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d9e0d2ab5fab72fa0ce45bc0a16cd13b755439b0 | subsection | 41 | 42 | Centroid Detection Methodologies | We initiate at the mean of the cluster samples because the spring-like pulling of each sample on the center is the same as the mean of the samples pulling on the center with a spring constant equal to the number of samples (proved below).
Proposition The combined spring-like pull of each sample, S_i, on a center, C, ... | {
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} | 1804.11195 | Revealing patterns in HIV viral load data and classifying patients via a
novel machine learning cluster summarization method | [
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"Robert J. White",
"Kristen Bush",
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b8739bd35e2108d6867d45aa3d5eb49c147e9e39 | abstract | 0 | 60 | Abstract | Extending ASP with constraints (CASP) enhances its expressiveness and
performance. This extension is not straightforward as the grounding phase,
present in most ASP systems, removes variables and the links among them, and
also causes a combinatorial explosion in the size of the program. Several
methods to overcome this... | {
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} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
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27a798f2e4320fc2a3e63b3dc5b72c5ed2930bd7 | subsection | 1 | 60 | Introduction | Answer Set Programming (ASP) has emerged as a successful paradigm for
developing intelligent applications. It uses the stable model
semantics for
programs with negation.
ASP has attracted much attention due to its
expressiveness, ability to incorporate
non-monotonicity, represent knowledge, and model combinatorial
pro... | {
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6ccfea7ed96c79edef07ad5c1d5917087f8bf2a4 | subsection | 2 | 60 | Introduction | Once the
input program is translated, they benefit from the features and
performance of the target ASP and CLP solvers. However, the translation may
result in a large propositional representation or weak propagation
strength.
Extensions of ASP systems with constraint
propagators ,
that generate and propagate new cons... | {
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9ffc78b0d219bdcdc908c6b4b8829af8cf64af49 | subsection | 3 | 60 | Background: ASP and s(ASP) | ASP , is a
logic programming and modelling language. An ASP program \Pi is a
finite set of rules. Each rule r \in \Pi is of the form:a \leftarrow b_1 \wedge \dots \wedge b_m \wedge not\ b_{m+1}
\wedge \dots \wedge not\ b_n.where a and b_1, \dots , b_n are atoms and not corresponds to
default negation. An atom is an exp... | {
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6f2f4dc9c12e33443dbfe0e09a2258646719e3bb | subsection | 4 | 60 | Background: ASP and s(ASP) | The top-down
evaluation makes the grounding phase unnecessary. The
execution of an s(ASP) program starts with a query, and each
answer is the resulting mgu of a successful derivation, its
justification, and a (partial) stable model. This partial stable model
is a subset of the ASP stable model
including only the liter... | {
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428455f188248f2ba1c71fa649854916e418176f | subsection | 5 | 60 | Dual of a Logic Program | The dual of a predicate [style=MyInline]p/1 is another predicate that returns
the [style=MyInline]X such that p(X) is not true. It is used to
give a constructive answer to a goal [style=MyInline]not p(X). The dual of
a logic program is another logic program containing the dual of each
predicate in the program .
To synt... | {
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bd998700fea95d92e361216de74d3cedb9c64efb | subsection | 6 | 60 | Dual of a Logic Program | At the end of the chain, unification has to be negated
to obtain disequality, e.g., x = y is transformed into x \ne y
(Section REF ).Example 2 Given the program below:2
[style=MyProlog]
p(0).
p(X) :- q(X), not t(X,Y).
q(1).
t(1,2).the resulting dual program is:2
[style=MyProlog]
not p(X) :- not p1(X), not p2(X).
not p1... | {
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} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
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be0414244682cbfb5edf5aab8f4c43e55fedaeae | subsection | 7 | 60 | Constructive Disequality | Unlike Prolog's negation as failure,
disequality in s(ASP),
denoted by “\=” , represents the constructive negation of the
unification and is used to construct answers from negative literals.
Intuitively, [style=MyInline]Xā means that
X can be any term not unifiable with a.
In the implementation reported in (marple2017b... | {
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} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
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4c0f10717352bc5e60c93d4806253a7051881c9a | subsection | 8 | 60 | Body | In (marple2017b 2017b) the universal quantifier is evaluated
by [style=MyInline]forall(V, Goal) which checks if [style=MyInline]Goal is true for all
the possible values of [style=MyInline]V. When [style=MyInline]forall/2 succeeds,
the evaluation continues with the quantified variable unbound. Multiple
quantified variab... | {
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} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
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"Manuel Carro",
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f81ddf299fc05ffd74e9a7eb3a3e065d97567a06 | subsection | 9 | 60 | Body | Note that leaving [style=MyInline]X unbound after the success of
[style=MyInline]forall(X, p(X)) is consistent with the interpretation that
the answer set [style=MyInline]{p(X)} corresponds to \forall x.p(x).Finally, in order to break infinite loops, s(ASP) uses three
techniques to deal with odd loops over negation, ev... | {
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4d13101658519b7db2a72284c1eb20f8a239e1cb | subsection | 10 | 60 | Body | The
call [style=MyInline]apply(NV,V,Store) takes the object [style=MyInline]Store and makes
it part of the global store but substituting f̭or [style=MyInline]NV so that
the execution of [style=MyInline]NGoal can further constrain [style=MyInline]NV while r̭emains untouched. Note, however, that in the first iteration,
[... | {
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} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
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9287484ff05508b8cb8d371349b17854bf0c4ee6 | subsection | 11 | 60 | Body | Note that
if a conjunction C_{i} \wedge \lnot A_{i.j} is inconsistent, it
means that \lnot A_{i.j} has already been (successfully) checked.Each of the resulting constraint stores will be re-evaluated by
[style=MyInline]evalforall/3, where [style=MyInline]apply/3 will apply them to a new
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3a889ba8f664126466ba81d897dac2dceac7e10b | subsection | 12 | 60 | Non-Monotonic Checking Rules | Non-monotonic rules are used by s(ASP) to ensure that partial
stable models are consistent with the global constraints of the program.
Given a consistency rule of the form
\forall \vec{x} ( p_i(\vec{x}) \leftarrow \exists \vec{y} \ B_i \wedge \lnot p_i(\vec{x}) ), and in order to avoid contradictory rules of
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6a409239a0d39cdf203e6ba9172d8bf71e39459a | subsection | 13 | 60 | s(CASP): Design and
Implementation | S(CASP) (available together with the benchmarks used in this paper at
https://gitlab.software.imdea.org/joaquin.arias/sCASP) extends
s(ASP) by computing partial stable models of programs with
constraints. This extension makes the following contributions:
[Table: Speed comparison: s(CASP) vs. s(ASP) (time in ms).]The in... | {
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f71eff7f0cb9944f014b176c35bca05b1f75ac44 | subsection | 14 | 60 | s(CASP) Programs | An s(CASP) program is a finite set of rules of the form:a \leftarrow c_a \wedge b_1 \wedge \dots \wedge b_m \wedge not\ b_{m+1}
\wedge \dots \wedge not\ b_n.where the difference w.r.t. an ASP program is c_a, a simple constraint or
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f6c61637230513738fc9e37ef6646d24a5907009 | subsection | 15 | 60 | The Interpreter and the Disequality
Constraint Solver | The s(CASP) interpreter carries the environment (the call path and the
model) implicitly and delegates to Prolog all operations that Prolog
can do natively, such as handling the bindings due to unification, the
unbinding due to backtracking, and the operations with constraints,
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6b64737773e2bac396a24a7ba9c0479b0bc4cd18 | subsection | 16 | 60 | Integration of Constraint Solvers in s(CASP) | Holzbaur's CLP({Q}) solver was integrated in the
current implementation of s(CASP). Since the interpreter already
deals with the CLP(\ne ) constraint solver, only two details have to be taken in consideration:The compiler is extended to support CLP({Q}) relations
\lbrace <, >, =, \ge , \le , \ne \rbrace during the con... | {
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{
"arxiv_id": "",
"doi": "",
"end": 84,
"openalex_id": "",
"raw": "Holzbaur, C. 1995. OFAI CLP(Q,R) Manual, Edition 1.3.3. Tech. Rep. TR-95-09, Austrian Research Institute for Artificial Intelligence, Vienna.",
"source_ref_id": "227603c1791e52ac306ca422589df3d... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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0... | |
b7e5525acd1b87018741beb334858327c0924046 | subsection | 17 | 60 | Extending | Extending s(ASP) to programs with constraints requires a
generalization of forall (Algorithm REF )
which we will call
C-forall (Algorithm REF ). A successful evaluation of
[style=MyInline]Goal in s(CASP) returns, on backtracking, a (potentially
infinite) sequence of models and answer constraint stores
A_1, A_2, \dots .... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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0... | |
252fcff7c818f133a8605e3122959341ccad8037 | subsection | 18 | 60 | Examples and Evaluation | The expressiveness of s(CASP) allows the programmer to write programs /
queries that cannot be written in [C]ASP without resorting to a
complex, unnatural encoding. Additionally, the answers given by
s(CASP) are also more expressive than those given by ASP. This arises
from several points:s(CASP) inherits from s(ASP) t... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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0.046570129692554474,
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0.03402732312679291,
0.0025940111372619867,
-0.00... | |
f38fcfdfc18d07440d68c69880f2af30b9680bc1 | subsection | 19 | 60 | Stream Data Reasoning | Let us assume that we deal with data streams, some of whose items may
be contradictory . Moreover, different data
sources may have a different degree of trustworthiness which we use
to prefer a given data item in case of inconsistency.
Let us assume that [style=MyInline]p(X) and [style=MyInline]q(X) are contradictory
a... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.4230/oasics.iclp.2016.17",
"end": 88,
"openalex_id": "https://openalex.org/W2574190931",
"raw": "Arias, J. 2016. Tabled CLP for Reasoning over Stream Data. In Technical Communications of the 32nd Int'l Conference on Logic Programming (IC... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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-0.023556562140583992,
-0.03866083547472954,
-0.01... | |
bbf220c36f7a8bf51ba814cd4c2be22112f585d1 | subsection | 20 | 60 | Stream Data Reasoning | For example, the rule [style=MyInline]incompt(p(X),q(X)) does not have to be
grounded w.r.t. the stream database, and if timestamps were used as
trustworthiness measure,
for a query such as [style=MyInline]?-T.>.10,validstream(T,p(A)) the reasoner
would validate streams
received after [style=MyInline]T=10 regardless ho... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.4230/oasics.iclp.2016.17",
"end": 439,
"openalex_id": "https://openalex.org/W2574190931",
"raw": "Arias, J. 2016. Tabled CLP for Reasoning over Stream Data. In Technical Communications of the 32nd Int'l Conference on Logic Programming (I... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.023929346352815628,
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-0.013658651150763035,
-0.026935776695609093,... | |
56a4ad7415b070e6159983f6e9d0f8b013fe4b33 | subsection | 21 | 60 | Stream Data Reasoning | For the query [style=MyInline]?- validstream(Pr,Data), it returns:
[style=MyInline]{Pr=1, Data=p(A), Aā, Ab̄} because [style=MyInline]q(a) and
[style=MyInline]q(b) are more reliable than [style=MyInline]p(X);
[style=MyInline]{Pr=2, Data=q(b)}; and [style=MyInline]{Pr=3, Data=p(a)}. The
justification tree and the model ... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.014170977286994457,
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0.040331609547138214,
0.014483684673905373,
-0.01361420564353466,
-0.02869279682636261,... | |
dfc968e694660348dc0617b64f66e86ee569af95 | subsection | 22 | 60 | Yale Shooting Scenario | In the spoiling Yale shooting scenario ,
there is a gun and three possible actions: load, shoot,
and wait.
If we load the gun and shoot within 35 minutes, the turkey is killed.
Otherwise, the gun powder is spoiled. The executable plan must ensure
that we kill the turkey within 100 minutes, assuming that we are not
allo... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/s1471068417000242",
"end": 106,
"openalex_id": "https://openalex.org/W2963268497",
"raw": "Janhunen, T., Kaminski, R., Ostrowski, M., Schellhorn, S., Wanko, P., and Schaub, T. 2017. Clingo goes Linear Constraints over Reals and Inte... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.02099490538239479,
-0.03164493665099144,
-0.02865438349545002,
0.016402268782258034,
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0.0024851346388459206,
0.003473085118457675,
0.027937261387705803,
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0.06106221675872803,
-0.0035150444600731134,
0.004897793754935265,
-0.05907868593931198,
0... | |
f2d7f9cf84ffe4252c92c0d02f3614b369439957 | subsection | 23 | 60 | Yale Shooting Scenario | The query [style=MyInline]?-T.<.100, holds(T,st(dead,,),Actions), sets an upper bound
to
the duration [style=MyInline]T of the plan, and returns in
[style=MyInline]Actions the plan with the actions in reverse chronological
order: [style=MyInline]{T=55, Actions=[shoot, load, load]}, [style=MyInline]{T=66, Actions=[shoot... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
0.0018931848462671041,
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0.018662892282009125,
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0.0029585184529423714,
-0.004722947720438242,
0.04648173972964287,
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0.03299196437001228,
-0.02136389911174774,
0.004581793211400509,
-0.032595206052064896,
... | |
dab0de9980bfa35e8341996ae4ca5ea56d3b0006 | subsection | 24 | 60 | The Traveling Salesman Problem (TSP) | Let us consider a variant of the traveling salesman problem (visiting
every city in a country only once, starting and ending in the same
city, and moving between cities using the existing connections) where
we want to find out only the Hamiltonian cycles whose length is less
than a given upper bound.
Solutions for this... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/11562931_8",
"end": 422,
"openalex_id": "https://openalex.org/W1572712485",
"raw": "Dovier, A., Formisano, A., and Pontelli, E. 2005. A Comparison of CLP(FD) and ASP Solutions to NP-Complete Problems. In International Conference on ... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.03637581318616867,
-0.03637581318616867,
-0.059080179780721664,
0.04968106001615524,
0.001654573017731309,
-0.014258891344070435,
-0.022139808163046837,
-0.010223068296909332,
-0.006267351098358631,
0.05807312950491905,
-0.02395554818212986,
0.038267843425273895,
-0.0035971468314528465,
... | |
18d6d1c2237e79e9671c396d85a4ef1b7a01ff7e | subsection | 25 | 60 | The Traveling Salesman Problem (TSP) | The
ASP encoding is more compact, even if the CLP(\mathit {FD}) version uses the non-trivial
library predicate [style=MyInline]circuit/1, which does the bulk of the work.
We will show that s(CASP) is more expressive also in this problem.Finding the (bounded) path length in ASP requires using a specific,
ad-hoc builtin ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 683,
"openalex_id": "",
"raw": "Hölldobler, S. and Schweizer, L. 2014. Answer Set Programming and clasp, a Tutorial. In Young Scientists’ International Workshop on Trends in Information Processing (YSIP). 77.",
"source_ref_i... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.02765805274248123,
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-0.03353138267993927,
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0.009900758974254131,
0.046376436948776245,
-0.010465208441019058,
0.021235525608062744,
-0.0034496288280934095,
... | |
57f04f72f7479e0a9ddfb2a653cf7ed0d57e1c71 | subsection | 26 | 60 | Towers of Hanoi | We will not explain this problem here as it is widely known.
Let us just remind the reader that solving the puzzle with three
towers (the standard setup) and n disks requires at least 2^n - 1
movements.
[Figure: s(CASP) code for the Towers of Hanoi.]Known ASP encodings, for a standard solver, set a bound to the
number ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 408,
"openalex_id": "",
"raw": "Gebser, M., Kaminski, R., Kaufmann, B., Ostrowski, M., Schaub, T., and Thiele, S. 2008. A User’s Guide to gringo, clasp, clingo, and iclingo.",
"source_ref_id": "ba13e661b868d8a012f6aa0ad6db22... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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0.012653267942368984,
0.021096406504511833,
-0.0028334471862763166,
0.0009142920607700944,
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0.029837001115083694,
-0.007260947022587061,
0.010418543592095375,
-0.041948746889829636,
... | |
44e5c0167a47e51d15ce0d490f542c6a1f30d545 | subsection | 27 | 60 | Towers of Hanoi | For reference, .3 shows the partial stable
model for [style=MyInline]?-hanoi(7,T).
[Table: Run time (ms) comparison for the Towers of Hanoi with ndisks.]Table REF compares execution time (in milliseconds)
needed to solve the Towers of Hanoi with [style=MyInline]n disks by s(CASP) and
clingo 5.2.0 with the standard and ... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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-0.0011092528002336621,
-0.04898643493652344,
... | |
581908c8d3ec8565fd70c3ddeee4d6dcd38df495 | subsection | 28 | 60 | Conclusion and Future Work | We have reported on the design and implementation of s(CASP), a
top-down system to evaluate constraint answer set programs, based on
s(ASP). Its ability to express non-monotonic programs à la ASP is
coupled with the possibility of expressing control in a way similar to
traditional logic programming — and, in fact, a si... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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0.041872501373291016,
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0.029618995264172554,
-0.018387887626886368,
0.... | |
91f7258f33c7028bda973c5b44b24e29e6c1c350 | subsection | 29 | 60 | s(CASP) interpreter | The next figure shows a sketch of the s(CASP) interpreter's code
implemented in Ciao Prolog.2
[style=MyProlog, basewidth=.51em]
??(Query) :-
solve(Query,[],Mid),
solvegoal(nmrcheck,Mid,Just),
printjustmodel(Just).solve([],In,['$success'|In]).
solve([Goal|Gs],In,Out) :-
solvegoal(Goal,In,Mid),
solve(Gs,Mid,Out).solvegoa... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.0236093457788229,
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0.01585656777024269,
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-0.013101889751851559,
-0.016787512227892876,
... | |
9e16b42d1ee1b3a9fa7520955f93287f879a74f3 | subsection | 30 | 60 | Handling Loops | Top-down evaluations may enter loops. Several techniques, notably
tabling,
have been used to enhance the termination properties of LP systems.
This is more relevant in s(ASP) because the presence of
negation introduces new types of loops:Odd loop over negation: it occurs when a cycle in the
call graph contains an odd n... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-540-74610-2_4",
"end": 1649,
"openalex_id": "https://openalex.org/W1963107927",
"raw": "Gupta, G., Bansal, A., Min, R., Simon, L., and Mallya, A. 2007. Coinductive Logic Programming and its Applications. Logic Programming, 27–... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.03330386057496071,
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0.05430574342608452,
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0.04539215564727783,
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-0.033853329718112946,
-0.02820601873099804,
0.02... | |
d5810f0a0cd6acaf2bd4d2f9c76434c011c5035c | subsection | 31 | 60 | Handling Loops | We work
around this by checking that the call and its ancestor are equal
(Section REF ).
Example 10 The next program generates infinitely many answers to the query
[style=MyInline]?-nat(X).
2
[style=MyProlog]
nat(0).
nat(X) :- nat(Y), X is Y+1.
However, if it fails, when the recursive call [style=MyInline]nat(Y)
u... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.034902095794677734,
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0.03914282098412514,
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-0.0014177069533616304,
0.0035619051195681095,
-0.02861727774143219,
0.041736070066690445,
-0.007192455232143402,
-0.020288368687033653,
-0.039356384426355... | |
12932e0115a6b8c3fa8f2f5f494e1123dd0767a1 | subsection | 32 | 60 | Handling Loops | 2
[style=MyProlog, basewidth=.52em]
validstream(P,Data) :-
stream(P,Data),
not cancelled(P,Data).
cancelled(P,Data) :-
higherprio(P1,P),
stream(P1,Data1),
incompt(Data,Data1).
higherprio(PHi,PLo) :-
PHi.>.PLo.
incompt(p(X),q(X)).
incompt(q(X),p(X)).
stream(1,p(X)).
stream(2,q(a)).
stream(2,q(b)).
stream(3,p(a)).
not in... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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0.027205029502511024,
0.01846219040453434,
-0.0426613911986351,
-0.03729057312011719,
0.00... | |
ba1410f0722b8667fac2ca41ee5cad01bad40060 | subsection | 33 | 60 | Handling Loops | The output to a query consists of:
(i) a justification tree with the successful derivation (note that
variables could be free, ground, or constrained); (ii)
a model with the positive atoms defined by the program that support
the successful derivation; and (iii) the bindings of variables in the
query (in this example, t... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.020993974059820175,
-0.036068134009838104,
-0.044062934815883636,
0.01489107497036457,
-0.013769666664302349,
-0.05089817941188812,
-0.023801308125257492,
0.03573247790336609,
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0.02612040936946869,
-0.03734974563121796,
-0.0004548567230813205,
-0.00018964282935485244... | |
90e8cdb3141649bc0f75c7792196f090f254e590 | subsection | 34 | 60 | Handling Loops | Answer 1 (in 18.907 ms):
validstream(1,p(A.[a,b])) :-
stream(1,p(A.[a,b])),
not cancelled(1,p(A.[a,b])) :-
not ocancelled1(1,p(A.[a,b])) :-
forall(B,forall(C,not ocancelled1(1,p(A.[a,b]),B,C))) :-
forall(C,not ocancelled1(1,p(A.[a,b]),D.=<.1,C)) :-
not ocancelled1(1,p(A.[a,b]),D.=<.1,C) :-
not higherprio(D.=<.1,1) :-
n... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/s1471068417000242",
"end": 7085,
"openalex_id": "https://openalex.org/W2963268497",
"raw": "Janhunen, T., Kaminski, R., Ostrowski, M., Schellhorn, S., Wanko, P., and Schaub, T. 2017. Clingo goes Linear Constraints over Reals and Int... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.03737388551235199,
-0.015552111901342869,
-0.030890658497810364,
-0.022439584136009216,
0.026558339595794678,
0.0006178132025524974,
0.01880897954106331,
0.031363554298877716,
-0.01774115301668644,
0.02872449904680252,
0.007894279435276985,
-0.04228588193655014,
-0.050340332090854645,
0... | |
f0379b721d5c76e81c746400bab9016122a73f6d | subsection | 35 | 60 | Handling Loops | 2
[style=MyASP]
include "incmodelc.lp".
program base.
action(load).
action(shoot).
action(wait).
duration(load,25).
duration(shoot,5).
duration(wait,36).
unloaded(0).
sum at(0) = 0.
sum armed(0) = 0.
program step(n).
1 do(X,n) : action(X) 1.
sum at(n),-1*at(N') = D :-
do(X,n),
duration(X,D),
N' = n - 1.
loaded(n) :-
lo... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/11562931_8",
"end": 1067,
"openalex_id": "https://openalex.org/W1572712485",
"raw": "Dovier, A., Formisano, A., and Pontelli, E. 2005. A Comparison of CLP(FD) and ASP Solutions to NP-Complete Problems. In International Conference on... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.018173137679696083,
-0.028747431933879852,
-0.06329318135976791,
0.019821079447865486,
-0.0026740909088402987,
-0.036315757781267166,
-0.025375254452228546,
0.011161754839122295,
-0.020568756386637688,
0.019302282482385635,
-0.02694690227508545,
0.010086014866828918,
-0.003648741869255900... | |
c4d753e510db62a3883d01d53edde2a083e98879 | subsection | 36 | 60 | Handling Loops | 2
[style=MyProlog, basewidth=0.52em]
hamiltonianpath(Path) :-
graph(Nodes,Edges),
length(Nodes,N),
length(Path,N),
domain(Path,1,N),
makedomains(Path,1,Edges,N),
circuit(Path),
labeling([ff],Path).
makedomains([],,,).
makedomains([X|Y],Node,Edges,N) :-
findall(Z,
member([Node,Z],Edges),Succs),
reducedomains(N,Succs,X),... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.008620279841125011,
-0.03539654612541199,
-0.022900955751538277,
0.00958910770714283,
-0.029797179624438286,
-0.0580076165497303,
0.021451527252793312,
-0.017332101240754128,
-0.0026432981248944998,
0.03524397686123848,
-0.029293693602085114,
0.021893983706831932,
-0.013372876681387424,
... | |
f8c78f3623cf9d315a1f006de4b7b8ecd76d1e22 | subsection | 37 | 60 | Handling Loops | Answer 1 (in [2346.489] ms):
[ travelpath(b,61/10,[b,[31/10],c,[1],a,[1],d,[1],b]), path(b,b,b,61/10,[],
[b,[31/10],c,[1],a,[1],d,[1],b]), cycledist(d,b,1), cycle(d,b), edge(d,b),
distance(d,b,1), node(d), node(b), node(a), edge(d,a), distance(d,a,1),
other(d,a), node(b), cycle(d,b), node(c), distance(d,b,1), path(b,b,... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1488,
"openalex_id": "",
"raw": "Gebser, M., Kaminski, R., Kaufmann, B., Ostrowski, M., Schaub, T., and Thiele, S. 2008. A User’s Guide to gringo, clasp, clingo, and iclingo.",
"source_ref_id": "ba13e661b868d8a012f6aa0ad6db2... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.011079645715653896,
0.010171603411436081,
-0.03769521415233612,
0.006230393424630165,
-0.005410102661699057,
-0.01671103574335575,
-0.009957946836948395,
-0.006234208587557077,
-0.029423633590340614,
0.010766791179776192,
-0.030201954767107964,
0.008622589521110058,
0.012369219213724136,
... | |
75f35b331ddfdcdccace766258f17cc754a25cca | subsection | 38 | 60 | Handling Loops | 1 move(D,P,T) : disk(D) : peg(P) 1 :- moves(M), T = 1..M.
move(D,T) :- move(D,,T).
on(D,P,0) :- initon(D,P).
on(D,P,T) :- move(D,P,T).
on(D,P,T+1) :- on(D,P,T), not move(D,T+1), not moves(T).
blocked(D-1,P,T+1) :- on(D,P,T), disk(D), not moves(T).
blocked(D-1,P,T) :- blocked(D,P,T), disk(D).
:- move(D,P,T), blocked(D-1... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
0.01174781285226345,
0.007948844693601131,
-0.02187839336693287,
0.023510882630944252,
-0.010389948263764381,
-0.04159030690789223,
0.006400269456207752,
0.02364819310605526,
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0.003982050810009241,
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-0.024288984015583992,
-0.018079426139593124,
... | |
3a8edfad7a4de42251687debc2c7588b7fa6a547 | subsection | 39 | 60 | Handling Loops | Answer 1 (in [420.343] ms):
[ hanoi(7,127), move(a,b,1), move(a,c,2), move(b,c,3), move(a,b,4),
move(c,a,5), move(c,b,6), move(a,b,7), move(a,c,8), move(b,c,9),
move(b,a,10), move(c,a,11), move(b,c,12), move(a,b,13), move(a,c,14),
move(b,c,15), move(a,b,16), move(c,a,17), move(c,b,18), move(a,b,19),
move(c,a,20), move(... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.018950743600726128,
0.04696488380432129,
-0.009612695313990116,
-0.007629123516380787,
-0.0119395786896348,
0.030150296166539192,
-0.041319333016872406,
0.017913181334733963,
-0.02996719628572464,
0.012633828446269035,
-0.016631489619612694,
-0.009238868951797485,
0.008277598768472672,
... | |
76de50eaf7659285c3d1ed83cd7e76651b839a6f | subsection | 40 | 60 | Handling Loops | Several techniques, notably
tabling,
have been used to enhance the termination properties of LP systems.
This is more relevant in s(ASP) because the presence of
negation introduces new types of loops:Odd loop over negation: it occurs when a cycle in the
call graph contains an odd number of intervening negations. These
... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-540-74610-2_4",
"end": 1611,
"openalex_id": "https://openalex.org/W1963107927",
"raw": "Gupta, G., Bansal, A., Min, R., Simon, L., and Mallya, A. 2007. Coinductive Logic Programming and its Applications. Logic Programming, 27–... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.0394546240568161,
0.008856693282723427,
-0.034541867673397064,
0.04427583888173103,
-0.020383363589644432,
-0.051873829215765,
0.031246352940797806,
-0.005290367640554905,
-0.015409578569233418,
0.06185191124677658,
-0.02467058226466179,
-0.02616577036678791,
-0.029720652848482132,
0.02... | |
026266fb51dfacb852c579cea3cba36cec1f1619 | subsection | 41 | 60 | Handling Loops | We work
around this by checking that the call and its ancestor are equal
(Section REF ).
Example 10 The next program generates infinitely many answers to the query
[style=MyInline]?-nat(X).
2
[style=MyProlog]
nat(0).
nat(X) :- nat(Y), X is Y+1.
However, if it fails, when the recursive call [style=MyInline]nat(Y)
u... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.034902095794677734,
-0.0008976267999969423,
-0.010540798306465149,
0.03914282098412514,
-0.013446764089167118,
-0.042590320110321045,
-0.0014177069533616304,
0.0035619051195681095,
-0.02861727774143219,
0.041736070066690445,
-0.007192455232143402,
-0.020288368687033653,
-0.039356384426355... | |
6c99060b1cc478b8817a4d4c471426f5ee9c59de | subsection | 42 | 60 | Handling Loops | 2
[style=MyProlog, basewidth=.52em]
validstream(P,Data) :-
stream(P,Data),
not cancelled(P,Data).
cancelled(P,Data) :-
higherprio(P1,P),
stream(P1,Data1),
incompt(Data,Data1).
higherprio(PHi,PLo) :-
PHi.>.PLo.
incompt(p(X),q(X)).
incompt(q(X),p(X)).
stream(1,p(X)).
stream(2,q(a)).
stream(2,q(b)).
stream(3,p(a)).
not in... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.022887013852596283,
-0.03368968516588211,
-0.02035418339073658,
-0.0018958075670525432,
-0.009818528778851032,
-0.04876459762454033,
0.01890467293560505,
0.007777770049870014,
-0.0332319438457489,
0.027205029502511024,
0.01846219040453434,
-0.0426613911986351,
-0.03729057312011719,
0.00... | |
5724822924466ffaacf61a64ac1332f3e252a967 | subsection | 43 | 60 | Handling Loops | The output to a query consists of:
(i) a justification tree with the successful derivation (note that
variables could be free, ground, or constrained); (ii)
a model with the positive atoms defined by the program that support
the successful derivation; and (iii) the bindings of variables in the
query (in this example, t... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.020993974059820175,
-0.036068134009838104,
-0.044062934815883636,
0.01489107497036457,
-0.013769666664302349,
-0.05089817941188812,
-0.023801308125257492,
0.03573247790336609,
-0.000691344088409096,
0.02612040936946869,
-0.03734974563121796,
-0.0004548567230813205,
-0.00018964282935485244... | |
9f2146ace546a232ee8e2f12efa598c89d62e740 | subsection | 44 | 60 | Handling Loops | Answer 1 (in 18.907 ms):
validstream(1,p(A.[a,b])) :-
stream(1,p(A.[a,b])),
not cancelled(1,p(A.[a,b])) :-
not ocancelled1(1,p(A.[a,b])) :-
forall(B,forall(C,not ocancelled1(1,p(A.[a,b]),B,C))) :-
forall(C,not ocancelled1(1,p(A.[a,b]),D.=<.1,C)) :-
not ocancelled1(1,p(A.[a,b]),D.=<.1,C) :-
not higherprio(D.=<.1,1) :-
n... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/s1471068417000242",
"end": 7085,
"openalex_id": "https://openalex.org/W2963268497",
"raw": "Janhunen, T., Kaminski, R., Ostrowski, M., Schellhorn, S., Wanko, P., and Schaub, T. 2017. Clingo goes Linear Constraints over Reals and Int... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.03737388551235199,
-0.015552111901342869,
-0.030890658497810364,
-0.022439584136009216,
0.026558339595794678,
0.0006178132025524974,
0.01880897954106331,
0.031363554298877716,
-0.01774115301668644,
0.02872449904680252,
0.007894279435276985,
-0.04228588193655014,
-0.050340332090854645,
0... | |
93306cbe9ed644706a1e030a816607b24b6cd719 | subsection | 45 | 60 | Handling Loops | 2
[style=MyASP]
include "incmodelc.lp".
program base.
action(load).
action(shoot).
action(wait).
duration(load,25).
duration(shoot,5).
duration(wait,36).
unloaded(0).
sum at(0) = 0.
sum armed(0) = 0.
program step(n).
1 do(X,n) : action(X) 1.
sum at(n),-1*at(N') = D :-
do(X,n),
duration(X,D),
N' = n - 1.
loaded(n) :-
lo... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/11562931_8",
"end": 1067,
"openalex_id": "https://openalex.org/W1572712485",
"raw": "Dovier, A., Formisano, A., and Pontelli, E. 2005. A Comparison of CLP(FD) and ASP Solutions to NP-Complete Problems. In International Conference on... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.018173137679696083,
-0.028747431933879852,
-0.06329318135976791,
0.019821079447865486,
-0.0026740909088402987,
-0.036315757781267166,
-0.025375254452228546,
0.011161754839122295,
-0.020568756386637688,
0.019302282482385635,
-0.02694690227508545,
0.010086014866828918,
-0.003648741869255900... | |
1a362459522b676123b5b51a34ad5be970a5b9cc | subsection | 46 | 60 | Handling Loops | 2
[style=MyProlog, basewidth=0.52em]
hamiltonianpath(Path) :-
graph(Nodes,Edges),
length(Nodes,N),
length(Path,N),
domain(Path,1,N),
makedomains(Path,1,Edges,N),
circuit(Path),
labeling([ff],Path).
makedomains([],,,).
makedomains([X|Y],Node,Edges,N) :-
findall(Z,
member([Node,Z],Edges),Succs),
reducedomains(N,Succs,X),... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.008620279841125011,
-0.03539654612541199,
-0.022900955751538277,
0.00958910770714283,
-0.029797179624438286,
-0.0580076165497303,
0.021451527252793312,
-0.017332101240754128,
-0.0026432981248944998,
0.03524397686123848,
-0.029293693602085114,
0.021893983706831932,
-0.013372876681387424,
... | |
456ccb5401c9252af5e1cbfec7bc5ee78950c678 | subsection | 47 | 60 | Handling Loops | Answer 1 (in [2346.489] ms):
[ travelpath(b,61/10,[b,[31/10],c,[1],a,[1],d,[1],b]), path(b,b,b,61/10,[],
[b,[31/10],c,[1],a,[1],d,[1],b]), cycledist(d,b,1), cycle(d,b), edge(d,b),
distance(d,b,1), node(d), node(b), node(a), edge(d,a), distance(d,a,1),
other(d,a), node(b), cycle(d,b), node(c), distance(d,b,1), path(b,b,... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1488,
"openalex_id": "",
"raw": "Gebser, M., Kaminski, R., Kaufmann, B., Ostrowski, M., Schaub, T., and Thiele, S. 2008. A User’s Guide to gringo, clasp, clingo, and iclingo.",
"source_ref_id": "ba13e661b868d8a012f6aa0ad6db2... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
-0.011079645715653896,
0.010171603411436081,
-0.03769521415233612,
0.006230393424630165,
-0.005410102661699057,
-0.01671103574335575,
-0.009957946836948395,
-0.006234208587557077,
-0.029423633590340614,
0.010766791179776192,
-0.030201954767107964,
0.008622589521110058,
0.012369219213724136,
... | |
2d57db5d0800f64d9462e9a9cd94322be324a22d | subsection | 48 | 60 | Handling Loops | 1 move(D,P,T) : disk(D) : peg(P) 1 :- moves(M), T = 1..M.
move(D,T) :- move(D,,T).
on(D,P,0) :- initon(D,P).
on(D,P,T) :- move(D,P,T).
on(D,P,T+1) :- on(D,P,T), not move(D,T+1), not moves(T).
blocked(D-1,P,T+1) :- on(D,P,T), disk(D), not moves(T).
blocked(D-1,P,T) :- blocked(D,P,T), disk(D).
:- move(D,P,T), blocked(D-1... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
0.01174781285226345,
0.007948844693601131,
-0.02187839336693287,
0.023510882630944252,
-0.010389948263764381,
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0.006400269456207752,
0.02364819310605526,
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0.003982050810009241,
-0.0030456585809588432,
-0.024288984015583992,
-0.018079426139593124,
... | |
2e726355688ce553b1c2d0b884567572f1535022 | subsection | 49 | 60 | Handling Loops | Answer 1 (in [420.343] ms):
[ hanoi(7,127), move(a,b,1), move(a,c,2), move(b,c,3), move(a,b,4),
move(c,a,5), move(c,b,6), move(a,b,7), move(a,c,8), move(b,c,9),
move(b,a,10), move(c,a,11), move(b,c,12), move(a,b,13), move(a,c,14),
move(b,c,15), move(a,b,16), move(c,a,17), move(c,b,18), move(a,b,19),
move(c,a,20), move(... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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... | |
40e59e4a5ceb953866180ab5692d82df60b60e94 | subsection | 50 | 60 | s(CASP) encoding of stream.pl | The next figure shows the code of stream.pl with the dual program and the NMR generated by the extended compiler of s(CASP).2
[style=MyProlog, basewidth=.52em]
validstream(P,Data) :-
stream(P,Data),
not cancelled(P,Data).cancelled(P,Data) :-
higherprio(P1,P),
stream(P1,Data1),
incompt(Data,Data1).higherprio(PHi,PLo) :-... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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... | |
2b23163971f4dfe19db68d140637807675195e82 | subsection | 51 | 60 | s(CASP) output of stream.pl | The next figure shows the output for the query
[style=MyInline]?-validstream(Pr,Data) when it is made to the program stream.pl
(see .1). The output to a query consists of:
(i) a justification tree with the successful derivation (note that
variables could be free, ground, or constrained); (ii)
a model with the positive ... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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... | |
a601db2e33da99cef05a18147c176e4baa5a049c | subsection | 52 | 60 | s(CASP) output of stream.pl | The
constraint store active at each call is shown close to each variable.[style=tree]
?- validstream(Pr, Data).Answer 1 (in 18.907 ms):validstream(1,p(A.[a,b])) :-
stream(1,p(A.[a,b])),
not cancelled(1,p(A.[a,b])) :-
not ocancelled1(1,p(A.[a,b])) :-
forall(B,forall(C,not ocancelled1(1,p(A.[a,b]),B,C))) :-
forall(C,not ... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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... | |
12e48f34c8d474ee491b0fe192f5cb856c9cfc8b | subsection | 53 | 60 | ASP + constraint encoding of yale_shooting_asp.pl | Nest figure shows the spoiled Yale shooting scenario model written in
clingo + constraints using multi-shot
solving .2
[style=MyASP]
include "incmodelc.lp".
program base.
action(load).
action(shoot).
action(wait).
duration(load,25).
duration(shoot,5).
duration(wait,36).
unloaded(0).
sum at(0) = 0.
sum armed(0) = 0.prog... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/s1471068417000242",
"end": 117,
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"raw": "Janhunen, T., Kaminski, R., Ostrowski, M., Schellhorn, S., Wanko, P., and Schaub, T. 2017. Clingo goes Linear Constraints over Reals and Inte... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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bb752edf4aee33c28e26907ff1a1ce0f3086cf3f | subsection | 54 | 60 | ASP encoding of hamicycle_asp.pl | The next figure shows an ASP program for the Travelling Salesman
Problem described in section REF . The encoding for
the Hamiltonian cycle part is from and the
code of #sum is adapted to run using clingo. The
bound on the total distance is one of the global constraints in the
program.[style=MyASP]
1 cycle(X,Y) : edge(... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/11562931_8",
"end": 206,
"openalex_id": "https://openalex.org/W1572712485",
"raw": "Dovier, A., Formisano, A., and Pontelli, E. 2005. A Comparison of CLP(FD) and ASP Solutions to NP-Complete Problems. In International Conference on ... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
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0... | |
f94f82fea01ccb775bdd75a98e024dce08b6d445 | subsection | 55 | 60 | CLP(FD) encoding of hamicycle_clpfd.pl | The next figure shows the program in CLP(\mathit {FD}) for the
Hamiltonian cycle problem presented in ,
using SICStus Prolog 3.11.2. Note that the library predicate
[style=MyInline]circuit/1 does the bulk of the work. Its implementation
is non-trivial and shares a lot of code with the
implementation of all_different, a... | {
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{
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"raw": "Dovier, A., Formisano, A., and Pontelli, E. 2005. A Comparison of CLP(FD) and ASP Solutions to NP-Complete Problems. In International Conference on ... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
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8dd2bb07c7331806177dba0736422cfd75581f34 | subsection | 56 | 60 | s(CASP) output of hamicycle_scasp.pl | The next figure shows the output to the query
[style=MyInline]?-D.<.10,cycle(a,D,Cycle) to the program hamicycle_scasp.pl
(Figure REF in Section REF ).[style=tree]
?- D.<.10, travelpath(b, D, Cycle).Answer 1 (in [2346.489] ms):[ travelpath(b,61/10,[b,[31/10],c,[1],a,[1],d,[1],b]), path(b,b,b,61/10,[],
[b,[31/10],c,[1],... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
"cs.LO"
] | 2,018 | en | Computer Science | [
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d126ec478c7c9f434565c9cc6f91b232014c7962 | subsection | 57 | 60 | ASP encoding of toh_asp.pl | The next program is part of :[style=MyASP, basewidth=0.48em]
peg(a;b;c).
disk(1..7).
initon(1..7,a).
goalon(1..7,b).
moves(127).
1 move(D,P,T) : disk(D) : peg(P) 1 :- moves(M), T = 1..M.
move(D,T) :- move(D,,T).
on(D,P,0) :- initon(D,P).
on(D,P,T) :- move(D,P,T).
on(D,P,T+1) :- on(D,P,T), not move(D,T+1), not moves(T).... | {
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"end": 81,
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"raw": "Gebser, M., Kaminski, R., Kaufmann, B., Ostrowski, M., Schaub, T., and Thiele, S. 2008. A User’s Guide to gringo, clasp, clingo, and iclingo.",
"source_ref_id": "ba13e661b868d8a012f6aa0ad6db221... | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
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39d1ed5d8c4105c8a939bda4e27d97d36599d84c | subsection | 58 | 60 | ASP incremental encoding of toh_aspI.pl | The next program is part of the clingo distribution and is
available at
https://github.com/potassco/clingo/tree/master/examples/gringo/toh2
[style=MyASP, basewidth=0.48em]
include <incmode>.program base.
peg(a;b;c).
disk(1..7).
initon(1..7,a).
goalon(1..7,b).on(D,P,0) :- initon(D,P).program step(t).
1 move(D,P,t): disk... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
"cs.PL",
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... | |
03a7d30d50cda475cca7851ec363acdc1665f658 | subsection | 59 | 60 | s(CASP) output of hanoi.pl | [style=tree]
?- hanoi(7,T).Answer 1 (in [420.343] ms):[ hanoi(7,127), move(a,b,1), move(a,c,2), move(b,c,3), move(a,b,4),
move(c,a,5), move(c,b,6), move(a,b,7), move(a,c,8), move(b,c,9),
move(b,a,10), move(c,a,11), move(b,c,12), move(a,b,13), move(a,c,14),
move(b,c,15), move(a,b,16), move(c,a,17), move(c,b,18), move(a,... | {
"cite_spans": []
} | 1804.11162 | Constraint Answer Set Programming without Grounding | [
"Joaquín Arias",
"Manuel Carro",
"Elmer Salazar",
"Kyle Marple",
"Gopal Gupta"
] | [
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c8023be5990a223250b9beb0e6e0f4902130062c | abstract | 0 | 16 | Abstract | With the recent success of embeddings in natural language processing,
research has been conducted into applying similar methods to code analysis.
Most works attempt to process the code directly or use a syntactic tree
representation, treating it like sentences written in a natural language.
However, none of the existin... | {
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} | 1806.07336 | Neural Code Comprehension: A Learnable Representation of Code Semantics | [
"Tal Ben-Nun",
"Alice Shoshana Jakobovits",
"Torsten Hoefler"
] | [
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ce7097cdd75db5420c6f58fac78399514847c8f5 | subsection | 1 | 16 | Introduction | The emergence of the “Big Data era” manifests in the form of a dramatic increase in accessible code. In the year 2017 alone, GitHub reports approximately 1 billion git commits (code modification uploads) written in 337 different programming languages. Sifting through, categorizing, and understanding code thus becomes ... | {
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} | 1806.07336 | Neural Code Comprehension: A Learnable Representation of Code Semantics | [
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a26ba25afc67c6c6e5ee83cb9dab642bd68a536d | subsection | 2 | 16 | Introduction | In turn, the XFG structure is used to train an embedding space for individual statements, called inst2vec (from the word “instruction”), which is fed to RNNs for a variety of high-level tasks.Neural Code Comprehension is evaluated on multiple levels, using clustering and analogies for inst2vec, as well as three differe... | {
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} | 1806.07336 | Neural Code Comprehension: A Learnable Representation of Code Semantics | [
"Tal Ben-Nun",
"Alice Shoshana Jakobovits",
"Torsten Hoefler"
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8482f2152565d6bc7f0c140c49053264ba5ce5fd | subsection | 3 | 16 | Related Work | Distributed representations of code were first suggested by Allamanis et al. , followed by several works leveraging embeddings to apply NLP techniques to programming languages , . | {
"cite_spans": []
} | 1806.07336 | Neural Code Comprehension: A Learnable Representation of Code Semantics | [
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04d91bc61417893cab9c302305d85eadb0513479 | subsection | 4 | 16 | Code Representation | Previous research focuses on embedding high-level programming languages such as Java , , C , or OpenCL in the form of tokens or statements, as well as lower level representations such as object code . To the best of our knowledge, however, no attempt has been made to train embeddings for compiler IRs prior to this wor... | {
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} | 1806.07336 | Neural Code Comprehension: A Learnable Representation of Code Semantics | [
"Tal Ben-Nun",
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b179cc3e2929b313009f1c7e7f3fd0790373f885 | subsection | 5 | 16 | Automated Tasks on Code | Learned representations of code are commonly used for two types of tasks: uncovering program semantics or optimizing programs. For the former task, code embeddings have been used to perform function or variable naming , , clone detection , code completion , ,
summarization , and algorithm classification .
As for progra... | {
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} | 1806.07336 | Neural Code Comprehension: A Learnable Representation of Code Semantics | [
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"Alice Shoshana Jakobovits",
"Torsten Hoefler"
] | [
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] | 2,018 | en | Computer Science | [
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aea957182a5a34be0ad1968a70ddc888ff1f139a | subsection | 6 | 16 | Embedding Evaluation | Previous works that use code embeddings do not evaluate the quality of the trained space on its own merit, but rather through the performance of subsequent (downstream) tasks. One exception is Allamanis et al. , who present empirical evidence of vector similarities for similar method names.
To the best of our knowledge... | {
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e61b5ed15d3138b4f808d207f4049d24915b9035 | subsection | 7 | 16 | A Robust Distributional Hypothesis of Code | The linguistic Distributional Hypothesis , is given by: Words that occur in the same contexts tend to have similar meanings. We stipulate that code, which describes a sequence of operations to a processor, behaves similarly, and paraphrase this hypothesis to:Statements that occur in the same contexts tend to have simil... | {
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} | 1806.07336 | Neural Code Comprehension: A Learnable Representation of Code Semantics | [
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"Alice Shoshana Jakobovits",
"Torsten Hoefler"
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944d4398ff6eab3f0d8562a5311bd7fdbd1b5621 | subsection | 8 | 16 | Statements | To choose the right abstraction for statements, we take two concerns into account: universality and uniformity. As stated above, source code comes in many languages and thus fixating on a single one would hinder universality. At the other extreme, machine code (assembly) is target-specific, containing specialized instr... | {
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8d23ea98f9f8a02272d8a46c105a21b2f4aecc6a | subsection | 9 | 16 | Context | The definition of a context for code statements should also be carefully considered.
We define context as statements whose execution directly depends on each other. Learning from consecutive statements in code does not necessarily fulfill this definition, as, for example, a programmer may use a variable in the first li... | {
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e39e6f1a6e2fda9f4e1269bcc9a02894e8081295 | subsection | 10 | 16 | Similarity | To define similarity, one first needs to define the semantics of a statement. We draw the definition of semantics from Operational Semantics in programming language theory, which refers to the effects (e.g., preconditions, postconditions) of each computational step in a given program. In this paper, we specifically ass... | {
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f7df941dc0ec5adebb7f1f3d757fea1bf8828ca1 | subsection | 11 | 16 | Contextual Flow Processing | The aforementioned statements and contexts cannot be directly extracted from source code, but rather require processing akin to partial compilation (e.g., dataflow extraction). In this section, we briefly describe a popular compilation pipeline and proposed modifications to create a learnable vocabulary of statements a... | {
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} | 1806.07336 | Neural Code Comprehension: A Learnable Representation of Code Semantics | [
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338d9b08ea3572e55d69005c6b2b7c1692fa9f4c | subsection | 12 | 16 | Compilation, Static Single Assignment, and LLVM IR | Major contemporary compilers, such as GCC and LLVM, support multiple programming languages and hardware targets. To avoid duplication in code optimization techniques, they enforce a strict separation between the source language (frontend), an Intermediate Representation (IR) that can be optimized, and the target machin... | {
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8e7d2ecd1033f653f00ee101e9c92f98353b5ef8 | subsection | 13 | 16 | Contextual Flow Graphs | To analyze dataflow for optimization, LLVM divides the IR statements into “basic blocks”, which contain no control-flow divergence, illustrated in Fig. REF . Within a basic block, statements naturally create traceable dataflow as SSA lists data dependencies in the form of input identifiers (even if conditional), and as... | {
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3f94cdc66f825c66e834a4fcc710efcc54b2fae7 | subsection | 14 | 16 | Body | With XFGs providing a notion of context, we can now train an embedding space for individual statements. To support learnability, desiderata for such a space include: (a) statements that are in close proximity should have similar artifacts on a system (i.e., use the same resources); and (b) changing the same attributes ... | {
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2820e46d8576e7d4048af62a2ecc4a9e1b60956b | subsection | 15 | 16 | Preprocessing | First, we filter out comments and metadata from statements. Then, identifiers and immediate values (numeric constants, strings) are replaced with %ID and <INT/FLOAT/STRING> respectively, where immediate values are fed separately to downstream RNNs. Lastly, data structures are “inlined”, that is, their contents are enco... | {
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d3b501aeefdba57581dfeaf0074308926ff8eb66 | abstract | 0 | 58 | Abstract | Let $G$ be a linearly reductive group acting on a vector space $V$, and $f$ a
(semi-)invariant polynomial on $V$. In this paper we study systematically
decompositions of the Bernstein-Sato polynomial of $f$ in parallel with some
representation-theoretic properties of the action of $G$ on $V$. We provide a
technique bas... | {
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"András Cristian Lőrincz"
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6e481088c743d9185d7d3419682b0626e976ed48 | subsection | 1 | 58 | Introduction | The classification of irreducible prehomogeneous vector spaces was achieved in . The computation of b-functions (i.e. Bernstein-Sato polynomials) of their semi-invariants has been completed using sophisticated methods such as microlocal calculus (for example, see , ). Extensive calculations have been done also in the c... | {
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6c63263399cb7ce08fe458a80a49c650e32dbdf5 | subsection | 2 | 58 | Introduction | In Section we explain how one can use the technique based on the multiplicity one property to derive this result in an elementary way.A simple, yet non-trivial example of interest is the following semi-invariant, coming from the quiver 4:\det \begin{pmatrix}
X & Y & 0 \\
0 & Y & Z
\end{pmatrix}.Here X,Y,Z are generic m... | {
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} | 1802.07760 | Decompositions of Bernstein-Sato polynomials and slices | [
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3d8bec343d337135ed954e5b9cad57154d9cf502 | subsection | 3 | 58 | Introduction | Then for any l-tuple (d_1,\dots ,d_l), we use the following notation in \mathbb {C}[s_1,\dots ,s_l]:[s]^{d_1,\dots ,d_l}_{a,b}=\prod _{i=a+1}^b \prod _{j=0}^{d-1} (d_1s_1+\dots +d_ls_l+i+j),where d=m_1d_1+\dots + m_ld_l. | {
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eac8afa4f1e7e1620080e87062c3092eb26fad87 | subsection | 4 | 58 | Definition | First we define and briefly recall some basic properties about Bernstein-Sato polynomials. We will interchangeably call them also b-functions, especially in the contexts of Theorem REF and Lemma REF from Section REF . For details on Bernstein-Sato polynomials, we refer the reader to , .Throughout this paper we work ove... | {
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34f0d655adfbef957b0deae72ad3f923b1cba5a6 | subsection | 5 | 58 | Body | Let G be a (connected) reductive algebraic group, acting rationally on V. That is, we have a morphism of algebraic groups \rho : G \rightarrow \operatorname{GL}(V). Then we have an action of G on \mathbb {C}[V] by (g\cdot f)(v)=f(g^{-1}\cdot v) for all v\in V, where g\in G, f\in \mathbb {C}[V]. We call a polynomial f\i... | {
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c2269fe077c897c23908fbd7cddc926f7cdf6de4 | subsection | 6 | 58 | Body | We have the \operatorname{GL}(V)-equivariant pairing between \mathbb {C}[V]_d and \mathbb {C}[V^*]_d by\langle P,P^*\rangle = P^*(\partial x) \cdot P(x).This gives a \operatorname{GL}(V)-equivariant isomorphism \mathbb {C}[V^*]_d \cong (\mathbb {C}[V]_d)^*.Let f\in \mathbb {C}[V] be a semi-invariant of weight \sigma , ... | {
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74a9e66c3e23047bc39e4c5f1aff05d6ae6d70f4 | subsection | 7 | 58 | Body | Assume that the product \sigma _1\cdots \sigma _l is a multiplicity-free weight in \mathbb {C}[V]. In this case we can take respective dual semi-invariants f^*_1,\dots , f^*_l\in \mathbb {C}[V^*]. Put \underline{f}=(f_1,\dots , f_l) and \underline{f}^*=(f_1^*,\dots , f_l^*). For a multi-variable \underline{s}=(s_1,\dot... | {
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"András Cristian Lőrincz"
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9b01288b4c8e9d4a9c50027a0a6e8fe8dc596ad3 | subsection | 8 | 58 | Bernstein-Sato polynomials of ideals | Now we consider tuples of polynomials \underline{f}=(f_1,\dots ,f_r) with f_i \in \mathbb {C}[V], from a different viewpoint. Following , we introduce (note that in the case of r=1 we recover Definition REF ):Definition 1.5
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183e96df26be6b1db9ed109a3e44f2f628532ea0 | subsection | 9 | 58 | Bernstein-Sato polynomials of ideals | If we let s=s_1+\dots +s_r then there exists a polynomial P_f(s)\in \mathbb {C}[s] such thatD_{\underline{f}}\cdot \underline{f}^{\underline{s}} = P_{\underline{f}}(s)\cdot \underline{f}^{\underline{s}},and the Bernstein-Sato polynomial b_{\underline{f}}(s) divides P_{\underline{f}}(s).As in the case r=1 (by Theorem RE... | {
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416c4512cbcb91a7ab46052075b9c69bed7b8dec | subsection | 10 | 58 | Slices and the multiplicity one property | In this section, we develop several techniques for calculating b-functions. These are similar to the methods used in , , . The slice method developed in Section REF will be used further in Section . | {
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425283167f8b1cb49e7e82d3b8b8fc583013222f | subsection | 11 | 58 | Slices | Let H be a connected affine algebraic group and V a rational H-module. Let f\in \mathbb {C}[V] be a non-zero H-semi-invariant of weight \sigma . Denote by \mathfrak {h} the Lie algebra of H. Fix an element v\in V and let H_v be the stabilizer of v. The tangent space at v to the orbit \mathcal {O}=G\cdot v of v is T_v(\... | {
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98e54f314d59e01fb3567d99780133a7aa051153 | subsection | 12 | 58 | Expansions and the multiplicity one property | We recall and generalize some considerations from . Let G be a (connected) reductive group with a Borel subgroup B that contains a maximal torus T. The irreducible rational G-modules are parameterized by dominant T-weights. Let V an algebraic G-module, and fix f\in \operatorname{SI}(G,V)_\sigma with \sigma multiplicity... | {
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79b9907c6b764d98a8a0b3c00c644b30aeea554b | subsection | 13 | 58 | Expansions and the multiplicity one property | Geometrically, if f_1^{(1)},\dots , f_p^{(1)} generate the (reduced) defining ideal of a closed subset of the zero-set Z(f), then we have an expansion (REF ) as above.Example 2.4
The case considered in is when V is reducible, that is, there is a non-trivial G-decomposition V=E\oplus F. Then \mathbb {C}[V]=\mathbb {C}[... | {
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7c344c97666d3cb4d0eb01bb612a3e2be89f495f | subsection | 14 | 58 | Expansions and the multiplicity one property | N_{\lambda \cdot \sigma ^{-1}}) that is G-dual to f_1^{(1)},\dots , f_p^{(1)} (resp. f_1^{(2)},\dots ,f_p^{(2)}) with respect to the pairing REF .As in , we assume that the following multiplicity one property is satisfied: \mathbb {C}[V]_{\lambda \cdot \sigma ^{d-k-1}}=M_\lambda \cdot f^{d-k-1}, or equivalently:\mbox{T... | {
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1585b34f41f432e47059971f27bce5b5f91daeec | subsection | 15 | 58 | Expansions and the multiplicity one property | Clearly, -1 is a root of b_2(s), and b_1(s) satisfies\left(\displaystyle \sum _{i=1}^n \partial _{i} x_{i}\right) \cdot f^s = b_{1}(s) \cdot f^s.The operator on the LHS equals E+n, where E denotes the usual Euler operator. Hence, we have b_1(s)=ds+n, proving our claim.We note that for all irreducible prehomogeneous spa... | {
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e7b259ca0dca7b66d0388d94f708db2120ab4ede | subsection | 16 | 58 | Examples of irreducible prehomogeneous spaces | As explained in , the decomposition technique as in Example REF can be used to obtain in an elementary way the b-functions of some classical (semi-)invariants such as the determinant and the Pfaffian. Previous proofs rely on sophisticated methods such as Capelli's identity (see , ) or microlocal calculus (see ). Howeve... | {
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14d567d1cf094c6f16ada1b4068fcdbbdfc27a1d | subsection | 17 | 58 | Examples of irreducible prehomogeneous spaces | By Theorem REF , the b-function of f decomposes as b_n(s)=b_{k,1}(s)\cdot b_{k,2}(s), and for any i=1,\dots ,p we have the equation\left(\dfrac{1}{f^{(1)}_i (x)} \, f^{*(2)}_i(\partial x) \right) \cdot f^{s+1}(x) = b_{k,2}(s) f^s(x).We can choose f_1^{(1)} (resp. f_1^{*(2)}) to be k\times k (resp. (n-k) \times (n-k)) m... | {
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188415f7e74d56c679c53738e7ae3d7c4ddb02ba | subsection | 18 | 58 | Examples of irreducible prehomogeneous spaces | By Lemma REF , we have b_{f,\mathcal {O}_k}=b_{f,v}(s) = b_{f_v}(s)=b_{n-k}(s), hence obtaining the desired equality. We will exploit techniques with slices more systematically in the next section.Example 2.9 (\operatorname{SO}(m)\times \operatorname{GL}(n), \Lambda _1 \otimes \Lambda _1), where m>n.This example is als... | {
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87175b911d12bd6c1ca420796f79ad58dd5645e9 | subsection | 19 | 58 | Examples of irreducible prehomogeneous spaces | Specializing the equation above atX=\begin{bmatrix}
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3136ae884c9ab13854797d82b23167a61f15d381 | subsection | 20 | 58 | Examples of irreducible prehomogeneous spaces | Putting r=1, we obtainb_{m,n}(s) =(s+2n-1)(s+2m)\cdot b_{m-1,n-1}(s) = \prod _{i=1}^n \left(s+ 2i-1\right)\left(s+2(m-n+i) \right).Example 2.11 (\operatorname{GL}(2), 3\Lambda _1), the space of binary cubics.This example appears also in . Here V=\operatorname{Sym}^3 \mathbb {C}^2 is the space of binary cubic forms with... | {
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437630fbc7dd728c0a8357de3300d08c265c9189 | subsection | 21 | 58 | Examples of irreducible prehomogeneous spaces | Hence (REF ) holds, and by Theorem REF we have a decomposition b(s) = b_{2,1}(s) \cdot b_{2,2}(s). We give more details for this case. A basis of M_\lambda =M_{\lambda ^*\cdot \sigma } (resp. basis of N_{\lambda ^*}=N_{\lambda \cdot \sigma ^{-1}}) is given by the 2\times 2 minors of\begin{bmatrix}
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65ed6a15193fa71292c6cd599491b341c3e438ce | subsection | 22 | 58 | Examples of irreducible prehomogeneous spaces | Using this, it is easy to see that the multiplicity one property (REF ) holds for all cases k=1,2,3, just as in the above example. Hence one can apply Theorem REF here as well and obtain decompositions of the b-function of f. | {
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