Contextual hypotheses and semantics of logic programs

Martin, Éric

doi:10.1017/s1471068411000378

Cited by 3 publications

(26 citation statements)

References 21 publications

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“…Given a set of rules P, we use [ P ] to denote the set of literals captured by the Kripke-Kleene semantics of P, that is, the set of all literals all of whose closed instances can be generated from P. In [23], we showed that semantics of P different to the Kripke-Kleene semantics can be captured by [ P + Ω E ] where E is a set of literals and P + Ω E is a set of rules obtained from P by selecting some occurrences of literals in the bodies of P's rules-Ω determines which occurrences of literals to select-and assuming that the members of E hold in the contexts selected by Ω in P. For instance, the well-founded semantics [11] can be obtained by:…”

Section: Can_fly(x) ← Bird(x) Not¬can_fly(x)mentioning

confidence: 99%

“…Our aim in this paper is to present a denotational semantics to logic programming as a complement to the operational semantics presented in [23], so that:…”

Section: Can_fly(x) ← Bird(x) Not¬can_fly(x)mentioning

confidence: 99%

“…Even though we have referred to and we will further refer to notions from logic programming that the reader might not know, e.g., the well founded semantics of a logic program, and even though we have referred to and will further refer a lot to [23], this paper is self contained and does not presuppose any knowledge of logic programming nor to have read [23]. This paper can be seen as a self-contained, particular presentation of logic programming, from the perspective of a denotation semantics, as [23] offers a self-contained, particular presentation of logic programming, from the perspective of an operational semantics. Parts of this paper will establish relationships between the denotational semantics presented here and the operational semantics described in [23], and for that purpose, all the relevant material from [23] will be reproduced here, again in a self-contained style.…”

Section: Can_fly(x) ← Bird(x) Not¬can_fly(x)mentioning

confidence: 99%

“…This paper can be seen as a self-contained, particular presentation of logic programming, from the perspective of a denotation semantics, as [23] offers a self-contained, particular presentation of logic programming, from the perspective of an operational semantics. Parts of this paper will establish relationships between the denotational semantics presented here and the operational semantics described in [23], and for that purpose, all the relevant material from [23] will be reproduced here, again in a self-contained style.…”

Section: Can_fly(x) ← Bird(x) Not¬can_fly(x)mentioning

confidence: 99%

“…Since the operational semantics presented in [23] interprets disjunction and existential quantification constructively, it should come as no surprise that the language used to define (P, Ω) * , E * and [ P + Ω E ] * is modal, with 2 interpreted as "can be proved", "can be derived". For instance, the rule p ← q ∨ ¬q can be mapped to the logical formula 2q ∨ 2¬q → 2p which can be interpreted classically as: if q can be proved or if ¬q can be proved then p can be proved.…”

Section: Can_fly(x) ← Bird(x) Not¬can_fly(x)mentioning

confidence: 99%

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Logic programming as classical inference

Martin

2015

Journal of Applied Logic

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We propose a denotational semantics for logic programming based on a classical notion of logical consequence which is apt to capture the main proposed semantics of logic programs. In other words, we show that any of those semantics can be viewed as a relation of the form T |= X where T is a theory which naturally represents the logic program under consideration together with a set of formulas playing the role of "hypotheses", in a way which is dictated by that semantics, |= is a notion of logical consequence which is classical because negation, disjunction and existential quantification receive their classical meaning, and X represents what can be inferred from the logic program, or an intended interpretation of that logic program (such as an answer-set, its well-founded model, etc.). The logical setting we propose extends the language of classical modal logic as it deals with modal operators indexed by ordinals. We make use of two kinds of basic modal formulas: 2 α ϕ which intuitively means that the logical program can generate ϕ by stage α of the generation process, and 3 α 2 β ϕ with α > β, which intuitively means that ϕ can be used as a hypothesis from stage β of the generation process onwards, possibly expecting to confirm ϕ by stage α (so expecting 2 α ϕ to be generated). This allows us to capture Rondogiannis and Wadge's version of the well-founded semantics [27] where a member of the well-founded model is a closed atom which receives an ordinal truth value of true α or false α for some ordinal α: in our framework, this corresponds to having T |= 2 α ϕ or T |= 2 α ¬ϕ, respectively, with T being the natural representation of the logic program under consideration and the right set of "hypotheses" as dictated by the well-founded semantics. The framework we present goes much beyond the proposed traditional semantics for logic programming, as it can for instance let us investigate under which conditions a set of hypotheses can be minimal, with each hypothesis being activated as late as possible and confirmed as soon as possible, setting the theoretical foundation to sophisticated ways of making local use of hypotheses in knowledge-based systems, while still being theoretically grounded in a classical notion of logical consequence.

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Section: Can_fly(x) ← Bird(x) Not¬can_fly(x)mentioning

confidence: 99%

“…Our aim in this paper is to present a denotational semantics to logic programming as a complement to the operational semantics presented in [23], so that:…”

Section: Can_fly(x) ← Bird(x) Not¬can_fly(x)mentioning

confidence: 99%

Section: Can_fly(x) ← Bird(x) Not¬can_fly(x)mentioning

confidence: 99%

Section: Can_fly(x) ← Bird(x) Not¬can_fly(x)mentioning

confidence: 99%

Section: Can_fly(x) ← Bird(x) Not¬can_fly(x)mentioning

confidence: 99%

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Logic programming as classical inference

Martin

2015

Journal of Applied Logic

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Disjunctive logic programs, answer sets, and the cut rule

Martin

2022

Arch. Math. Logic

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In Minker and Rajasekar (J Log Program 9(1):45–74, 1990), Minker proposed a semantics for negation-free disjunctive logic programs that offers a natural generalisation of the fixed point semantics for definite logic programs. We show that this semantics can be further generalised for disjunctive logic programs with classical negation, in a constructive modal-theoretic framework where rules are built from claims and hypotheses, namely, formulas of the form $$\Box \varphi $$ □ φ and $$\Diamond \Box \varphi $$ ◊ □ φ where $$\varphi $$ φ is a literal, respectively, yielding a “base semantics” for general disjunctive logic programs. Model-theoretically, this base semantics is expressed in terms of a classical notion of logical consequence. It has a complete proof procedure based on a general form of the cut rule. Usually, alternative semantics of logic programs amount to a particular interpretation of nonclassical negation as “failure to derive.” The counterpart in our framework is to complement the original program with a set of hypotheses required to satisfy specific conditions, and apply the base semantics to the resulting set. We demonstrate the approach for the answer set semantics. The proposed framework is purely classical in mainly three ways. First, it uses classical negation as unique form of negation. Second, it advocates the computation of logical consequences rather than of particular models. Third, it makes no reference to a notion of preferred or minimal interpretation.

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Nonmonotonicity in the Framework of Parametric Logic

Martin

2018

Stud Logica

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Contextual hypotheses and semantics of logic programs

Cited by 3 publications

References 21 publications

Logic programming as classical inference

Logic programming as classical inference

Disjunctive logic programs, answer sets, and the cut rule

Nonmonotonicity in the Framework of Parametric Logic

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