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Three Semantics for Modular Systems
[article]

2014
*
arXiv
*
pre-print

and

arXiv:1405.1229v1
fatcat:mvvpx37hl5drfhsxop2f5mqbfy
*Ternovska*2010a) . ... ILP, ASP-CP, DPLL(T)-based) (Tasharrofi, Wu, and*Ternovska*2011; Tasharrofi, Wu, and*Ternovska*2012) . ...##
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A Logic for Non-Monotone Inductive Definitions
[article]

2005
*
arXiv
*
pre-print

,∆i) = · Marc Denecker and

arXiv:cs/0501025v1
fatcat:uswqrvsanrcepgpev7kf6ea3gi
*Eugenia**Ternovska*(I o (∆1,...,∆i−1) ) ∆i , I extends I o (∆1,...,∆i−1) and I |= ∆ i . By the induction hypothesis, I satisfies also ∆ 1 ∧ . . . ∧ ∆ i−1 . ... · Marc Denecker and*Eugenia**Ternovska*The anti-monotonicity of Γ ∆ implies that the sequence (I ξ ) ξ≥0 is increasing and (J ξ ) ξ≥0 is decreasing. Moreover, for each ξ, I ξ ⊑ J ξ . ...##
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Solving Modular Model Expansion Tasks
[article]

2011
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arXiv
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pre-print

The work we describe here is a part of a research program of developing foundations of declarative solving of search problems. We consider the model expansion task as the task representing the essence of search problems where we are given an instance of a problem and are searching for a solution satisfying certain properties. Such tasks are common in artificial intelligence, formal verification, computational biology. Recently, the model expansion framework was extended to deal with multiple

arXiv:1109.0583v1
fatcat:lq5p4ouwx5clxarwiyhcchdrli
## more »

... ules. In the current paper, inspired by practical combined solvers, we introduce an algorithm to solve model expansion tasks for modular systems. We show that our algorithm closely corresponds to what is done in practice in different areas such as Satisfiability Modulo Theories (SMT), Integer Linear Programming (ILP), Answer Set Programming (ASP).##
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Lifted Relational Algebra with Recursion and Connections to Modal Logic
[article]

2016
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arXiv
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pre-print

We propose a new formalism for specifying and reasoning about problems that involve heterogeneous "pieces of information" -- large collections of data, decision procedures of any kind and complexity and connections between them. The essence of our proposal is to lift Codd's relational algebra from operations on relational tables to operations on classes of structures (with recursion), and to add a direction of information propagation. We observe the presence of information propagation in

arXiv:1612.09251v1
fatcat:ulstwcfr35c3nlqinxykjptnii
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... formalisms for efficient reasoning and use it to express unary negation and operations used in graph databases. We carefully analyze several reasoning tasks and establish a precise connection between a generalized query evaluation and temporal logic model checking. Our development allows us to reveal a general correspondence between classical and modal logics and may shed a new light on the good computational properties of modal logics and related formalisms.##
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Inductive situation calculus

2007
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Artificial Intelligence
*

For more detail, we refer to (Denecker &

doi:10.1016/j.artint.2007.02.002
fatcat:eshson5fzbflvpen4gyiwjz2tm
*Ternovska*2004) . ... A more detailed exposition can be found in (Denecker &*Ternovska*2004) . A new binary connective ← is called the definitional implication. ...##
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Expressive power and abstraction in Essence

2008
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Constraints
*

Development of languages for specifying or modelling problems is an important direction in constraint modelling. To provide greater abstraction and modelling convenience, these languages are becoming more syntactically rich, leading to a variety of questions about their expressive power. In this paper, we consider the expressiveness of Essence, a specification language with a rich variety of syntactic features. We identify natural fragments of Essence that capture the complexity classes P, NP,

doi:10.1007/s10601-008-9050-3
fatcat:dg64v3pux5gs5pxxxljiqgwl5y
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... ll levels Σ p i of the polynomial hierarchy, and all levels k-NEXP of the nondeterministic exponential-time hierarchy. The union of these classes is the very large complexity class ELEMENTARY. One goal is to begin to understand which features play a role in the high expressive power of the language and which are purely features of convenience. We also discuss the expressive limit of Essence, a notion of capturing NP-search which is slightly different than that of capturing NP decision problems, and the formalization of arithmetic in Essence and related languages. Our study of arithmetic raises some, perhaps surprising, issues. Our study is an application of descriptive complexity theory, and illustrates the value of taking a logic-based view of modelling and specification languages.##
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Clause-Learning for Modular Systems
[chapter]

2015
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Lecture Notes in Computer Science
*

We present an algorithm, CDCL-AMS, for solving Modular Systems consisting of a set of modules where, for each module, we have a simple "black-box" solver. The algorithm is based on the Conflict-Directed Clause Learning algorithm for SAT, and communicates asynchronously with the black-box solvers to accommodate high variability in response latencies. this setting, queries are essentially propositional, so for simplicity we present our algorithms in purely propositional form. Our setting is as follows.

doi:10.1007/978-3-319-23264-5_37
fatcat:er7nrzyaxfe3lczoexegtjxpwy
##
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Towards Capturing PTIME with no Counting Construct (but with a Choice Operator)
[article]

2021
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arXiv
*
pre-print

The central open question in Descriptive Complexity is whether there is a logic that characterizes deterministic polynomial time (PTIME) on relational structures. Towards this goal, we define a logic that is obtained from first-order logic with fixed points, FO(FP), by a series of transformations that include restricting logical connectives and adding a dynamic version of Hilbert's Choice operator Epsilon. The formalism can be viewed, simultaneously, as an algebra of binary relations and as a

arXiv:2111.07978v1
fatcat:s2jiasyeafd7hb2kp4uka5ugii
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... near-time modal dynamic logic, where algebraic expressions describing "proofs" or "programs" appear inside the modalities. We show how counting, reachability and "mixed" examples (that include linear equations modulo two) are axiomatized in the logic, and how an arbitrary PTIME Turing machine can be encoded. For each fixed Choice function, the data complexity of model checking is in PTIME. However, there can be exponentially many such functions. A crucial question is under what syntactic conditions on algebraic terms checking just one Choice function is sufficient. Answering this question requires a study of symmetries among computations. This paper sets mathematical foundations towards such a study via algebraic and automata-theoretic techniques.##
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Solving Modular Model Expansion: Case Studies
[chapter]

2013
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Lecture Notes in Computer Science
*

Model expansion task is the task representing the essence of search problems where we are given an instance of a problem and are searching for a solution satisfying certain properties. Such tasks are common in AI planning, scheduling, logistics, supply chain management, etc., and are inherently modular. Recently, the model expansion framework was extended to deal with multiple modules to represent e.g. the task of constructing a logistics service provider relying on local service providers. In

doi:10.1007/978-3-642-41524-1_12
fatcat:h5u5o5vvkvdivfxsvs4ec3r3rm
## more »

... he current paper, we study existing systems that operate in a modular way in order to obtain general principles of solving modular model expansion tasks. We introduce a general algorithm to solve model expansion tasks for modular systems. We demonstrate, through several case studies, that our algorithm closely corresponds to what is done in practice in different areas such as Satisfiability Modulo Theories (SMT), Integer Linear Programming (ILP), Answer Set Programming (ASP). We make our framework language-independent through a model-theoretic development.##
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Reducing Inductive Definitions to Propositional Satisfiability
[chapter]

2005
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Lecture Notes in Computer Science
*

The FO(ID) logic is an extension of classical first-order logic with a uniform representation of various forms of inductive definitions. The definitions are represented as sets of rules and they are interpreted by two-valued well-founded models. For a large class of combinatorial and search problems, knowledge representation in FO(ID) offers a viable alternative to the paradigm of Answer Set Programming. The main reasons are that (i) the logic is an extension of classical logic and (ii) the

doi:10.1007/11562931_18
fatcat:j2hthptbrnffpo42kv5n322gne
## more »

... ntics of the language is based on well-understood principles of mathematical induction. In this paper, we define a reduction from the propositional fragment of FO(ID) to SAT. The reduction is based on a novel characterization of two-valued well-founded models using a set of inequality constraints on level mappings associated with the atoms. We also show how the reduction to SAT can be adapted for logic programs under the stable model semantics. Our experiments show that when using a state of the art SAT solver both reductions are competitive with other answer set programming systems -both direct implementations and SAT based.##
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Grounding Formulas with Complex Terms
[chapter]

2011
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Lecture Notes in Computer Science
*

Given a finite domain, grounding is the the process of creating a variablefree first-order formula equivalent to a first-order sentence. The first-order sentences can be used to describe a combinatorial search problem. Efficient grounding algorithms would help in solving such problems effectively and make advanced solver technology (such as SAT) accessible to a wider variety of users. One promising method for grounding is based on the relational algebra from the field of Database research. We

doi:10.1007/978-3-642-21043-3_2
fatcat:iysh54zqfzaqpl3yq3uggda2me
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... scribe the extension of this method to ground formulas of first-order logic extended with arithmetic and aggregate operators. The method allows choice of particular CNF representations of complex constraints to be parameterized easily. We have implemented the methods we describe, and demonstrated that they can be effective in practice.##
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A logic of nonmonotone inductive definitions

2008
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ACM Transactions on Computational Logic
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· Marc Denecker and

doi:10.1145/1342991.1342998
fatcat:265gngzzhnealdwfqk6v3mjqre
*Eugenia**Ternovska*... In [Denecker and*Ternovska*2004a; 2007] , this solution was integrated in situation calculus. ...##
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On the Complexity of Model Expansion
[chapter]

2010
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Lecture Notes in Computer Science
*

We study the complexity of model expansion (MX), which is the problem of expanding a given finite structure with additional relations to produce a finite model of a given formula. This is the logical task underlying many practical constraint languages and systems for representing and solving search problems, and our work is motivated by the need to provide theoretical foundations for these. We present results on both data and combined complexity of MX for several fragments and extensions of FO

doi:10.1007/978-3-642-16242-8_32
fatcat:3rbwgsab3ba6fifzkw4v6d4vmq
## more »

... hat are relevant for this purpose, in particular the guarded fragment GF k of FO and extensions of FO and GF k with inductive definitions. We present these in the context of the two closely related, but more studied, problems of model checking and finite satisfiability. To obtain results on FO(ID), the extension of FO with inductive definitions, we provide translations between FO(ID) with FO(LFP), which are of independent interest.##
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Algebra of Modular Systems: Containment and Equivalence

2021
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AAAI Conference on Artificial Intelligence
*

The algebra of (

dblp:conf/aaai/BulatovT21
fatcat:wa7zfwcmjbffzck63mttvr7aly
*Ternovska*2019 ) is similar to ours, but it also does not differentiate between abstract and concrete modules. ... Early work on combining heterogeneous modules, with a range of operators, is (Järvisalo et al. 2009 ) and (Tasharrofi and*Ternovska*2011) . ... We would also like to prove a Containment property for an efficient restriction of a dynamic version of the algebra (*Ternovska*2019) , where inputs and outputs of modules are specified, similarly to ...##
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Inputs, Outputs, and Composition in the Logic of Information Flows
[article]

2022
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arXiv
*
pre-print

The logic of information flows (LIF) is a general framework in which tasks of a procedural nature can be modeled in a declarative, logic-based fashion. The first contribution of this paper is to propose semantic and syntactic definitions of inputs and outputs of LIF expressions. We study how the two relate and show that our syntactic definition is optimal in a sense that is made precise. The second contribution is a systematic study of the expressive power of sequential composition in LIF. Our

arXiv:2209.06448v1
fatcat:kiqt3erm4fgt5fcyot7x7usy2q
## more »

... esults on composition tie in the results on inputs and outputs, and relate LIF to first-order logic (FO) and bounded-variable LIF to bounded-variable FO.
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