ON MONADIC LOGIC OF RECURSIVE PROGRAMS WITH PARAMETERS

Transcription

1 Bulletin of the Section of Logic Volume 18/2 (1989), pp reedition 2006 [original edition, pp ] A. L. Rastsvetaev ON MONADIC LOGIC OF RECURSIVE PROGRAMS WITH PARAMETERS 1. Introduction The model of computation considered here, i.e. schemes of programs, was introduced in 1958 by Iu. Ianov. In 60-70ties its generalizations were studied by a number of authors Glushkov s regular expressions [2] and others. In 1979 the filtration method of modal logic was applied by M. Fisher and R. Ladner to Propositional Dynamic Logic of regular programs PDL, and later enlarged to PDL with data constants [5]. In [8] using finite tree automata, an exponential decision procedure was obtained for PDL with deterministic atomic programs and converse (DPDL) with looping construct of R. Sherman, and in [7] an exponential procedure for DPDL with data constants was announced. On the other hand, the context-free form of recursion leads to undecidability of the logic. In this paper a logic L P of recursive programs with parameters is defined, and the notion of interpretability of program logics, a generalization of translability, is used to establish decidability of L P and some of its variants. The obtained results show that nondeterministic Ianov schemes with looping are powerful enough to reflect in monadic case, at least at interpretability (and hence reducibility) level, the recursion with parameters, constant and invertible operators (that is not case for commutative operators, see [6]).

2 58 A. L. Rastsvetaev 2. Definitions 1) Two-valued propositional logic is a pair (Γ, T ), where the language Γ consists of formulas and semantics T is a set of models t which represent the truth-valued functions, t : Γ {true, false}. The language is built up by induction from atomic formulas Γ 0 Γ using propositional connectives,,..., with standard interpretation. If tα = true, then α is satisfied in model t, t = α. α is valid, = α, iff t T (t = α) and α is satisfiable iff = α. 2) An effective mapping tr : Γ Γ is a translation of logic L = (Γ, T ) into logic L = (Γ, T ) iff α Γ t T (t = α t = trα). L is translatable in L iff there is a translation from L to L. L reduces to L iff the algorithmic problem of satisfiability for L reduces to problem for L. If L is translatable to L then, obviously, L is reducible to L. The logics L and L may have, however, different semantics (e.g. the class of L-models may strictly contain the class of L -models), hence we shall need some more general notion for the establishing the reduction. Definition. Let us consider two logics L = (Γ, T ) and L = (Γ, T ). A mapping i : Γ Γ is an interpretation iff it commutes with propositional connectives and α Γ( t T t T : for each atomic subformula β of α we have t = β iff t = iβ, and conversely, t T t T : for each atomic subformula β of α, t = β iff t = iβ). If L is interpretable in L, then L is reducible to L. While in the language of Dynamic Logic [3] the terms are defined independently from formulas and notion of scheme is a generalization of the notion of term, we shall admit below occurrences of formulas in terms and schemes will essentially generalize the formulas. Let us consider pairwise disjoint sets of monadic symbols: functional and predicate constants F and P and functional and predicate variables Φ and Π. Our definitions will include also the context-free schemes. F T 1. Individual variable x is a functional term. F T 2. If τ functional term and ɛ functional symbol, then ɛ(τ) functional term. F T 3. If τ and τ 1 functional terms and α logical term, then if α then τ else τ 1 is a (conditional) functional term.

3 On Monadic Logic of Recursive Programs with Parameters 59 LT 1. Symbols true and false are logical terms. LT 2. If α, β and β 1 - logical terms, then if α then β else β 1 logical term. LT 3. If τ functional term and ɛ predicate symbol, then ɛ(τ) logical term. LT 4. Every formula is a logical term. Df. An expression ɛ(x) τ(x) is a functional (predicate) term, is called a definition of functional (predicate) variable ɛ. Sch. A pair S = (S d, S i ), where S d is a nonempty finite set of definitions and S i distinguished initial variable, is called an (U ) scheme. A scheme is functional (Boolean) iff its initial variable is functional (predicate). AF 1. Every Boolean scheme is an atomic formula. AF 2. If S is a scheme, then loop(s) is an atomic formula. The size of a formula (or scheme) is the number of occurrences of symbols in it. Remind that an interpretation of signature < F, P > is a pair I = (D, γ), where D is a set, the domain of interpretation, and γ is a mapping, which assigns to every functional constant f F an unary total function f I : D D and to every predicate constant p P unary total predicate P I : D {true, false}. A term over D is a term with individual variables substituted by elements of D. Any pair t = (I, d), where d D, we shall call a model. A sequence c of terms over D (finite or infinite) is called a computation of scheme S in a model t = (I, d i ), iff (1) The first member of c is the term S i (d i ). (2) If c i is not last member of c, then the next member c i+1 can be gotten from it by: 1. evaluation all terms in interpretation I using the following substitution rules: f(d) f I (d); p(d) p I (d); (true τ...), (false... τ) τ; α(d) true, if (I, d) = α, and false otherwise (α is a formula), and then 2. substituting the leftmost innermost occurrence of functional or predicate variable by the term in the right side of the definition of this variable. (3) The sequence c is finite iff it contains an element of D or a truth-

4 60 A. L. Rastsvetaev value. Note that if a computation ended, then the term in item (3) will be final; it is called the value of computation. An atomic formula is satisfied in a model t iff - (the case of Boolean scheme) it has in t a finite computation with value true; - (the case of looping formula) scheme under the loop construction has in t an infinite computation. This completes the definition of logic L. Every subclass K U of schemes determines a sublogic L K L d = L U. In particular the logic L P is the logic of P -schemes with no occurrences of functional variables. The set of syntactically continuous formulas of the propositional mu-calculus of D. Kozen is translatable in logic L P, as well as the logic loop-dp DL, DP DL with looping construct. 3. Theorems The following theorem is proved using in particular the idea of control predicates of Z. Manna. Theorem 1. Corollary. Logic L P is interpretable in logic loop-dp DL. L P is decidable with an exponential time complexity. A monadic functional symbol f is called constant symbol and g and h invertible symbols, if the axiom scheme fp true fp false, accordingly λ gh hg, where denotes program equivalence, λ identity function and p stands for arbitrary formula, is fulfilled in every model. Theorem 2. The set of L P -formulas with occurrences of constant and invertible functional symbols in outermost programs only, is interpretable in L P. The proof uses the result from [1] about the equivalence between the P -schemes and schemes with additional stack memory [4]. While for a long time the stack memory is used as the standard device for implementing the recursion, the result shows that essentially this is all that can be made using it in monadic case. (But programs with stack memory can be exponentially

5 On Monadic Logic of Recursive Programs with Parameters 61 more succinct.) Note also the exponential growth of formulas size in the interpretation constructed, if constant symbols appear in them. References [1] A. L. Rastsvetaev, Ob algoritmiceskih svoistvah nekotorych klassov monadiceskih shem programm (Russ.) in print. [2] A. L. Semenov, Nekotorye algoritmiceskie problemy dla Sistem Algoritmiceskih Algebr, DAN USSR 239/5 (1978, pp [3] A. P. Stolboushkin and M. A. Taiclin, Dinamiceskie logiki, [in:] Kibernetika i vycislitelnaia technika 2, M (1986), pp [4] A. V. Feklicev, Shemy program s magazinom i markerami i ih razreshimye svoistva, Izvestia AS USSR. Math series 5 (1984), pp [5] S. Passy and T. Tinchev, PDL with data constants, Information Processing Letters 20 no 1 (1985), pp [6] A. L. Rastsvetaev, About recognizability of some properties of monadic schemes of programs with commutative functions, Absracts of VIII-th International Colloquium on Logic, Methodology and Philosophy of Science, Moscow August 1987 vol. 5, pp [7] M. K. Valiev, On deterministic PDL with converse operator and constants, Absracts of VIII-th International Colloquium on Logic, Methodology and Philosophy of Science, Moscow August 1987 vol. i, p [8] Vardi and P. Wolper, Automata theoretic techniques of monadic logics of programs, JCSS 32 (1986), pp Moscow State University Mathematical Department