Logical Foundations of Well-Founded ∗

Pedro Cabalar Sergei Odintsov David Pearce Department of Computer Science, Sobolev Institute of Mathematics, Universidad Rey Juan Carlos, Corunna University, Spain. Novosibirsk, Russia. Madrid, Spain. [email protected] [email protected] [email protected]

Abstract arbitrary propositional theories.2 A major challenge of the paper is to axiomatise the base . We propose a solution to a long-standing problem in the foun- Well-founded semantics defines a nonmonotonic infer- dations of well-founded semantics (WFS) for logic programs. ence relation whose properties have been keenly studied. The problem addressed is this: which (non-modal) logic can But compared with other logical formalisms popular in be considered adequate for well-founded semantics in the sense that its minimal models (appropriately defined) coin- knowledge representation and reasoning, the logical founda- cide with the partial stable models of a logic program? We tions of WFS remain largely uncharted territory. One cause approach this problem by identifying the HT 2 frames pre- seems to be that an adequate underlying logic for WFS has viously proposed by Cabalar to capture WFS as structures simply not been identified and studied. While well-founded of a kind used by Doˇsen to characterise a family of models are easy to describe using a three-valued semantics, weaker than intuitionistic and minimal logic. We define a it is much harder to say under which logic or logics is well- notion of minimal, total HT 2 model which we call partial founded closed. Also WFS is still closely tied to a equilibrium model. Since for normal logic programs these very restricted , that of normal logic programs. Vari- models coincide with partial stable models, we propose the ous attempts have been made to extend this syntax, mainly resulting partial equilibrium logic as a logical foundation for in two directions, by adding a second, explicit op- well-founded semantics. In addition we axiomatise the logic of HT 2-models and prove that it captures the strong equiva- erator and, secondly, by permitting disjunctive rules, ie rules lence of theories in partial equilibrium logic. whose heads may comprise disjunctions of atoms. While the first avenue has seen some success in terms of implemented Keywords: well-founded semantics, partial stable mod- systems and practical applications (Pereira & Alferes 1992), els, equilibrium logic progress on the second path has been limited. The main problem is that there have been several different proposals for how to extend WFS with disjunction and virtually no Introduction agreement on which is more adequate or even on the general Of the various proposals for dealing with default negation in criteria by which adequacy can be assessed. logic programming that go beyond the methods of ordinary A natural way to identify a logical foundation for a logic Prolog, the well-founded semantics (WFS) of Van Gelder, programming semantics or inference relation is to represent Ross and Schlipf (van Gelder, Ross, & Schlipf 1991) has the intended models of the semantics as minimal models proved to be one of the most attractive and resilient. Partic- in a suitable logic. In the case of stable models and an- ularly its favourable computational properties have made it swer sets a solution of this sort was found several years ago. popular among system developers and the well-known im- The logic of here-and-there, HT , (also known as G¨odel’s plementation XSB-Prolog1 is now extensively used in AI 3-valued logic) was used in (Pearce 1997) to represent sta- problem solving and applications in knowledge represen- ble models as minimal models and was identified as a max- tation and reasoning. The present paper studies the logi- imal logic with the property that equivalent theories have cal foundations of WFS and proposes a solution to a long- the same (stable model) semantics. The weakest extension standing open problem: how to characterise partial stable of the logic containing strong negation and its axioms plays models as minimal models in a suitable nonclassical, non- the same role for answer set programming with two nega- modal logic. We thereby obtain a deductive base logic (in tions. Later, in (Lifschitz, Pearce, & Valverde 2001), it was the sense of (Dietrich 1994)) for well-founded inference as shown that HT characterises the strong equivalence of pro- well as a means to extend WFS to disjunctive programs and grams (see below) under stable semantics, and subsequently (de Jongh & Hendriks 2002) identified the family of all su- ∗Partially supported by CICyT project TIC-2003-9001-C02 and WASP (IST-2001-37004) 2Without going into technicalities, very roughly if is some Copyright c 2006, American Association for Artificial Intelli- nonmonotonic inference relation then a monotonic logic|∼ forms gence (www.aaai.org). All rights reserved. a deductive base for if extends -inference and -equivalentL 1See http://www.cs.sunysb.edu/ sbprolog/xsb-page.html theories are also equivalent|∼ |∼ under .L L ∼ |∼

25 perintuitionistic logics with this property. This work has form q p, not r is treated simply as a logical formula led to a flourishing research programme in the area of logic ← p r q. programming and nonmonotonic reasoning, influencing re- ∧ ¬ → search topics such as how to: (i) extend the syntax of an- Now consider the simple rule p not p, or p p written swer set programs eg. to allow rules with boolean formulas as a logical formula, and let us suppose← that¬ the→ non-logical in heads and bodies (Lifschitz, Tang, & Turner 1999), or constants are p and q. The well-founded model of this pro- even to arbitrary propositional formulas (Pearce 1997; Fer- gram makes the atom p undecided and q false. Now suppose raris 2005); (ii) provide a semantics for answer set programs we enlarge our program by adding a new rule q not p. with additional constructs such as cardinality constraints, Viewed as a formula p q, from the standpoint← of intu- weight constraints (Ferraris & Lifschitz 2005) or aggregates itionistic logic we have¬ added→ no new ‘content’, since intu- (Ferraris 2005); (iii) characterise strong equivalence for itionistically p q follows logically from p p, so other nonmonotonic inference relations and LP semantics the second program¬ → is equivalent to the first. But¬ → consider (Turner 2001; 2004; Odintsov & Pearce 2005); (iv) define the well-founded model of the enlarged program: now the variants of strong equivalence such as partial equivalence atom q has changed value and become undecided; the (Woltran 2004), equivalence wrt particular classes of formu- two programs are not equivalent under WFS. So intuition- las (Eiter & Fink 2003; Pearce & Valverde 2004b) or equiv- istically equivalent formulas need not have the same well- alence across different vocabularies (Pearce & Valverde founded semantics. 2004a); (v) use monotonic base logics to verify valid pro- The paper is organised as follows. We first review Doˇsen gram transformations (Osorio, Navarro, & Arrazola 2001; semantics (Doˇsen 1986) for a logic called N, a general Pearce 2004). frameworkfor dealing with weak negation, and then proceed to study a particular case N ∗ that results from combining The approach and main results of the paper this approach with Routley semantics (Routley & Routley Our aim here is to initiate a similar foundational study for 1972) (also used in (Odintsov & Pearce 2005) for paracon- 2 WFS following the same approach adopted for stable mod- sistent answer sets). We then consider the semantics of HT els and answer sets: find a minimal model characterisation frames introduced by (Cabalar 2001) to model partial stable of the intended logic programming structures and axioma- models and we show that they are a particular case of N ∗ tise the resulting base logic. In the case of stable models the frames. We also present in this section a 6-valued character- 2 corresponding minimal models in the logic HT are called isation of HT that sheds some new light on the compari- equilibrium models. For that will become clear, in son with Przymusinski’s 3-valued definition of partial stable 2 the case of partial stable semantics and WFS it is natural to models. Next we define a concept of minimal and total HT call the resulting minimal models partial equilibrium mod- model called partial equilibrium model and we show that on els. logic programs these coincide with partial stable models, so There are several possible approaches to understanding that partial equilibrium logic provides a foundation for and and extending WFS. For example the technique of (Brass extension of well-founded inference. Two of the main con- 2 & Dix 1994) to capture WFS via a set of program trans- tributions of the paper follow. First we axiomatise the HT formations has been much discussed in the literature. Our logic and provea completenesstheorem, then we establish in approach by contrast proceeds via partial stable (p-stable) the following section a strong equivalence theorem to show 2 models. There are several reasons for this. First, p-stable how the logic HT captures strongly equivalent theories in models, though defined via program reducts, are not too partial equilibrium logic. The paper concludes with a brief far removed conceptually from ordinary model theoretic se- discussion of related work and with some topics for future mantics. Thus we can hope to analyse them via logical and investigation. model-theoreticmethods. Second, p-stable models are a nat- Our study is still at a preliminary stage and many issues ural generalisation of ordinary (2-valued) stable models to a are left open for future work. For instance, it is evident that multi-valued setting; and as we have seen stable models do partial equilibrium logic provides a means to extend well- admit a natural logical foundation. Third, the well-founded founded and p-stable semantics beyond the syntax of normal model of a normal logic program coincides with the unique programs, even to arbitrary propositional theories. However, minimal p-stable model. So if we can capture p-stable se- a more detailed analysis of the behaviour of this extension mantics for normal programsin terms of minimal models for and a comparison with other extensions of WFS in the lit- some logic, then a further minimisation process will yield erature are topics we are currently exploring and hope to the well-founded model. present in future work. Before proceeding to the technical details of the paper let us see briefly why, unlike in the case of stable model - Dosenˇ and Routley semantics ing, we cannot use superintuitionistic logics as a basis for The logic we are going to investigate is an extension of a well-founded semantics. When dealing with deductive bases logic introduced by Doˇsen in (Doˇsen 1986) (see also (Doˇsen for logic programmingwe are assuming the direct or ‘trivial’ 1999)) which he denotes by N. Doˇsen’s aim was to study translation of program rules into logical formulas. In this pa- logics weaker than Johansson’s minimal logic. We recall per we use the negationsymbol ‘ ’ correspondingto the op- here the main definitions and facts regarding N. Formu- erator not of logic programs, and¬ ordinary implication‘ ’ las of N are built-up in the usual way using atoms from a replaces the inverted arrow ‘ ’. So a program rule of→ the given propositional signature At and the standard logical ←

26 constants: , , , , respectively standing for conjunc- An N model = W, ,R,V is called an N ∗ model tion, disjunction,∧ ∨ → implication¬ and negation. We write F or if it satisfies M h ≤ i to stand for the set of all well-formed formulae of this lan- x x∗(xRx∗ y(xRy y x∗)) (6) guage. The rules of inference for N are and ∀ ∃ ∧ ∀ ⇒ ≤ the contraposition rule ie, for any world x of there exists a -greatest world M ≤ α β x∗ among those accessible from x by R. It can be easily → (RC) checked that the new axioms of N are valid in all N mod- β α ∗ ∗ ¬ → ¬ els. Moreover, N ∗ is complete wrt to that class of models. The set of axioms contains the axiom schemata of positive The persistence of of formulas (3) plus the exis- logic plus: tence of a greatest R-accessible world (6) implies that the validity of negated formulas at x is equivalent to α β (α β) (1) ¬ ∧ ¬ → ¬ ∨ V ( ϕ, x) = 1 V (ϕ, x∗) = 1. Definition 1 (N model) A model for N is a quadruple ¬ ⇔ This observation allows us to define a Routley style seman- = W, ,R,V such that: (i) W, is a partial or- tics (Routley & Routley 1972) for extensions of . Mdering (ofh worlds),≤ (ii)i R W 2 is anh accessibility≤i relation N ∗ among worlds verifying ( ⊆R) R, (iii) and finally, V is Definition 2 A Routley frame is a triple W, , , where a valuation function from≤At W⊆ 0, 1 satisfying: W is a set, a partial order on W and :hW ≤ W∗i is such × −→ { } ≤ ∗ → that x y iff y∗ x∗. A Routley model is a Routley frame V (p, u) = 1 & u w V (p, w) = 1 (2) ≤ ≤ ≤ ⇒ together with a valuation V : At W 0, 1 as for N models. × −→ { } V is extendedto a valuation on all formulasvia the following conditions. As usual, a formula ϕ is valid in a Routley model if it is valid at every world of this model. V (ϕ ψ, w) = 1 iff V (ϕ, w) = V (ψ, w) = 1 • ∧ Completeness proofs for N and N ∗ can be obtained via V (ϕ ψ, w) = 1 iff V (ϕ, w) = 1 or V (ψ, w) = 1 the method of canonical models. We now sketch this for • ∨ N ∗. Let S be some logic extending N ∗, ie some set of for- V (ϕ ψ, w) = 1 iff for every w0 such that w w0, • → ≤ mulas containing N ∗ and closed under , (RC), V (ϕ, w0) = 1 V (ψ, w0) = 1 ⇒ and modus ponens. First we say that a set of formulas Γ V ( ϕ, w) = 1 iff for every w0 such that wRw0 is a theory wrt S (S theory) if it contains S and is closed • ¬ V (ϕ, w0) = 0 under modus ponens and a prime S theory if it additionally As the reader may have already observed, the main differ- satisfies the disjunction property: ence with respect to intuitionistic frames is the presence of a α β Γ α Γ or β Γ new accessibility relation R used for interpreting negation, ∨ ∈ ⇒ ∈ ∈ while remains for implication. Extending valuation V to Next we note a standard extension lemma. Let Σ and ∆ be ≤ all formulas, we use for positive connectives the same con- sets of formulas. A relation Σ S ∆ means that for some ` ditions as in the case of , but for negation ϕ0, . . . , ϕn ∆ the disjunction ϕ0 ... ϕn can be ob- ∈ ∨ ∨ we use instead the condition involving the relation R. tained from elements of S and Σ using the rule of modus A proposition ϕ is said to be true in an N model = ponens. W, ,R,V , if V (ϕ, v) = 1, for all v W . A formulaM ϕ Lemma 1 (Extension lemma) For any extension S of N ∗, his valid≤ , in symbolsi = ϕ, if it is true in∈ every N model. It any sets of formulas Σ and ∆, if Σ S ∆, then there is a is easy to prove by induction| that condition (2) above holds 6` prime S theory Γ Σ such that Γ S ∆. for any formula ϕ, ie ⊇ 6` On this basis one defines canonical models as follows. V (ϕ, u) = 1 & u w V (ϕ, w) = 1 (3) ≤ ⇒ Definition 3 (Canonical model) Let S be any extension of c c c Moreover, N is complete, that is, a formula is valid iff it is a N ∗. The canonical S frame is the triple W , , where c h ≤ ∗ ci theorem of N. (i) W is the set of prime theories wrt S, (ii) Γ ∆ := c ≤ Γ ∆, (iii) Γ∗ := α α Γ . The canonical S model Let us consider now the logic N ∗ obtained by adding to ⊆ { | ¬ 6∈ } N the following axioms is the canonical S frame together with the valuation function V c such that V c(p, Γ) = 1 iff p Γ. (α α) β (4) ∈ ¬ → → It is not hard to check that the canonicalmodel S is indeed (α β) α β (5) a Routley model. The only non-trivial item is to prove that ¬ ∧ → ¬ ∨ ¬ c the -function is well defined, ie that Γ is a prime theory. Thus, both De Morgan laws are provable in N ∗ ∗ ∗ c Lemma 2 For any prime S theory Γ, the set Γ∗ is also a N ∗ (α β) α β, (α β) α β ` ¬ ∧ ↔ ¬ ∨ ¬ ¬ ∨ ↔ ¬ ∧ ¬ prime S theory. since axioms (1) and (5) explicitly provide one direction of Proof. For ϕ S, we have ϕ Γ, otherwise Γ is trivial ∈ ¬ 6∈ each law, whereas the opposite implications can be inferred by axiom (4). Thus, S Γ∗. Let ϕ and ϕ ψ be in Γ∗, via the rule (RC). ie ϕ, (ϕ ψ) Γ⊆. By the disjunction→ property of Γ ¬ ¬ → 6∈

27 and De Morgan laws (ϕ (ϕ ψ)) Γ, equivalently, An interesting observation is that when we force h = h0 ¬ ∧ → 6∈ (ϕ ψ) Γ. Since ψ (ϕ ψ) N ∗, ψ and t = t0 we actually obtain that and R collapse into the ¬ ∧ 6∈ ¬ → ¬ ∧ ∈ ¬ 6∈ ≤ Γ, i.e. ψ Γ∗. We have proved that Γ∗ is closed under same relation and, in fact, the whole structure becomes an ∈ modus ponens. The disjunction property of Γ∗ follows by HT frame. Thus, it is easy to see that: De Morgan laws. 2 Proposition 2 Any valid formula in HT 2 is also a valid for- Now the completeness for N ∗ follows from the lemma for mula in HT . the canonical model. Proposition 3 In HT 2 models the following formulas are Lemma 3 In the canonical S model, for every Γ W and ∈ c valid: α α, α α, (α α) β, every ϕ, (α β)→ ¬¬α ¬ β↔, α ¬¬¬α ¬ β→ →β. V c(ϕ, Γ) = 1 ϕ Γ. ¬ ∧ → ¬ ∨ ¬ ∧ ¬ → ¬ ∨ ¬¬ ⇔ ∈ Proof. Since HT 2 frames are finite, the validity of formulas Proof. By induction on the complexity of ϕ. 2 can be verified directly. 2 The completeness property follows by noting that if 2 N ∗ According to Proposition 3, the HT frame defines then by the extension lemma there is a prime theory6` ϕ N ∗ an extension of N ∗ and we can replace the above de- Γ such that ϕ Γ. It follows from Lemma 3 that ϕ does 6∈ fined models by models based on the Routley frame not hold in the canonical N ∗ model and therefore is not N ∗- HT 2 HT 2 HT 2 = W , , , where W = h, h0, t, t0 and valid. Wthe orderingh and≤ the∗i action of are represented{ in} the Theorem 1 For any formula ϕ, ϕ iff ϕ is valid in every following diagram.≤ ∗ `N ∗ Routley model. t0 Finally, we note that an intuitionistic negation can be de- ? •O  That is, h = t = t , (h ) = (t ) = fined in N . Fix some propositional variable p and put   ∗ ∗ 0 0 ∗ 0 ∗ ∗ 0 t o h t and u < v iff v is strictly higher •• 0 := (p p ) and α := α . than u in the diagram. ⊥ ¬ 0 → 0 − → ⊥ From axiom (4) it follows that for any Routley model = h• M W, , ,V , the constant is not satisfied at any world 2 HT 2 h ≤ ∗ i ⊥ Now, fix some HT model = ,V . For w w W . Therefore, the satisfaction of the derived expres- HT 2 M hW i ∈ ∈ W let us set ∆M := ϕ : V (ϕ, w) = 1 . sion α coincides exactly with the interpretation of negation w { } − 2 in intuitionistic logic: Lemma 4 For an arbitrary HT 2 model = HT ,V M2 hW i the following hold. (i) ∆w is a prime HT theory for any iff 0 such that 0 0 V ( α, w) = 1 w w w ,V (α, w ) = 0. HT 2 − ∀ ≤ w W . (ii) ∆ ∆ iff u v. (iii) ∆ = ∆∗ and ∈ u ⊆ v ≤ t0 h Since satisfaction of positive connectives is also defined ∆ = ∆∗ . (iv) ϕ ψ ∆ iff for all v w either t h0 w as in intuitionistic logic, we actually have ϕ ∆ or ψ ∆ . → ∈ ≥ 6∈ v ∈ v Proposition 1 The , , , -fragment of N ∗ coincides Proof. All these properties follow straightforwardly from with intuitionistic logic.h∨ ∧ → −i the definition of validity of formulas in Routley models and the structure of an HT 2 frame. 2 HT 2-models Lemma 5 Let Q = ∆h, ∆h0 , ∆t, ∆t0 be a quadruple of As mentioned in the introduction, it was shown in (Pearce prime HT 2 theoriesh satisfying all conditionsi of Lemma 4. 2 HT 2 1997) that the so-called logic of here-and-there, HT , can Define an HT model Q = ,VQ as follows: be used as a foundation for the stable model semantics for M hW i HT 2 logic programs. In the semantics for intermediate or super- VQ(p, w) = 1 iff p ∆w, w W . 2 ∈ ∈ intuitionistic logics, HT can be captured by rooted Kripke Then for all w W HT we have frames with two elements, commonly denoted by h and t ∈ Q and called ‘here’ and ‘there’, with h t. In (Cabalar 2001) ∆wM = ∆w. a notion of HT 2 model was introduced≤ and studied in or- Proof. By induction on the structure of formulas. 2 der to capture partial stable models for logic programs. The Due to the last two lemmas, the Routley HT 2 model motivation for the notation is that HT 2 models are based on can be identified with the quadruple of prime HT 2 theoriesM frames that include for each world w in an HT -model an ∆M, ∆M, ∆M, ∆M . Now, let us say that a prime theory h h h0 t t0 i additional world w0 accessible from w via the relation. In ∆ is: consistent if ϕ ϕ ∆ for any ϕ; inconsistent if it ≤ addition, just as we have h t in an HT -model, we have is not consistent; complete∧ ¬ 6∈if ϕ ∆ implies ϕ ∆; and 2 ≤ also h0 t0 in an HT -model. More precisely we define weakly complete if ϕ ∆ implies6∈ ϕ ∆¬. ∈ 2 ≤ HT in terms of N models as follows. ¬ 6∈ 2 ¬¬ ∈ Lemma 6 Let ∆ be a prime HT theory. (i) ∆∗∗ = ϕ : 2 2 { Definition 4 (HT model) An HT model is an N model ϕ ∆ ; (ii) ∆ ∆∗∗ and ∆∗ = ∆∗∗∗; (iii) ∆ = Γ∗ = W, ,R,V such that (i) W comprises 4 worlds iff¬¬∆ is∈ closed} under⊆ the rule ϕ/ϕ; (iv) if ∆ is inconsis- M h ≤ i ¬¬ denoted by h, h0, t, t0, (ii) is a partial ordering on W sat- tent, then ∆ is weakly complete; (v) ∆∗ is consistent iff ∆ ≤ 2 isfying h t, h h0, h0 t0 and t t0, (iii) R W is is weakly complete; (vi) ∆ ∆∗ iff ∆ is consistent; (vii) ≤ ≤ ≤ ≤ ⊆ ⊆ given by hRh0, h0Rh, tRt0, t0Rt, hRt0, h0Rt. (iv) V is an ∆∗ ∆ iff ∆ is complete; (viii) ∆ = ∆∗ iff ∆ is consistent N-valuation. and⊆ complete.

28 + Proof. (i). By definition ϕ ∆∗∗ iff ϕ ∆∗ iff ϕ a pair of sets (T ,T −), respectively denoting true and ∈ ¬ 6∈ ¬¬ ∈ • + ∆. false atoms, with T T − = , ∩ ∅ (ii) The inclusion ∆ ∆∗∗ follows from the previous a set L of literals which is consistent (it contains no pair item and the formula ϕ ⊆ ϕ HT 2. By definition ϕ • → ¬¬ ∈ ∈ p, p). ∆∗∗∗ iff ϕ ∆. The latter is equivalent to ϕ ∆ ¬ ¬¬¬ 6∈ 2 ¬ 6∈ but it is clear that we may equivalently use any of the three due to ϕ ϕ HT , ie ϕ ∆∗. ¬ ↔ ¬¬¬ ∈ ∈ representations. (iii) If ∆ = Γ∗, then ∆ = ∆∗∗ by the previous item. Let Two ordering relations among 3-valued interpretations are ϕ ∆, then ϕ ∆∗∗ = ∆ by item (i). Conversely, if ∆ ¬¬ ∈ ∈ defined such that, if and , then: is closed under the rule ϕ/ϕ, then ∆∗∗ = ∆ by item (i). T1 = (T1,T10) T2 = (T2,T20) ¬¬ (iv) This follows from the formula ϕ ϕ ψ ψ i) T1 T2 iff T1 T2 and T10 T20, HT 2 and the disjunction property of ∆∧¬. → ¬ ∨¬¬ ∈ ≤ ⊆ ⊆ ii) T1 T2 iff T1 T2 and T20 T10. (v) The consistency of ∆∗ means that for every ϕ, either  ⊆ ⊆ ϕ ∆∗ or ϕ ∆∗. By definition of ∆∗ this is equiva- In (Przymusinski 1994), these relations receive the lent6∈ to ϕ ¬ ∆6∈or ϕ ∆ for every ϕ, ie to the weak of standard and Fitting’s ordering respectively. The re- ≤ completeness¬ ∈ of ∆. ¬¬ ∈ lation intuitively represents that one interpretation contains (vi) Let ∆ ∆∗. If ϕ ∆, then ϕ ∆∗, ie ϕ ∆. “less truth” than the other. It is equivalent to the condition: ⊆ ∈ ∈ ¬ 6∈ Assume ∆ is consistent, then ϕ ∆ implies ϕ ∆, ie p V, T1(p) T2(p), where denotes now the inte- ∈ ¬ 6∈ ∀ ∈ ≤ ≤ ϕ ∆∗. ger ordering for values. The other relation, , measures the ∈  (vii) Let ∆∗ ∆. If ϕ ∆, then ϕ ∆∗, and so degree of knowledge in terms of undefined atoms. Interpre- ϕ ∆. ⊆ ¬ 6∈ ∈ tations with shape (T,T ) are called complete (they have no ∈ Let ∆ be complete. If ϕ ∆∗, then ϕ ∆ and ϕ ∆ undefined atoms). by completeness. ∈ ¬ 6∈ ∈ Given a 3-valued interpretation T, Przymusinski’s valua- (viii) Follows from items (vi) and (vii). 2 tion of formulas is defined so that conjunction is the mini- mum, disjunction the maximum, and negation and implica- Proposition 4 If ϕ HT 2, then 6∈ tion are defined as: 2 ϕ ψ ψ : ψ F or . 6`HT { } ∪ { ∧ ¬ ∈ } T( ϕ) = 2 T(ϕ) 2 Proof. Let ϕ HT and ϕ (ψ0 ψ0) ... • ¬ − 6∈ 2 ∨ ∧ ¬ ∨ ∨ 2 if T(ϕ) T(ψ) (ψn ψn) HT . The latter means by the exten- T(ϕ ψ) = ≤ sion∧ lemma ¬ that∈ for any prime theory ∆ either ϕ ∆ or • → 0 otherwise ∈ 2  ∆ is inconsistent. However, ϕ is refutable at some HT Additionally, valuation of truth constants is fixed as T( ) = > model ∆h, ∆h0 , ∆t, ∆t0 , which means ϕ ∆h. Since 2, T( ) = 0 and T(u) = 1 (the latter is a new constant h i 6∈ ⊥ ∆h ∆t0 = ∆∗ , ∆h is consistent by item (vi) of the previ- representing undefinedness). ⊆ h ous lemma. 2 The definition of partial stable model relies on a gener- By contraposition we obtain the following consequence. alisation of the program reduct (Gelfond & Lifschitz 1988) Corollary 1 HT 2 is closed under the rule to the 3-valued case. Given a 3-valued interpretation T, the reduct T is formed by replacing each negated literal in α (β β) Π p ∨ ∧ ¬ . program Π by , u or depending on whether T(p) is¬0, 1 α or 2 respectively.> ⊥ Minimal HT 2 models and partial stable models Definition 5 (Partial stable model) A 3-valued interpreta- The correspondence between HT 2 and WFS is established tion T is a partial stable model if it is the -minimal model by the fact that, as we prove in this section, some mini- of ΠT. ≤ 2 mal HT models coincide with Przymusinki’s partial sta- In (Przymusinski1994) it is shown that a positive program ble models (Przymusinski 1994), when we restrict the syn- (like ΠT when Π is normal) has a unique -minimal model. tax to that of normal logic programs. We recall next some It was also shown that the well founded model≤ T of a normal basic definitions from Przymusinski’s 3-valued setting, and program Π is the -minimal partial stable model. Again, proceed later to introduce a related multi-valued characteri-  2 for the case of normal programs, it has also been provedthat sation of HT that will be very useful for comparison pur- there exists a -minimum partial stable model, ie, a unique poses. well founded model. A 3-valued interpretation T is a mappingfrom the propo- Now, let us return to HT 2 and consider a model = sitional signature At to the set of truth values3 0, 1, 2 re- M { } W, ,∗ ,V denoting by H,H0,T,T 0 the four sets of atoms spectively standing for false, undefined and true. We can respectivelyh ≤ i verified at each corresponding point or world also represent the interpretation T as a pair of sets of atoms h, h0, t, t0. Since, by construction, H H0 and T T 0, it (T,T 0) satisfying T T 0 where T(p) = 0 iff p T 0, ⊆ ⊆ ⊆ 6∈ is clear that we can represent as a pair H, T of 3-valued T(p) = 2 iff p T and T(p) = 1 otherwise (ie, p T 0 T ). M h i ∈ ∈ \ interpretations H = (H,H0) and T = (T,T 0). In this way, Notice that in the literature, it is perhaps more usual to find we could define the possible “situations” of a formula in the alternative forms: HT 2 by using a pair of values xy with x, y 0, 1, 2 . 3 ∈ { } In (Przymusinski 1994), 1 and 2 are respectively represented Condition (3) restricts the number of these situations to the as 1/2 and 1 instead. following six 00 := , 01 := t0 , 11 := h0, t0 , 02 := ∅ { } { }

29 t, t0 , 12 := h0, t, t0 , 22 := h, h0, t, t0 , where each In other words a partial equilibrium model of Π has the form set{ shows} the worlds{ at} which the formula{ is satisfied.} Thus, T, T and is such that if H, T is any model of Π with an alternative way of describing HT 2 is by providing its hH Ti , then H = T. Partialh equilibriumi logic is the logic logical matrix in terms of a 6-valued logic. As a result, determined≤ by truth in all partial equilibrium models of a 2 the above setting becomes an algebra of 6 cones: HT := theory. Formally we can define a nonmonotonic inference 00, 01, 11, 02, 12, 22 , , , , where andA are set relation by: h{theoretical join and meet,} ∨ whereas∧ → ¬i and ∨are defined∧ as → ¬ Definition 7 (entailment) Let Π be a theory, ϕ a formula follows: and (Π) the collection of all partial equilibrium mod- els ofPEMΠ. We say that Π entails ϕ, in symbols Π ϕ, if either x y := w : w w0 (w0 x w0 y) , → { ≤ ⇒ ∈ ⇒ ∈ } (i) or (ii) holds: (i) (Π) = and =|∼ϕ for every x := w : w∗ x . PEM 6 ∅ M | ¬ { 6∈ } (Π); (ii) (Π) = and ϕ is true in all M2 ∈ PEM PEM ∅ The only distinguished element is 22. The lattice structure HT -models of Π. of this algebra can be described by the condition xy zt In this definition we consider the skeptical or cautious entail- x z & y t and is shown in Figure 1, together with≤ the⇔ ment relation; a credulous variant is easily given if needed. resulting≤ truth-tables.≤ Clause (ii) is needed because not all consistent theories have partial equilibrium models. Again (ii) represents one pos- ϕ ϕ sible route to understanding entailment in the absence of 00 ¬22 intended models; other possibilities may be considered de- 01 11 pending on context. 22 11 11 Finally, we proceed now to use the representation based 02 00 on pairs of 3-valued interpretations to establish a straight- 12 12 00 forward correspondence to partial stable models. We be- z DDD zzz D 22 00 gin by noting a property we will use below: examining 11D 02 the table for implication in Figure 1, it is easy to see that DD zz 00 01 11 02 12 22 D zz → M(ϕ ψ) = 22 iff, given M(ϕ) = xy and M(ψ) = uv, 01 00 22 22 22 22 22 22 we have→ both x u and y v. We also fix the HT 2 01 00 22 22 22 22 22 valuation of constants≤ as M( ≤) = 22, M(u) = 11 and 00 11 00 02 22 02 22 22 M( ) = 00. > 02 00 11 11 22 22 22 ⊥ 12 00 01 11 02 22 22 Lemma 7 For any M = H, T and any atom p: 22 00 01 11 02 12 22 M( p) = M(( p)T). h i ¬ ¬ Proof. Assume T has the form (T,T 0). We have three Given V (φ) = xy and V (ψ) = zt : cases, depending on T(p). (i) For T(p) = 2 the reduct is ( p)T = , but we also V (φ ψ) = uv u = min(x, z)& v = min(y, t) ¬ ⊥ ∧ ⇔ have p T , ie, M(p) 02, 12, 22 and so M( p) = V (φ ψ) = uv u = max(x, z)& v = max(y, t) ∈ ∈ { } ¬ ∨ ⇔ 00 = M( ). ⊥ 2 (ii) If T(p) = 1 the reduct is ( p)T = u and we also have Figure 1: Lattice structure and truth tables for the 6-valued HT ¬ . p T 0 T , ie, M(p) 01, 11 , which means M( p) = 11∈= M\(u). ∈ { } ¬ (iii) If T(p) = 0 the reduct is ( p)T = and we also An interesting observation is that, by the semantics, if ¬ > get p T 0, ie, M(p) = 00, which means M( p) = 22 = (H, T) is a model then necessarily H T, since it is 6∈ ¬ easy to check that this condition is equivalent≤ to H T M( ). 2 ⊆ > and H0 T 0. Moreover, for any theory Π note that if 2 ⊆ Corollary 2 For any HT interpretation M = H, T and H, T = Π then also T, T = Π. any normal logic program Π: M = Π iff M = Πh T. i 2 h Thei ordering | canh be extendedi | to a partial ordering ¢ | | among models as≤ follows. We set H , T ¢ H , T if Lemma 8 Let Π be a positive logic program (possibly con- h 1 1i h 2 2i (i) T1 = T2; (ii) H1 H2. A model H, T in which H = taining constants in the body) and let T be a 3-valued model T is said to be total.≤ Note that the termh totali model does of Π. Then, for any M = H, T and any rule r Π: not refer to the absence of undefined atoms. To represent M(r) = 22 iff H(r) = 2. h i ∈ this, we further say that a total partial equilibrium model is Proof. First, we note that for any atom or constant ϕ, complete if T has the form (T,T ). M(ϕ) = xy iff H(ϕ) = x and T(ϕ) = y. Now, let r We are interested here in a special kind of minimal model have the form (A A B) and let M(A ) = x y that we call a partial equilibrium model. i n i i i and M(B) = uv.∧ Condition · · · ∧ M→(r) = 22 means that both Definition 6 (Partial equilibrium model) A model of a min xi u and min yi v. However, the former is theory Π is said to be a partial equilibrium model ofMΠ if (i) equivalent{ } ≤ to H(r) = 2,{ whereas} ≤ the latter means T(r) = 2 is total; (ii) is minimal among models of Π under the that in our case is always true, as T is a 3-valued model of Mordering ¢. M Π. 2

30 Theorem 2 Let Π be a normal logic program. Then T, T Let ∆ be a prime HT ∗ theory such that ϕ ∆. By the h i 0 6∈ is a partial equilibrium model of Π if and only if T is a rule (EC) if HT ∗ ϕ0, then HT ∗ ϕ0 β β β partial stable model of Π. F or . Thus we may6` assume that ∆6`is { consistent.} ∪ { ∧ ¬ | ∈ 1.} Assume that ∆ is consistent and complete. We prove Proof. From Corollary 2, we can safely replace program Π that for any ϕ and ψ, by ΠT in the claim, provided that for determining if T, T h i is in partial equilibrium, we fix the second component of ϕ ψ ∆ ϕ ∆ or ψ ∆. (7) HT 2 models to T. But now, as ΠT is positive, we can apply → ∈ ⇔ 6∈ ∈ Lemma 8. In particular, we get first that is an 2 The direct implication is obvious. If ψ ∆, then ϕ T, T HT ∈ → model iff is a 3-valued model. And similarly,h i we also get ψ ∆ by the positive axiom ψ (ϕ ψ). Let ϕ ∆ T ∈ → → 6∈ that for any , is an 2 model if is a and ϕ ψ ∆. By completeness ϕ, (ϕ ψ) ∆, H < T H, T HT H → 6∈ ¬ ¬ → ∈ 3-valued model. h i 2 whence ϕ ∆ by A6. Consistency and completeness of ∆ imply¬¬ in this∈ case ϕ ∆. This proves the Following (Przymusinski 1994), once partial stable mod- ∈ els are captured, we can further minimise among them desired equivalence. wrt the amount of information (ie, defined atoms) to ob- By Lemma 6 we have ∆ = ∆∗. Thus, the quadruple 2 ∆, ∆, ∆, ∆ is an HT 2 model refuting ϕ . Note that we tain a well-founded model. Thus, an HT well-founded h i 0 model would just be a partial equilibrium model, -minimal have established also the following fact  among the partial equilibrium models of Γ. Lemma 9 If ∆ is a complete prime HT ∗ theory closed un- der the rule ϕ/ϕ, then ∆ is a maximal element of HT ∗ . Axiomatisation of HT 2 ¬¬ W Proof. In the above reasoning the consistency of ∆ was Although we have described HT 2 via a class of frames, it used to establish that ∆ is closed under the rule ϕ/ϕ. can be considered a logic in the sense that it defines a set of Therefore, equivalence (7) holds for ∆. Assume ∆¬¬is not formulas: those valid on these frames. A more constructive HT ∗ 2 maximal. Let ∆0 and ψ be such that ∆ ∆0 and (and perhaps more standard) definition of HT is also pos- ∈ W 0 ⊂ sible using a calculus, that is, a set of axioms and inference ψ0 ∆0 ∆. On one hand, ψ0 ∆ by (7). On the other ∈ \ − ∈ 2 rules. hand, ψ0 ∆ by canonical model lemma. 2. Assume− 6∈ that ∆ is not complete, but is consistent and Let HT ∗ be an N ∗ extension obtained by adding the fol- lowing axioms: closed under ϕ/ϕ. Item (vi) of Lemma 6 implies ∆ , where as¬¬ item (iii) implies that . Thus ⊆ A1. α α ∆∗ ∆ = Γ∗ ∆ = ∆∗∗ by item (ii) of the same lemma. Note that is complete A2. −α ∨( −α − (β (β (γ γ)))) ∆∗ − 2 ∨ → ∨ → ∨ − 2 since ∆∗∗ ∆∗. By the last lemma ∆∗ is a maximal prime A3. i=0((αi j=i αj) j=i αj) i=0 αi ⊆ → 6 → 6 → HT ∗ theory. A4. α α We claim that A5. αV→ ¬¬α βW β W W ∧ ¬ → ¬ ∨ ¬¬ A6. α (α β) α ϕ ψ ∆ (ϕ ∆ ψ ∆) (ϕ ∆∗ ψ ∆∗). A7. ¬ α∧ ¬ →β (→α ¬¬β) (α β) → ∈ ⇔ 6∈ ∨ ∈ ∧ 6∈ ∨ ∈ The direct implication is obvious. We prove the converse A8. ¬¬α ∨ ¬¬β ∨ ¬(α →β) ∨( ¬¬β α→), and the¬¬ elimination∧ ¬¬ → of → ∨ → rule implication. Since ∆∗ is maximal, the second conjunctive term means exactly that ϕ ψ ∆∗. Therefore, (ϕ α (β β) ψ) ∆. → ∈ ¬ → ∨ ∧ ¬ (EC) 6∈ α If ψ ∆, then ϕ ψ ∆. Assume ϕ ∆ and ψ ∆. By the∈ rule ϕ/ϕ→we have∈ ϕ ∆ 6∈and ψ 6∈ ∆. Proposition 5 The canonical frame HT ∗ satisfies the fol- ¬¬ ¬¬ 6∈ ¬¬ 6∈ W By A7 at least one of formulas ϕ, ψ, (ϕ ψ) or lowing properties: (i) HT ∗ is strongly directed; (ii) (ϕ ψ) belongs to ∆. Therefore,¬¬ ¬¬ (¬ϕ →ψ) ∆ W HT ∗ is of depth 3; (iii) each element of HT ∗ has at ¬¬and so →ϕ ψ ∆. ¬¬ → ∈ W W most two immediate successors; (iv) elements of HT ∗ sat- One can→ see∈ that all conditions of Lemma 4 hold for the W isfy all properties listed in Lemma 6. quadruple ∆, ∆∗, ∆, ∆∗ . We have thus constructed the following countermodelh fori ϕ . Proof. Items (i)-(iii) can be inferred from axioms A1, A2 0 and A3 respectively in the same way as for superintuitionis- ∆∗ tic logics determined by these axioms. Item (iv) holds since ? •O only axioms of HT ∗ were used in the proof of Lemma 6. 2   2  Theorem 3 HT ∗ = HT . ∆o ∆ •• ∗ 2 Proof. The inclusion HT ∗ HT follows from the def- ⊆ 2 inition of HT ∗. All its axioms are HT tautologies, which can be verified directly. Moreover, HT 2 is closed under the ∆• rule (EC) by Corollary 1. 3. Now suppose ϕ0 ∆, where ∆ is a consistent weakly 2 6∈ We prove the nontrivial inclusion HT ∗ HT . Take complete prime theory not closed under ϕ/ϕ. By item ⊆ 2 ¬¬ some ϕ non-provable in HT ∗ and construct an HT model (vi) of Lemma 6 we have ∆ ∆∗, whence ∆∗ is also 0 ⊆ refuting ϕ0. weakly complete. By item (iii) of the same lemma ∆∗ is

31 closed under the rule ϕ/ϕ. Therefore, ∆∗ is complete We show that ∆∗ is complete. For any formula ϕ we have ¬¬ andit is maximalin HT ∗ by Lemma9. Item(v)ofLemma either ϕ ∆ or ϕ ∆, consequently, ϕ ∆ or ϕ ¬ 6∈ ¬¬ 6∈ ∈ ¬ ∈ 6 implies that ∆ isW consistent. Consistency and complete- ∆ by the definition of the -operation. ∗ ∗ ness imply the equality ∆∗∗ = ∆∗ by item (viii). The inclu- Completeness of ∆∗ implies its maximality and the in- sion ∆ ∆ is proper since ∆ is not closed under ϕ/ϕ. clusion ∆∗∗ ∆∗. Since ∆∗ = ∆∗∗∗, we conclude that ∗ ⊆ If there⊆ is no other proper extension of ∆, we obtain¬¬ a coun- ∆∗∗ is consistent. By this fact and inconsistency of ∆∗ the termodel ∆, ∆ , ∆ , ∆ . inclusion ∆∗∗ ∆∗ is proper. By item (ii) of Lemma 6. ∗ ∗ ∗ ⊆ h i ∆ ∆∗∗. This inclusion is also proper since ∆ is not closed ⊆ ∆∗ under ϕ/ϕ. If there is no other extension of ∆ we obtain ¬¬ ? O the countermodel ∆, ∆∗, ∆∗∗, ∆∗ . •  h i   ∆ ∆ o ∆ ∗ ∗ •• ∗ ? •O    ∆ o ∆ ∆• ∗∗ •• ∗

HT ∗ Assume that there is Γ such that Γ ∆, ∆∗ ∆• ∈ W HT ∗ 6∈ { } and ∆ Γ. Since ∆∗ is maximal and is strongly ⊆ W If there is one more such that , it is unique directed, we have Γ ∆∗. By the antimonotonicity of the Γ ∆ Γ ⊆ by item (iii) of Proposition 5. The strong⊆ directedness of -operation we obtain Γ∗ = ∆∗. HT ∗ ∗ By item (iii) of Proposition 5 there exists at most two and the maximality of ∆∗ imply Γ ∆∗. Since WHT ∗ ⊆ prime theories Γ and Γ between ∆ and ∆ . Assume is of depth 3, theories ∆∗∗ and Γ are incompara- 1 2 ∗ W Γ = Γ . Then Γ and Γ are mutually incomparable by ble. By antimonotonicity of we have Γ∗ ∆∗ and 1 6 2 1 2 ∗ ⊆ item (ii) of Proposition 5. ∆∗∗ Γ∗. Thus, Γ∗ ∆∗, ∆∗∗ . Taking into account ⊆ ∈ { } Γ Γ∗∗ we obtain Γ∗ = ∆∗∗. We arrive at the counter- ⊆ ∆∗ = ∆∗∗ = Γ1∗ = Γ2∗ model ∆, Γ, ∆∗∗(= Γ∗), ∆∗ , and we are done.  ?? h i  ??  ?? ∆  ? ∗ Γ1 ? Γ2 ? •O ??   ??   ??    ∆ = Γ o Γ ∆ ∗∗ ∗ ••

Let ϕ Γ1 Γ2 and ψ Γ2 Γ1. In thiscase ϕ, ψ ∆∗∗ • ∈ \ ∈ \ ∈ ∆ and ϕ, ψ ∆ by item (i) of Lemma 6. By axiom A8 2 we obtain¬¬ ¬¬(ϕ ∈ψ) (ψ ϕ) ∆. Since ∆ possesses the → ∨ → ∈ disjunction property we have ϕ ψ ∆ or ψ ϕ ∆. Strong equivalence wrt partial equilibrium Both cases contradict the choice of→ϕ and∈ ψ. Inthefirst→ ∈ case, there is an extension Γ of ∆ such that ϕ Γ and ψ Γ . logic 1 ∈ 1 6∈ 1 In the second case, we have ψ Γ2 and ϕ Γ2. We now establish a strong equivalence theorem for partial Thus, if there is a proper extension∈ Γ of 6∈∆ different from equilibrium logic. The notion of strong equivalence is im- ∆∗, it is unique and we obtain for ϕ0 the countermodel portant both conceptually and as a potential tool for sim- ∆, Γ, ∆∗, ∆∗ . plifying nonmonotonic programs and theories and optimis- h i ing their computation. For stable semantics strong equiva- ∆∗ lence can be completely captured in the logic HT (Lifschitz, ? •O Pearce, & Valverde 2001) and in ASP this fact has given rise   to a lively programme of research into defining and com-  puting different equivalence concepts, see eg (Eiter, Fink & ∆∗ o Γ •• Woltran 2006; Woltran 2004). In the case of WFS and p- stable semantics, however, to our knowledge until now, with • the exception of (Nomikos, Rondogiannis & Wadge 2005), ∆ there have been no studies of strong equivalence and related notions. 4. Consider the last case. Let ϕ0 ∆, where ∆ is We begin by noting that, when considering logic pro- consistent but is neither weakly complete6∈ nor closed under grams, equivalence under Przymusinki’s 3-valued logic is ϕ/ϕ. Again by Lemma 6 we have ∆ ∆∗ and ∆∗ is not adequate for testing strong equivalence, much in the ¬¬ ⊆ closed under ϕ/ϕ. Since ∆ is not weakly complete, ∆∗ same way as is not suitable for strong equiv- is not consistent¬¬ by item (v) of the same lemma. alence under stable models. In fact, as happens in that

32 case, it is not even suitable for checking regular equiva- It is routine to check that both H, T1 and T1, T1 are lence. As an example, the programs p q, p q models of Γ, which proves thath T , Ti ish not a partiali { → ¬ → } h 1 1i and p q, q p are equivalent under Przymusinski’s equilibrium model of Γ1 Γ. 3-valued{ → logic¬ although→ } they clearly have different well- Let us prove that T ,∪T is a partial equilibrium model h 1 1i founded models – the first one makes p false and q true, of Γ2 Γ. Assume that J, T1 = Γ2 Γ. Since H Γ while the second leaves both atoms undefined. we have∪ H J, on theh other handi | the∪ inclusion Π ⊆ Γ ⊆ 0 ⊆ Returning to arbitrary theories, in the present context we guarantees that H0 J 0. One of these inclusions must be ⊆ say that two propositional theories Γ1 and Γ2 are strongly proper, because H, T is not a model of Γ . h 1i 2 equivalent if for any theory Γ, theories Γ1 Γ and Γ2 Γ If H = J, the satisfiability of Π implies the equality ∪ ∪ 6 1 have the same partial equilibrium models. J = T since p0 is false at h. At the same time, formulas of Π3 imply J 0 = T 0 p0 . Indeed, let p1 J H. Since Proposition 6 Theories Γ1 and Γ2 are strongly equivalent ∪ { } ∈ \ 2 both p0 and p1 are true at h0, the validity of p0 p1 q iff Γ1 and Γ2 are equivalent in HT . ∧ → means that q J for all q T 0 H0. ∈ ∈ \ Proof. We consider the non-trivial direction. Let us as- Assume now J 0 H0 = and p J 0 H0. All implica- \ 6 ∅ 2 ∈ \ sume that Γ1 and Γ2 have different models and construct a tions p0 p2 q, q T 0 H0, are in Π2, whence J 0 = T 0. set of formulas Γ such that Γ1 Γ and Γ2 Γ have different At the same∧ time→ J,∈T \= p p . Now the equality ∪ ∪ h 1i | 0 → 2 partial equilibrium models. H = T follows from the fact that (p0 p2) q Π3 for 2 Let H, T be an HT -model such that H, T = Γ1 and all q T H. → → ∈ 2 H, Th = Γi. Note that in this case we haveh T,iT | = Γ . ∈ \ 2 h i 6| 2 h i | 1 The above result can be extended to show that HT also Case 1. Assume T, T = Γ2. If T = T 0, put Γ := captures strong equivalence wrt well-founded models (ie, - T . In this case it canh be easilyi 6| seen that T, T is a partial minimal partial equilibrium models)5.  equilibrium model of Γ Γ,butitis notamodelofh i Γ Γ. Note that unlike in the case of strong equivalence under 1 ∪ 2 ∪ Assume T = T 0 and p0 T 0 T . Now we put stable model semantics, we cannot assume in the general 6 ∈ \ case that the formulas in Γ have the syntax of logic program Γ := T p0 q q T 0 . ∪ {¬ → | ∈ } rules. So when Γ1 and Γ2 have the form of logic programs, it is clear that HT 2 equivalence is a sufficient condition for Clearly, T, T = Γ2 Γ. Let J, T = Γ1 Γ. Since h i 6| ∪ h i | ∪ strong equivalence, but it is an open question whether can T Γ, we have J = T . The atom p0 is refuted at t (by Γ ⊆ be taken to be a logic program (of whatever kind) in the case choice p0 T ), therefore, p0 ∆h by h0∗ = t. From the 6∈ ¬ ∈ 0 of non-equivalence. definition of Γ we have that T 0 ∆h0 , whence J 0 = T 0. We have thus proved that T, T is⊆ a partial equilibrium model of Γ Γ. h i Related work 1 ∪ Case 2. Let T, T = Γ2. Fix some atom p0 such that There has been a number of attempts to providea foundation h i | p0 T 0 and p0 does not occur in formulas of Γ1 and Γ2. We for well-founded semantics; some are more or less logical in put6∈ nature, others employ alternative mathematical methods. Of the former kind, we should mention: Γ := H p0 p0 Π0 Π1 Π2 Π3, ∪ { ↔ ¬ } ∪ ∪ ∪ ∪ The approach of (Bochman 1998a; 1998b) which analy- where • ses several logic programming semantics, including WFS, Π0 = p0 p p H0 in a generalised framework of Gentzen-style deduction. A { → | ∈ } Π1 = p q p0 p, q T H strong point of Bochman’s method of bi-consequence re- { → ∨ | ∈ \ } Π2 = (p0 p) q p T 0 H0, q T H lations is its ability to capture different semantics within { → → | ∈ \ ∈ \ } Π3 = p0 p q p T 0 H, q T 0 H0 the same framework. The method is somewhat removed { ∧ → | ∈ \ ∈ \ } Note that for any model H, T , validity of p p at from ordinary logic and model theory and does not pro- h i 0 ↔ ¬ 0 vide Hilbert-style axiomatisations. It remains to be seen this model means that p0 H0 T , ie p0 is true exactly at 4 ∈ \ whether it might complement the methods described here. h0 and t0. Consider the new models H, T1 and T1, T1 , where Another type of technique can be found in (Rondogiannis h i h i T1 := (T,T 0 p0 ). Since p0 is not involved in the com- • & Wadge 2002) which proposes an infinite-valued logic ∪ { } putation of validity of formulas from Γ1 and Γ2 we still have to capture WFS. However it is unclear how this logic re- lates to other known multi-valuedlogics and how it can be H, T = Γ , H, T = Γ , h 1i | 1 h 1i 6| 2 used to extend the semantics beyond the format of normal T , T = Γ , T , T = Γ . programs. h 1 1i | 1 h 1 1i | 2 4 Another recent approach is that of (Alcˆantara, Dam´asio, If in the model under consideration T = T 0, instead of adding 6 • & Pereira 2004) which studies WFS and variants using to Γ the formula p0 p0 for a new atom p0 we can take p1 ↔ ¬ ∈ semantical frames. These are closely related to the HT 2- T 0 T and replace p0 by p1 in Π1,..., Π4. The negation p1 is \ ¬ ¬ true exactly at h0 and t0. frames described here and in (Cabalar 2001). However no Alternatively, we could pass to the conservative extension of logical axiomatisation of the semantics is presented. HT 2 obtained by adding a new constant u together with axiom 5 u u. In this case, we replace p0 by u. The details will be included in a sequel to the present paper. ↔ ¬

33 The method of representing WFS via embeddings into strong equivalence results for special classes of models • nonmonotonic modal logics. This has been explored no- • such as the well-founded models defined above; tably in (Przymusinski 1995; Bonatti 1995). Though how to add strong or explicit negation to partial equilib- this is quite different from our aim to study WFS in a • rium logic and compare this with the well-known system language close to the original logical syntax, we also WFSX with explicit negation (Pereira & Alferes 1992). hope in the future to examine modal counterparts of partial equilibrium logic and thereby make comparisons prooftheory and implementation methods for partial equi- • with frameworks such as those of (Przymusinski 1995; librium logic. Bonatti 1995). We hope to present some of this ongoing work in a sequel to Among efforts to capture well-founded reasoning using the present paper. other mathematical methods, we should mention: as applied in (Bondarenko, Dung, • Kowalski, & Toni 1997). This method has proved flexible Alcˆantara, J.; Dam´a-sio, C.; and Pereira, L. M. 2004. enough to model several kinds of semantics for logic pro- A frame-based characterisation of the paraconsistent well- gramming and nonmonotonic reasoning and implementa- founded semantics with explicit negation. tions are now being developed (Dung, Kowalski, & Toni Bochman, A. 1998a. A logical foundation for logic pro- 2006). This approach can provide ways to enlarge the gramming I: Biconsequence relations and nonmonotonic syntactical scope of well-founded reasoning. It remains completion. Journal of Logic Programming 35:151–170. to be seen how they relate to logical systems such as par- Bochman, A. 1998b. A logical foundation for logic pro- tial equilibrium logic. gramming II: Semantics for logic programs. Journal of The infinite-game semantics recently proposed by (Ron- Logic Programming 35:171–194. • dogiannis & Wadge 2005). This appears at present to be Bonatti, P. 1995. Autoepistemic logics as a unifying frame- restricted to the syntax of normal programs. work for the semantic of logic programs. Journal of Logic Programming 22(2):91–149. Conclusions and future work Bondarenko, A, Dung, P.M., Kowalski, R.A., Toni, F. We have proposed partial equilibrium logic as a general sys- 1997. An abstract, argumentation-theoreticapproach to de- tem of nonmonotonic logic to act as a foundation for the fault reasoning. Artificial Intelligence 93(1-2) , 63-101. semantics of partial stable models and thereby for well- Brass, S., and Dix, J. 1994. A disjunctive semantics founded inference. Our approach has been to identify an based on unfolding and bottom-up evaluation. In IFIP’94 underlying monotonic logical framework to be used as a ba- Congress, Workshop FG2: Disjunctive Logic Program- sis. The natural choice is a logic in which partial stability ming and Disjunctive Databases, 83–91. can be expressed as a simple minimality condition with well- foundedness as a special case. The condition of equilibrium Cabalar, P. 2001. Well-founded semantics as two- that captures stable models in the logic of here-and-therecan dimensional here-and-there. In Proceedings of ASP 01, be readily generalised to a minimality condition that cap- 2001 AAAI Spring Symposium Series. tures partial stability in a logic HT 2 which corresponds in de Jongh, D., and Hendriks, A. 2002. Characterization of a natural way to HT . In this paper we have shown how the strongly equivalent logic programs in intermediate logics. resulting logic has a six-valued truth matrix and can be ax- Theory and Practice of Logic Programming 3(3):259–270. iomatised as an extension of Doˇsen’s logic N. Although the Dietrich, J. 1994. Deductive bases of nonmonotonic infer- 2 negation of HT , corresponding to the well-founded nega- ence operations. Technical report, NTZ Report, Universit¨at tion, is rather weak, intuitionistic negation is actually de- Leipzig. finable in HT 2. We have seen also that HT 2 captures the Doˇsen, K. 1986. Negation as a modal operator. Rep. Math. strong equivalence of theories in partial equilibrium logic. Logic 20:15–28. The present paper reports on ongoing work that will con- tinue to investigate many more issues in the foundations of Doˇsen, K. 1999. Negation in the light of modal logic. In WFS and p-stable semantics. Work currently in progress is Gabbay, D & Wansing, H. (eds), What is negation? Appl. examining a series of further topics including: Log. Ser. 13. 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