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....the programs. They can change their values during a program execution. The values of the static variables do not depend on the time points. Quantification is allowed only over the static variables. We provide a proof system of the logic and prove its completeness by the Rasiowa Sikorski method [24]. The proof system contains infinitary rules for temporal operators. In order to show how these rules work, we consider a toy concurrent program for which the corresponding models are exhibited, the temporal semantics axioms are defined and the correctness of the program is formally proved within ....

....i.e. Ax 6 oe. We build a model for Ax and :oe. That is, we construct a model M = F ; A; I; S) with M j= Ax and (M;w) j= oe, for some w 2 W M . We follow the idea of Rasiowa and Sikorski for constructing models on ultrafilters in the Lindenbaum Tarski algebra of a given theory. See, e.g. [24] or [2] By axiom A15 and the generalization rule R6, the quantifiers correspond to certain sups and infs in Lindenbaum Tarski algebra: ffl [8v ] inff[ t) t 2 T s g, ffl [9v ] supf[ t) t 2 T s g. By a temporal ultrafilter we mean a maximal proper filter U in the Lindenbaum Tarski ....

H. Rasiowa and R. Sikorski, The mathematics of metamathematics, North-Holland, 1970.

....and studied, it seems that topological methods and results have so far been under utilized for solving purely logical problems. Besides extensive research on abstract model theory involving topological machinery (see [Barwise and Feferman, 1985] I am aware of not many other publications, such as [Rasiowa and Sikorski, 1963]and[Goldblatt, 1985] which more explicitly pursue that direction. In this paper we begin systematic exploration of the idea of using basic topological techniques and results to obtain relative completeness results in logic. The preliminary section 1 contains some background from logic and ....

....of Baire s category theorem, we obtain a general relative completeness result (theorem 3.5, and theorem 4. 6 as a particular case in first order logic) which seemingly has so far been unnoticed, or certainly not popular, despite the well known relationship of Baire s theorem to logic (see [Rasiowa and Sikorski, 1963]and[Goldblatt, 1985] In section 3 we discuss logical topologies and logical approximation in classical logic. Not surprisingly, we show that a topology on the set of all complete theories in a first order language is logical i# it contains Stone topology (proposition 4.1) and briefly study a ....

Rasiowa H. and R. Sikorski, The Mathematics of Metamathematics, PWN, Warsaw, 1963. 12

....operation P : Pow(P) Gamma ; P (where Pow(P) refers to the set of all subsets of P) defined by P 0 7 P P P P 0 . Our major concern in this paper is with this latter operation: alternative quantification over data. We have given it a universal algebraic definition in the same way as Rasiowa and Sikorski (1963) generalise the binary joins ( and meets ( of boolean algebras to obtain existential (9) and universal (8) quantification. In our signatures alternative quantification is present as an operation symbol that binds a data variable. That is, we shall write P x:s p (p some process term and x a ....

....to obtain a ground complete axiomatisation of the strong bisimulation algebra from an complete axiomatisation of the data, provided that it has a built in equality predicate and Skolem functions. This paper is organised as follows. We adapt the notion of algebra with generalised operations of Rasiowa and Sikorski (1963) to a many sorted setting and we extend equational logic with a rule for generalised operations (x2) This enables us to provide the class of process algebras with alternative quantification over data with a precise universal algebraic definition (x3) In x4 we single out specific members of this ....

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Rasiowa, H. and Sikorski, R. (1963). The mathematics of metamathematics . Pa'nstowe wydawnictwo naukowe, Warszawa, Poland.

....on the state space of a frame. From the 1944 work by McKinsey and Tarski ( MT44] the axioms for the # operator of the wellknown modal logic called S4 correspond exactly to the properties of the topological interior operator. While this landmark result is well known in the modal logic literature ([RS63]) very little use has been made of it until recently. Because we want to express the continuity of relations and the definitions of semi continuity are couched in terms of general topology, we need an interior operator. Hence, this use of S4 is crucial to the formalisation of hybrid dynamical ....

....the classical propositional logic and the fundamental concepts from general topology. The chief contribution of this work is that it draws together and consolidates material from very recent literature (for example [DN00] and [PBV99] together with much older work (for example [MT44] JT51] and [RS63]) The main result of the essay is in Chapter 5: a completeness proof for the Hilbert style proof system for the family of logics TopPML. While the result can essentially be found in Fischer Servi ( FS81] the contribution here is to lift it to the more general setting in which the S4 # is ....

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H. Rasiowa and R. Sikorski. Mathematics of Metamathematics. PWNWarsaw, 1963.

....mix axiom scheme: #a##lsc : #a### # ##a## where #a## def = a]#. In the intended semantics, the plain S4 box # is given the McKinsey Tarski interpretation as the interior operator in an arbitrary topological space (S, T ) see (McKinsey 1941) and also (McKinsey and Tarski 1944) and (Rasiowa and Sikorski 1963). So ## denotes the largest open set in T contained in the denotation set of #; dually, 3 is the topological closure operator. The labelled S4 modalities [a] and #a# are given the standard relational Kripke semantics in terms of reflexive and transitive relations, or preorders, R a # S S; ....

....the case that #HLSC (# 1 # # #n ) # # for any finite subset # 1 , #n # #. The #S4 axiom schemes and the # monotonicity rule as presented correspond precisely to Kuratowski s (1966) axiomatisation of the topological interior operator (McKinsey 1941, McKinsey and Tarski 1944) (Rasiowa and Sikorski 1963). Note that the usual box necessitation rules can be derived from the monotonicity rules using the box K# axioms. 10 Likewise, the [a]S4 axiom schemata capture the properties of the operator Pre # [R a ] for preordered relations R a . The verification that F # #a### # ##a## for all frames F ....

Rasiowa, H., and R. Sikorski. 1963. The Mathematics of Metamathematics.

....follows from the fact that Kleene s three valued connectives have all the basic properties of the classical connectives except the excluded middle law. The rules for 2 and exhibit the typical pattern of handling modalities, or more generally quanti cation, 7 in R S systems (see [Ko97,98] [RS63]) and their soundness can be easily established from the de nition of the interpretation. The same is true for the more involved rules concerning U. The only expansion rules of our logic are (tr ) and (as ) Rule (tr ) expresses transitivity of the accessibility relation, and rule (as ) ....

Rasiowa, H., Sikorski, R. The mathematics of metamathematics, Warsaw, PWN 1963.

....of Sciences, Warsaw, ul. Ordona 21. 1 For instance, one can use them in the formal justification of any derivation rules that are useful in proving partial order properties of concurrent programs or systems. The method of proving completeness is strongly based on the technique of Q filters [RS70] tuned to our logics. This technique is traditional in Algorithmic Logic [MS87] and it has been also used for proving completeness of an infinitary axiomatization of first order linear time temporal logic [Sz87] The preliminary version of this paper appeared in [Pe93b] The rest of the paper is ....

....algebra LTA = F orm= Gamma; true] false] where ffl [ ffl [ ffl Gamma[ is a non degenerate Boolean algebra, 2. iff [ true] 3. 6 iff [ 6= false] The proof of the above theorem is standard and can be found in [RS70] p. 257) Let be a partial ordering in F orm= Theta F orm= defined as follows: iff ) Lemma 5.1 In the algebra LTA the following conditions hold: a) EX u EG ] inf i2 f[EX u EX i ]g, for u 2 Sigma , b) EX u E( U ) sup i2 f[EX u EX i ( g, for u 2 ....

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: Rasiowa, H., and Sikorski, R., The mathematics of metamathematics, PWN, Warszawa, 1970.

.... and a , together with an elementary cross or mix axiom scheme 3 a lsc : 3 a # # 3 a # where 3 a # def = a #. In the intended semantics, the plain S4 box is given the McKinsey Tarski interpretation as the interior operator in an arbitrary topological space (S, T ) [16, 17, 18], so # denotes the largest open set in T contained in #, and dually, 3 is the topological closure operator. The labelled S4 modalities a and 3 a are given the standard relational Kripke semantics, with the S4 axioms corresponding to reflexive and transitive relations, or preorders, R a ....

.... a # # a # We write # H LSC #, or say # is LSC provable, if the formula # # L( a ) has an LSC Hilbert style derivation. The S4 axiom schemes and the monotonicity rule are presented so as to correspond precisely to Kuratowski s axiomatization of the topological interior operator [15, 16, 17, 18]. Note that the usual box necessitation rules can be derived from the monotonicity rules using the box Ktt axioms. Likewise, the a S4 axiom schemes and the a monotonicity rule capture the properties of the Pre # [R a ] operator for preordered relations R a . The verification that F ....

H. Rasiowa and R. Sikorski. The Mathematics of Metamathematics. PWN, Warsaw, 1963.

....2. pref hwhile OE 0 do ff odiOE] sup i [pref h(if OE 0 then ff fi) i i( OE 0 OE) where pref is an arbitrary prefix. By this theorem the Lindenbaum algebra can be considered as as a Boolean algebra with an at most enumerable set of infinite operations. Hence the algebra is a Q algebra [30]. By a Q filter in the algebra L with a set of infinite operations Q, we shall understand a maximal filter that preserves all Q operations. That is, a maximal filter F such that sup i [pref ffi (if OE 0 then ff fi) i ( OE 0 OE) 2 F implies that there exists an i 0 such that [pref ffi (if ....

....shall understand a maximal filter that preserves all Q operations. That is, a maximal filter F such that sup i [pref ffi (if OE 0 then ff fi) i ( OE 0 OE) 2 F implies that there exists an i 0 such that [pref ffi (if OE 0 then ff fi) i 0 ( OE 0 OE) 2 F . The following theorems from [30] are important. Theorem 6.7 For every non zero element a in a Boolean algebra B with an at most enumerable set of infinite operations Q, there exists a Q filter F such that a 2 F . 56 Theorem 6.8 If the theory T is consistent, then the Lindenbaum algebra L of that theory is a non degenerate ....

Rasiowa, H. and R. Sirkorski, The Mathematics of Metamathematics, PWN, Warszaw, 1963.

....q # ; q # can be collected into one global property. There is a long standing tradition to develop algebraic semantics for various logics, the most famous example being Boolean algebras as semantic structures for classical propositional logic, Heyting algebras for intuitionistic logic [62], and Boolean algebras with operators for modal logics [41] thus, we shall put special emphasis on the algebraic structures arising from various rule systems, whenever this is appropriate. Deterministic dependence relations along with various fields of application are introduced in Section 3. In ....

....[12] and [13] 6.3 Relational proof systems A valuable tool in the formalisation and implementation of information logics are the relational proof systems introduced by Orl owska [52] which are sound and complete for standard modal logics. Such systems are in the style of Rasiowa Sikorski [62], and consist of decomposition rules, specific rules and (sequences of) axiomatic expressions. The application of a decomposition rule syntactically simplifies a formula, while the specific rules are the counterparts of the properties satisfied by the accessibility relations. Axiomatic sequences ....

Rasiowa, H. & Sikorski, R. (1963). The Mathematics of Metamathematics, vol. 41 of Polska Akademia Nauk. Monografie matematyczne. Warsaw: PWN.

....relation algebraic setting, and has exhibited a sound and complete proof system for the logic. Furthermore, various classes of algebras can be equivalently represented as suitable relational theories with relational proof systems [10, 15, 16] Such systems are in the style of Rasiowa Sikorski [21], and consist of decomposition rules, specific rules and (sequences of) axiomatic expressions. A decomposition rule when applied to an expression of the theory returns a set of expressions which are syntactically simpler than the original one. These rules provide definitions of relational ....

Rasiowa, H. & Sikorski, R. (1963). The Mathematics of Metamathematics, vol. 41 of Polska Akademia Nauk. Monografie matematyczne. Warsaw: PWN. 10

....Poland orlowska itl.waw.pl Abstract Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa Sikorski (1963) for relation algebras generated by a contact relation. 1 Introduction Contact relations arise in the context of qualitative geometry and spatial reasoning, going back to the work of de Laguna (1922) Nicod (1924) Whitehead (1929) and, more recently, of Clarke (1981) Cohn et al. 1997) ....

....It is our aim to present in this communication a sound and complete logic for contact relation algebras in which we can prove general facts about contact relation algebras. The semantics of this logic is relational as introduced by Orl owska (1991, 1996) while the proof system is in the style of Rasiowa Sikorski (1963). The rest of the paper is structured as follows: We start with a definition of the language # and its semantics, followed by the proof system. Before we embark on the proofs of soundness and completeness of the system, we shall give an example of a proof, namely, we show that P as defined by ....

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Rasiowa, H. & Sikorski, R. (1963). The Mathematics of Metamathematics, vol. 41 of Po l ska Akademia Nauk. Monografie matematyczne. Warsaw: PWN.

....relational converse, we can also modally express the convexity property. Adding topological structure Within modal logic, there is a well known way of representing a topology T on the state space X of an LTS or Kripke model. From McKinsey and Tarski s work in the 1940 s (McKinsey Tarski 1944) (Rasiowa Sikorski 1963), the axioms for the box modality of the modal logic S4 correspond exactly to those of the Kuratowski axioms for the topological interior operator, and dually, the S4 diamond corresponds to topological closure. The logic S4 is better known by its relational Kripke semantics in terms of pre orders: ....

Rasiowa, H., and Sikorski, R. 1963. The Mathematics of Metamathematics. Warsaw: PWN.

....Buchi and Streett automata. Often propositional temporal logic are expressive enough. This gives a clue to (low level) decidability. 8 Conclusion and perspectives In the next part we will present the versions of tableaux systems due to Mostowski and to Rasiowa and Sikorski. Rasiowa and Sikorski [10] presented their version calling it diagrams of formulae . Following their intension, as well as the modern terminology presented above, it seems reasonable to treat their diagrams as a version of the tableaux method, especially useful for verifying satisfiability or tautologyhood of formulae. ....

H.Rasiowa i R.Sikorski, The mathematics of metamathematics, PWN, 1970

....back to Mostowski [4] building up from earlier ideas for the topological interpretation of modal and intuitionistic propositional calculi. Mostowski used this method to prove the non derivability of some formulae, and conjectured the completness of this method. This was established by Rasiowa [5]. We differ from these works by the fact that our metalanguage is constructive. Actually our metalanguage can be taken to be Intuitionistic Type Theory [3] A systematic analysis of model theory using a constructive metatheory can also be found in the work of Dragalin (see for instance [2] 1 ....

....Let 9x A be a provable formula, with A quantifier free. Theorem: There exists a finite number of terms t 1 ; t k such that A(t 1 ) A(t k ) is provable. Let C be the boolean algebra of all formulae, modulo equivalence. This algebra is usually called (Tarski )Lindenbaum algebra [5]. Let I the ideal of C generated by the set of all instances A(t) for t arbitrary term and B be the quotient algebra C=I : We consider the boolean model obtained by taking as domain the set D of all terms and B as boolean algebra. We interpret an atomic predicate P (x 1 ; xn ) by the ....

H. Rasiowa and R. Sikorski. The Mathematics of Metamathematics. Warszawa, 1963.

....ones is known as sectioning . For example, the Shunting Rule can be rendered as (q ) q ) A lattice that has an implication operator satisfying the Shunting Rule is called a Heyting algebra. Heyting algebras are to intuitionistic logic as Boolean algebras are to classical logic z (Rasiowa 1963; Vickers 1988) A complete lattice L is a Heyting algebra if and only if (p ) exists for every p 2 L, in other words if conjunction is universally disjunctive. The proof of the following proposition shows the Shunting Rule in action. Proposition 2.4.1. Distributivity over Implication) The ....

Rasiowa, H. and Sikorski, R. (1963) Mathematics of Metamathematics. Polish Scientific Publishers, Warsaw.

....as a subset of B. Moreover, we will denote the equivalence class of an atom A by A itself. Note that interpretations of definition 3.3 are not Herbrand interpretations, yet are interpretations defined on the Herbrand universe. These interpretations were called canonical realizations in [100, 79]. Theorem 3.4 shows that O actually models computed answer substitutions and that it is fully abstract, since P 1 P 2 implies O(P 1 ) O(P 2 ) Theorem 3.4 [47] Let P 1 ; P 2 be positive programs. P 1 P 2 iff O(P 1 ) O(P 2 ) The following theorem asserts that the observable behavior of ....

H. Rasiowa and R. Sikorski. The Mathematics of Metamathematics. North-Holland, Amsterdam, 1963.

....of some sorts is developed. 1 Introduction An issue in computer science logics that has gained much popularity lately are the so called labelled deductive systems [5] The predecessors of this type of deductive systems were Beth s tableau systems [1] and Rasiowa Sikorski (R S) deduction systems [12], both developed over thirty years ago. Their important feature is that they give the proof a structure of a nitely branching (usually binary) tree with vertices labelled by formulae (tableau) or sequences of formulae (R S decomposition tree) Moreover, they are both in a sense dual. First, in a ....

....any decomposition rule is applied to the leftmost decomposable formula or subsequence . Moreover, all indecomposable formulae are inherited by the next node in the decomposition tree. As an example, below we give the full set of decomposition rules for rst order classical logic (quoted from [12]) DECOMPOSITION RULES FOR CLASSICAL LOGIC: 0 ; 00 0 ; 00 0 ; 00 0 ; 00 0 ; 00 0 ; 00 j 0 ; 00 0 ; 00 0 ; 00 j 0 ; 00 0 ; 00 0 ; 00 0 ; 8x: 00 0 ; z) ....

Rasiowa, H., Sikorski, R. The mathematics of metamathematics, Warsaw, PWN 1963.

....distinct formalizations of deduction systems. One of them he called natural deduction. The second he called the logistic calculus. In this presentation we distinguish the second by calling it the sequent calculus. In a sense both formalizations are very close to each other. Rasiowa and Sikorski [7] present the sequent calculus in terms of refutation diagrams (trees) following Schutte [8] and [9] See also Lyndon [6] The tableaux method is a version of the sequent calculus developed by Beth [1] Hintikka [5] Smullyann [11] and others. Fundamental analogies and differences between these ....

....day mathematician s practice allow more comfortable definition of the concept of proof. The initial formulae of the tree (the leaves) are called the assumptions of this proof. All the other formulae of the proof tree follow from the higher ones. 4 The Rasiowa Sikorski system Rasiowa and Sikorski [7] presented their version of Gentzen s sequent calculus calling it diagrams of formulae . In one formalisation they do not use sequents but finite sets of formulae as labels of the nodes of the proof trees, with ( 1) 0) and ( as the derivation rules. In their exposition one can ....

H.Rasiowa i R.Sikorski, The mathematics of metamathematics, PWN, 1970

....of some sorts is developed. 1 Introduction An issue in computer science logics that has gained much popularity lately are the so called labelled deductive systems [5] The predecessors of this type of deductive systems were Beth s tableau systems [1] and Rasiowa Sikorski (R S) deduction systems [12], both developed over thirty years ago. Their important feature is that they give the proof a structure of a finitely branching (usually binary) tree with vertices labelled by formulae (tableau) or sequences of formulae (R S decomposition tree) Moreover, they are both in a sense dual. First, in ....

....decomposition rule is applied to the leftmost decomposable formula or subsequence Sigma . Moreover, all indecomposable formulae are inherited by the next node in the decomposition tree. As an example, below we give the full set of decomposition rules for first order classical logic (quoted from [12]) DECOMPOSITION RULES FOR CLASSICAL LOGIC: Omega 0 ; ff; Omega 00 Omega 0 ; ff; Omega 00 Omega 0 ; ff fi; Omega 00 Omega 0 ; ff; fi; Omega 00 Omega 0 ; ff fi) Omega 00 Omega 0 ; ff; Omega 00 j Omega 0 ; fi; Omega 00 Omega 0 ; ff fi; Omega 00 ....

Rasiowa, H., Sikorski, R. The mathematics of metamathematics, Warsaw, PWN 1963.

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Rasiowa and Sikorski, Mathematics of metamathematics, Warsaw, 1963.

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Rasiowa, H. and Sikorski, R. (1968). The Mathematics of Metamathematics. Warsaw: Panstwowe Wydawn. Naukowe.

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: Rasiowa, H., and Sikorski, R., The mathematics of metamathematics, PWN, Warszawa, 1970.

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H.Rasiowa, R.Sikorski. The Mathematics of Metamathematics. Warszawa: PWN, 1963.

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H. Rasiowa and R. Sikorski. The Mathematics of Metamathematics. Warsaw, 1963.

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H. Rasiowa and R. Sikorski. The Mathematics of Metamathematics. PWN, Warszawa, 1968.

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H. Rasiowa and R. Sikorski. The Mathematics of Metamathematics. Polish Academy, 1963.

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Rasiowa, H., Sikorski, R. The mathematics of metamathematics. Warszawa 1963

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