This paper presents the denitions of several complexity classes of non-deterministic search queri... more This paper presents the denitions of several complexity classes of non-deterministic search queries. Classes are dened via two generalizations of the query output tuple problem (used for deterministic queries) to the context of non-deterministic queries. Several taxonomic results are then established. Furthermore, non-deterministic queries are shown not to be in general renable (i.e., they cannot be replaced by) de- terministic queries unless they loose \genericity" (i.e., the independence from data encoding). In other words, either the genericity is to be with- drawn or any 'meaningful' query language must be non-deterministic. An other important issue is to recognize which queries classes can be ecien tly implemented, i.e., one of solutions can be computed using a polynomial time single-valued function. We analyze two classes enjoying this property: the well-known class NQPTIME and a new class, NQPF. NQPTIME is shown to have a strange behavior: nding a solution of a query in the class can be done in polynomial time but testing a possible solution is not. NQPF does not suer from this anomaly and, besides, it is captured by a query language.
Answer Set Programming (ASP) is a novel programming paradigm, which allows to solve problems in a... more Answer Set Programming (ASP) is a novel programming paradigm, which allows to solve problems in a simple and highly declarative way. The language of ASP (function-free disjunctive logic programming) is very expressive, and allows to represent even problems of high complexity (every problem in the complexity class ¦ È ¾ AEÈ AEÈ ).
Answer set programming with aggregates (ASP A ) allows for modelling many problems in a more conc... more Answer set programming with aggregates (ASP A ) allows for modelling many problems in a more concise way than with standard answer set programming (ASP ). Previous works have focused on semantical and theoretical issues, and only few works have addressed computational issues, such as presenting algorithms and methods for implementing ASP A . In this paper, we fix a rich ASP language supporting recursive aggregates, and study means for implementing a reasoning engine for it. In particular, we leverage work on unfounded sets for ASP A , and show how they can be used for characterizing and computing answer sets. Furthermore, we introduce an operator for effectively computing the greatest unfounded set (GUS), and study how it can be evaluated in a modular way, providing a means both for pruning the search space and for answer set checking. We have implemented these ideas and provide a brief sketch of the prototype architecture.
... The well known Nixon Diamond example, shown next, illus-trates a case of ambiguity determined... more ... The well known Nixon Diamond example, shown next, illus-trates a case of ambiguity determined by multiple in-herit ance. Example 2.5(De feating). BQM program Pnix is shown below: The Nixon Diamond quaker = Ipacif?! c} republican = Tpacz f zst e} ...
The paper proposes a knowledge representation language which extends logic programming with disju... more The paper proposes a knowledge representation language which extends logic programming with disjunction, inheritance, true negation and modularization. The resulting language is called Disjunctive Ordered Logic (DOL for short). A model-theoretic semantics for DOL is given, and it is shown to extend the stable model semantics of disjunctive logic programs. A number of examples show the suitability of DOL for knowledge representation and commonsense reasoning. Among other things, the proposed language appears to be a powerful tool for the description of diagnostic processes which are based on stepwise refinements. The complexity and the expressiveness of the language are carefully analyzed. The analysis pays particular attention to the relative power and complexity of inheritance, negation and disjunction. An interesting result in this course concerns the role played by inheritance. Indeed, our results show that inheritance does not increase at all the complexity of any fragment of the language, while it does increase the expressive power of some DOL fragments.
... 55-76. [21] Ross. KA, The Well Founded Semantics for Disjunctive Logic Programs, In W. Kim, J... more ... 55-76. [21] Ross. KA, The Well Founded Semantics for Disjunctive Logic Programs, In W. Kim, J.-M. Nicolas and S. Nishio, ed., Deductive and Object-Oriented Databases, pages 385-402, Elsevier Science Publishers BV, 1990. ... LPNMR'95 (to appear). [25] Van Gelder, A., Ross ...
... 290 Nicola Leone et al. on a collection of benchmark programs taken from different domains. W... more ... 290 Nicola Leone et al. on a collection of benchmark programs taken from different domains. We mainly selected programs where the instantiation process is hard, and it takes a rele-vant part of the entire computation (like, eg, CRISTAL, HANOI, RAMSEY), but we considered ...
Abstract. The paper presents OntoDLV a system based on an extension of Disjunctive Logic Programm... more Abstract. The paper presents OntoDLV a system based on an extension of Disjunctive Logic Programming (DLP) which combines the expres-sive power of DLP with the modeling capabilities of the object-oriented languages. In particular, the OntoDLV language supports the most ...
This volume contains the papers presented at the third international workshop on Applications of ... more This volume contains the papers presented at the third international workshop on Applications of Logic Programming to the (Semantic) Web and Web Services (ALPSWS2008) held on the 12h of December 2008 in Udine, Italy, as part of the 24 th International Conference on Logic Programming (ICLP 2008).
Logic Based Agents for E-learning Alfredo Garro, Nicola Leone and Francesco Ricca University of C... more Logic Based Agents for E-learning Alfredo Garro, Nicola Leone and Francesco Ricca University of Calabria, 87030 Rende (CS), Italy garro@ si. deis. unical. it, leone@ mat. unical. it, ricca@ mat. unical. it Abstract Informally, a learning path is a sequence of learn-ing objects ...
Abstract. In this paper we present the IDUM system, a successful application of logic programming... more Abstract. In this paper we present the IDUM system, a successful application of logic programming to e-tourism. IDUM exploits two technologies that are based on the state-of-the-art ASP system DLV: (i) a system for ontology representation and reasoning, called OntoDLV; and, (ii) ...
Translation of Aggregate Programs to Normal Logic Programs .............. ... Graphs and coloring... more Translation of Aggregate Programs to Normal Logic Programs .............. ... Graphs and colorings for answer set programming with preferences: Preliminary Report ................................................ . ... Properties of maximal cliques of a pair-wise compatibility graph for three ...
One of the most significant language extensions to Answer Set Program- ming (ASP) has been the in... more One of the most significant language extensions to Answer Set Program- ming (ASP) has been the introduction of aggregates. A signifi cant amount of theoretical and practical work on aggregates in ASP has been published in recent years. In spite of these developments, aggregates are treated in a quite straightforward and ad-hoc way in most ASP systems. For the system
... Step 1. Translate P in an equivalent program P ; Step 2. Translate each rule of P in a corres... more ... Step 1. Translate P in an equivalent program P ; Step 2. Translate each rule of P in a corresponding SQL statement; Step 3. Compose and execute the query plan of statements generated in Step 2 on the DBMS; Step 4. Generate the residual program and load it in the Model ...
ABSTRACT The addition of aggregates has been one of the most relevant enhancements to the languag... more ABSTRACT The addition of aggregates has been one of the most relevant enhancements to the language of answer set programming (ASP). They strengthen the modelling power of ASP in terms of natural and concise problem representations. In this paper, we carry out an in-depth study of the computational complexity of the language. The analysis pays particular attention to the impact of syntactical restrictions on programs in the form of limited use of aggregates, disjunction, and negation. While the addition of aggregates does not affect the complexity of the full language with negation and disjunction, it turns out that their presence does increase the complexity of non-disjunctive ASP programs up to the second level of the polynomial hierarchy. Interestingly, under cautious reasoning nonmonotone aggregates are even harder than disjunction ( Pp2{\it \Pi}{^{p}_{2}} -complete vs co-NP-complete on positive programs). However, we show that there are large classes of aggregates the addition of which does not cause any complexity gap even for normal programs, including the fragment allowing for arbitrary monotone, arbitrary antimonotone, and stratified (i.e., non-recursive) nonmonotone aggregates. Moreover, we also prove that for positive programs with arbitrary monotone, stratified antimonotone, and stratified nonmonotone aggregates the complexity remains polynomial. This analysis provides some useful indications on the possibility to implement aggregates in existing reasoning engines.
Second order logic over finite structures is well-known to capture the levels of the polynomial h... more Second order logic over finite structures is well-known to capture the levels of the polynomial hierarchy PH. Recently, it has been shown that Θ
This paper presents the denitions of several complexity classes of non-deterministic search queri... more This paper presents the denitions of several complexity classes of non-deterministic search queries. Classes are dened via two generalizations of the query output tuple problem (used for deterministic queries) to the context of non-deterministic queries. Several taxonomic results are then established. Furthermore, non-deterministic queries are shown not to be in general renable (i.e., they cannot be replaced by) de- terministic queries unless they loose \genericity" (i.e., the independence from data encoding). In other words, either the genericity is to be with- drawn or any 'meaningful' query language must be non-deterministic. An other important issue is to recognize which queries classes can be ecien tly implemented, i.e., one of solutions can be computed using a polynomial time single-valued function. We analyze two classes enjoying this property: the well-known class NQPTIME and a new class, NQPF. NQPTIME is shown to have a strange behavior: nding a solution of a query in the class can be done in polynomial time but testing a possible solution is not. NQPF does not suer from this anomaly and, besides, it is captured by a query language.
Answer Set Programming (ASP) is a novel programming paradigm, which allows to solve problems in a... more Answer Set Programming (ASP) is a novel programming paradigm, which allows to solve problems in a simple and highly declarative way. The language of ASP (function-free disjunctive logic programming) is very expressive, and allows to represent even problems of high complexity (every problem in the complexity class ¦ È ¾ AEÈ AEÈ ).
Answer set programming with aggregates (ASP A ) allows for modelling many problems in a more conc... more Answer set programming with aggregates (ASP A ) allows for modelling many problems in a more concise way than with standard answer set programming (ASP ). Previous works have focused on semantical and theoretical issues, and only few works have addressed computational issues, such as presenting algorithms and methods for implementing ASP A . In this paper, we fix a rich ASP language supporting recursive aggregates, and study means for implementing a reasoning engine for it. In particular, we leverage work on unfounded sets for ASP A , and show how they can be used for characterizing and computing answer sets. Furthermore, we introduce an operator for effectively computing the greatest unfounded set (GUS), and study how it can be evaluated in a modular way, providing a means both for pruning the search space and for answer set checking. We have implemented these ideas and provide a brief sketch of the prototype architecture.
... The well known Nixon Diamond example, shown next, illus-trates a case of ambiguity determined... more ... The well known Nixon Diamond example, shown next, illus-trates a case of ambiguity determined by multiple in-herit ance. Example 2.5(De feating). BQM program Pnix is shown below: The Nixon Diamond quaker = Ipacif?! c} republican = Tpacz f zst e} ...
The paper proposes a knowledge representation language which extends logic programming with disju... more The paper proposes a knowledge representation language which extends logic programming with disjunction, inheritance, true negation and modularization. The resulting language is called Disjunctive Ordered Logic (DOL for short). A model-theoretic semantics for DOL is given, and it is shown to extend the stable model semantics of disjunctive logic programs. A number of examples show the suitability of DOL for knowledge representation and commonsense reasoning. Among other things, the proposed language appears to be a powerful tool for the description of diagnostic processes which are based on stepwise refinements. The complexity and the expressiveness of the language are carefully analyzed. The analysis pays particular attention to the relative power and complexity of inheritance, negation and disjunction. An interesting result in this course concerns the role played by inheritance. Indeed, our results show that inheritance does not increase at all the complexity of any fragment of the language, while it does increase the expressive power of some DOL fragments.
... 55-76. [21] Ross. KA, The Well Founded Semantics for Disjunctive Logic Programs, In W. Kim, J... more ... 55-76. [21] Ross. KA, The Well Founded Semantics for Disjunctive Logic Programs, In W. Kim, J.-M. Nicolas and S. Nishio, ed., Deductive and Object-Oriented Databases, pages 385-402, Elsevier Science Publishers BV, 1990. ... LPNMR'95 (to appear). [25] Van Gelder, A., Ross ...
... 290 Nicola Leone et al. on a collection of benchmark programs taken from different domains. W... more ... 290 Nicola Leone et al. on a collection of benchmark programs taken from different domains. We mainly selected programs where the instantiation process is hard, and it takes a rele-vant part of the entire computation (like, eg, CRISTAL, HANOI, RAMSEY), but we considered ...
Abstract. The paper presents OntoDLV a system based on an extension of Disjunctive Logic Programm... more Abstract. The paper presents OntoDLV a system based on an extension of Disjunctive Logic Programming (DLP) which combines the expres-sive power of DLP with the modeling capabilities of the object-oriented languages. In particular, the OntoDLV language supports the most ...
This volume contains the papers presented at the third international workshop on Applications of ... more This volume contains the papers presented at the third international workshop on Applications of Logic Programming to the (Semantic) Web and Web Services (ALPSWS2008) held on the 12h of December 2008 in Udine, Italy, as part of the 24 th International Conference on Logic Programming (ICLP 2008).
Logic Based Agents for E-learning Alfredo Garro, Nicola Leone and Francesco Ricca University of C... more Logic Based Agents for E-learning Alfredo Garro, Nicola Leone and Francesco Ricca University of Calabria, 87030 Rende (CS), Italy garro@ si. deis. unical. it, leone@ mat. unical. it, ricca@ mat. unical. it Abstract Informally, a learning path is a sequence of learn-ing objects ...
Abstract. In this paper we present the IDUM system, a successful application of logic programming... more Abstract. In this paper we present the IDUM system, a successful application of logic programming to e-tourism. IDUM exploits two technologies that are based on the state-of-the-art ASP system DLV: (i) a system for ontology representation and reasoning, called OntoDLV; and, (ii) ...
Translation of Aggregate Programs to Normal Logic Programs .............. ... Graphs and coloring... more Translation of Aggregate Programs to Normal Logic Programs .............. ... Graphs and colorings for answer set programming with preferences: Preliminary Report ................................................ . ... Properties of maximal cliques of a pair-wise compatibility graph for three ...
One of the most significant language extensions to Answer Set Program- ming (ASP) has been the in... more One of the most significant language extensions to Answer Set Program- ming (ASP) has been the introduction of aggregates. A signifi cant amount of theoretical and practical work on aggregates in ASP has been published in recent years. In spite of these developments, aggregates are treated in a quite straightforward and ad-hoc way in most ASP systems. For the system
... Step 1. Translate P in an equivalent program P ; Step 2. Translate each rule of P in a corres... more ... Step 1. Translate P in an equivalent program P ; Step 2. Translate each rule of P in a corresponding SQL statement; Step 3. Compose and execute the query plan of statements generated in Step 2 on the DBMS; Step 4. Generate the residual program and load it in the Model ...
ABSTRACT The addition of aggregates has been one of the most relevant enhancements to the languag... more ABSTRACT The addition of aggregates has been one of the most relevant enhancements to the language of answer set programming (ASP). They strengthen the modelling power of ASP in terms of natural and concise problem representations. In this paper, we carry out an in-depth study of the computational complexity of the language. The analysis pays particular attention to the impact of syntactical restrictions on programs in the form of limited use of aggregates, disjunction, and negation. While the addition of aggregates does not affect the complexity of the full language with negation and disjunction, it turns out that their presence does increase the complexity of non-disjunctive ASP programs up to the second level of the polynomial hierarchy. Interestingly, under cautious reasoning nonmonotone aggregates are even harder than disjunction ( Pp2{\it \Pi}{^{p}_{2}} -complete vs co-NP-complete on positive programs). However, we show that there are large classes of aggregates the addition of which does not cause any complexity gap even for normal programs, including the fragment allowing for arbitrary monotone, arbitrary antimonotone, and stratified (i.e., non-recursive) nonmonotone aggregates. Moreover, we also prove that for positive programs with arbitrary monotone, stratified antimonotone, and stratified nonmonotone aggregates the complexity remains polynomial. This analysis provides some useful indications on the possibility to implement aggregates in existing reasoning engines.
Second order logic over finite structures is well-known to capture the levels of the polynomial h... more Second order logic over finite structures is well-known to capture the levels of the polynomial hierarchy PH. Recently, it has been shown that Θ
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