016/0022-0000(87)90005-5 fatcat:sgpxwc4ypfg5ngpcsxpcldo5be

Choiceless polynomial time

Andreas Blass, Yuri Gurevich, Saharon Shelah
1999 Annals of Pure and Applied Logic  
The resulting logic expresses all properties expressible in any other PTime logic in the literature.  ...  Turing machines deÿne polynomial time (PTime) on strings but cannot deal with structures like graphs directly, and there is no known, easily computable string encoding of isomorphism classes of structures  ...  Bipartite Matching is not inCPTime + . Proof. We can use exactly the same proof as for Theorem 43, because the two structures used in that proof had the same cardinality.  ... 
doi:10.1016/s0168-0072(99)00005-6 fatcat:gqnccerwmfd5pdyvg6k33q72ue

Page 4089 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews  
By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure.  ...  This implies in particular that every extension of fixed-point logic by means of monadic Lindstrém quantifiers which stays within PTime must be strictly contained in Fixed-Point + Counting.” 98g:03084  ... 

Canonization for two variables and puzzles on the square

Martin Otto
1997 Annals of Pure and Applied Logic  
of their equivalence class with respect to indistinguishability in either of these logics.  ...  In fact we exhibit PTIME inverses to the natural PTIME invariants that characterize structures up to L2or ?-equivalence, respectively.  ...  The relational part of I@) with linearly ordered universe Ak/w can in a canonical way be encoded over the ordered standard universe of size IAl if k-tuples (in an initial segment with respect to the lexicographic  ... 
doi:10.1016/s0168-0072(96)00047-4 fatcat:ezv2uqeasjgwljsddviacuzium

Division in idealized unit cost RAMs

Janos Simon
1981 Journal of computer and system sciences (Print)  
Again, the availability of integer division seems to play a crucial role in these results. 421  ...  We study the power of RAM acceptors with several instruction sets. We exhibit several instances where the availability of the division operator increases the power of the acceptors.  ...  ACKNOWLEDGMENTS Joel Seiferas was a friendly source of help and encouragement for the results in Section 3.  ... 
doi:10.1016/0022-0000(81)90041-6 fatcat:m32rqzb6q5aglcdqidwewjca3q

Choiceless polynomial time [article]

Andreas Blass, Yuri Gurevich, Saharon Shelah
1997 arXiv   pre-print
The resulting logic is more expressive than other PTime logics in the literature. A more difficult theorem shows that the logic does not capture all PTime.  ...  Turing machines define polynomial time (PTime) on strings but cannot deal with structures like graphs directly, and there is no known, easily computable string encoding of isomorphism classes of structures  ...  standard encoding of an ordered version of some structure in K.  ... 
arXiv:math/9705225v1 fatcat:5i3bafda25ezvnjyqwxd47e2uy
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