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PTIME Computation of Transitive Closures of Octagonal Relations
[article]

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 ). ...

##
###
PTIME Computation of Transitive Closures of Octagonal Relations
[chapter]

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 ). ...

##
###
Bisimulation-invariant PTIME and higher-dimensional μ-calculus

1999
*
Theoretical Computer Science
*

Bisimulation-invariant

doi:10.1016/s0304-3975(98)00314-4
fatcat:klwctsikb5fhpm45567db6vn6y
*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). ...##
###
On the theory of the PTIME degrees of the recursive sets

1990
*
Journal of computer and system sciences (Print)
*

There

doi:10.1016/0022-0000(90)90024-f
fatcat:x6rmsefnxreepkfhrvyveejcnq
*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*...##
###
Automorphisms in the PTIME-Turing degrees of recursive sets

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. ...

##
###
On the theory of the PTIME degrees of the recursive sets

1988
*
[1988] Proceedings. Structure in Complexity Theory Third Annual Conference
*

There

doi:10.1109/sct.1988.5285
dblp:conf/coco/ShinodaS88
fatcat:n5hbeedjmbfnzgnfoxrw7wdq64
*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. ...##
###
On polynomial time computation over unordered structures

2002
*
Journal of Symbolic Logic (JSL)
*

We consider several algorithmic problems near the border

doi:10.2178/jsl/1190150152
fatcat:xojjlyg65rbplcz4gc7gbhmxr4
*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*≤. ...##
###
On Polynomial Time Computation Over Unordered Structures
[article]

2001
*
arXiv
*
pre-print

We consider several algorithmic problems near the border

arXiv:math/0102059v1
fatcat:io6j7jst3zfbvjlmrqj4hqpuzi
*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 ...##
###
XPath Query Satisfiability is in PTIME for Real-World DTDs
[chapter]

*
Lecture Notes in Computer Science
*

However,

doi:10.1007/978-3-540-75288-2_3
fatcat:yd6j6vfpxvcjlnw7ylkpx6wncu
*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*...##
###
An analysis of the nonemptiness problem for classes of reversal-bounded multicounter machines

1987
*
Journal of computer and system sciences (Print)
*

Furthermore, we show that

doi:10.1016/0022-0000(87)90005-5
fatcat:sgpxwc4ypfg5ngpcsxpcldo5be
*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. ...##
###
Choiceless polynomial time

1999
*
Annals of Pure and Applied Logic
*

The resulting logic expresses all properties expressible

doi:10.1016/s0168-0072(99)00005-6
fatcat:gqnccerwmfd5pdyvg6k33q72ue
*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. ...##
###
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

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 ...

##
###
Division in idealized unit cost RAMs

1981
*
Journal of computer and system sciences (Print)
*

Again, the availability

doi:10.1016/0022-0000(81)90041-6
fatcat:m32rqzb6q5aglcdqidwewjca3q
*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. ...##
###
Choiceless polynomial time
[article]

1997
*
arXiv
*
pre-print

The resulting logic

arXiv:math/9705225v1
fatcat:5i3bafda25ezvnjyqwxd47e2uy
*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. ...
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